Theorem mvmumamul1 20360

Description: The multiplication of an MxN matrix with an N-dimensional vector corresponds to the matrix multiplication of an MxN matrix with an Nx1 matrix. (Contributed by AV, 14-Mar-2019.)

Hypotheses

Ref	Expression
mvmumamul1.x	|- .X. = ( R maMul <. M , N , { (/) } >. )
mvmumamul1.t	|- .x. = ( R maVecMul <. M , N >. )
mvmumamul1.b	|- B = ( Base ` R )
mvmumamul1.r	|- ( ph -> R e. Ring )
mvmumamul1.m	|- ( ph -> M e. Fin )
mvmumamul1.n	|- ( ph -> N e. Fin )
mvmumamul1.a	|- ( ph -> A e. ( B ^m ( M X. N ) ) )
mvmumamul1.y	|- ( ph -> Y e. ( B ^m N ) )
mvmumamul1.z	|- ( ph -> Z e. ( B ^m ( N X. { (/) } ) ) )

Assertion

Ref	Expression
mvmumamul1	|- ( ph -> ( A. j e. N ( Y ` j ) = ( j Z (/) ) -> A. i e. M ( ( A .x. Y ) ` i ) = ( i ( A .X. Z ) (/) ) ) )

Distinct variable groups:   i, j, N   i, Y, j   i, Z, j   ph, i, j

Allowed substitution hints:   A( i, j)   B( i, j)   R( i, j)   .x. ( i, j)   .X. ( i, j)   M( i, j)

Proof of Theorem mvmumamul1

Dummy variable k is distinct from all other variables.

