Theorem mpt2matmul 20252

Description: Multiplication of two N x N matrices given in maps-to notation. (Contributed by AV, 29-Oct-2019.)

Hypotheses

Ref	Expression
mpt2matmul.a	|- A = ( N Mat R )
mpt2matmul.b	|- B = ( Base ` R )
mpt2matmul.m	|- .X. = ( .r ` A )
mpt2matmul.t	|- .x. = ( .r ` R )
mpt2matmul.r	|- ( ph -> R e. V )
mpt2matmul.n	|- ( ph -> N e. Fin )
mpt2matmul.x	|- X = ( i e. N , j e. N |-> C )
mpt2matmul.y	|- Y = ( i e. N , j e. N |-> E )
mpt2matmul.c	|- ( ( ph /\ i e. N /\ j e. N ) -> C e. B )
mpt2matmul.e	|- ( ( ph /\ i e. N /\ j e. N ) -> E e. B )
mpt2matmul.d	|- ( ( ph /\ ( k = i /\ m = j ) ) -> D = C )
mpt2matmul.f	|- ( ( ph /\ ( m = i /\ l = j ) ) -> F = E )
mpt2matmul.1	|- ( ( ph /\ k e. N /\ m e. N ) -> D e. U )
mpt2matmul.2	|- ( ( ph /\ m e. N /\ l e. N ) -> F e. W )

Assertion

Ref	Expression
mpt2matmul	|- ( ph -> ( X .X. Y ) = ( k e. N , l e. N |-> ( R gsumg ( m e. N |-> ( D .x. F ) ) ) ) )

Distinct variable groups:   D, i, j   i, F, j   i, N, j, k, l, m   R, i, j, k, l, m   k, X, l, m   k, Y, l, m   ph, i, j, k, l, m   .x. , k, l

Allowed substitution hints:   A( i, j, k, m, l)   B( i, j, k, m, l)   C( i, j, k, m, l)   D( k, m, l)   .x. ( i, j, m)   .X. ( i, j, k, m, l)   U( i, j, k, m, l)   E( i, j, k, m, l)   F( k, m, l)   V( i, j, k, m, l)   W( i, j, k, m, l)   X( i, j)   Y( i, j)

