Metamath Proof Explorer

Theorem marepvval

Description: Third 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 marepvval ⊢ M ∈ B ∧ C ∈ V ∧ K ∈ N → M Q C ⁡ K = 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 3 4 marepvval0 ⊢ M ∈ B ∧ C ∈ V → M Q C = k ∈ N ⟼ i ∈ N , j ∈ N ⟼ if j = k C ⁡ i i M j
6 5 3adant3 ⊢ M ∈ B ∧ C ∈ V ∧ K ∈ N → M Q C = k ∈ N ⟼ i ∈ N , j ∈ N ⟼ if j = k C ⁡ i i M j
7 6 fveq1d ⊢ M ∈ B ∧ C ∈ V ∧ K ∈ N → M Q C ⁡ K = k ∈ N ⟼ i ∈ N , j ∈ N ⟼ if j = k C ⁡ i i M j ⁡ K
8 eqid ⊢ 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
9 eqeq2 ⊢ k = K → j = k ↔ j = K
10 9 ifbid ⊢ k = K → if j = k C ⁡ i i M j = if j = K C ⁡ i i M j
11 10 mpoeq3dv ⊢ k = K → i ∈ N , j ∈ N ⟼ if j = k C ⁡ i i M j = i ∈ N , j ∈ N ⟼ if j = K C ⁡ i i M j
12 simp3 ⊢ M ∈ B ∧ C ∈ V ∧ K ∈ N → K ∈ N
13 1 2 matrcl ⊢ M ∈ B → N ∈ Fin ∧ R ∈ V
14 13 simpld ⊢ M ∈ B → N ∈ Fin
15 14 14 jca ⊢ M ∈ B → N ∈ Fin ∧ N ∈ Fin
16 15 3ad2ant1 ⊢ M ∈ B ∧ C ∈ V ∧ K ∈ N → N ∈ Fin ∧ N ∈ Fin
17 mpoexga ⊢ N ∈ Fin ∧ N ∈ Fin → i ∈ N , j ∈ N ⟼ if j = K C ⁡ i i M j ∈ V
18 16 17 syl ⊢ M ∈ B ∧ C ∈ V ∧ K ∈ N → i ∈ N , j ∈ N ⟼ if j = K C ⁡ i i M j ∈ V
19 8 11 12 18 fvmptd3 ⊢ M ∈ B ∧ C ∈ V ∧ K ∈ N → k ∈ N ⟼ i ∈ N , j ∈ N ⟼ if j = k C ⁡ i i M j ⁡ K = i ∈ N , j ∈ N ⟼ if j = K C ⁡ i i M j
20 7 19 eqtrd ⊢ M ∈ B ∧ C ∈ V ∧ K ∈ N → M Q C ⁡ K = i ∈ N , j ∈ N ⟼ if j = K C ⁡ i i M j