Step	Hyp	Ref	Expression
1		mvmumamul1.t	. . . . . 6 |- .x. = ( R maVecMul <. M , N >. )
2		mvmumamul1.b	. . . . . 6 |- B = ( Base ` R )
3		eqid 2622	. . . . . 6 |- ( .r ` R ) = ( .r ` R )
4		mvmumamul1.r	. . . . . . 7 |- ( ph -> R e. Ring )
5	4	adantr 481	. . . . . 6 |- ( ( ph /\ i e. M ) -> R e. Ring )
6		mvmumamul1.m	. . . . . . 7 |- ( ph -> M e. Fin )
7	6	adantr 481	. . . . . 6 |- ( ( ph /\ i e. M ) -> M e. Fin )
8		mvmumamul1.n	. . . . . . 7 |- ( ph -> N e. Fin )
9	8	adantr 481	. . . . . 6 |- ( ( ph /\ i e. M ) -> N e. Fin )
10		mvmumamul1.a	. . . . . . 7 |- ( ph -> A e. ( B ^m ( M X. N ) ) )
11	10	adantr 481	. . . . . 6 |- ( ( ph /\ i e. M ) -> A e. ( B ^m ( M X. N ) ) )
12		mvmumamul1.y	. . . . . . 7 |- ( ph -> Y e. ( B ^m N ) )
13	12	adantr 481	. . . . . 6 |- ( ( ph /\ i e. M ) -> Y e. ( B ^m N ) )
14		simpr 477	. . . . . 6 |- ( ( ph /\ i e. M ) -> i e. M )
15	1, 2, 3, 5, 7, 9, 11, 13, 14	mvmulfv 20350	. . . . 5 |- ( ( ph /\ i e. M ) -> ( ( A .x. Y ) ` i ) = ( R gsumg ( k e. N |-> ( ( i A k ) ( .r ` R ) ( Y ` k ) ) ) ) )
16	15	adantlr 751	. . . 4 |- ( ( ( ph /\ A. j e. N ( Y ` j ) = ( j Z (/) ) ) /\ i e. M ) -> ( ( A .x. Y ) ` i ) = ( R gsumg ( k e. N |-> ( ( i A k ) ( .r ` R ) ( Y ` k ) ) ) ) )
17		fveq2 6191	. . . . . . . . . . . 12 |- ( j = k -> ( Y ` j ) = ( Y ` k ) )
18		oveq1 6657	. . . . . . . . . . . 12 |- ( j = k -> ( j Z (/) ) = ( k Z (/) ) )
19	17, 18	eqeq12d 2637	. . . . . . . . . . 11 |- ( j = k -> ( ( Y ` j ) = ( j Z (/) ) <-> ( Y ` k ) = ( k Z (/) ) ) )
20	19	rspccv 3306	. . . . . . . . . 10 |- ( A. j e. N ( Y ` j ) = ( j Z (/) ) -> ( k e. N -> ( Y ` k ) = ( k Z (/) ) ) )
21	20	adantl 482	. . . . . . . . 9 |- ( ( ph /\ A. j e. N ( Y ` j ) = ( j Z (/) ) ) -> ( k e. N -> ( Y ` k ) = ( k Z (/) ) ) )
22	21	imp 445	. . . . . . . 8 |- ( ( ( ph /\ A. j e. N ( Y ` j ) = ( j Z (/) ) ) /\ k e. N ) -> ( Y ` k ) = ( k Z (/) ) )
23	22	oveq2d 6666	. . . . . . 7 |- ( ( ( ph /\ A. j e. N ( Y ` j ) = ( j Z (/) ) ) /\ k e. N ) -> ( ( i A k ) ( .r ` R ) ( Y ` k ) ) = ( ( i A k ) ( .r ` R ) ( k Z (/) ) ) )
24	23	mpteq2dva 4744	. . . . . 6 |- ( ( ph /\ A. j e. N ( Y ` j ) = ( j Z (/) ) ) -> ( k e. N |-> ( ( i A k ) ( .r ` R ) ( Y ` k ) ) ) = ( k e. N |-> ( ( i A k ) ( .r ` R ) ( k Z (/) ) ) ) )
25	24	oveq2d 6666	. . . . 5 |- ( ( ph /\ A. j e. N ( Y ` j ) = ( j Z (/) ) ) -> ( R gsumg ( k e. N |-> ( ( i A k ) ( .r ` R ) ( Y ` k ) ) ) ) = ( R gsumg ( k e. N |-> ( ( i A k ) ( .r ` R ) ( k Z (/) ) ) ) ) )
26	25	adantr 481	. . . 4 |- ( ( ( ph /\ A. j e. N ( Y ` j ) = ( j Z (/) ) ) /\ i e. M ) -> ( R gsumg ( k e. N |-> ( ( i A k ) ( .r ` R ) ( Y ` k ) ) ) ) = ( R gsumg ( k e. N |-> ( ( i A k ) ( .r ` R ) ( k Z (/) ) ) ) ) )
27		mvmumamul1.x	. . . . . . 7 |- .X. = ( R maMul <. M , N , { (/) } >. )
28		snfi 8038	. . . . . . . 8 |- { (/) } e. Fin
29	28	a1i 11	. . . . . . 7 |- ( ( ph /\ i e. M ) -> { (/) } e. Fin )
30		mvmumamul1.z	. . . . . . . 8 |- ( ph -> Z e. ( B ^m ( N X. { (/) } ) ) )
31	30	adantr 481	. . . . . . 7 |- ( ( ph /\ i e. M ) -> Z e. ( B ^m ( N X. { (/) } ) ) )
32		0ex 4790	. . . . . . . . 9 |- (/) e. _V
33	32	snid 4208	. . . . . . . 8 |- (/) e. { (/) }
34	33	a1i 11	. . . . . . 7 |- ( ( ph /\ i e. M ) -> (/) e. { (/) } )
35	27, 2, 3, 5, 7, 9, 29, 11, 31, 14, 34	mamufv 20193	. . . . . 6 |- ( ( ph /\ i e. M ) -> ( i ( A .X. Z ) (/) ) = ( R gsumg ( k e. N |-> ( ( i A k ) ( .r ` R ) ( k Z (/) ) ) ) ) )
36	35	eqcomd 2628	. . . . 5 |- ( ( ph /\ i e. M ) -> ( R gsumg ( k e. N |-> ( ( i A k ) ( .r ` R ) ( k Z (/) ) ) ) ) = ( i ( A .X. Z ) (/) ) )
37	36	adantlr 751	. . . 4 |- ( ( ( ph /\ A. j e. N ( Y ` j ) = ( j Z (/) ) ) /\ i e. M ) -> ( R gsumg ( k e. N |-> ( ( i A k ) ( .r ` R ) ( k Z (/) ) ) ) ) = ( i ( A .X. Z ) (/) ) )
38	16, 26, 37	3eqtrd 2660	. . 3 |- ( ( ( ph /\ A. j e. N ( Y ` j ) = ( j Z (/) ) ) /\ i e. M ) -> ( ( A .x. Y ) ` i ) = ( i ( A .X. Z ) (/) ) )
39	38	ralrimiva 2966	. 2 |- ( ( ph /\ A. j e. N ( Y ` j ) = ( j Z (/) ) ) -> A. i e. M ( ( A .x. Y ) ` i ) = ( i ( A .X. Z ) (/) ) )
40	39	ex 450	1 |- ( ph -> ( A. j e. N ( Y ` j ) = ( j Z (/) ) -> A. i e. M ( ( A .x. Y ) ` i ) = ( i ( A .X. Z ) (/) ) ) )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   A.wral 2912   (/)c0 3915   {csn 4177   <.cop 4183   <.cotp 4185   |-> cmpt 4729   X. cxp 5112   ` cfv 5888  (class class class)co 6650   ^m cmap 7857   Fincfn 7955   Basecbs 15857   .rcmulr 15942   gsum_g cgsu 16101   Ringcrg 18547   maMul cmmul 20189   maVecMul cmvmul 20346

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-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-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-1o 7560  df-en 7956  df-fin 7959  df-mamu 20190  df-mvmul 20347

This theorem is referenced by:  mavmumamul1  20361