Proof of Theorem mpt2matmul

Step	Hyp	Ref	Expression
1		mpt2matmul.n	. . 3 |- ( ph -> N e. Fin )
2		mpt2matmul.r	. . 3 |- ( ph -> R e. V )
3		mpt2matmul.a	. . . . . . 7 |- A = ( N Mat R )
4		eqid 2622	. . . . . . 7 |- ( R maMul <. N , N , N >. ) = ( R maMul <. N , N , N >. )
5	3, 4	matmulr 20244	. . . . . 6 |- ( ( N e. Fin /\ R e. V ) -> ( R maMul <. N , N , N >. ) = ( .r ` A ) )
6		mpt2matmul.m	. . . . . 6 |- .X. = ( .r ` A )
7	5, 6	syl6eqr 2674	. . . . 5 |- ( ( N e. Fin /\ R e. V ) -> ( R maMul <. N , N , N >. ) = .X. )
8	7	oveqd 6667	. . . 4 |- ( ( N e. Fin /\ R e. V ) -> ( X ( R maMul <. N , N , N >. ) Y ) = ( X .X. Y ) )
9	8	eqcomd 2628	. . 3 |- ( ( N e. Fin /\ R e. V ) -> ( X .X. Y ) = ( X ( R maMul <. N , N , N >. ) Y ) )
10	1, 2, 9	syl2anc 693	. 2 |- ( ph -> ( X .X. Y ) = ( X ( R maMul <. N , N , N >. ) Y ) )
11		eqid 2622	. . 3 |- ( Base ` R ) = ( Base ` R )
12		mpt2matmul.t	. . 3 |- .x. = ( .r ` R )
13		mpt2matmul.x	. . . . 5 |- X = ( i e. N , j e. N |-> C )
14		eqid 2622	. . . . . 6 |- ( Base ` A ) = ( Base ` A )
15		mpt2matmul.c	. . . . . . 7 |- ( ( ph /\ i e. N /\ j e. N ) -> C e. B )
16		mpt2matmul.b	. . . . . . 7 |- B = ( Base ` R )
17	15, 16	syl6eleq 2711	. . . . . 6 |- ( ( ph /\ i e. N /\ j e. N ) -> C e. ( Base ` R ) )
18	3, 11, 14, 1, 2, 17	matbas2d 20229	. . . . 5 |- ( ph -> ( i e. N , j e. N |-> C ) e. ( Base ` A ) )
19	13, 18	syl5eqel 2705	. . . 4 |- ( ph -> X e. ( Base ` A ) )
20	3, 11	matbas2 20227	. . . . 5 |- ( ( N e. Fin /\ R e. V ) -> ( ( Base ` R ) ^m ( N X. N ) ) = ( Base ` A ) )
21	1, 2, 20	syl2anc 693	. . . 4 |- ( ph -> ( ( Base ` R ) ^m ( N X. N ) ) = ( Base ` A ) )
22	19, 21	eleqtrrd 2704	. . 3 |- ( ph -> X e. ( ( Base ` R ) ^m ( N X. N ) ) )
23		mpt2matmul.y	. . . . 5 |- Y = ( i e. N , j e. N |-> E )
24		mpt2matmul.e	. . . . . . 7 |- ( ( ph /\ i e. N /\ j e. N ) -> E e. B )
25	24, 16	syl6eleq 2711	. . . . . 6 |- ( ( ph /\ i e. N /\ j e. N ) -> E e. ( Base ` R ) )
26	3, 11, 14, 1, 2, 25	matbas2d 20229	. . . . 5 |- ( ph -> ( i e. N , j e. N |-> E ) e. ( Base ` A ) )
27	23, 26	syl5eqel 2705	. . . 4 |- ( ph -> Y e. ( Base ` A ) )
28	27, 21	eleqtrrd 2704	. . 3 |- ( ph -> Y e. ( ( Base ` R ) ^m ( N X. N ) ) )
29	4, 11, 12, 2, 1, 1, 1, 22, 28	mamuval 20192	. 2 |- ( ph -> ( X ( R maMul <. N , N , N >. ) Y ) = ( k e. N , l e. N |-> ( R gsumg ( m e. N |-> ( ( k X m ) .x. ( m Y l ) ) ) ) ) )
30	13	a1i 11	. . . . . . 7 |- ( ( ( ph /\ k e. N /\ l e. N ) /\ m e. N ) -> X = ( i e. N , j e. N |-> C ) )
31		equcom 1945	. . . . . . . . . . . . . 14 |- ( i = k <-> k = i )
32		equcom 1945	. . . . . . . . . . . . . 14 |- ( j = m <-> m = j )
33	31, 32	anbi12i 733	. . . . . . . . . . . . 13 |- ( ( i = k /\ j = m ) <-> ( k = i /\ m = j ) )
34		mpt2matmul.d	. . . . . . . . . . . . 13 |- ( ( ph /\ ( k = i /\ m = j ) ) -> D = C )
35	33, 34	sylan2b 492	. . . . . . . . . . . 12 |- ( ( ph /\ ( i = k /\ j = m ) ) -> D = C )
36	35	eqcomd 2628	. . . . . . . . . . 11 |- ( ( ph /\ ( i = k /\ j = m ) ) -> C = D )
37	36	ex 450	. . . . . . . . . 10 |- ( ph -> ( ( i = k /\ j = m ) -> C = D ) )
38	37	3ad2ant1 1082	. . . . . . . . 9 |- ( ( ph /\ k e. N /\ l e. N ) -> ( ( i = k /\ j = m ) -> C = D ) )
39	38	adantr 481	. . . . . . . 8 |- ( ( ( ph /\ k e. N /\ l e. N ) /\ m e. N ) -> ( ( i = k /\ j = m ) -> C = D ) )
40	39	imp 445	. . . . . . 7 |- ( ( ( ( ph /\ k e. N /\ l e. N ) /\ m e. N ) /\ ( i = k /\ j = m ) ) -> C = D )
41		simpl2 1065	. . . . . . 7 |- ( ( ( ph /\ k e. N /\ l e. N ) /\ m e. N ) -> k e. N )
42		simpr 477	. . . . . . 7 |- ( ( ( ph /\ k e. N /\ l e. N ) /\ m e. N ) -> m e. N )
43		simpl1 1064	. . . . . . . 8 |- ( ( ( ph /\ k e. N /\ l e. N ) /\ m e. N ) -> ph )
44		mpt2matmul.1	. . . . . . . 8 |- ( ( ph /\ k e. N /\ m e. N ) -> D e. U )
45	43, 41, 42, 44	syl3anc 1326	. . . . . . 7 |- ( ( ( ph /\ k e. N /\ l e. N ) /\ m e. N ) -> D e. U )
46	30, 40, 41, 42, 45	ovmpt2d 6788	. . . . . 6 |- ( ( ( ph /\ k e. N /\ l e. N ) /\ m e. N ) -> ( k X m ) = D )
47	23	a1i 11	. . . . . . 7 |- ( ( ( ph /\ k e. N /\ l e. N ) /\ m e. N ) -> Y = ( i e. N , j e. N |-> E ) )
48		equcomi 1944	. . . . . . . . . . . . . 14 |- ( i = m -> m = i )
49		equcomi 1944	. . . . . . . . . . . . . 14 |- ( j = l -> l = j )
50	48, 49	anim12i 590	. . . . . . . . . . . . 13 |- ( ( i = m /\ j = l ) -> ( m = i /\ l = j ) )
51		mpt2matmul.f	. . . . . . . . . . . . 13 |- ( ( ph /\ ( m = i /\ l = j ) ) -> F = E )
52	50, 51	sylan2 491	. . . . . . . . . . . 12 |- ( ( ph /\ ( i = m /\ j = l ) ) -> F = E )
53	52	ex 450	. . . . . . . . . . 11 |- ( ph -> ( ( i = m /\ j = l ) -> F = E ) )
54	53	3ad2ant1 1082	. . . . . . . . . 10 |- ( ( ph /\ k e. N /\ l e. N ) -> ( ( i = m /\ j = l ) -> F = E ) )
55	54	adantr 481	. . . . . . . . 9 |- ( ( ( ph /\ k e. N /\ l e. N ) /\ m e. N ) -> ( ( i = m /\ j = l ) -> F = E ) )
56	55	imp 445	. . . . . . . 8 |- ( ( ( ( ph /\ k e. N /\ l e. N ) /\ m e. N ) /\ ( i = m /\ j = l ) ) -> F = E )
57	56	eqcomd 2628	. . . . . . 7 |- ( ( ( ( ph /\ k e. N /\ l e. N ) /\ m e. N ) /\ ( i = m /\ j = l ) ) -> E = F )
58		simpl3 1066	. . . . . . 7 |- ( ( ( ph /\ k e. N /\ l e. N ) /\ m e. N ) -> l e. N )
59		mpt2matmul.2	. . . . . . . 8 |- ( ( ph /\ m e. N /\ l e. N ) -> F e. W )
60	43, 42, 58, 59	syl3anc 1326	. . . . . . 7 |- ( ( ( ph /\ k e. N /\ l e. N ) /\ m e. N ) -> F e. W )
61	47, 57, 42, 58, 60	ovmpt2d 6788	. . . . . 6 |- ( ( ( ph /\ k e. N /\ l e. N ) /\ m e. N ) -> ( m Y l ) = F )
62	46, 61	oveq12d 6668	. . . . 5 |- ( ( ( ph /\ k e. N /\ l e. N ) /\ m e. N ) -> ( ( k X m ) .x. ( m Y l ) ) = ( D .x. F ) )
63	62	mpteq2dva 4744	. . . 4 |- ( ( ph /\ k e. N /\ l e. N ) -> ( m e. N |-> ( ( k X m ) .x. ( m Y l ) ) ) = ( m e. N |-> ( D .x. F ) ) )
64	63	oveq2d 6666	. . 3 |- ( ( ph /\ k e. N /\ l e. N ) -> ( R gsumg ( m e. N |-> ( ( k X m ) .x. ( m Y l ) ) ) ) = ( R gsumg ( m e. N |-> ( D .x. F ) ) ) )
65	64	mpt2eq3dva 6719	. 2 |- ( ph -> ( k e. N , l e. N |-> ( R gsumg ( m e. N |-> ( ( k X m ) .x. ( m Y l ) ) ) ) ) = ( k e. N , l e. N |-> ( R gsumg ( m e. N |-> ( D .x. F ) ) ) ) )
66	10, 29, 65	3eqtrd 2660	1 |- ( ph -> ( X .X. Y ) = ( k e. N , l e. N |-> ( R gsumg ( m e. N |-> ( D .x. F ) ) ) ) )

