Datasets:

Symbolica

/

mm_dataset_random_gpt2_512

like 0

2nd0

[[ ]] |- ( 2nd ` (/) ) = (/)

[[ |- A = B |- B = C |- C = D ]] |- A = D \ {{ A : ( 2nd ` (/) ) }} \ {{ B : U. ran { (/) } }} \ {{ C : U. (/) }} \ {{ D : (/) }}

3eqtri

aj/SCJOl7SM=

2nd0

[[ ]] |- ( 2nd ` (/) ) = U. ran { (/) }

[[ ]] |- ( 2nd ` A ) = U. ran { A } \ {{ A : (/) }}

2ndval

YHuk2J4iQSY=

aj/SCJOl7SM=

2nd0

[[ ]] |- U. ran { (/) } = U. (/)

[[ |- A = B ]] |- U. A = U. B \ {{ A : ran { (/) } }} \ {{ B : (/) }}

unieqi

3VhaS8vYCiY=

aj/SCJOl7SM=

2nd0

[[ ]] |- ran { (/) } = (/)

[[ |- ph |- ( ph <-> ps ) ]] |- ps \ {{ ph : dom { (/) } = (/) }} \ {{ ps : ran { (/) } = (/) }}

mpbi

8CqNRvsPgEw=

3VhaS8vYCiY=

2nd0

[[ ]] |- dom { (/) } = (/)

[[ ]] |- dom { (/) } = (/)

dmsn0

6JlZ8yozxjw=

8CqNRvsPgEw=

2nd0

[[ ]] |- ( dom { (/) } = (/) <-> ran { (/) } = (/) )

[[ ]] |- ( dom A = (/) <-> ran A = (/) ) \ {{ A : { (/) } }}

dm0rn0

pMI6c2Y1gCM=

8CqNRvsPgEw=

2nd0

[[ ]] |- U. (/) = (/)

[[ ]] |- U. (/) = (/)

uni0

QR+jnn+mWj4=

aj/SCJOl7SM=

rlimo1

[[ ]] |- ( F ~~>r A -> F e. O(1) )

[[ |- ( ph -> ch ) |- ( ph -> ( ps <-> ch ) ) ]] |- ( ph -> ps ) \ {{ ph : F ~~>r A }} \ {{ ps : F e. O(1) }} \ {{ ch : E. y e. RR E. w e. RR A. z e. dom F ( y <_ z -> ( abs ` ( F ` z ) ) <_ w ) }}

mpbird

A3OCjA4eDEU=

rlimo1

[[ ]] |- ( F ~~>r A -> E. y e. RR E. w e. RR A. z e. dom F ( y <_ z -> ( abs ` ( F ` z ) ) <_ w ) )

[[ |- ( ph -> ps ) |- ( ph -> ( ps -> ch ) ) ]] |- ( ph -> ch ) \ {{ ph : F ~~>r A }} \ {{ ps : E. y e. RR A. z e. dom F ( y <_ z -> ( abs ` ( ( F ` z ) - A ) ) < 1 ) }} \ {{ ch : E. y e. RR E. w e. RR A. z e. dom F ( y <_ z -> ( abs ` ( F ` z ) ) <_ w ) }}

mpd

OYkP9gFdaV0=

A3OCjA4eDEU=

rlimo1

[[ ]] |- ( F ~~>r A -> E. y e. RR A. z e. dom F ( y <_ z -> ( abs ` ( ( F ` z ) - A ) ) < 1 ) )

[[ |- ( ph -> A. z e. A B e. V ) |- ( ph -> R e. RR+ ) |- ( ph -> ( z e. A |-> B ) ~~>r C ) ]] |- ( ph -> E. y e. RR A. z e. A ( y <_ z -> ( abs ` ( B - C ) ) < R ) ) \ {{ ph : F ~~>r A }} \ {{ y : y }} \ {{ z : z }} \ {{ A : dom F }} \ {{ B : ( F ` z ) }} \ {{ C : A }} \ {{ R : 1 }} \ {{ V : CC }}

rlimi

m9JYZb+fV0Q=

OYkP9gFdaV0=

rlimo1

[[ ]] |- ( F ~~>r A -> A. z e. dom F ( F ` z ) e. CC )

[[ |- ( ( ph /\ x e. A ) -> ps ) ]] |- ( ph -> A. x e. A ps ) \ {{ ph : F ~~>r A }} \ {{ ps : ( F ` z ) e. CC }} \ {{ x : z }} \ {{ A : dom F }}

ralrimiva

Jz9TWR9zcxs=

m9JYZb+fV0Q=

rlimo1

[[ ]] |- ( ( F ~~>r A /\ z e. dom F ) -> ( F ` z ) e. CC )

[[ |- ( ph -> F : A --> B ) ]] |- ( ( ph /\ C e. A ) -> ( F ` C ) e. B ) \ {{ ph : F ~~>r A }} \ {{ A : dom F }} \ {{ B : CC }} \ {{ C : z }} \ {{ F : F }}

ffvelrnda

W/Zyl9YMgjM=

Jz9TWR9zcxs=

rlimo1

[[ ]] |- ( F ~~>r A -> F : dom F --> CC )

[[ ]] |- ( F ~~>r A -> F : dom F --> CC ) \ {{ A : A }} \ {{ F : F }}

rlimf

xEEtZyyoKnA=

W/Zyl9YMgjM=

rlimo1

[[ ]] |- ( F ~~>r A -> 1 e. RR+ )

[[ |- ph ]] |- ( ps -> ph ) \ {{ ph : 1 e. RR+ }} \ {{ ps : F ~~>r A }}

a1i

z4KWmaY880I=

m9JYZb+fV0Q=

rlimo1

[[ ]] |- 1 e. RR+

[[ ]] |- 1 e. RR+

1rp

RfAiE8gg4XA=

z4KWmaY880I=

rlimo1

[[ ]] |- ( F ~~>r A -> ( z e. dom F |-> ( F ` z ) ) ~~>r A )

[[ |- ( ph -> A = B ) |- ( ph -> A R C ) ]] |- ( ph -> B R C ) \ {{ ph : F ~~>r A }} \ {{ A : F }} \ {{ B : ( z e. dom F |-> ( F ` z ) ) }} \ {{ C : A }} \ {{ R : ~~>r }}

eqbrtrrd

QhdsBwfXMgk=

m9JYZb+fV0Q=

rlimo1

[[ ]] |- ( F ~~>r A -> F = ( z e. dom F |-> ( F ` z ) ) )

