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 ) ↑m 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 ) ↑m 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 ) ) ) )