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Metamath Proof Explorer

Theorem minmar1val0

Description: Second substitution for the definition of a matrix for a minor. (Contributed by AV, 31-Dec-2018)

Ref Expression
Hypotheses minmar1fval.a ⊢ 𝐴 = ( 𝑁 Mat 𝑅 )
minmar1fval.b ⊢ 𝐵 = ( Base ‘ 𝐴 )
minmar1fval.q ⊢ 𝑄 = ( 𝑁 minMatR1 𝑅 )
minmar1fval.o ⊢ 1 = ( 1r ‘ 𝑅 )
minmar1fval.z ⊢ 0 = ( 0g ‘ 𝑅 )
Assertion minmar1val0 ( 𝑀 ∈ 𝐵 → ( 𝑄 ‘ 𝑀 ) = ( 𝑘 ∈ 𝑁 , 𝑙 ∈ 𝑁 ↦ ( 𝑖 ∈ 𝑁 , 𝑗 ∈ 𝑁 ↦ if ( 𝑖 = 𝑘 , if ( 𝑗 = 𝑙 , 1 , 0 ) , ( 𝑖 𝑀 𝑗 ) ) ) ) )

Proof

Step Hyp Ref Expression
1 minmar1fval.a ⊢ 𝐴 = ( 𝑁 Mat 𝑅 )
2 minmar1fval.b ⊢ 𝐵 = ( Base ‘ 𝐴 )
3 minmar1fval.q ⊢ 𝑄 = ( 𝑁 minMatR1 𝑅 )
4 minmar1fval.o ⊢ 1 = ( 1r ‘ 𝑅 )
5 minmar1fval.z ⊢ 0 = ( 0g ‘ 𝑅 )
6 1 2 matrcl ⊢ ( 𝑀 ∈ 𝐵 → ( 𝑁 ∈ Fin ∧ 𝑅 ∈ V ) )
7 6 simpld ⊢ ( 𝑀 ∈ 𝐵 → 𝑁 ∈ Fin )
8 mpoexga ⊢ ( ( 𝑁 ∈ Fin ∧ 𝑁 ∈ Fin ) → ( 𝑘 ∈ 𝑁 , 𝑙 ∈ 𝑁 ↦ ( 𝑖 ∈ 𝑁 , 𝑗 ∈ 𝑁 ↦ if ( 𝑖 = 𝑘 , if ( 𝑗 = 𝑙 , 1 , 0 ) , ( 𝑖 𝑀 𝑗 ) ) ) ) ∈ V )
9 7 7 8 syl2anc ⊢ ( 𝑀 ∈ 𝐵 → ( 𝑘 ∈ 𝑁 , 𝑙 ∈ 𝑁 ↦ ( 𝑖 ∈ 𝑁 , 𝑗 ∈ 𝑁 ↦ if ( 𝑖 = 𝑘 , if ( 𝑗 = 𝑙 , 1 , 0 ) , ( 𝑖 𝑀 𝑗 ) ) ) ) ∈ V )
10 oveq ⊢ ( 𝑚 = 𝑀 → ( 𝑖 𝑚 𝑗 ) = ( 𝑖 𝑀 𝑗 ) )
11 10 ifeq2d ⊢ ( 𝑚 = 𝑀 → if ( 𝑖 = 𝑘 , if ( 𝑗 = 𝑙 , 1 , 0 ) , ( 𝑖 𝑚 𝑗 ) ) = if ( 𝑖 = 𝑘 , if ( 𝑗 = 𝑙 , 1 , 0 ) , ( 𝑖 𝑀 𝑗 ) ) )
12 11 mpoeq3dv ⊢ ( 𝑚 = 𝑀 → ( 𝑖 ∈ 𝑁 , 𝑗 ∈ 𝑁 ↦ if ( 𝑖 = 𝑘 , if ( 𝑗 = 𝑙 , 1 , 0 ) , ( 𝑖 𝑚 𝑗 ) ) ) = ( 𝑖 ∈ 𝑁 , 𝑗 ∈ 𝑁 ↦ if ( 𝑖 = 𝑘 , if ( 𝑗 = 𝑙 , 1 , 0 ) , ( 𝑖 𝑀 𝑗 ) ) ) )
13 12 mpoeq3dv ⊢ ( 𝑚 = 𝑀 → ( 𝑘 ∈ 𝑁 , 𝑙 ∈ 𝑁 ↦ ( 𝑖 ∈ 𝑁 , 𝑗 ∈ 𝑁 ↦ if ( 𝑖 = 𝑘 , if ( 𝑗 = 𝑙 , 1 , 0 ) , ( 𝑖 𝑚 𝑗 ) ) ) ) = ( 𝑘 ∈ 𝑁 , 𝑙 ∈ 𝑁 ↦ ( 𝑖 ∈ 𝑁 , 𝑗 ∈ 𝑁 ↦ if ( 𝑖 = 𝑘 , if ( 𝑗 = 𝑙 , 1 , 0 ) , ( 𝑖 𝑀 𝑗 ) ) ) ) )
14 1 2 3 4 5 minmar1fval ⊢ 𝑄 = ( 𝑚 ∈ 𝐵 ↦ ( 𝑘 ∈ 𝑁 , 𝑙 ∈ 𝑁 ↦ ( 𝑖 ∈ 𝑁 , 𝑗 ∈ 𝑁 ↦ if ( 𝑖 = 𝑘 , if ( 𝑗 = 𝑙 , 1 , 0 ) , ( 𝑖 𝑚 𝑗 ) ) ) ) )
15 13 14 fvmptg ⊢ ( ( 𝑀 ∈ 𝐵 ∧ ( 𝑘 ∈ 𝑁 , 𝑙 ∈ 𝑁 ↦ ( 𝑖 ∈ 𝑁 , 𝑗 ∈ 𝑁 ↦ if ( 𝑖 = 𝑘 , if ( 𝑗 = 𝑙 , 1 , 0 ) , ( 𝑖 𝑀 𝑗 ) ) ) ) ∈ V ) → ( 𝑄 ‘ 𝑀 ) = ( 𝑘 ∈ 𝑁 , 𝑙 ∈ 𝑁 ↦ ( 𝑖 ∈ 𝑁 , 𝑗 ∈ 𝑁 ↦ if ( 𝑖 = 𝑘 , if ( 𝑗 = 𝑙 , 1 , 0 ) , ( 𝑖 𝑀 𝑗 ) ) ) ) )
16 9 15 mpdan ⊢ ( 𝑀 ∈ 𝐵 → ( 𝑄 ‘ 𝑀 ) = ( 𝑘 ∈ 𝑁 , 𝑙 ∈ 𝑁 ↦ ( 𝑖 ∈ 𝑁 , 𝑗 ∈ 𝑁 ↦ if ( 𝑖 = 𝑘 , if ( 𝑗 = 𝑙 , 1 , 0 ) , ( 𝑖 𝑀 𝑗 ) ) ) ) )