R✝ : Type u₁
inst✝⁴ : Semiring R✝
S : Type u₂
inst✝³ : CommSemiring S
A : Type u₃
inst✝² : Semiring A
inst✝¹ : Algebra S A
r✝ : R✝ → R✝ → Prop
R : Type u₁
inst✝ : Ring R
r : R → R → Prop
a b : R
⊢ RingQuot.sub r { toQuot := Quot.mk (Rel r) a } { toQuot := Quot.mk (Rel r) b } = { toQuot := Quot.mk (Rel r) (a - b) }
R✝ : Type u₁
inst✝⁴ : Semiring R✝
S : Type u₂
inst✝³ : CommSemiring S
A : Type u₃
inst✝² : Semiring A
inst✝¹ : Algebra S A
r✝ : R✝ → R✝ → Prop
R : Type u₁
inst✝ : Ring R
r : R → R → Prop
a b : R
⊢ { toQuot := Quot.mk (Rel r) a } - { toQuot := Quot.mk (Rel r) b } = { toQuot := Quot.mk (Rel r) (a - b) }
show sub r _ _ = _
R✝ : Type u₁
inst✝⁴ : Semiring R✝
S : Type u₂
inst✝³ : CommSemiring S
A : Type u₃
inst✝² : Semiring A
inst✝¹ : Algebra S A
r✝ : R✝ → R✝ → Prop
R : Type u₁
inst✝ : Ring R
r : R → R → Prop
a b : R
⊢ (match { toQuot := Quot.mk (Rel r) a }, { toQuot := Quot.mk (Rel r) b } with
    | { toQuot := a }, { toQuot := b } =>
      {
        toQuot :=
          Quot.map₂ Sub.sub (_ : ∀ ⦃a b c : R⦄, Rel r b c → Rel r (a - b) (a - c))
            (_ : ∀ ⦃a b c : R⦄, Rel r a b → Rel r (a - c) (b - c)) a b }) =
    { toQuot := Quot.mk (Rel r) (a - b) }
R✝ : Type u₁
inst✝⁴ : Semiring R✝
S : Type u₂
inst✝³ : CommSemiring S
A : Type u₃
inst✝² : Semiring A
inst✝¹ : Algebra S A
r✝ : R✝ → R✝ → Prop
R : Type u₁
inst✝ : Ring R
r : R → R → Prop
a b : R
⊢ RingQuot.sub r { toQuot := Quot.mk (Rel r) a } { toQuot := Quot.mk (Rel r) b } = { toQuot := Quot.mk (Rel r) (a - b) }
rw [sub_def]
no goals
R✝ : Type u₁
inst✝⁴ : Semiring R✝
S : Type u₂
inst✝³ : CommSemiring S
A : Type u₃
inst✝² : Semiring A
inst✝¹ : Algebra S A
r✝ : R✝ → R✝ → Prop
R : Type u₁
inst✝ : Ring R
r : R → R → Prop
a b : R
⊢ (match { toQuot := Quot.mk (Rel r) a }, { toQuot := Quot.mk (Rel r) b } with
    | { toQuot := a }, { toQuot := b } =>
      {
        toQuot :=
          Quot.map₂ Sub.sub (_ : ∀ ⦃a b c : R⦄, Rel r b c → Rel r (a - b) (a - c))
            (_ : ∀ ⦃a b c : R⦄, Rel r a b → Rel r (a - c) (b - c)) a b }) =
    { toQuot := Quot.mk (Rel r) (a - b) }
rfl