Theorem mamures 20196

Description: Rows in a matrix product are functions only of the corresponding rows in the left argument. (Contributed by SO, 9-Jul-2018.)

Hypotheses

Ref	Expression
mamures.f	|- F = ( R maMul <. M , N , P >. )
mamures.g	|- G = ( R maMul <. I , N , P >. )
mamures.b	|- B = ( Base ` R )
mamures.r	|- ( ph -> R e. V )
mamures.m	|- ( ph -> M e. Fin )
mamures.n	|- ( ph -> N e. Fin )
mamures.p	|- ( ph -> P e. Fin )
mamures.i	|- ( ph -> I C_ M )
mamures.x	|- ( ph -> X e. ( B ^m ( M X. N ) ) )
mamures.y	|- ( ph -> Y e. ( B ^m ( N X. P ) ) )

Assertion

Ref	Expression
mamures	|- ( ph -> ( ( X F Y ) |` ( I X. P ) ) = ( ( X |` ( I X. N ) ) G Y ) )

Proof of Theorem mamures

Dummy variables i j k are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		mamures.i	. . . 4 |- ( ph -> I C_ M )
2		ssid 3624	. . . . 5 |- P C_ P
3	2	a1i 11	. . . 4 |- ( ph -> P C_ P )
4		resmpt2 6758	. . . 4 |- ( ( I C_ M /\ P C_ P ) -> ( ( i e. M , j e. P |-> ( R gsumg ( k e. N |-> ( ( i X k ) ( .r ` R ) ( k Y j ) ) ) ) ) |` ( I X. P ) ) = ( i e. I , j e. P |-> ( R gsumg ( k e. N |-> ( ( i X k ) ( .r ` R ) ( k Y j ) ) ) ) ) )
5	1, 3, 4	syl2anc 693	. . 3 |- ( ph -> ( ( i e. M , j e. P |-> ( R gsumg ( k e. N |-> ( ( i X k ) ( .r ` R ) ( k Y j ) ) ) ) ) |` ( I X. P ) ) = ( i e. I , j e. P |-> ( R gsumg ( k e. N |-> ( ( i X k ) ( .r ` R ) ( k Y j ) ) ) ) ) )
6		ovres 6800	. . . . . . . . 9 |- ( ( i e. I /\ k e. N ) -> ( i ( X |` ( I X. N ) ) k ) = ( i X k ) )
7	6	3ad2antl2 1224	. . . . . . . 8 |- ( ( ( ph /\ i e. I /\ j e. P ) /\ k e. N ) -> ( i ( X |` ( I X. N ) ) k ) = ( i X k ) )
8	7	eqcomd 2628	. . . . . . 7 |- ( ( ( ph /\ i e. I /\ j e. P ) /\ k e. N ) -> ( i X k ) = ( i ( X |` ( I X. N ) ) k ) )
9	8	oveq1d 6665	. . . . . 6 |- ( ( ( ph /\ i e. I /\ j e. P ) /\ k e. N ) -> ( ( i X k ) ( .r ` R ) ( k Y j ) ) = ( ( i ( X |` ( I X. N ) ) k ) ( .r ` R ) ( k Y j ) ) )
10	9	mpteq2dva 4744	. . . . 5 |- ( ( ph /\ i e. I /\ j e. P ) -> ( k e. N |-> ( ( i X k ) ( .r ` R ) ( k Y j ) ) ) = ( k e. N |-> ( ( i ( X |` ( I X. N ) ) k ) ( .r ` R ) ( k Y j ) ) ) )
11	10	oveq2d 6666	. . . 4 |- ( ( ph /\ i e. I /\ j e. P ) -> ( R gsumg ( k e. N |-> ( ( i X k ) ( .r ` R ) ( k Y j ) ) ) ) = ( R gsumg ( k e. N |-> ( ( i ( X |` ( I X. N ) ) k ) ( .r ` R ) ( k Y j ) ) ) ) )
12	11	mpt2eq3dva 6719	. . 3 |- ( ph -> ( i e. I , j e. P |-> ( R gsumg ( k e. N |-> ( ( i X k ) ( .r ` R ) ( k Y j ) ) ) ) ) = ( i e. I , j e. P |-> ( R gsumg ( k e. N |-> ( ( i ( X |` ( I X. N ) ) k ) ( .r ` R ) ( k Y j ) ) ) ) ) )
13	5, 12	eqtrd 2656	. 2 |- ( ph -> ( ( i e. M , j e. P |-> ( R gsumg ( k e. N |-> ( ( i X k ) ( .r ` R ) ( k Y j ) ) ) ) ) |` ( I X. P ) ) = ( i e. I , j e. P |-> ( R gsumg ( k e. N |-> ( ( i ( X |` ( I X. N ) ) k ) ( .r ` R ) ( k Y j ) ) ) ) ) )