[[ |- ( ph -> F : A --> B ) ]] |- ( ph -> F = ( x e. A |-> ( F ` x ) ) ) \ {{ ph : F ~~>r A }} \ {{ x : z }} \ {{ A : dom F }} \ {{ B : CC }} \ {{ F : F }}

feqmptd

K2yfoQXjkGg=

QhdsBwfXMgk=

rlimo1

[[ ]] |- ( F ~~>r A -> F : dom F --> CC )

[[ ]] |- ( F ~~>r A -> F : dom F --> CC ) \ {{ A : A }} \ {{ F : F }}

rlimf

cdqV/xm97AI=

K2yfoQXjkGg=

rlimo1

[[ ]] |- ( F ~~>r A -> F ~~>r A )

[[ ]] |- ( ph -> ph ) \ {{ ph : F ~~>r A }}

id

PCqgrVVDUWg=

QhdsBwfXMgk=

rlimo1

[[ ]] |- ( F ~~>r A -> ( E. y e. RR A. z e. dom F ( y <_ z -> ( abs ` ( ( F ` z ) - A ) ) < 1 ) -> E. y e. RR E. w e. RR A. z e. dom F ( y <_ z -> ( abs ` ( F ` z ) ) <_ w ) ) )

[[ |- ( ( ph /\ x e. A ) -> ( ps -> ch ) ) ]] |- ( ph -> ( E. x e. A ps -> E. x e. A ch ) ) \ {{ ph : F ~~>r A }} \ {{ ps : A. z e. dom F ( y <_ z -> ( abs ` ( ( F ` z ) - A ) ) < 1 ) }} \ {{ ch : E. w e. RR A. z e. dom F ( y <_ z -> ( abs ` ( F ` z ) ) <_ w ) }} \ {{ x : y }} \ {{ A : RR }}

reximdva

xiedZ79/hRE=

OYkP9gFdaV0=

rlimo1

[[ ]] |- ( ( F ~~>r A /\ y e. RR ) -> ( A. z e. dom F ( y <_ z -> ( abs ` ( ( F ` z ) - A ) ) < 1 ) -> E. w e. RR A. z e. dom F ( y <_ z -> ( abs ` ( F ` z ) ) <_ w ) ) )

[[ |- ( ph -> ps ) |- ( ph -> ( ch -> th ) ) |- ( ( ps /\ th ) -> ta ) ]] |- ( ph -> ( ch -> ta ) ) \ {{ ph : ( F ~~>r A /\ y e. RR ) }} \ {{ ps : ( ( abs ` A ) + 1 ) e. RR }} \ {{ ch : A. z e. dom F ( y <_ z -> ( abs ` ( ( F ` z ) - A ) ) < 1 ) }} \ {{ th : A. z e. dom F ( y <_ z -> ( abs ` ( F ` z ) ) <_ ( ( abs ` A ) + 1 ) ) }} \ {{ ta : E. w e. RR A. z e. dom F ( y <_ z -> ( abs ` ( F ` z ) ) <_ w ) }}

syl6an

zmLZMdnk4Ak=

xiedZ79/hRE=

rlimo1

[[ ]] |- ( ( F ~~>r A /\ y e. RR ) -> ( ( abs ` A ) + 1 ) e. RR )

[[ |- ( ph -> ps ) |- ( ps -> ch ) ]] |- ( ph -> ch ) \ {{ ph : ( F ~~>r A /\ y e. RR ) }} \ {{ ps : ( abs ` A ) e. RR }} \ {{ ch : ( ( abs ` A ) + 1 ) e. RR }}

syl

Pru3fHfOT1c=

zmLZMdnk4Ak=

rlimo1

[[ ]] |- ( ( F ~~>r A /\ y e. RR ) -> ( abs ` A ) e. RR )

[[ |- ( ph -> A e. CC ) ]] |- ( ph -> ( abs ` A ) e. RR ) \ {{ ph : ( F ~~>r A /\ y e. RR ) }} \ {{ A : A }}

abscld

Kl8/oLckaTI=

Pru3fHfOT1c=

rlimo1

[[ ]] |- ( ( F ~~>r A /\ y e. RR ) -> A e. CC )

[[ |- ( ph -> ps ) ]] |- ( ( ph /\ ch ) -> ps ) \ {{ ph : F ~~>r A }} \ {{ ps : A e. CC }} \ {{ ch : y e. RR }}

adantr

ZwXNPo5LbwA=

Kl8/oLckaTI=

rlimo1

[[ ]] |- ( F ~~>r A -> A e. CC )

[[ ]] |- ( F ~~>r A -> A e. CC ) \ {{ A : A }} \ {{ F : F }}

rlimcl

7Y52U/jhNzg=

ZwXNPo5LbwA=

rlimo1

[[ ]] |- ( ( abs ` A ) e. RR -> ( ( abs ` A ) + 1 ) e. RR )

[[ ]] |- ( A e. RR -> ( A + 1 ) e. RR ) \ {{ A : ( abs ` A ) }}

peano2re

oNx9ci8/B24=

Pru3fHfOT1c=

rlimo1

[[ ]] |- ( ( F ~~>r A /\ y e. RR ) -> ( A. z e. dom F ( y <_ z -> ( abs ` ( ( F ` z ) - A ) ) < 1 ) -> A. z e. dom F ( y <_ z -> ( abs ` ( F ` z ) ) <_ ( ( abs ` A ) + 1 ) ) ) )

[[ |- ( ( ph /\ x e. A ) -> ( ps -> ch ) ) ]] |- ( ph -> ( A. x e. A ps -> A. x e. A ch ) ) \ {{ ph : ( F ~~>r A /\ y e. RR ) }} \ {{ ps : ( y <_ z -> ( abs ` ( ( F ` z ) - A ) ) < 1 ) }} \ {{ ch : ( y <_ z -> ( abs ` ( F ` z ) ) <_ ( ( abs ` A ) + 1 ) ) }} \ {{ x : z }} \ {{ A : dom F }}

ralimdva

691RlIAMKXo=

zmLZMdnk4Ak=

rlimo1

[[ ]] |- ( ( ( F ~~>r A /\ y e. RR ) /\ z e. dom F ) -> ( ( y <_ z -> ( abs ` ( ( F ` z ) - A ) ) < 1 ) -> ( y <_ z -> ( abs ` ( F ` z ) ) <_ ( ( abs ` A ) + 1 ) ) ) )

