-- Mathlib/RingTheory/MvPolynomial/WeightedHomogeneous.lean
theorem weightedHomogeneousComponent_of_mem [DecidableEq M] {m n : M}
    {p : MvPolynomial σ R} (h : p ∈ weightedHomogeneousSubmodule R w n) :
    weightedHomogeneousComponent w m p = if m = n then p else 0

Proof state:
case pos
R : Type u_1
M : Type u_2
inst✝² : CommSemiring R
σ : Type u_3
inst✝¹ : AddCommMonoid M
w : σ → M
inst✝ : DecidableEq M
m n : M
p : MvPolynomial σ R
h : IsWeightedHomogeneous w p n
x : σ →₀ ℕ
zero_coeff : coeff x p = 0
⊢ (if (weight w) x = m then coeff x p else 0) = coeff x (if m = n then p else 0)

Tactic:
split_ifs <;>
simp only [zero_coeff, coeff_zero]

Result:
no goals

Proof state:
case neg
R : Type u_1
M : Type u_2
inst✝² : CommSemiring R
σ : Type u_3
inst✝¹ : AddCommMonoid M
w : σ → M
inst✝ : DecidableEq M
m n : M
p : MvPolynomial σ R
h : IsWeightedHomogeneous w p n
x : σ →₀ ℕ
zero_coeff : ¬coeff x p = 0
⊢ (if n = m then coeff x p else 0) = coeff x (if m = n then p else 0)

Tactic:
simp only [show n = m ↔ m = n from eq_comm]

Result:
case neg
R : Type u_1
M : Type u_2
inst✝² : CommSemiring R
σ : Type u_3
inst✝¹ : AddCommMonoid M
w : σ → M
inst✝ : DecidableEq M
m n : M
p : MvPolynomial σ R
h : IsWeightedHomogeneous w p n
x : σ →₀ ℕ
zero_coeff : ¬coeff x p = 0
⊢ (if m = n then coeff x p else 0) = coeff x (if m = n then p else 0)

Proof state:
case neg
R : Type u_1
M : Type u_2
inst✝² : CommSemiring R
σ : Type u_3
inst✝¹ : AddCommMonoid M
w : σ → M
inst✝ : DecidableEq M
m n : M
p : MvPolynomial σ R
h : IsWeightedHomogeneous w p n
x : σ →₀ ℕ
zero_coeff : ¬coeff x p = 0
⊢ (if (weight w) x = m then coeff x p else 0) = coeff x (if m = n then p else 0)

Tactic:
rw [h zero_coeff]

Result:
case neg
R : Type u_1
M : Type u_2
inst✝² : CommSemiring R
σ : Type u_3
inst✝¹ : AddCommMonoid M
w : σ → M
inst✝ : DecidableEq M
m n : M
p : MvPolynomial σ R
h : IsWeightedHomogeneous w p n
x : σ →₀ ℕ
zero_coeff : ¬coeff x p = 0
⊢ (if n = m then coeff x p else 0) = coeff x (if m = n then p else 0)

Proof state:
case a
R : Type u_1
M : Type u_2
inst✝² : CommSemiring R
σ : Type u_3
inst✝¹ : AddCommMonoid M
w : σ → M
inst✝ : DecidableEq M
m n : M
p : MvPolynomial σ R
h : IsWeightedHomogeneous w p n
x : σ →₀ ℕ
⊢ (if (weight w) x = m then coeff x p else 0) = coeff x (if m = n then p else 0)

Tactic:
by_cases zero_coeff : coeff x p = 0

Result:
case pos
R : Type u_1
M : Type u_2
inst✝² : CommSemiring R
σ : Type u_3
inst✝¹ : AddCommMonoid M
w : σ → M
inst✝ : DecidableEq M
m n : M
p : MvPolynomial σ R
h : IsWeightedHomogeneous w p n
x : σ →₀ ℕ
zero_coeff : coeff x p = 0
⊢ (if (weight w) x = m then coeff x p else 0) = coeff x (if m = n then p else 0)

case neg
R : Type u_1
M : Type u_2
inst✝² : CommSemiring R
σ : Type u_3
inst✝¹ : AddCommMonoid M
w : σ → M
inst✝ : DecidableEq M
m n : M
p : MvPolynomial σ R
h : IsWeightedHomogeneous w p n
x : σ →₀ ℕ
zero_coeff : ¬coeff x p = 0
⊢ (if (weight w) x = m then coeff x p else 0) = coeff x (if m = n then p else 0)

Proof state:
case neg
R : Type u_1
M : Type u_2
inst✝² : CommSemiring R
σ : Type u_3
inst✝¹ : AddCommMonoid M
w : σ → M
inst✝ : DecidableEq M
m n : M
p : MvPolynomial σ R
h : IsWeightedHomogeneous w p n
x : σ →₀ ℕ
zero_coeff : ¬coeff x p = 0
⊢ (if m = n then coeff x p else 0) = coeff x (if m = n then p else 0)

