Metamath Proof Explorer

Theorem marrepfval

Description: First substitution for the definition of the matrix row replacement function. (Contributed by AV, 12-Feb-2019) (Proof shortened by AV, 2-Mar-2024)

Ref Expression
Hypotheses marrepfval.a 
|- A = ( N Mat R )
marrepfval.b 
|- B = ( Base ` A )
marrepfval.q 
|- Q = ( N matRRep R )
marrepfval.z 
|- .0. = ( 0g ` R )
Assertion marrepfval 
|- Q = ( m e. B , s e. ( Base ` R ) |-> ( k e. N , l e. N |-> ( i e. N , j e. N |-> if ( i = k , if ( j = l , s , .0. ) , ( i m j ) ) ) ) )

Proof

Step Hyp Ref Expression
1 marrepfval.a 
 |-  A = ( N Mat R )
2 marrepfval.b 
 |-  B = ( Base ` A )
3 marrepfval.q 
 |-  Q = ( N matRRep R )
4 marrepfval.z 
 |-  .0. = ( 0g ` R )
5 2 fvexi 
 |-  B e. _V
6 fvexd 
 |-  ( ( N e. _V /\ R e. _V ) -> ( Base ` R ) e. _V )
7 mpoexga 
 |-  ( ( B e. _V /\ ( Base ` R ) e. _V ) -> ( m e. B , s e. ( Base ` R ) |-> ( k e. N , l e. N |-> ( i e. N , j e. N |-> if ( i = k , if ( j = l , s , .0. ) , ( i m j ) ) ) ) ) e. _V )
8 5 6 7 sylancr 
 |-  ( ( N e. _V /\ R e. _V ) -> ( m e. B , s e. ( Base ` R ) |-> ( k e. N , l e. N |-> ( i e. N , j e. N |-> if ( i = k , if ( j = l , s , .0. ) , ( i m j ) ) ) ) ) e. _V )
9 oveq12 
 |-  ( ( n = N /\ r = R ) -> ( n Mat r ) = ( N Mat R ) )
10 9 fveq2d 
 |-  ( ( n = N /\ r = R ) -> ( Base ` ( n Mat r ) ) = ( Base ` ( N Mat R ) ) )
11 1 fveq2i 
 |-  ( Base ` A ) = ( Base ` ( N Mat R ) )
12 2 11 eqtri 
 |-  B = ( Base ` ( N Mat R ) )
13 10 12 eqtr4di 
 |-  ( ( n = N /\ r = R ) -> ( Base ` ( n Mat r ) ) = B )
14 fveq2 
 |-  ( r = R -> ( Base ` r ) = ( Base ` R ) )
15 14 adantl 
 |-  ( ( n = N /\ r = R ) -> ( Base ` r ) = ( Base ` R ) )
16 simpl 
 |-  ( ( n = N /\ r = R ) -> n = N )
17 fveq2 
 |-  ( r = R -> ( 0g ` r ) = ( 0g ` R ) )
18 17 4 eqtr4di 
 |-  ( r = R -> ( 0g ` r ) = .0. )
19 18 ifeq2d 
 |-  ( r = R -> if ( j = l , s , ( 0g ` r ) ) = if ( j = l , s , .0. ) )
20 19 ifeq1d 
 |-  ( r = R -> if ( i = k , if ( j = l , s , ( 0g ` r ) ) , ( i m j ) ) = if ( i = k , if ( j = l , s , .0. ) , ( i m j ) ) )
21 20 adantl 
 |-  ( ( n = N /\ r = R ) -> if ( i = k , if ( j = l , s , ( 0g ` r ) ) , ( i m j ) ) = if ( i = k , if ( j = l , s , .0. ) , ( i m j ) ) )
22 16 16 21 mpoeq123dv 
 |-  ( ( n = N /\ r = R ) -> ( i e. n , j e. n |-> if ( i = k , if ( j = l , s , ( 0g ` r ) ) , ( i m j ) ) ) = ( i e. N , j e. N |-> if ( i = k , if ( j = l , s , .0. ) , ( i m j ) ) ) )
23 16 16 22 mpoeq123dv 