[[ |- ( ph -> ( ps -> ch ) ) ]] |- ( ph -> ( ( th -> ps ) -> ( th -> ch ) ) ) \ {{ ph : ( ( F ~~>r A /\ y e. RR ) /\ z e. dom F ) }} \ {{ ps : ( abs ` ( ( F ` z ) - A ) ) < 1 }} \ {{ ch : ( abs ` ( F ` z ) ) <_ ( ( abs ` A ) + 1 ) }} \ {{ th : y <_ z }}

imim2d

R5Bi11VB4lg=

691RlIAMKXo=

rlimo1

[[ ]] |- ( ( ( F ~~>r A /\ y e. RR ) /\ z e. dom F ) -> ( ( abs ` ( ( F ` z ) - A ) ) < 1 -> ( abs ` ( F ` z ) ) <_ ( ( abs ` A ) + 1 ) ) )

[[ |- ( ph -> ( ps -> ch ) ) |- ( ph -> ( ch -> th ) ) ]] |- ( ph -> ( ps -> th ) ) \ {{ ph : ( ( F ~~>r A /\ y e. RR ) /\ z e. dom F ) }} \ {{ ps : ( abs ` ( ( F ` z ) - A ) ) < 1 }} \ {{ ch : ( abs ` ( F ` z ) ) < ( ( abs ` A ) + 1 ) }} \ {{ th : ( abs ` ( F ` z ) ) <_ ( ( abs ` A ) + 1 ) }}

syld

p1hZtvl8Fl4=

R5Bi11VB4lg=

rlimo1

[[ ]] |- ( ( ( F ~~>r A /\ y e. RR ) /\ z e. dom F ) -> ( ( abs ` ( ( F ` z ) - A ) ) < 1 -> ( abs ` ( F ` z ) ) < ( ( abs ` A ) + 1 ) ) )

[[ |- ( ph -> ( ps -> ch ) ) |- ( ph -> ( ch <-> th ) ) ]] |- ( ph -> ( ps -> th ) ) \ {{ ph : ( ( F ~~>r A /\ y e. RR ) /\ z e. dom F ) }} \ {{ ps : ( abs ` ( ( F ` z ) - A ) ) < 1 }} \ {{ ch : ( ( abs ` ( F ` z ) ) - ( abs ` A ) ) < 1 }} \ {{ th : ( abs ` ( F ` z ) ) < ( ( abs ` A ) + 1 ) }}

sylibd

JlFIy77jzHU=

p1hZtvl8Fl4=

rlimo1

[[ ]] |- ( ( ( F ~~>r A /\ y e. RR ) /\ z e. dom F ) -> ( ( abs ` ( ( F ` z ) - A ) ) < 1 -> ( ( abs ` ( F ` z ) ) - ( abs ` A ) ) < 1 ) )

[[ |- ( ph -> ps ) |- ( ph -> ( ( ps /\ ch ) -> th ) ) ]] |- ( ph -> ( ch -> th ) ) \ {{ ph : ( ( F ~~>r A /\ y e. RR ) /\ z e. dom F ) }} \ {{ ps : ( ( abs ` ( F ` z ) ) - ( abs ` A ) ) <_ ( abs ` ( ( F ` z ) - A ) ) }} \ {{ ch : ( abs ` ( ( F ` z ) - A ) ) < 1 }} \ {{ th : ( ( abs ` ( F ` z ) ) - ( abs ` A ) ) < 1 }}

mpand

0VyDvYg0ygQ=

JlFIy77jzHU=

rlimo1

[[ ]] |- ( ( ( F ~~>r A /\ y e. RR ) /\ z e. dom F ) -> ( ( abs ` ( F ` z ) ) - ( abs ` A ) ) <_ ( abs ` ( ( F ` z ) - A ) ) )

[[ |- ( ph -> A e. CC ) |- ( ph -> B e. CC ) ]] |- ( ph -> ( ( abs ` A ) - ( abs ` B ) ) <_ ( abs ` ( A - B ) ) ) \ {{ ph : ( ( F ~~>r A /\ y e. RR ) /\ z e. dom F ) }} \ {{ A : ( F ` z ) }} \ {{ B : A }}

abs2difd

A+ijYbeSyh8=

0VyDvYg0ygQ=

rlimo1

[[ ]] |- ( ( ( F ~~>r A /\ y e. RR ) /\ z e. dom F ) -> ( F ` z ) e. CC )

[[ |- ( ( ph /\ ps ) -> ch ) ]] |- ( ( ( ph /\ th ) /\ ps ) -> ch ) \ {{ ph : F ~~>r A }} \ {{ ps : z e. dom F }} \ {{ ch : ( F ` z ) e. CC }} \ {{ th : y e. RR }}

adantlr

VI66rlcPmw8=

A+ijYbeSyh8=

rlimo1

[[ ]] |- ( ( F ~~>r A /\ z e. dom F ) -> ( F ` z ) e. CC )

[[ |- ( ph -> F : A --> B ) ]] |- ( ( ph /\ C e. A ) -> ( F ` C ) e. B ) \ {{ ph : F ~~>r A }} \ {{ A : dom F }} \ {{ B : CC }} \ {{ C : z }} \ {{ F : F }}

ffvelrnda

6LMo5bmsTB0=

VI66rlcPmw8=

rlimo1

[[ ]] |- ( F ~~>r A -> F : dom F --> CC )

[[ ]] |- ( F ~~>r A -> F : dom F --> CC ) \ {{ A : A }} \ {{ F : F }}

rlimf

c1lYdK4nAjI=

6LMo5bmsTB0=

rlimo1

[[ ]] |- ( ( ( F ~~>r A /\ y e. RR ) /\ z e. dom F ) -> A e. CC )

[[ |- ( ph -> ps ) ]] |- ( ( ph /\ ch ) -> ps ) \ {{ ph : ( F ~~>r A /\ y e. RR ) }} \ {{ ps : A e. CC }} \ {{ ch : z e. dom F }}

adantr

Ceubj5bESz0=

A+ijYbeSyh8=

rlimo1

[[ ]] |- ( ( F ~~>r A /\ y e. RR ) -> A e. CC )

[[ |- ( ph -> ps ) ]] |- ( ( ph /\ ch ) -> ps ) \ {{ ph : F ~~>r A }} \ {{ ps : A e. CC }} \ {{ ch : y e. RR }}

adantr

oa2IiAhnLQ4=

Ceubj5bESz0=

rlimo1

[[ ]] |- ( F ~~>r A -> A e. CC )

[[ ]] |- ( F ~~>r A -> A e. CC ) \ {{ A : A }} \ {{ F : F }}

rlimcl

zDV7lKVyxmQ=

oa2IiAhnLQ4=

rlimo1

[[ ]] |- ( ( ( F ~~>r A /\ y e. RR ) /\ z e. dom F ) -> ( ( ( ( abs ` ( F ` z ) ) - ( abs ` A ) ) <_ ( abs ` ( ( F ` z ) - A ) ) /\ ( abs ` ( ( F ` z ) - A ) ) < 1 ) -> ( ( abs ` ( F ` z ) ) - ( abs ` A ) ) < 1 ) )