Syntax hints:   -> wi 4   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   <.cotp 4185   |-> cmpt 4729   X. cxp 5112   ` cfv 5888  (class class class)co 6650   |-> cmpt2 6652   ^m cmap 7857   Fincfn 7955   Basecbs 15857   .rcmulr 15942   gsum_g cgsu 16101   maMul cmmul 20189   Mat cmat 20213

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  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  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-nel 2898  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-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-ot 4186  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  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-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  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-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-supp 7296  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-oadd 7564  df-er 7742  df-map 7859  df-ixp 7909  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-fsupp 8276  df-sup 8348  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-nn 11021  df-2 11079  df-3 11080  df-4 11081  df-5 11082  df-6 11083  df-7 11084  df-8 11085  df-9 11086  df-n0 11293  df-z 11378  df-dec 11494  df-uz 11688  df-fz 12327  df-struct 15859  df-ndx 15860  df-slot 15861  df-base 15863  df-sets 15864  df-ress 15865  df-plusg 15954  df-mulr 15955  df-sca 15957  df-vsca 15958  df-ip 15959  df-tset 15960  df-ple 15961  df-ds 15964  df-hom 15966  df-cco 15967  df-0g 16102  df-prds 16108  df-pws 16110  df-sra 19172  df-rgmod 19173  df-dsmm 20076  df-frlm 20091  df-mamu 20190  df-mat 20214

This theorem is referenced by:  mat2pmatmul  20536