# 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 = ( 1r𝑅 )
minmar1fval.z 0 = ( 0g𝑅 )
Assertion minmar1val0 ( 𝑀𝐵 → ( 𝑄𝑀 ) = ( 𝑘𝑁 , 𝑙𝑁 ↦ ( 𝑖𝑁 , 𝑗𝑁 ↦ 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 = ( 1r𝑅 )
5 minmar1fval.z 0 = ( 0g𝑅 )
6 1 2 matrcl ( 𝑀𝐵 → ( 𝑁 ∈ Fin ∧ 𝑅 ∈ V ) )
7 6 simpld ( 𝑀𝐵𝑁 ∈ Fin )
8 mpoexga ( ( 𝑁 ∈ Fin ∧ 𝑁 ∈ Fin ) → ( 𝑘𝑁 , 𝑙𝑁 ↦ ( 𝑖𝑁 , 𝑗𝑁 ↦ if ( 𝑖 = 𝑘 , if ( 𝑗 = 𝑙 , 1 , 0 ) , ( 𝑖 𝑀 𝑗 ) ) ) ) ∈ V )
9 7 7 8 syl2anc ( 𝑀𝐵 → ( 𝑘𝑁 , 𝑙𝑁 ↦ ( 𝑖𝑁 , 𝑗𝑁 ↦ 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 → ( 𝑖𝑁 , 𝑗𝑁 ↦ if ( 𝑖 = 𝑘 , if ( 𝑗 = 𝑙 , 1 , 0 ) , ( 𝑖 𝑚 𝑗 ) ) ) = ( 𝑖𝑁 , 𝑗𝑁 ↦ if ( 𝑖 = 𝑘 , if ( 𝑗 = 𝑙 , 1 , 0 ) , ( 𝑖 𝑀 𝑗 ) ) ) )
13 12 mpoeq3dv ( 𝑚 = 𝑀 → ( 𝑘𝑁 , 𝑙𝑁 ↦ ( 𝑖𝑁 , 𝑗𝑁 ↦ if ( 𝑖 = 𝑘 , if ( 𝑗 = 𝑙 , 1 , 0 ) , ( 𝑖 𝑚 𝑗 ) ) ) ) = ( 𝑘𝑁 , 𝑙𝑁 ↦ ( 𝑖𝑁 , 𝑗𝑁 ↦ if ( 𝑖 = 𝑘 , if ( 𝑗 = 𝑙 , 1 , 0 ) , ( 𝑖 𝑀 𝑗 ) ) ) ) )
14 1 2 3 4 5 minmar1fval 𝑄 = ( 𝑚𝐵 ↦ ( 𝑘𝑁 , 𝑙𝑁 ↦ ( 𝑖𝑁 , 𝑗𝑁 ↦ if ( 𝑖 = 𝑘 , if ( 𝑗 = 𝑙 , 1 , 0 ) , ( 𝑖 𝑚 𝑗 ) ) ) ) )
15 13 14 fvmptg ( ( 𝑀𝐵 ∧ ( 𝑘𝑁 , 𝑙𝑁 ↦ ( 𝑖𝑁 , 𝑗𝑁 ↦ if ( 𝑖 = 𝑘 , if ( 𝑗 = 𝑙 , 1 , 0 ) , ( 𝑖 𝑀 𝑗 ) ) ) ) ∈ V ) → ( 𝑄𝑀 ) = ( 𝑘𝑁 , 𝑙𝑁 ↦ ( 𝑖𝑁 , 𝑗𝑁 ↦ if ( 𝑖 = 𝑘 , if ( 𝑗 = 𝑙 , 1 , 0 ) , ( 𝑖 𝑀 𝑗 ) ) ) ) )
16 9 15 mpdan ( 𝑀𝐵 → ( 𝑄𝑀 ) = ( 𝑘𝑁 , 𝑙𝑁 ↦ ( 𝑖𝑁 , 𝑗𝑁 ↦ if ( 𝑖 = 𝑘 , if ( 𝑗 = 𝑙 , 1 , 0 ) , ( 𝑖 𝑀 𝑗 ) ) ) ) )