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

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 e. B -> ( Q ` M ) = ( k e. N , l e. N |-> ( i e. N , j e. 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 e. B -> ( N e. Fin /\ R e. _V ) )
7 6 simpld 
 |-  ( M e. B -> N e. Fin )
8 mpoexga 
 |-  ( ( N e. Fin /\ N e. Fin ) -> ( k e. N , l e. N |-> ( i e. N , j e. N |-> if ( i = k , if ( j = l , .1. , .0. ) , ( i M j ) ) ) ) e. _V )
9 7 7 8 syl2anc 
 |-  ( M e. B -> ( k e. N , l e. N |-> ( i e. N , j e. N |-> if ( i = k , if ( j = l , .1. , .0. ) , ( i M j ) ) ) ) e. _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 e. N , j e. N |-> if ( i = k , if ( j = l , .1. , .0. ) , ( i m j ) ) ) = ( i e. N , j e. N |-> if ( i = k , if ( j = l , .1. , .0. ) , ( i M j ) ) ) )
13 12 mpoeq3dv 
 |-  ( m = M -> ( k e. N , l e. N |-> ( i e. N , j e. N |-> if ( i = k , if ( j = l , .1. , .0. ) , ( i m j ) ) ) ) = ( k e. N , l e. N |-> ( i e. N , j e. N |-> if ( i = k , if ( j = l , .1. , .0. ) , ( i M j ) ) ) ) )
14 1 2 3 4 5 minmar1fval 
 |-  Q = ( m e. B |-> ( k e. N , l e. N |-> ( i e. N , j e. N |-> if ( i = k , if ( j = l , .1. , .0. ) , ( i m j ) ) ) ) )
15 13 14 fvmptg 
 |-  ( ( M e. B /\ ( k e. N , l e. N |-> ( i e. N , j e. N |-> if ( i = k , if ( j = l , .1. , .0. ) , ( i M j ) ) ) ) e. _V ) -> ( Q ` M ) = ( k e. N , l e. N |-> ( i e. N , j e. N |-> if ( i = k , if ( j = l , .1. , .0. ) , ( i M j ) ) ) ) )
16 9 15 mpdan 
 |-  ( M e. B -> ( Q ` M ) = ( k e. N , l e. N |-> ( i e. N , j e. N |-> if ( i = k , if ( j = l , .1. , .0. ) , ( i M j ) ) ) ) )