Theorem marrepval 20368

Description: Third 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
marrepval	|- ( ( ( M e. B /\ S e. ( Base ` R ) ) /\ ( K e. N /\ L e. N ) ) -> ( K ( M Q S ) L ) = ( i e. N , j e. N |-> if ( i = K , if ( j = L , S , .0. ) , ( i M j ) ) ) )

Distinct variable groups:   i, N, j   R, i, j   i, M, j   S, i, j   i, K, j   i, L, j

Allowed substitution hints:   A( i, j)   B( i, j)   Q( i, j)   .0. ( i, j)

Proof of Theorem marrepval

Dummy variables k l are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		marrepfval.a	. . . 4 |- A = ( N Mat R )
2		marrepfval.b	. . . 4 |- B = ( Base ` A )
3		marrepfval.q	. . . 4 |- Q = ( N matRRep R )
4		marrepfval.z	. . . 4 |- .0. = ( 0g ` R )
5	1, 2, 3, 4	marrepval0 20367	. . 3 |- ( ( M e. B /\ S e. ( Base ` R ) ) -> ( M Q S ) = ( k e. N , l e. N |-> ( i e. N , j e. N |-> if ( i = k , if ( j = l , S , .0. ) , ( i M j ) ) ) ) )
6	5	adantr 481	. 2 |- ( ( ( M e. B /\ S e. ( Base ` R ) ) /\ ( K e. N /\ L e. N ) ) -> ( M Q S ) = ( k e. N , l e. N |-> ( i e. N , j e. N |-> if ( i = k , if ( j = l , S , .0. ) , ( i M j ) ) ) ) )
7		simprl 794	. . 3 |- ( ( ( M e. B /\ S e. ( Base ` R ) ) /\ ( K e. N /\ L e. N ) ) -> K e. N )
8		simplrr 801	. . 3 |- ( ( ( ( M e. B /\ S e. ( Base ` R ) ) /\ ( K e. N /\ L e. N ) ) /\ k = K ) -> L e. N )
9	1, 2	matrcl 20218	. . . . . . 7 |- ( M e. B -> ( N e. Fin /\ R e. _V ) )
10	9	simpld 475	. . . . . 6 |- ( M e. B -> N e. Fin )
11	10, 10	jca 554	. . . . 5 |- ( M e. B -> ( N e. Fin /\ N e. Fin ) )
12	11	ad3antrrr 766	. . . 4 |- ( ( ( ( M e. B /\ S e. ( Base ` R ) ) /\ ( K e. N /\ L e. N ) ) /\ ( k = K /\ l = L ) ) -> ( N e. Fin /\ N e. Fin ) )
13		mpt2exga 7246	. . . 4 |- ( ( N e. Fin /\ N e. Fin ) -> ( i e. N , j e. N |-> if ( i = k , if ( j = l , S , .0. ) , ( i M j ) ) ) e. _V )
14	12, 13	syl 17	. . 3 |- ( ( ( ( M e. B /\ S e. ( Base ` R ) ) /\ ( K e. N /\ L e. N ) ) /\ ( k = K /\ l = L ) ) -> ( i e. N , j e. N |-> if ( i = k , if ( j = l , S , .0. ) , ( i M j ) ) ) e. _V )
15		eqeq2 2633	. . . . . . 7 |- ( k = K -> ( i = k <-> i = K ) )
16	15	adantr 481	. . . . . 6 |- ( ( k = K /\ l = L ) -> ( i = k <-> i = K ) )
17		eqeq2 2633	. . . . . . . 8 |- ( l = L -> ( j = l <-> j = L ) )
18	17	ifbid 4108	. . . . . . 7 |- ( l = L -> if ( j = l , S , .0. ) = if ( j = L , S , .0. ) )
19	18	adantl 482	. . . . . 6 |- ( ( k = K /\ l = L ) -> if ( j = l , S , .0. ) = if ( j = L , S , .0. ) )
20	16, 19	ifbieq1d 4109	. . . . 5 |- ( ( k = K /\ l = L ) -> if ( i = k , if ( j = l , S , .0. ) , ( i M j ) ) = if ( i = K , if ( j = L , S , .0. ) , ( i M j ) ) )
21	20	mpt2eq3dv 6721	. . . 4 |- ( ( k = K /\ l = L ) -> ( i e. N , j e. N |-> if ( i = k , if ( j = l , S , .0. ) , ( i M j ) ) ) = ( i e. N , j e. N |-> if ( i = K , if ( j = L , S , .0. ) , ( i M j ) ) ) )
22	21	adantl 482	. . 3 |- ( ( ( ( M e. B /\ S e. ( Base ` R ) ) /\ ( K e. N /\ L e. N ) ) /\ ( k = K /\ l = L ) ) -> ( i e. N , j e. N |-> if ( i = k , if ( j = l , S , .0. ) , ( i M j ) ) ) = ( i e. N , j e. N |-> if ( i = K , if ( j = L , S , .0. ) , ( i M j ) ) ) )
23	7, 8, 14, 22	ovmpt2dv2 6794	. 2 |- ( ( ( M e. B /\ S e. ( Base ` R ) ) /\ ( K e. N /\ L e. N ) ) -> ( ( M Q S ) = ( k e. N , l e. N |-> ( i e. N , j e. N |-> if ( i = k , if ( j = l , S , .0. ) , ( i M j ) ) ) ) -> ( K ( M Q S ) L ) = ( i e. N , j e. N |-> if ( i = K , if ( j = L , S , .0. ) , ( i M j ) ) ) ) )
24	6, 23	mpd 15	1 |- ( ( ( M e. B /\ S e. ( Base ` R ) ) /\ ( K e. N /\ L e. N ) ) -> ( K ( M Q S ) L ) = ( i e. N , j e. N |-> if ( i = K , if ( j = L , S , .0. ) , ( i M j ) ) ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   _Vcvv 3200   ifcif 4086   ` cfv 5888  (class class class)co 6650   |-> cmpt2 6652   Fincfn 7955   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:  marrepeval  20369  marrepcl  20370  1marepvmarrepid  20381  smadiadetglem1  20477  smadiadetglem2  20478  madjusmdetlem1  29893