[[ |- ( ph -> ps ) |- ( ph -> ch ) |- ( ph -> th ) |- ( ( ps /\ ch /\ th ) -> ta ) ]] |- ( ph -> ta ) \ {{ ph : ( ( F ~~>r A /\ y e. RR ) /\ z e. dom F ) }} \ {{ ps : ( ( abs ` ( F ` z ) ) - ( abs ` A ) ) e. RR }} \ {{ ch : ( abs ` ( ( F ` z ) - A ) ) e. RR }} \ {{ th : 1 e. RR }} \ {{ ta : ( ( ( ( abs ` ( F ` z ) ) - ( abs ` A ) ) <_ ( abs ` ( ( F ` z ) - A ) ) /\ ( abs ` ( ( F ` z ) - A ) ) < 1 ) -> ( ( abs ` ( F ` z ) ) - ( abs ` A ) ) < 1 ) }}

syl3anc

Kfnan8JRXl0=

0VyDvYg0ygQ=

rlimo1

[[ ]] |- ( ( ( F ~~>r A /\ y e. RR ) /\ z e. dom F ) -> ( ( abs ` ( F ` z ) ) - ( abs ` A ) ) e. RR )

[[ |- ( ph -> A e. RR ) |- ( ph -> B e. RR ) ]] |- ( ph -> ( A - B ) e. RR ) \ {{ ph : ( ( F ~~>r A /\ y e. RR ) /\ z e. dom F ) }} \ {{ A : ( abs ` ( F ` z ) ) }} \ {{ B : ( abs ` A ) }}

resubcld

QatjwSB2pSk=

Kfnan8JRXl0=

rlimo1

[[ ]] |- ( ( ( F ~~>r A /\ y e. RR ) /\ z e. dom F ) -> ( abs ` ( F ` z ) ) e. RR )

[[ |- ( ph -> A e. CC ) ]] |- ( ph -> ( abs ` A ) e. RR ) \ {{ ph : ( ( F ~~>r A /\ y e. RR ) /\ z e. dom F ) }} \ {{ A : ( F ` z ) }}

abscld

jIQPvg5Vj3c=

QatjwSB2pSk=

rlimo1

[[ ]] |- ( ( ( F ~~>r A /\ y e. RR ) /\ z e. dom F ) -> ( F ` z ) e. CC )

[[ |- ( ( ph /\ ps ) -> ch ) ]] |- ( ( ( ph /\ th ) /\ ps ) -> ch ) \ {{ ph : F ~~>r A }} \ {{ ps : z e. dom F }} \ {{ ch : ( F ` z ) e. CC }} \ {{ th : y e. RR }}

adantlr

mrF5lvFyM1A=

jIQPvg5Vj3c=

rlimo1

[[ ]] |- ( ( F ~~>r A /\ z e. dom F ) -> ( F ` z ) e. CC )

[[ |- ( ph -> F : A --> B ) ]] |- ( ( ph /\ C e. A ) -> ( F ` C ) e. B ) \ {{ ph : F ~~>r A }} \ {{ A : dom F }} \ {{ B : CC }} \ {{ C : z }} \ {{ F : F }}

ffvelrnda

DA0sbMnCuVQ=

mrF5lvFyM1A=

rlimo1

[[ ]] |- ( F ~~>r A -> F : dom F --> CC )

[[ ]] |- ( F ~~>r A -> F : dom F --> CC ) \ {{ A : A }} \ {{ F : F }}

rlimf

sOKLDo6f7Cs=

DA0sbMnCuVQ=

rlimo1

[[ ]] |- ( ( ( F ~~>r A /\ y e. RR ) /\ z e. dom F ) -> ( abs ` A ) e. RR )

[[ |- ( ph -> ps ) ]] |- ( ( ph /\ ch ) -> ps ) \ {{ ph : ( F ~~>r A /\ y e. RR ) }} \ {{ ps : ( abs ` A ) e. RR }} \ {{ ch : z e. dom F }}

adantr

CMHdvNQVYFE=

QatjwSB2pSk=

rlimo1

[[ ]] |- ( ( F ~~>r A /\ y e. RR ) -> ( abs ` A ) e. RR )

[[ |- ( ph -> A e. CC ) ]] |- ( ph -> ( abs ` A ) e. RR ) \ {{ ph : ( F ~~>r A /\ y e. RR ) }} \ {{ A : A }}

abscld

S9PvvYTgpjM=

CMHdvNQVYFE=

rlimo1

[[ ]] |- ( ( F ~~>r A /\ y e. RR ) -> A e. CC )

[[ |- ( ph -> ps ) ]] |- ( ( ph /\ ch ) -> ps ) \ {{ ph : F ~~>r A }} \ {{ ps : A e. CC }} \ {{ ch : y e. RR }}

adantr

LIGsxt0GcF4=

S9PvvYTgpjM=

rlimo1

[[ ]] |- ( F ~~>r A -> A e. CC )

[[ ]] |- ( F ~~>r A -> A e. CC ) \ {{ A : A }} \ {{ F : F }}

rlimcl

axL7CC1pbgk=

LIGsxt0GcF4=

rlimo1

[[ ]] |- ( ( ( F ~~>r A /\ y e. RR ) /\ z e. dom F ) -> ( abs ` ( ( F ` z ) - A ) ) e. RR )

[[ |- ( ph -> A e. CC ) ]] |- ( ph -> ( abs ` A ) e. RR ) \ {{ ph : ( ( F ~~>r A /\ y e. RR ) /\ z e. dom F ) }} \ {{ A : ( ( F ` z ) - A ) }}

abscld

L3UJFsEI8QY=

Kfnan8JRXl0=

rlimo1

[[ ]] |- ( ( ( F ~~>r A /\ y e. RR ) /\ z e. dom F ) -> ( ( F ` z ) - A ) e. CC )

[[ |- ( ph -> A e. CC ) |- ( ph -> B e. CC ) ]] |- ( ph -> ( A - B ) e. CC ) \ {{ ph : ( ( F ~~>r A /\ y e. RR ) /\ z e. dom F ) }} \ {{ A : ( F ` z ) }} \ {{ B : A }}

subcld

LWdeORnbWVw=

L3UJFsEI8QY=

rlimo1

[[ ]] |- ( ( ( F ~~>r A /\ y e. RR ) /\ z e. dom F ) -> ( F ` z ) e. CC )

