Metamath Proof Explorer

Theorem marepvval0

Description: Second substitution for the definition of the function replacing a column of a matrix by a vector. (Contributed by AV, 14-Feb-2019) (Revised by AV, 26-Feb-2019)

Ref Expression
Hypotheses marepvfval.a ⊢ A = N Mat R
marepvfval.b ⊢ B = Base A
marepvfval.q ⊢ Q = N matRepV R
marepvfval.v ⊢ V = Base R N
Assertion marepvval0 ⊢ M ∈ B ∧ C ∈ V → M Q C = k ∈ N ⟼ i ∈ N , j ∈ N ⟼ if j = k C ⁡ i i M j

Proof

Step Hyp Ref Expression
1 marepvfval.a ⊢ A = N Mat R
2 marepvfval.b ⊢ B = Base A
3 marepvfval.q ⊢ Q = N matRepV R
4 marepvfval.v ⊢ V = Base R N
5 1 2 matrcl ⊢ M ∈ B → N ∈ Fin ∧ R ∈ V
6 5 simpld ⊢ M ∈ B → N ∈ Fin
7 6 adantr ⊢ M ∈ B ∧ C ∈ V → N ∈ Fin
8 7 mptexd ⊢ M ∈ B ∧ C ∈ V → k ∈ N ⟼ i ∈ N , j ∈ N ⟼ if j = k C ⁡ i i M j ∈ V
9 fveq1 ⊢ c = C → c ⁡ i = C ⁡ i
10 9 adantl ⊢ m = M ∧ c = C → c ⁡ i = C ⁡ i
11 oveq ⊢ m = M → i m j = i M j
12 11 adantr ⊢ m = M ∧ c = C → i m j = i M j
13 10 12 ifeq12d ⊢ m = M ∧ c = C → if j = k c ⁡ i i m j = if j = k C ⁡ i i M j
14 13 mpoeq3dv ⊢ m = M ∧ c = C → i ∈ N , j ∈ N ⟼ if j = k c ⁡ i i m j = i ∈ N , j ∈ N ⟼ if j = k C ⁡ i i M j
15 14 mpteq2dv ⊢ m = M ∧ c = C → k ∈ N ⟼ i ∈ N , j ∈ N ⟼ if j = k c ⁡ i i m j = k ∈ N ⟼ i ∈ N , j ∈ N ⟼ if j = k C ⁡ i i M j
16 1 2 3 4 marepvfval ⊢ Q = m ∈ B , c ∈ V ⟼ k ∈ N ⟼ i ∈ N , j ∈ N ⟼ if j = k c ⁡ i i m j
17 15 16 ovmpoga ⊢ M ∈ B ∧ C ∈ V ∧ k ∈ N ⟼ i ∈ N , j ∈ N ⟼ if j = k C ⁡ i i M j ∈ V → M Q C = k ∈ N ⟼ i ∈ N , j ∈ N ⟼ if j = k C ⁡ i i M j
18 8 17 mpd3an3 ⊢ M ∈ B ∧ C ∈ V → M Q C = k ∈ N ⟼ i ∈ N , j ∈ N ⟼ if j = k C ⁡ i i M j