 |-  ( ( n = N /\ r = R ) -> ( k e. n , l e. n |-> ( i e. n , j e. n |-> if ( i = k , if ( j = l , s , ( 0g ` r ) ) , ( i m j ) ) ) ) = ( k e. N , l e. N |-> ( i e. N , j e. N |-> if ( i = k , if ( j = l , s , .0. ) , ( i m j ) ) ) ) )
24 13 15 23 mpoeq123dv 
 |-  ( ( n = N /\ r = R ) -> ( m e. ( Base ` ( n Mat r ) ) , s e. ( Base ` r ) |-> ( k e. n , l e. n |-> ( i e. n , j e. n |-> if ( i = k , if ( j = l , s , ( 0g ` r ) ) , ( i m j ) ) ) ) ) = ( m e. B , s e. ( Base ` R ) |-> ( k e. N , l e. N |-> ( i e. N , j e. N |-> if ( i = k , if ( j = l , s , .0. ) , ( i m j ) ) ) ) ) )
25 df-marrep 
 |-  matRRep = ( n e. _V , r e. _V |-> ( m e. ( Base ` ( n Mat r ) ) , s e. ( Base ` r ) |-> ( k e. n , l e. n |-> ( i e. n , j e. n |-> if ( i = k , if ( j = l , s , ( 0g ` r ) ) , ( i m j ) ) ) ) ) )
26 24 25 ovmpoga 
 |-  ( ( N e. _V /\ R e. _V /\ ( m e. B , s e. ( Base ` R ) |-> ( k e. N , l e. N |-> ( i e. N , j e. N |-> if ( i = k , if ( j = l , s , .0. ) , ( i m j ) ) ) ) ) e. _V ) -> ( N matRRep R ) = ( m e. B , s e. ( Base ` R ) |-> ( k e. N , l e. N |-> ( i e. N , j e. N |-> if ( i = k , if ( j = l , s , .0. ) , ( i m j ) ) ) ) ) )
27 8 26 mpd3an3 
 |-  ( ( N e. _V /\ R e. _V ) -> ( N matRRep R ) = ( m e. B , s e. ( Base ` R ) |-> ( k e. N , l e. N |-> ( i e. N , j e. N |-> if ( i = k , if ( j = l , s , .0. ) , ( i m j ) ) ) ) ) )
28 25 mpondm0 
 |-  ( -. ( N e. _V /\ R e. _V ) -> ( N matRRep R ) = (/) )
29 matbas0pc 
 |-  ( -. ( N e. _V /\ R e. _V ) -> ( Base ` ( N Mat R ) ) = (/) )
30 12 29 syl5eq 
 |-  ( -. ( N e. _V /\ R e. _V ) -> B = (/) )
31 30 orcd 
 |-  ( -. ( N e. _V /\ R e. _V ) -> ( B = (/) \/ ( Base ` R ) = (/) ) )
32 0mpo0 
 |-  ( ( B = (/) \/ ( Base ` R ) = (/) ) -> ( m e. B , s e. ( Base ` R ) |-> ( k e. N , l e. N |-> ( i e. N , j e. N |-> if ( i = k , if ( j = l , s , .0. ) , ( i m j ) ) ) ) ) = (/) )
33 31 32 syl 
 |-  ( -. ( N e. _V /\ R e. _V ) -> ( m e. B , s e. ( Base ` R ) |-> ( k e. N , l e. N |-> ( i e. N , j e. N |-> if ( i = k , if ( j = l , s , .0. ) , ( i m j ) ) ) ) ) = (/) )
34 28 33 eqtr4d 
 |-  ( -. ( N e. _V /\ R e. _V ) -> ( N matRRep R ) = ( m e. B , s e. ( Base ` R ) |-> ( k e. N , l e. N |-> ( i e. N , j e. N |-> if ( i = k , if ( j = l , s , .0. ) , ( i m j ) ) ) ) ) )
35 27 34 pm2.61i 
 |-  ( N matRRep R ) = ( m e. B , s e. ( Base ` R ) |-> ( k e. N , l e. N |-> ( i e. N , j e. N |-> if ( i = k , if ( j = l , s , .0. ) , ( i m j ) ) ) ) )
36 3 35 eqtri 
 |-  Q = ( m e. B , s e. ( Base ` R ) |-> ( k e. N , l e. N |-> ( i e. N , j e. N |-> if ( i = k , if ( j = l , s , .0. ) , ( i m j ) ) ) ) )