[[ |- ( ( ph /\ ps ) -> ch ) ]] |- ( ( ( ph /\ th ) /\ ps ) -> ch ) \ {{ ph : F ~~>r A }} \ {{ ps : z e. dom F }} \ {{ ch : ( F ` z ) e. CC }} \ {{ th : y e. RR }}

adantlr

nXQxBuXAqFU=

LWdeORnbWVw=

rlimo1

[[ ]] |- ( ( F ~~>r A /\ z e. dom F ) -> ( F ` z ) e. CC )

[[ |- ( ph -> F : A --> B ) ]] |- ( ( ph /\ C e. A ) -> ( F ` C ) e. B ) \ {{ ph : F ~~>r A }} \ {{ A : dom F }} \ {{ B : CC }} \ {{ C : z }} \ {{ F : F }}

ffvelrnda

D/g4rPGd7Dg=

nXQxBuXAqFU=

rlimo1

[[ ]] |- ( F ~~>r A -> F : dom F --> CC )

[[ ]] |- ( F ~~>r A -> F : dom F --> CC ) \ {{ A : A }} \ {{ F : F }}

rlimf

IN5rUW8AAHs=

D/g4rPGd7Dg=

rlimo1

[[ ]] |- ( ( ( F ~~>r A /\ y e. RR ) /\ z e. dom F ) -> A e. CC )

[[ |- ( ph -> ps ) ]] |- ( ( ph /\ ch ) -> ps ) \ {{ ph : ( F ~~>r A /\ y e. RR ) }} \ {{ ps : A e. CC }} \ {{ ch : z e. dom F }}

adantr

dgdfZ9/+KR8=

LWdeORnbWVw=

rlimo1

[[ ]] |- ( ( F ~~>r A /\ y e. RR ) -> A e. CC )

[[ |- ( ph -> ps ) ]] |- ( ( ph /\ ch ) -> ps ) \ {{ ph : F ~~>r A }} \ {{ ps : A e. CC }} \ {{ ch : y e. RR }}

adantr

Vmr1I8HgRy8=

dgdfZ9/+KR8=

rlimo1

[[ ]] |- ( F ~~>r A -> A e. CC )

[[ ]] |- ( F ~~>r A -> A e. CC ) \ {{ A : A }} \ {{ F : F }}

rlimcl

jaU71cQc2Gc=

Vmr1I8HgRy8=

rlimo1

[[ ]] |- ( ( ( F ~~>r A /\ y e. RR ) /\ z e. dom F ) -> 1 e. RR )

[[ ]] |- ( ph -> 1 e. RR ) \ {{ ph : ( ( F ~~>r A /\ y e. RR ) /\ z e. dom F ) }}

1red

Ol1zK7wb20Q=

Kfnan8JRXl0=

rlimo1

[[ ]] |- ( ( ( ( abs ` ( F ` z ) ) - ( abs ` A ) ) e. RR /\ ( abs ` ( ( F ` z ) - A ) ) e. RR /\ 1 e. RR ) -> ( ( ( ( abs ` ( F ` z ) ) - ( abs ` A ) ) <_ ( abs ` ( ( F ` z ) - A ) ) /\ ( abs ` ( ( F ` z ) - A ) ) < 1 ) -> ( ( abs ` ( F ` z ) ) - ( abs ` A ) ) < 1 ) )

[[ ]] |- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( ( A <_ B /\ B < C ) -> A < C ) ) \ {{ A : ( ( abs ` ( F ` z ) ) - ( abs ` A ) ) }} \ {{ B : ( abs ` ( ( F ` z ) - A ) ) }} \ {{ C : 1 }}

lelttr

LwM83PqLrwc=

Kfnan8JRXl0=

rlimo1

[[ ]] |- ( ( ( F ~~>r A /\ y e. RR ) /\ z e. dom F ) -> ( ( ( abs ` ( F ` z ) ) - ( abs ` A ) ) < 1 <-> ( abs ` ( F ` z ) ) < ( ( abs ` A ) + 1 ) ) )

[[ |- ( ph -> A e. RR ) |- ( ph -> B e. RR ) |- ( ph -> C e. RR ) ]] |- ( ph -> ( ( A - B ) < C <-> A < ( B + C ) ) ) \ {{ ph : ( ( F ~~>r A /\ y e. RR ) /\ z e. dom F ) }} \ {{ A : ( abs ` ( F ` z ) ) }} \ {{ B : ( abs ` A ) }} \ {{ C : 1 }}

ltsubadd2d

TQZILblXmjA=

JlFIy77jzHU=

rlimo1

[[ ]] |- ( ( ( F ~~>r A /\ y e. RR ) /\ z e. dom F ) -> ( abs ` ( F ` z ) ) e. RR )

[[ |- ( ph -> A e. CC ) ]] |- ( ph -> ( abs ` A ) e. RR ) \ {{ ph : ( ( F ~~>r A /\ y e. RR ) /\ z e. dom F ) }} \ {{ A : ( F ` z ) }}

abscld

wKiB0Wq1Eno=

TQZILblXmjA=

rlimo1

[[ ]] |- ( ( ( F ~~>r A /\ y e. RR ) /\ z e. dom F ) -> ( F ` z ) e. CC )

[[ |- ( ( ph /\ ps ) -> ch ) ]] |- ( ( ( ph /\ th ) /\ ps ) -> ch ) \ {{ ph : F ~~>r A }} \ {{ ps : z e. dom F }} \ {{ ch : ( F ` z ) e. CC }} \ {{ th : y e. RR }}

adantlr

x2z7fNJagk0=

wKiB0Wq1Eno=

rlimo1

[[ ]] |- ( ( F ~~>r A /\ z e. dom F ) -> ( F ` z ) e. CC )

[[ |- ( ph -> F : A --> B ) ]] |- ( ( ph /\ C e. A ) -> ( F ` C ) e. B ) \ {{ ph : F ~~>r A }} \ {{ A : dom F }} \ {{ B : CC }} \ {{ C : z }} \ {{ F : F }}

ffvelrnda

wpYK+oD3bl8=

x2z7fNJagk0=

rlimo1

[[ ]] |- ( F ~~>r A -> F : dom F --> CC )

[[ ]] |- ( F ~~>r A -> F : dom F --> CC ) \ {{ A : A }} \ {{ F : F }}

rlimf

KB8h7SDbRRc=

wpYK+oD3bl8=

rlimo1

[[ ]] |- ( ( ( F ~~>r A /\ y e. RR ) /\ z e. dom F ) -> ( abs ` A ) e. RR )

