This is an inofficial mirror of http://metamath.tirix.org for personal testing of a visualizer extension only.

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 ) ↑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 ) ) ) ) )