Theorem mvmulfval 20348

Description: Functional value of the matrix vector multiplication operator. (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 )

Assertion

Ref	Expression
mvmulfval	|- ( 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 ) ) ) ) ) ) )

Distinct variable groups:   i, j, x, y, ph   i, M, j, x, y   i, N, j, x, y   R, i, j, x, y   x, B, y   x, .x. , y, i

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

Proof of Theorem mvmulfval

Dummy variables m n o r are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		mvmulfval.x	. 2 |- .X. = ( R maVecMul <. M , N >. )
2		df-mvmul 20347	. . . 4 |- maVecMul = ( r e. _V , o e. _V |-> [_ ( 1st ` o ) / m ]_ [_ ( 2nd ` o ) / n ]_ ( x e. ( ( Base ` r ) ^m ( m X. n ) ) , y e. ( ( Base ` r ) ^m n ) |-> ( i e. m |-> ( r gsumg ( j e. n |-> ( ( i x j ) ( .r ` r ) ( y ` j ) ) ) ) ) ) )
3	2	a1i 11	. . 3 |- ( ph -> maVecMul = ( r e. _V , o e. _V |-> [_ ( 1st ` o ) / m ]_ [_ ( 2nd ` o ) / n ]_ ( x e. ( ( Base ` r ) ^m ( m X. n ) ) , y e. ( ( Base ` r ) ^m n ) |-> ( i e. m |-> ( r gsumg ( j e. n |-> ( ( i x j ) ( .r ` r ) ( y ` j ) ) ) ) ) ) ) )
4		fvex 6201	. . . . 5 |- ( 1st ` o ) e. _V
5		fvex 6201	. . . . 5 |- ( 2nd ` o ) e. _V
6		xpeq12 5134	. . . . . . 7 |- ( ( m = ( 1st ` o ) /\ n = ( 2nd ` o ) ) -> ( m X. n ) = ( ( 1st ` o ) X. ( 2nd ` o ) ) )
7	6	oveq2d 6666	. . . . . 6 |- ( ( m = ( 1st ` o ) /\ n = ( 2nd ` o ) ) -> ( ( Base ` r ) ^m ( m X. n ) ) = ( ( Base ` r ) ^m ( ( 1st ` o ) X. ( 2nd ` o ) ) ) )
8		oveq2 6658	. . . . . . 7 |- ( n = ( 2nd ` o ) -> ( ( Base ` r ) ^m n ) = ( ( Base ` r ) ^m ( 2nd ` o ) ) )
9	8	adantl 482	. . . . . 6 |- ( ( m = ( 1st ` o ) /\ n = ( 2nd ` o ) ) -> ( ( Base ` r ) ^m n ) = ( ( Base ` r ) ^m ( 2nd ` o ) ) )
10		simpl 473	. . . . . . 7 |- ( ( m = ( 1st ` o ) /\ n = ( 2nd ` o ) ) -> m = ( 1st ` o ) )
11		simpr 477	. . . . . . . . 9 |- ( ( m = ( 1st ` o ) /\ n = ( 2nd ` o ) ) -> n = ( 2nd ` o ) )
12	11	mpteq1d 4738	. . . . . . . 8 |- ( ( m = ( 1st ` o ) /\ n = ( 2nd ` o ) ) -> ( j e. n |-> ( ( i x j ) ( .r ` r ) ( y ` j ) ) ) = ( j e. ( 2nd ` o ) |-> ( ( i x j ) ( .r ` r ) ( y ` j ) ) ) )
13	12	oveq2d 6666	. . . . . . 7 |- ( ( m = ( 1st ` o ) /\ n = ( 2nd ` o ) ) -> ( r gsumg ( j e. n |-> ( ( i x j ) ( .r ` r ) ( y ` j ) ) ) ) = ( r gsumg ( j e. ( 2nd ` o ) |-> ( ( i x j ) ( .r ` r ) ( y ` j ) ) ) ) )
14	10, 13	mpteq12dv 4733	. . . . . 6 |- ( ( m = ( 1st ` o ) /\ n = ( 2nd ` o ) ) -> ( i e. m |-> ( r gsumg ( j e. n |-> ( ( i x j ) ( .r ` r ) ( y ` j ) ) ) ) ) = ( i e. ( 1st ` o ) |-> ( r gsumg ( j e. ( 2nd ` o ) |-> ( ( i x j ) ( .r ` r ) ( y ` j ) ) ) ) ) )