[[ |- ( ph -> ps ) ]] |- ( ( ph /\ ch ) -> ps ) \ {{ ph : ( F ~~>r A /\ y e. RR ) }} \ {{ ps : ( abs ` A ) e. RR }} \ {{ ch : z e. dom F }}

adantr

Obtdw9zYV1I=

TQZILblXmjA=

rlimo1

[[ ]] |- ( ( F ~~>r A /\ y e. RR ) -> ( abs ` A ) e. RR )

[[ |- ( ph -> A e. CC ) ]] |- ( ph -> ( abs ` A ) e. RR ) \ {{ ph : ( F ~~>r A /\ y e. RR ) }} \ {{ A : A }}

abscld

vk/KoBzmvhg=

Obtdw9zYV1I=

rlimo1

[[ ]] |- ( ( F ~~>r A /\ y e. RR ) -> A e. CC )

[[ |- ( ph -> ps ) ]] |- ( ( ph /\ ch ) -> ps ) \ {{ ph : F ~~>r A }} \ {{ ps : A e. CC }} \ {{ ch : y e. RR }}

adantr

2gMNzaq02jA=

vk/KoBzmvhg=

rlimo1

[[ ]] |- ( F ~~>r A -> A e. CC )

[[ ]] |- ( F ~~>r A -> A e. CC ) \ {{ A : A }} \ {{ F : F }}

rlimcl

5iJTb1uDnW8=

2gMNzaq02jA=

rlimo1

[[ ]] |- ( ( ( F ~~>r A /\ y e. RR ) /\ z e. dom F ) -> 1 e. RR )

[[ ]] |- ( ph -> 1 e. RR ) \ {{ ph : ( ( F ~~>r A /\ y e. RR ) /\ z e. dom F ) }}

1red

v2/Y7JkRTnQ=

TQZILblXmjA=

rlimo1

[[ ]] |- ( ( ( F ~~>r A /\ y e. RR ) /\ z e. dom F ) -> ( ( abs ` ( F ` z ) ) < ( ( abs ` A ) + 1 ) -> ( abs ` ( F ` z ) ) <_ ( ( abs ` A ) + 1 ) ) )

[[ |- ( ph -> ps ) |- ( ph -> ch ) |- ( ( ps /\ ch ) -> th ) ]] |- ( ph -> th ) \ {{ ph : ( ( F ~~>r A /\ y e. RR ) /\ z e. dom F ) }} \ {{ ps : ( abs ` ( F ` z ) ) e. RR }} \ {{ ch : ( ( abs ` A ) + 1 ) e. RR }} \ {{ th : ( ( abs ` ( F ` z ) ) < ( ( abs ` A ) + 1 ) -> ( abs ` ( F ` z ) ) <_ ( ( abs ` A ) + 1 ) ) }}

syl2anc

hw9LGqwk1lk=

p1hZtvl8Fl4=

rlimo1

[[ ]] |- ( ( ( F ~~>r A /\ y e. RR ) /\ z e. dom F ) -> ( abs ` ( F ` z ) ) e. RR )

[[ |- ( ph -> A e. CC ) ]] |- ( ph -> ( abs ` A ) e. RR ) \ {{ ph : ( ( F ~~>r A /\ y e. RR ) /\ z e. dom F ) }} \ {{ A : ( F ` z ) }}

abscld

p4nwHxoT9jo=

hw9LGqwk1lk=

rlimo1

[[ ]] |- ( ( ( F ~~>r A /\ y e. RR ) /\ z e. dom F ) -> ( F ` z ) e. CC )

[[ |- ( ( ph /\ ps ) -> ch ) ]] |- ( ( ( ph /\ th ) /\ ps ) -> ch ) \ {{ ph : F ~~>r A }} \ {{ ps : z e. dom F }} \ {{ ch : ( F ` z ) e. CC }} \ {{ th : y e. RR }}

adantlr

XPEN6f4l8SY=

p4nwHxoT9jo=

rlimo1

[[ ]] |- ( ( F ~~>r A /\ z e. dom F ) -> ( F ` z ) e. CC )

[[ |- ( ph -> F : A --> B ) ]] |- ( ( ph /\ C e. A ) -> ( F ` C ) e. B ) \ {{ ph : F ~~>r A }} \ {{ A : dom F }} \ {{ B : CC }} \ {{ C : z }} \ {{ F : F }}

ffvelrnda

uv2q8skp400=

XPEN6f4l8SY=

rlimo1

[[ ]] |- ( F ~~>r A -> F : dom F --> CC )

[[ ]] |- ( F ~~>r A -> F : dom F --> CC ) \ {{ A : A }} \ {{ F : F }}

rlimf

tm+p259c2CY=

uv2q8skp400=

rlimo1

[[ ]] |- ( ( ( F ~~>r A /\ y e. RR ) /\ z e. dom F ) -> ( ( abs ` A ) + 1 ) e. RR )

[[ |- ( ph -> ps ) ]] |- ( ( ph /\ ch ) -> ps ) \ {{ ph : ( F ~~>r A /\ y e. RR ) }} \ {{ ps : ( ( abs ` A ) + 1 ) e. RR }} \ {{ ch : z e. dom F }}

adantr

YfH1X5D2Ghc=

hw9LGqwk1lk=

rlimo1

[[ ]] |- ( ( F ~~>r A /\ y e. RR ) -> ( ( abs ` A ) + 1 ) e. RR )

[[ |- ( ph -> ps ) |- ( ps -> ch ) ]] |- ( ph -> ch ) \ {{ ph : ( F ~~>r A /\ y e. RR ) }} \ {{ ps : ( abs ` A ) e. RR }} \ {{ ch : ( ( abs ` A ) + 1 ) e. RR }}

syl

QOP3BQnj2gE=

YfH1X5D2Ghc=

rlimo1

[[ ]] |- ( ( F ~~>r A /\ y e. RR ) -> ( abs ` A ) e. RR )

[[ |- ( ph -> A e. CC ) ]] |- ( ph -> ( abs ` A ) e. RR ) \ {{ ph : ( F ~~>r A /\ y e. RR ) }} \ {{ A : A }}

abscld

NvmTYE3Y20E=

QOP3BQnj2gE=

rlimo1

[[ ]] |- ( ( F ~~>r A /\ y e. RR ) -> A e. CC )

[[ |- ( ph -> ps ) ]] |- ( ( ph /\ ch ) -> ps ) \ {{ ph : F ~~>r A }} \ {{ ps : A e. CC }} \ {{ ch : y e. RR }}

adantr

rDEuz7z7azI=

NvmTYE3Y20E=

rlimo1

[[ ]] |- ( F ~~>r A -> A e. CC )

