Theorem: LO.FirstOrder.Arith.Model.Exponential.range_exists_unique (Arithmetization/ISigmaZero/Exponential/Exp.lean)

State before:
M : Type u_1
inst✝⁵ : Zero M
inst✝⁴ : One M
inst✝³ : Add M
inst✝² : Mul M
inst✝¹ : LT M
inst✝ : M ⊧ₘ* 𝐈𝚺₁
x : M
⊢ ∃! y, Exponential x y
Tactic:
rcases range_exists x with ⟨y, h⟩
State after:
case intro
M : Type u_1
inst✝⁵ : Zero M
inst✝⁴ : One M
inst✝³ : Add M
inst✝² : Mul M
inst✝¹ : LT M
inst✝ : M ⊧ₘ* 𝐈𝚺₁
x y : M
h : Exponential x y
⊢ ∃! y, Exponential x y

State before:
case intro
M : Type u_1
inst✝⁵ : Zero M
inst✝⁴ : One M
inst✝³ : Add M
inst✝² : Mul M
inst✝¹ : LT M
inst✝ : M ⊧ₘ* 𝐈𝚺₁
x y : M
h : Exponential x y
⊢ ∃! y, Exponential x y
Tactic:
exact ExistsUnique.intro y h (by intro y' h'; exact h'.uniq h)
State after:
no goals

State before:
M : Type u_1
inst✝⁵ : Zero M
inst✝⁴ : One M
inst✝³ : Add M
inst✝² : Mul M
inst✝¹ : LT M
inst✝ : M ⊧ₘ* 𝐈𝚺₁
x y : M
h : Exponential x y
⊢ ∀ (y_1 : M), Exponential x y_1 → y_1 = y
Tactic:
intro y' h'
State after:
M : Type u_1
inst✝⁵ : Zero M
inst✝⁴ : One M
inst✝³ : Add M
inst✝² : Mul M
inst✝¹ : LT M
inst✝ : M ⊧ₘ* 𝐈𝚺₁
x y : M
h : Exponential x y
y' : M
h' : Exponential x y'
⊢ y' = y

State before:
M : Type u_1
inst✝⁵ : Zero M
inst✝⁴ : One M
inst✝³ : Add M
inst✝² : Mul M
inst✝¹ : LT M
inst✝ : M ⊧ₘ* 𝐈𝚺₁
x y : M
h : Exponential x y
y' : M
h' : Exponential x y'
⊢ y' = y
Tactic:
exact h'.uniq h
State after:
no goals