Theorem: MvPowerSeries.monomial_mul_monomial
File: Mathlib/RingTheory/MvPowerSeries/Basic.lean

tactic: ext k
state_before: σ : Type u_1
R : Type u_2
inst✝ : Semiring R
m n : σ →₀ ℕ
a b : R
⊢ (monomial R m) a * (monomial R n) b = (monomial R (m + n)) (a * b)
state_after: case h
σ : Type u_1
R : Type u_2
inst✝ : Semiring R
m n : σ →₀ ℕ
a b : R
k : σ →₀ ℕ
⊢ (coeff R k) ((monomial R m) a * (monomial R n) b) = (coeff R k) ((monomial R (m + n)) (a * b))

tactic: simp only [coeff_mul_monomial, coeff_monomial]
state_before: case h
σ : Type u_1
R : Type u_2
inst✝ : Semiring R
m n : σ →₀ ℕ
a b : R
k : σ →₀ ℕ
⊢ (coeff R k) ((monomial R m) a * (monomial R n) b) = (coeff R k) ((monomial R (m + n)) (a * b))
state_after: case h
σ : Type u_1
R : Type u_2
inst✝ : Semiring R
m n : σ →₀ ℕ
a b : R
k : σ →₀ ℕ
⊢ (if n ≤ k then (if k - n = m then a else 0) * b else 0) = if k = m + n then a * b else 0

tactic: split_ifs with h₁ h₂ h₃ h₃ h₂ <;> try rfl
state_before: case h
σ : Type u_1
R : Type u_2
inst✝ : Semiring R
m n : σ →₀ ℕ
a b : R
k : σ →₀ ℕ
⊢ (if n ≤ k then (if k - n = m then a else 0) * b else 0) = if k = m + n then a * b else 0
state_after: case neg
σ : Type u_1
R : Type u_2
inst✝ : Semiring R
m n : σ →₀ ℕ
a b : R
k : σ →₀ ℕ
h₁ : n ≤ k
h₂ : k - n = m
h₃ : ¬k = m + n
⊢ a * b = 0

case pos
σ : Type u_1
R : Type u_2
inst✝ : Semiring R
m n : σ →₀ ℕ
a b : R
k : σ →₀ ℕ
h₁ : n ≤ k
h₂ : ¬k - n = m
h₃ : k = m + n
⊢ 0 * b = a * b

case neg
σ : Type u_1
R : Type u_2
inst✝ : Semiring R
m n : σ →₀ ℕ
a b : R
k : σ →₀ ℕ
h₁ : n ≤ k
h₂ : ¬k - n = m
h₃ : ¬k = m + n
⊢ 0 * b = 0

case pos
σ : Type u_1
R : Type u_2
inst✝ : Semiring R
m n : σ →₀ ℕ
a b : R
k : σ →₀ ℕ
h₁ : ¬n ≤ k
h₂ : k = m + n
⊢ 0 = a * b

tactic: rfl
state_before: case neg
σ : Type u_1
R : Type u_2
inst✝ : Semiring R
m n : σ →₀ ℕ
a b : R
k : σ →₀ ℕ
h₁ : ¬n ≤ k
h₂ : ¬k = m + n
⊢ 0 = 0
state_after: no goals

tactic: rw [← h₂, tsub_add_cancel_of_le h₁] at h₃
state_before: case neg
σ : Type u_1
R : Type u_2
inst✝ : Semiring R
m n : σ →₀ ℕ
a b : R
k : σ →₀ ℕ
h₁ : n ≤ k
h₂ : k - n = m
h₃ : ¬k = m + n
⊢ a * b = 0
state_after: case neg
σ : Type u_1
R : Type u_2
inst✝ : Semiring R
m n : σ →₀ ℕ
a b : R
k : σ →₀ ℕ
h₁ : n ≤ k
h₂ : k - n = m
h₃ : ¬k = k
⊢ a * b = 0

tactic: exact (h₃ rfl).elim
state_before: case neg
σ : Type u_1
R : Type u_2
inst✝ : Semiring R
m n : σ →₀ ℕ
a b : R
k : σ →₀ ℕ
h₁ : n ≤ k
h₂ : k - n = m
h₃ : ¬k = k
⊢ a * b = 0
state_after: no goals

tactic: rw [h₃, add_tsub_cancel_right] at h₂
state_before: case pos
σ : Type u_1
R : Type u_2
inst✝ : Semiring R
m n : σ →₀ ℕ
a b : R
k : σ →₀ ℕ
h₁ : n ≤ k
h₂ : ¬k - n = m
h₃ : k = m + n
⊢ 0 * b = a * b
state_after: case pos
σ : Type u_1
R : Type u_2
inst✝ : Semiring R
m n : σ →₀ ℕ
a b : R
k : σ →₀ ℕ
h₁ : n ≤ k
h₂ : ¬m = m
h₃ : k = m + n
⊢ 0 * b = a * b

tactic: exact (h₂ rfl).elim
state_before: case pos
σ : Type u_1
R : Type u_2
inst✝ : Semiring R
m n : σ →₀ ℕ
a b : R
k : σ →₀ ℕ
h₁ : n ≤ k
h₂ : ¬m = m
h₃ : k = m + n
⊢ 0 * b = a * b
state_after: no goals

tactic: exact zero_mul b
state_before: case neg
σ : Type u_1
R : Type u_2
inst✝ : Semiring R
m n : σ →₀ ℕ
a b : R
k : σ →₀ ℕ
h₁ : n ≤ k
h₂ : ¬k - n = m
h₃ : ¬k = m + n
⊢ 0 * b = 0
state_after: no goals

tactic: rw [h₂] at h₁
state_before: case pos
σ : Type u_1
R : Type u_2
inst✝ : Semiring R
m n : σ →₀ ℕ
a b : R
k : σ →₀ ℕ
h₁ : ¬n ≤ k
h₂ : k = m + n
⊢ 0 = a * b
state_after: case pos
σ : Type u_1
R : Type u_2
inst✝ : Semiring R
m n : σ →₀ ℕ
a b : R
k : σ →₀ ℕ
h₁ : ¬n ≤ m + n
h₂ : k = m + n
⊢ 0 = a * b

tactic: exact (h₁ <| le_add_left le_rfl).elim
state_before: case pos
σ : Type u_1
R : Type u_2
inst✝ : Semiring R
m n : σ →₀ ℕ
a b : R
k : σ →₀ ℕ
h₁ : ¬n ≤ m + n
h₂ : k = m + n
⊢ 0 = a * b
state_after: no goals