[[ ]] |- ( F ~~>r A -> A e. CC ) \ {{ A : A }} \ {{ F : F }}

rlimcl

mYdBFNWdxSA=

rDEuz7z7azI=

rlimo1

[[ ]] |- ( ( abs ` A ) e. RR -> ( ( abs ` A ) + 1 ) e. RR )

[[ ]] |- ( A e. RR -> ( A + 1 ) e. RR ) \ {{ A : ( abs ` A ) }}

peano2re

/C9vCoc9XTA=

QOP3BQnj2gE=

rlimo1

[[ ]] |- ( ( ( abs ` ( F ` z ) ) e. RR /\ ( ( abs ` A ) + 1 ) e. RR ) -> ( ( abs ` ( F ` z ) ) < ( ( abs ` A ) + 1 ) -> ( abs ` ( F ` z ) ) <_ ( ( abs ` A ) + 1 ) ) )

[[ ]] |- ( ( A e. RR /\ B e. RR ) -> ( A < B -> A <_ B ) ) \ {{ A : ( abs ` ( F ` z ) ) }} \ {{ B : ( ( abs ` A ) + 1 ) }}

ltle

IicRNd17qx8=

hw9LGqwk1lk=

rlimo1

[[ ]] |- ( ( ( ( abs ` A ) + 1 ) e. RR /\ A. z e. dom F ( y <_ z -> ( abs ` ( F ` z ) ) <_ ( ( abs ` A ) + 1 ) ) ) -> E. w e. RR A. z e. dom F ( y <_ z -> ( abs ` ( F ` z ) ) <_ w ) )

[[ |- ( x = A -> ( ph <-> ps ) ) ]] |- ( ( A e. B /\ ps ) -> E. x e. B ph ) \ {{ ph : A. z e. dom F ( y <_ z -> ( abs ` ( F ` z ) ) <_ w ) }} \ {{ ps : A. z e. dom F ( y <_ z -> ( abs ` ( F ` z ) ) <_ ( ( abs ` A ) + 1 ) ) }} \ {{ x : w }} \ {{ A : ( ( abs ` A ) + 1 ) }} \ {{ B : RR }}

rspcev

iileqz179gw=

zmLZMdnk4Ak=

rlimo1

[[ ]] |- ( w = ( ( abs ` A ) + 1 ) -> ( A. z e. dom F ( y <_ z -> ( abs ` ( F ` z ) ) <_ w ) <-> A. z e. dom F ( y <_ z -> ( abs ` ( F ` z ) ) <_ ( ( abs ` A ) + 1 ) ) ) )

[[ |- ( ph -> ( ps <-> ch ) ) ]] |- ( ph -> ( A. x e. A ps <-> A. x e. A ch ) ) \ {{ ph : w = ( ( abs ` A ) + 1 ) }} \ {{ ps : ( y <_ z -> ( abs ` ( F ` z ) ) <_ w ) }} \ {{ ch : ( y <_ z -> ( abs ` ( F ` z ) ) <_ ( ( abs ` A ) + 1 ) ) }} \ {{ x : z }} \ {{ A : dom F }}

ralbidv

mIVETtdvJB0=

iileqz179gw=

rlimo1

[[ ]] |- ( w = ( ( abs ` A ) + 1 ) -> ( ( y <_ z -> ( abs ` ( F ` z ) ) <_ w ) <-> ( y <_ z -> ( abs ` ( F ` z ) ) <_ ( ( abs ` A ) + 1 ) ) ) )

[[ |- ( ph -> ( ps <-> ch ) ) ]] |- ( ph -> ( ( th -> ps ) <-> ( th -> ch ) ) ) \ {{ ph : w = ( ( abs ` A ) + 1 ) }} \ {{ ps : ( abs ` ( F ` z ) ) <_ w }} \ {{ ch : ( abs ` ( F ` z ) ) <_ ( ( abs ` A ) + 1 ) }} \ {{ th : y <_ z }}

imbi2d

zh4psF8aaVY=

mIVETtdvJB0=

rlimo1

[[ ]] |- ( w = ( ( abs ` A ) + 1 ) -> ( ( abs ` ( F ` z ) ) <_ w <-> ( abs ` ( F ` z ) ) <_ ( ( abs ` A ) + 1 ) ) )

[[ ]] |- ( A = B -> ( C R A <-> C R B ) ) \ {{ A : w }} \ {{ B : ( ( abs ` A ) + 1 ) }} \ {{ C : ( abs ` ( F ` z ) ) }} \ {{ R : <_ }}

breq2

4Kj/y/L43Bc=

zh4psF8aaVY=

rlimo1

[[ ]] |- ( F ~~>r A -> ( F e. O(1) <-> E. y e. RR E. w e. RR A. z e. dom F ( y <_ z -> ( abs ` ( F ` z ) ) <_ w ) ) )

[[ |- ( ph -> ps ) |- ( ph -> ch ) |- ( ( ps /\ ch ) -> th ) ]] |- ( ph -> th ) \ {{ ph : F ~~>r A }} \ {{ ps : F : dom F --> CC }} \ {{ ch : dom F C_ RR }} \ {{ th : ( F e. O(1) <-> E. y e. RR E. w e. RR A. z e. dom F ( y <_ z -> ( abs ` ( F ` z ) ) <_ w ) ) }}

syl2anc

tvmN5LTOLmY=

A3OCjA4eDEU=

rlimo1

[[ ]] |- ( F ~~>r A -> F : dom F --> CC )

[[ ]] |- ( F ~~>r A -> F : dom F --> CC ) \ {{ A : A }} \ {{ F : F }}

rlimf

TWAAZrsuiBA=

tvmN5LTOLmY=

rlimo1

[[ ]] |- ( F ~~>r A -> dom F C_ RR )

[[ ]] |- ( F ~~>r A -> dom F C_ RR ) \ {{ A : A }} \ {{ F : F }}

rlimss

SgBd384cCyo=

tvmN5LTOLmY=

rlimo1

[[ ]] |- ( ( F : dom F --> CC /\ dom F C_ RR ) -> ( F e. O(1) <-> E. y e. RR E. w e. RR A. z e. dom F ( y <_ z -> ( abs ` ( F ` z ) ) <_ w ) ) )

[[ ]] |- ( ( F : A --> CC /\ A C_ RR ) -> ( F e. O(1) <-> E. x e. RR E. m e. RR A. y e. A ( x <_ y -> ( abs ` ( F ` y ) ) <_ m ) ) ) \ {{ x : y }} \ {{ y : z }} \ {{ A : dom F }} \ {{ m : w }} \ {{ F : F }}

elo12

4Imjsem6lV8=

tvmN5LTOLmY=

ndmioo

[[ ]] |- ( -. ( A e. RR* /\ B e. RR* ) -> ( A (,) B ) = (/) )