14		mamures.f	. . . 4 |- F = ( R maMul <. M , N , P >. )
15		mamures.b	. . . 4 |- B = ( Base ` R )
16		eqid 2622	. . . 4 |- ( .r ` R ) = ( .r ` R )
17		mamures.r	. . . 4 |- ( ph -> R e. V )
18		mamures.m	. . . 4 |- ( ph -> M e. Fin )
19		mamures.n	. . . 4 |- ( ph -> N e. Fin )
20		mamures.p	. . . 4 |- ( ph -> P e. Fin )
21		mamures.x	. . . 4 |- ( ph -> X e. ( B ^m ( M X. N ) ) )
22		mamures.y	. . . 4 |- ( ph -> Y e. ( B ^m ( N X. P ) ) )
23	14, 15, 16, 17, 18, 19, 20, 21, 22	mamuval 20192	. . 3 |- ( ph -> ( X F Y ) = ( i e. M , j e. P |-> ( R gsumg ( k e. N |-> ( ( i X k ) ( .r ` R ) ( k Y j ) ) ) ) ) )
24	23	reseq1d 5395	. 2 |- ( ph -> ( ( X F Y ) |` ( I X. P ) ) = ( ( i e. M , j e. P |-> ( R gsumg ( k e. N |-> ( ( i X k ) ( .r ` R ) ( k Y j ) ) ) ) ) |` ( I X. P ) ) )
25		mamures.g	. . 3 |- G = ( R maMul <. I , N , P >. )
26		ssfi 8180	. . . 4 |- ( ( M e. Fin /\ I C_ M ) -> I e. Fin )
27	18, 1, 26	syl2anc 693	. . 3 |- ( ph -> I e. Fin )
28		elmapi 7879	. . . . . 6 |- ( X e. ( B ^m ( M X. N ) ) -> X : ( M X. N ) --> B )
29	21, 28	syl 17	. . . . 5 |- ( ph -> X : ( M X. N ) --> B )
30		xpss1 5228	. . . . . 6 |- ( I C_ M -> ( I X. N ) C_ ( M X. N ) )
31	1, 30	syl 17	. . . . 5 |- ( ph -> ( I X. N ) C_ ( M X. N ) )
32	29, 31	fssresd 6071	. . . 4 |- ( ph -> ( X |` ( I X. N ) ) : ( I X. N ) --> B )
33		fvex 6201	. . . . . . 7 |- ( Base ` R ) e. _V
34	15, 33	eqeltri 2697	. . . . . 6 |- B e. _V
35	34	a1i 11	. . . . 5 |- ( ph -> B e. _V )
36		xpfi 8231	. . . . . 6 |- ( ( I e. Fin /\ N e. Fin ) -> ( I X. N ) e. Fin )
37	27, 19, 36	syl2anc 693	. . . . 5 |- ( ph -> ( I X. N ) e. Fin )
38	35, 37	elmapd 7871	. . . 4 |- ( ph -> ( ( X |` ( I X. N ) ) e. ( B ^m ( I X. N ) ) <-> ( X |` ( I X. N ) ) : ( I X. N ) --> B ) )
39	32, 38	mpbird 247	. . 3 |- ( ph -> ( X |` ( I X. N ) ) e. ( B ^m ( I X. N ) ) )
40	25, 15, 16, 17, 27, 19, 20, 39, 22	mamuval 20192	. 2 |- ( ph -> ( ( X |` ( I X. N ) ) G Y ) = ( i e. I , j e. P |-> ( R gsumg ( k e. N |-> ( ( i ( X |` ( I X. N ) ) k ) ( .r ` R ) ( k Y j ) ) ) ) ) )
41	13, 24, 40	3eqtr4d 2666	1 |- ( ph -> ( ( X F Y ) |` ( I X. P ) ) = ( ( X |` ( I X. N ) ) G Y ) )

Syntax hints:   -> wi 4   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   _Vcvv 3200   C_ wss 3574   <.cotp 4185   |-> cmpt 4729   X. cxp 5112   |` cres 5116   -->wf 5884   ` 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

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-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-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-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-oadd 7564  df-er 7742  df-map 7859  df-en 7956  df-fin 7959  df-mamu 20190

This theorem is referenced by:  mdetmul  20429