Theorem mvmulval 20349

Description: Multiplication of a vector with a matrix. (Contributed by AV, 23-Feb-2019.)

Hypotheses

Ref	Expression
mvmulfval.x	|- .X. = ( R maVecMul <. M , N >. )
mvmulfval.b	|- B = ( Base ` R )
mvmulfval.t	|- .x. = ( .r ` R )
mvmulfval.r	|- ( ph -> R e. V )
mvmulfval.m	|- ( ph -> M e. Fin )
mvmulfval.n	|- ( ph -> N e. Fin )
mvmulval.x	|- ( ph -> X e. ( B ^m ( M X. N ) ) )
mvmulval.y	|- ( ph -> Y e. ( B ^m N ) )

Assertion

Ref	Expression
mvmulval	|- ( ph -> ( X .X. Y ) = ( i e. M |-> ( R gsumg ( j e. N |-> ( ( i X j ) .x. ( Y ` j ) ) ) ) ) )

Distinct variable groups:   i, j, ph   i, M, j   i, N, j   R, i, j   .x. , i   i, X, j   i, Y, j

Allowed substitution hints:   B( i, j)   .x. ( j)   .X. ( i, j)   V( i, j)

Proof of Theorem mvmulval

Dummy variables x y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		mvmulfval.x	. . 3 |- .X. = ( R maVecMul <. M , N >. )
2		mvmulfval.b	. . 3 |- B = ( Base ` R )
3		mvmulfval.t	. . 3 |- .x. = ( .r ` R )
4		mvmulfval.r	. . 3 |- ( ph -> R e. V )
5		mvmulfval.m	. . 3 |- ( ph -> M e. Fin )
6		mvmulfval.n	. . 3 |- ( ph -> N e. Fin )
7	1, 2, 3, 4, 5, 6	mvmulfval 20348	. 2 |- ( ph -> .X. = ( x e. ( B ^m ( M X. N ) ) , y e. ( B ^m N ) |-> ( i e. M |-> ( R gsumg ( j e. N |-> ( ( i x j ) .x. ( y ` j ) ) ) ) ) ) )
8		oveq 6656	. . . . . . 7 |- ( x = X -> ( i x j ) = ( i X j ) )
9		fveq1 6190	. . . . . . 7 |- ( y = Y -> ( y ` j ) = ( Y ` j ) )
10	8, 9	oveqan12d 6669	. . . . . 6 |- ( ( x = X /\ y = Y ) -> ( ( i x j ) .x. ( y ` j ) ) = ( ( i X j ) .x. ( Y ` j ) ) )
11	10	adantl 482	. . . . 5 |- ( ( ph /\ ( x = X /\ y = Y ) ) -> ( ( i x j ) .x. ( y ` j ) ) = ( ( i X j ) .x. ( Y ` j ) ) )
12	11	mpteq2dv 4745	. . . 4 |- ( ( ph /\ ( x = X /\ y = Y ) ) -> ( j e. N |-> ( ( i x j ) .x. ( y ` j ) ) ) = ( j e. N |-> ( ( i X j ) .x. ( Y ` j ) ) ) )
13	12	oveq2d 6666	. . 3 |- ( ( ph /\ ( x = X /\ y = Y ) ) -> ( R gsumg ( j e. N |-> ( ( i x j ) .x. ( y ` j ) ) ) ) = ( R gsumg ( j e. N |-> ( ( i X j ) .x. ( Y ` j ) ) ) ) )
14	13	mpteq2dv 4745	. 2 |- ( ( ph /\ ( x = X /\ y = Y ) ) -> ( i e. M |-> ( R gsumg ( j e. N |-> ( ( i x j ) .x. ( y ` j ) ) ) ) ) = ( i e. M |-> ( R gsumg ( j e. N |-> ( ( i X j ) .x. ( Y ` j ) ) ) ) ) )
15		mvmulval.x	. 2 |- ( ph -> X e. ( B ^m ( M X. N ) ) )
16		mvmulval.y	. 2 |- ( ph -> Y e. ( B ^m N ) )
17		mptexg 6484	. . 3 |- ( M e. Fin -> ( i e. M |-> ( R gsumg ( j e. N |-> ( ( i X j ) .x. ( Y ` j ) ) ) ) ) e. _V )
18	5, 17	syl 17	. 2 |- ( ph -> ( i e. M |-> ( R gsumg ( j e. N |-> ( ( i X j ) .x. ( Y ` j ) ) ) ) ) e. _V )
19	7, 14, 15, 16, 18	ovmpt2d 6788	1 |- ( ph -> ( X .X. Y ) = ( i e. M |-> ( R gsumg ( j e. N |-> ( ( i X j ) .x. ( Y ` j ) ) ) ) ) )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   _Vcvv 3200   <.cop 4183   |-> 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   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-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-mvmul 20347

This theorem is referenced by:  mvmulfv  20350  mavmulval  20351