Theorem: LO.FirstOrder.Arith.Model.Pow2.mul_iff (Arithmetization/ISigmaZero/Exponential/Pow2.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* 𝐈𝚫₀
a b : M
h : Pow2 (a * b)
⊢ a ∣ a * b
Tactic:
simp
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* 𝐈𝚫₀
a b : M
h : Pow2 (a * b)
⊢ b ∣ a * b
Tactic:
simp
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* 𝐈𝚫₀
a b : M
⊢ Pow2 a ∧ Pow2 b → Pow2 (a * b)
Tactic:
rintro ⟨ha, hb⟩
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* 𝐈𝚫₀
a b : M
ha : Pow2 a
hb : Pow2 b
⊢ Pow2 (a * b)

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* 𝐈𝚫₀
a b : M
ha : Pow2 a
hb : Pow2 b
⊢ Pow2 (a * b)
Tactic:
exact ha.mul hb
State after:
no goals