Theorem marrepfval 20366

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

Hypotheses

Ref	Expression
marrepfval.a	|- A = ( N Mat R )
marrepfval.b	|- B = ( Base ` A )
marrepfval.q	|- Q = ( N matRRep R )
marrepfval.z	|- .0. = ( 0g ` R )

Assertion

Ref	Expression
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 ) ) ) ) )

Distinct variable groups:   B, m, s   i, N, j, k, l, m, s   R, i, j, k, l, m, s

Allowed substitution hints:   A( i, j, k, m, s, l)   B( i, j, k, l)   Q( i, j, k, m, s, l)   .0. ( i, j, k, m, s, l)

Proof of Theorem marrepfval

Dummy variables n r are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		marrepfval.q	. 2 |- Q = ( N matRRep R )
2		marrepfval.b	. . . . . 6 |- B = ( Base ` A )
3		fvex 6201	. . . . . 6 |- ( Base ` A ) e. _V
4	2, 3	eqeltri 2697	. . . . 5 |- B e. _V
5		fvexd 6203	. . . . 5 |- ( ( N e. _V /\ R e. _V ) -> ( Base ` R ) e. _V )
6		mpt2exga 7246	. . . . 5 |- ( ( 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 )
7	4, 5, 6	sylancr 695	. . . 4 |- ( ( 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 )
8		oveq12 6659	. . . . . . . 8 |- ( ( n = N /\ r = R ) -> ( n Mat r ) = ( N Mat R ) )
9	8	fveq2d 6195	. . . . . . 7 |- ( ( n = N /\ r = R ) -> ( Base ` ( n Mat r ) ) = ( Base ` ( N Mat R ) ) )
10		marrepfval.a	. . . . . . . . 9 |- A = ( N Mat R )
11	10	fveq2i 6194	. . . . . . . 8 |- ( Base ` A ) = ( Base ` ( N Mat R ) )
12	2, 11	eqtri 2644	. . . . . . 7 |- B = ( Base ` ( N Mat R ) )
13	9, 12	syl6eqr 2674	. . . . . 6 |- ( ( n = N /\ r = R ) -> ( Base ` ( n Mat r ) ) = B )
14		fveq2 6191	. . . . . . 7 |- ( r = R -> ( Base ` r ) = ( Base ` R ) )
15	14	adantl 482	. . . . . 6 |- ( ( n = N /\ r = R ) -> ( Base ` r ) = ( Base ` R ) )
16		simpl 473	. . . . . . 7 |- ( ( n = N /\ r = R ) -> n = N )
17		fveq2 6191	. . . . . . . . . . . 12 |- ( r = R -> ( 0g ` r ) = ( 0g ` R ) )
18		marrepfval.z	. . . . . . . . . . . 12 |- .0. = ( 0g ` R )
19	17, 18	syl6eqr 2674	. . . . . . . . . . 11 |- ( r = R -> ( 0g ` r ) = .0. )
20	19	ifeq2d 4105	. . . . . . . . . 10 |- ( r = R -> if ( j = l , s , ( 0g ` r ) ) = if ( j = l , s , .0. ) )
21	20	ifeq1d 4104	. . . . . . . . 9 |- ( 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	21	adantl 482	. . . . . . . 8 |- ( ( 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 ) ) )
23	16, 16, 22	mpt2eq123dv 6717	. . . . . . 7 |- ( ( 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 ) ) ) )
24	16, 16, 23	mpt2eq123dv 6717	. . . . . 6 |- ( ( 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 ) ) ) ) )
25	13, 15, 24	mpt2eq123dv 6717	. . . . 5 |- ( ( 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 ) ) ) ) ) )
26		df-marrep 20364	. . . . 5 |- 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 ) ) ) ) ) )
27	25, 26	ovmpt2ga 6790	. . . 4 |- ( ( 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 ) ) ) ) ) )
28	7, 27	mpd3an3 1425	. . 3 |- ( ( 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 ) ) ) ) ) )
29	26	mpt2ndm0 6875	. . . . 5 |- ( -. ( N e. _V /\ R e. _V ) -> ( N matRRep R ) = (/) )
30		mpt20 6725	. . . . 5 |- ( m e. (/) , 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 ) ) ) ) ) = (/)
31	29, 30	syl6eqr 2674	. . . 4 |- ( -. ( N e. _V /\ R e. _V ) -> ( N matRRep R ) = ( m e. (/) , 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 ) ) ) ) ) )
32		matbas0pc 20215	. . . . . 6 |- ( -. ( N e. _V /\ R e. _V ) -> ( Base ` ( N Mat R ) ) = (/) )
33	12, 32	syl5eq 2668	. . . . 5 |- ( -. ( N e. _V /\ R e. _V ) -> B = (/) )
34		eqidd 2623	. . . . 5 |- ( -. ( N e. _V /\ R e. _V ) -> ( Base ` R ) = ( Base ` R ) )
35		mpt2eq12 6715	. . . . 5 |- ( ( B = (/) /\ ( Base ` R ) = ( 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 ) ) ) ) ) = ( m e. (/) , 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	33, 34, 35	syl2anc 693	. . . 4 |- ( -. ( 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 ) ) ) ) ) = ( m e. (/) , 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 ) ) ) ) ) )
37	31, 36	eqtr4d 2659	. . 3 |- ( -. ( 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 ) ) ) ) ) )
38	28, 37	pm2.61i 176	. 2 |- ( 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 ) ) ) ) )
39	1, 38	eqtri 2644	1 |- 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 ) ) ) ) )

Syntax hints:   -. wn 3   /\ wa 384   = wceq 1483   e. wcel 1990   _Vcvv 3200   (/)c0 3915   ifcif 4086   ` cfv 5888  (class class class)co 6650   |-> cmpt2 6652   Basecbs 15857   0gc0g 16100   Mat cmat 20213   matRRep cmarrep 20362

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-1st 7168  df-2nd 7169  df-slot 15861  df-base 15863  df-mat 20214  df-marrep 20364

This theorem is referenced by:  marrepval0  20367