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 ) ↑m 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 ) ↑m 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 ) ) ) ) )