[[ |- dom F = ( S X. S ) ]] |- ( -. ( A e. S /\ B e. S ) -> ( A F B ) = (/) ) \ {{ A : A }} \ {{ B : B }} \ {{ S : RR* }} \ {{ F : (,) }}

ndmov

s6j2BRhxnlE=

ndmioo

[[ ]] |- dom (,) = ( RR* X. RR* )

[[ |- F : A --> B ]] |- dom F = A \ {{ A : ( RR* X. RR* ) }} \ {{ B : ~P RR* }} \ {{ F : (,) }}

fdmi

95s07LnlyUw=

s6j2BRhxnlE=

ndmioo

[[ ]] |- (,) : ( RR* X. RR* ) --> ~P RR*

[[ |- O = ( x e. RR* , y e. RR* |-> { z e. RR* | ( x R z /\ z S y ) } ) ]] |- O : ( RR* X. RR* ) --> ~P RR* \ {{ x : x }} \ {{ y : y }} \ {{ z : z }} \ {{ R : < }} \ {{ S : < }} \ {{ O : (,) }}

ixxf

fe5qZ4TeaF4=

95s07LnlyUw=

ndmioo

[[ ]] |- (,) = ( x e. RR* , y e. RR* |-> { z e. RR* | ( x < z /\ z < y ) } )

[[ ]] |- (,) = ( x e. RR* , y e. RR* |-> { z e. RR* | ( x < z /\ z < y ) } ) \ {{ x : x }} \ {{ y : y }} \ {{ z : z }}

df-ioo

bosrlJF/OXU=

fe5qZ4TeaF4=

acongid

[[ ]] |- ( ( A e. ZZ /\ B e. ZZ ) -> ( A || ( B - B ) \/ A || ( B - -u B ) ) )

[[ |- ( ph -> ps ) ]] |- ( ph -> ( ps \/ ch ) ) \ {{ ph : ( A e. ZZ /\ B e. ZZ ) }} \ {{ ps : A || ( B - B ) }} \ {{ ch : A || ( B - -u B ) }}

orcd

nuovS4wNBQs=

acongid

[[ ]] |- ( ( A e. ZZ /\ B e. ZZ ) -> A || ( B - B ) )

[[ ]] |- ( ( A e. ZZ /\ B e. ZZ ) -> A || ( B - B ) ) \ {{ A : A }} \ {{ B : B }}

congid

PiK1exGQeH0=

nuovS4wNBQs=

zartop

[[ |- S = ( Spec ` R ) |- J = ( TopOpen ` S ) ]] |- ( R e. CRing -> J e. Top )

[[ |- ( ph -> ps ) |- ( ps -> ch ) ]] |- ( ph -> ch ) \ {{ ph : R e. CRing }} \ {{ ps : J e. ( TopOn ` ( PrmIdeal ` R ) ) }} \ {{ ch : J e. Top }}

syl

f0Hpm4GmbjE=

zartop

[[ |- S = ( Spec ` R ) |- J = ( TopOpen ` S ) ]] |- ( R e. CRing -> J e. ( TopOn ` ( PrmIdeal ` R ) ) )

[[ |- ( ph -> ( ps /\ ch ) ) ]] |- ( ph -> ps ) \ {{ ph : R e. CRing }} \ {{ ps : J e. ( TopOn ` ( PrmIdeal ` R ) ) }} \ {{ ch : ran ( i e. ( LIdeal ` R ) |-> { j e. ( PrmIdeal ` R ) | i C_ j } ) = ( Clsd ` J ) }}

simpld

iYokK9s5aAA=

f0Hpm4GmbjE=

zartop

[[ |- S = ( Spec ` R ) |- J = ( TopOpen ` S ) ]] |- ( R e. CRing -> ( J e. ( TopOn ` ( PrmIdeal ` R ) ) /\ ran ( i e. ( LIdeal ` R ) |-> { j e. ( PrmIdeal ` R ) | i C_ j } ) = ( Clsd ` J ) ) )

[[ |- S = ( Spec ` R ) |- J = ( TopOpen ` S ) |- P = ( PrmIdeal ` R ) |- V = ( i e. ( LIdeal ` R ) |-> { j e. P | i C_ j } ) ]] |- ( R e. CRing -> ( J e. ( TopOn ` P ) /\ ran V = ( Clsd ` J ) ) ) \ {{ P : ( PrmIdeal ` R ) }} \ {{ R : R }} \ {{ S : S }} \ {{ i : k }} \ {{ j : j }} \ {{ J : J }} \ {{ V : ( i e. ( LIdeal ` R ) |-> { j e. ( PrmIdeal ` R ) | i C_ j } ) }}

zartopn

RedKm5ZyPS0=

iYokK9s5aAA=

zartop

[[ ]] |- ( PrmIdeal ` R ) = ( PrmIdeal ` R )

[[ ]] |- A = A \ {{ A : ( PrmIdeal ` R ) }}

eqid

vF/a2Jx+ays=

RedKm5ZyPS0=

zartop

[[ ]] |- ( i e. ( LIdeal ` R ) |-> { j e. ( PrmIdeal ` R ) | i C_ j } ) = ( k e. ( LIdeal ` R ) |-> { j e. ( PrmIdeal ` R ) | k C_ j } )

[[ |- ( x = y -> B = C ) ]] |- ( x e. A |-> B ) = ( y e. A |-> C ) \ {{ x : i }} \ {{ y : k }} \ {{ A : ( LIdeal ` R ) }} \ {{ B : { j e. ( PrmIdeal ` R ) | i C_ j } }} \ {{ C : { j e. ( PrmIdeal ` R ) | k C_ j } }}

cbvmptv

YJjsn4fbEXk=

RedKm5ZyPS0=

zartop

[[ ]] |- ( i = k -> { j e. ( PrmIdeal ` R ) | i C_ j } = { j e. ( PrmIdeal ` R ) | k C_ j } )

[[ |- ( ph -> ( ps <-> ch ) ) ]] |- ( ph -> { x e. A | ps } = { x e. A | ch } ) \ {{ ph : i = k }} \ {{ ps : i C_ j }} \ {{ ch : k C_ j }} \ {{ x : j }} \ {{ A : ( PrmIdeal ` R ) }}

rabbidv

WYJHXbbS+2A=

YJjsn4fbEXk=