Tactic:
split_ifs with h1

Result:
case pos
R : Type u_1
M : Type u_2
inst✝² : CommSemiring R
σ : Type u_3
inst✝¹ : AddCommMonoid M
w : σ → M
inst✝ : DecidableEq M
m n : M
p : MvPolynomial σ R
h : IsWeightedHomogeneous w p n
x : σ →₀ ℕ
zero_coeff : ¬coeff x p = 0
h1 : m = n
⊢ coeff x p = coeff x p

case neg
R : Type u_1
M : Type u_2
inst✝² : CommSemiring R
σ : Type u_3
inst✝¹ : AddCommMonoid M
w : σ → M
inst✝ : DecidableEq M
m n : M
p : MvPolynomial σ R
h : IsWeightedHomogeneous w p n
x : σ →₀ ℕ
zero_coeff : ¬coeff x p = 0
h1 : ¬m = n
⊢ 0 = coeff x 0

Proof state:
case neg
R : Type u_1
M : Type u_2
inst✝² : CommSemiring R
σ : Type u_3
inst✝¹ : AddCommMonoid M
w : σ → M
inst✝ : DecidableEq M
m n : M
p : MvPolynomial σ R
h : IsWeightedHomogeneous w p n
x : σ →₀ ℕ
zero_coeff : ¬coeff x p = 0
h1 : ¬m = n
⊢ 0 = coeff x 0

Tactic:
simp only [coeff_zero]

Result:
no goals

Proof state:
R : Type u_1
M : Type u_2
inst✝² : CommSemiring R
σ : Type u_3
inst✝¹ : AddCommMonoid M
w : σ → M
inst✝ : DecidableEq M
m n : M
p : MvPolynomial σ R
h : IsWeightedHomogeneous w p n
⊢ (weightedHomogeneousComponent w m) p = if m = n then p else 0

Tactic:
ext x

Result:
case a
R : Type u_1
M : Type u_2
inst✝² : CommSemiring R
σ : Type u_3
inst✝¹ : AddCommMonoid M
w : σ → M
inst✝ : DecidableEq M
m n : M
p : MvPolynomial σ R
h : IsWeightedHomogeneous w p n
x : σ →₀ ℕ
⊢ coeff x ((weightedHomogeneousComponent w m) p) = coeff x (if m = n then p else 0)

Proof state:
case pos
R : Type u_1
M : Type u_2
inst✝² : CommSemiring R
σ : Type u_3
inst✝¹ : AddCommMonoid M
w : σ → M
inst✝ : DecidableEq M
m n : M
p : MvPolynomial σ R
h : IsWeightedHomogeneous w p n
x : σ →₀ ℕ
zero_coeff : ¬coeff x p = 0
h1 : m = n
⊢ coeff x p = coeff x p

Tactic:
rfl

Result:
no goals

Proof state:
case a
R : Type u_1
M : Type u_2
inst✝² : CommSemiring R
σ : Type u_3
inst✝¹ : AddCommMonoid M
w : σ → M
inst✝ : DecidableEq M
m n : M
p : MvPolynomial σ R
h : IsWeightedHomogeneous w p n
x : σ →₀ ℕ
⊢ coeff x ((weightedHomogeneousComponent w m) p) = coeff x (if m = n then p else 0)

Tactic:
rw [coeff_weightedHomogeneousComponent]

Result:
case a
R : Type u_1
M : Type u_2
inst✝² : CommSemiring R
σ : Type u_3
inst✝¹ : AddCommMonoid M
w : σ → M
inst✝ : DecidableEq M
m n : M
p : MvPolynomial σ R
h : IsWeightedHomogeneous w p n
x : σ →₀ ℕ
⊢ (if (weight w) x = m then coeff x p else 0) = coeff x (if m = n then p else 0)

Proof state:
R : Type u_1
M : Type u_2
inst✝² : CommSemiring R
σ : Type u_3
inst✝¹ : AddCommMonoid M
w : σ → M
inst✝ : DecidableEq M
m n : M
p : MvPolynomial σ R
h : p ∈ weightedHomogeneousSubmodule R w n
⊢ (weightedHomogeneousComponent w m) p = if m = n then p else 0

Tactic:
simp only [mem_weightedHomogeneousSubmodule] at h

Result:
R : Type u_1
M : Type u_2
inst✝² : CommSemiring R
σ : Type u_3
inst✝¹ : AddCommMonoid M
w : σ → M
inst✝ : DecidableEq M
m n : M
p : MvPolynomial σ R
h : IsWeightedHomogeneous w p n
⊢ (weightedHomogeneousComponent w m) p = if m = n then p else 0