case pos
R : Type u_1
inst✝¹ : CommRing R
P : Prop
inst✝ : CharZero R
h_equal : Algebra ℚ R → P
h_mixed : ∀ (p : ℕ), Nat.Prime p → MixedCharZero R p → P
h : ∃ p, p > 0 ∧ MixedCharZero R p
⊢ P

case neg
R : Type u_1
inst✝¹ : CommRing R
P : Prop
inst✝ : CharZero R
h_equal : Algebra ℚ R → P
h_mixed : ∀ (p : ℕ), Nat.Prime p → MixedCharZero R p → P
h : ¬∃ p, p > 0 ∧ MixedCharZero R p
⊢ P
R : Type u_1
inst✝¹ : CommRing R
P : Prop
inst✝ : CharZero R
h_equal : Algebra ℚ R → P
h_mixed : ∀ (p : ℕ), Nat.Prime p → MixedCharZero R p → P
⊢ P
by_cases h : ∃ p > 0, MixedCharZero R p
case pos.intro.intro
R : Type u_1
inst✝¹ : CommRing R
P : Prop
inst✝ : CharZero R
h_equal : Algebra ℚ R → P
h_mixed : ∀ (p : ℕ), Nat.Prime p → MixedCharZero R p → P
p : ℕ
H : p > 0
hp : MixedCharZero R p
⊢ P
case pos
R : Type u_1
inst✝¹ : CommRing R
P : Prop
inst✝ : CharZero R
h_equal : Algebra ℚ R → P
h_mixed : ∀ (p : ℕ), Nat.Prime p → MixedCharZero R p → P
h : ∃ p, p > 0 ∧ MixedCharZero R p
⊢ P
rcases h with ⟨p, ⟨H, hp⟩⟩
case pos.intro.intro
R : Type u_1
inst✝¹ : CommRing R
P : Prop
inst✝ : CharZero R
h_equal : Algebra ℚ R → P
h_mixed : ∀ (p : ℕ), p > 0 → MixedCharZero R p → P
p : ℕ
H : p > 0
hp : MixedCharZero R p
⊢ P
case pos.intro.intro
R : Type u_1
inst✝¹ : CommRing R
P : Prop
inst✝ : CharZero R
h_equal : Algebra ℚ R → P
h_mixed : ∀ (p : ℕ), Nat.Prime p → MixedCharZero R p → P
p : ℕ
H : p > 0
hp : MixedCharZero R p
⊢ P
rw [← MixedCharZero.reduce_to_p_prime] at h_mixed
no goals
case pos.intro.intro
R : Type u_1
inst✝¹ : CommRing R
P : Prop
inst✝ : CharZero R
h_equal : Algebra ℚ R → P
h_mixed : ∀ (p : ℕ), p > 0 → MixedCharZero R p → P
p : ℕ
H : p > 0
hp : MixedCharZero R p
⊢ P
exact h_mixed p H hp
case neg
R : Type u_1
inst✝¹ : CommRing R
P : Prop
inst✝ : CharZero R
h_equal : Algebra ℚ R → P
h_mixed : ∀ (p : ℕ), Nat.Prime p → MixedCharZero R p → P
h : ¬∃ p, p > 0 ∧ MixedCharZero R p
⊢ Algebra ℚ R
case neg
R : Type u_1
inst✝¹ : CommRing R
P : Prop
inst✝ : CharZero R
h_equal : Algebra ℚ R → P
h_mixed : ∀ (p : ℕ), Nat.Prime p → MixedCharZero R p → P
h : ¬∃ p, p > 0 ∧ MixedCharZero R p
⊢ P
apply h_equal
case neg
R : Type u_1
inst✝¹ : CommRing R
P : Prop
inst✝ : CharZero R
h_equal : Algebra ℚ R → P
h_mixed : ∀ (p : ℕ), Nat.Prime p → MixedCharZero R p → P
h : Nonempty (Algebra ℚ R)
⊢ Algebra ℚ R
case neg
R : Type u_1
inst✝¹ : CommRing R
P : Prop
inst✝ : CharZero R
h_equal : Algebra ℚ R → P
h_mixed : ∀ (p : ℕ), Nat.Prime p → MixedCharZero R p → P
h : ¬∃ p, p > 0 ∧ MixedCharZero R p
⊢ Algebra ℚ R
rw [← isEmpty_algebraRat_iff_mixedCharZero, not_isEmpty_iff] at h
no goals
case neg
R : Type u_1
inst✝¹ : CommRing R
P : Prop
inst✝ : CharZero R
h_equal : Algebra ℚ R → P
h_mixed : ∀ (p : ℕ), Nat.Prime p → MixedCharZero R p → P
h : Nonempty (Algebra ℚ R)
⊢ Algebra ℚ R
exact h.some