Theorem: LO.FirstOrder.Arith.Model.le_of_dvd (Arithmetization/Basic/PeanoMinus.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* 𝐏𝐀⁻
a b c : M
h : 0 < b
⊢ a ∣ b → a ≤ b
Tactic:
rintro ⟨c, rfl⟩
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* 𝐏𝐀⁻
a c✝ c : M
h : 0 < a * c
⊢ a ≤ a * c

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* 𝐏𝐀⁻
a c✝ c : M
h : 0 < a * c
⊢ a ≤ a * c
Tactic:
exact le_mul_self_of_pos_right
  (pos_iff_ne_zero.mpr (show c ≠ 0 from by rintro rfl; simp at 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* 𝐏𝐀⁻
a c✝ c : M
h : 0 < a * c
⊢ c ≠ 0
Tactic:
rintro rfl
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* 𝐏𝐀⁻
a c : M
h : 0 < a * 0
⊢ False

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* 𝐏𝐀⁻
a c : M
h : 0 < a * 0
⊢ False
Tactic:
simp at h
State after:
no goals