Theorem: LO.FirstOrder.Arith.Model.mod_mul (Arithmetization/Basic/IOpen.lean)

State before:
M : Type u_1
inst✝⁶ : Zero M
inst✝⁵ : One M
inst✝⁴ : Add M
inst✝³ : Mul M
inst✝² : LT M
inst✝¹ : M ⊧ₘ* 𝐏𝐀⁻
inst✝ : M ⊧ₘ* 𝐈open
a b m : M
pos : 0 < m
⊢ a * b % m = (m * (a / m) + a % m) * (m * (b / m) + b % m) % m
Tactic:
simp [div_add_mod]
State after:
no goals

State before:
M : Type u_1
inst✝⁶ : Zero M
inst✝⁵ : One M
inst✝⁴ : Add M
inst✝³ : Mul M
inst✝² : LT M
inst✝¹ : M ⊧ₘ* 𝐏𝐀⁻
inst✝ : M ⊧ₘ* 𝐈open
a b m : M
pos : 0 < m
⊢ (m * (a / m) + a % m) * (m * (b / m) + b % m) % m = a % m * (b % m) % m
Tactic:
simp [add_mul, mul_add, pos, mul_left_comm _ m, add_assoc, mul_assoc]
State after:
no goals