15	7, 9, 14	mpt2eq123dv 6717	. . . . 5 |- ( ( m = ( 1st ` o ) /\ n = ( 2nd ` o ) ) -> ( x e. ( ( Base ` r ) ^m ( m X. n ) ) , y e. ( ( Base ` r ) ^m n ) |-> ( i e. m |-> ( r gsumg ( j e. n |-> ( ( i x j ) ( .r ` r ) ( y ` j ) ) ) ) ) ) = ( x e. ( ( Base ` r ) ^m ( ( 1st ` o ) X. ( 2nd ` o ) ) ) , y e. ( ( Base ` r ) ^m ( 2nd ` o ) ) |-> ( i e. ( 1st ` o ) |-> ( r gsumg ( j e. ( 2nd ` o ) |-> ( ( i x j ) ( .r ` r ) ( y ` j ) ) ) ) ) ) )
16	4, 5, 15	csbie2 3563	. . . 4 |- [_ ( 1st ` o ) / m ]_ [_ ( 2nd ` o ) / n ]_ ( x e. ( ( Base ` r ) ^m ( m X. n ) ) , y e. ( ( Base ` r ) ^m n ) |-> ( i e. m |-> ( r gsumg ( j e. n |-> ( ( i x j ) ( .r ` r ) ( y ` j ) ) ) ) ) ) = ( x e. ( ( Base ` r ) ^m ( ( 1st ` o ) X. ( 2nd ` o ) ) ) , y e. ( ( Base ` r ) ^m ( 2nd ` o ) ) |-> ( i e. ( 1st ` o ) |-> ( r gsumg ( j e. ( 2nd ` o ) |-> ( ( i x j ) ( .r ` r ) ( y ` j ) ) ) ) ) )
17		simprl 794	. . . . . . . 8 |- ( ( ph /\ ( r = R /\ o = <. M , N >. ) ) -> r = R )
18	17	fveq2d 6195	. . . . . . 7 |- ( ( ph /\ ( r = R /\ o = <. M , N >. ) ) -> ( Base ` r ) = ( Base ` R ) )
19		mvmulfval.b	. . . . . . 7 |- B = ( Base ` R )
20	18, 19	syl6eqr 2674	. . . . . 6 |- ( ( ph /\ ( r = R /\ o = <. M , N >. ) ) -> ( Base ` r ) = B )
21		fveq2 6191	. . . . . . . . 9 |- ( o = <. M , N >. -> ( 1st ` o ) = ( 1st ` <. M , N >. ) )
22	21	ad2antll 765	. . . . . . . 8 |- ( ( ph /\ ( r = R /\ o = <. M , N >. ) ) -> ( 1st ` o ) = ( 1st ` <. M , N >. ) )
23		mvmulfval.m	. . . . . . . . . 10 |- ( ph -> M e. Fin )
24		mvmulfval.n	. . . . . . . . . 10 |- ( ph -> N e. Fin )
25		op1stg 7180	. . . . . . . . . 10 |- ( ( M e. Fin /\ N e. Fin ) -> ( 1st ` <. M , N >. ) = M )
26	23, 24, 25	syl2anc 693	. . . . . . . . 9 |- ( ph -> ( 1st ` <. M , N >. ) = M )
27	26	adantr 481	. . . . . . . 8 |- ( ( ph /\ ( r = R /\ o = <. M , N >. ) ) -> ( 1st ` <. M , N >. ) = M )
28	22, 27	eqtrd 2656	. . . . . . 7 |- ( ( ph /\ ( r = R /\ o = <. M , N >. ) ) -> ( 1st ` o ) = M )
29		fveq2 6191	. . . . . . . . 9 |- ( o = <. M , N >. -> ( 2nd ` o ) = ( 2nd ` <. M , N >. ) )
30	29	ad2antll 765	. . . . . . . 8 |- ( ( ph /\ ( r = R /\ o = <. M , N >. ) ) -> ( 2nd ` o ) = ( 2nd ` <. M , N >. ) )
31		op2ndg 7181	. . . . . . . . . 10 |- ( ( M e. Fin /\ N e. Fin ) -> ( 2nd ` <. M , N >. ) = N )
32	23, 24, 31	syl2anc 693	. . . . . . . . 9 |- ( ph -> ( 2nd ` <. M , N >. ) = N )
33	32	adantr 481	. . . . . . . 8 |- ( ( ph /\ ( r = R /\ o = <. M , N >. ) ) -> ( 2nd ` <. M , N >. ) = N )
34	30, 33	eqtrd 2656	. . . . . . 7 |- ( ( ph /\ ( r = R /\ o = <. M , N >. ) ) -> ( 2nd ` o ) = N )
35	28, 34	xpeq12d 5140	. . . . . 6 |- ( ( ph /\ ( r = R /\ o = <. M , N >. ) ) -> ( ( 1st ` o ) X. ( 2nd ` o ) ) = ( M X. N ) )
36	20, 35	oveq12d 6668	. . . . 5 |- ( ( ph /\ ( r = R /\ o = <. M , N >. ) ) -> ( ( Base ` r ) ^m ( ( 1st ` o ) X. ( 2nd ` o ) ) ) = ( B ^m ( M X. N ) ) )
37	20, 34	oveq12d 6668	. . . . 5 |- ( ( ph /\ ( r = R /\ o = <. M , N >. ) ) -> ( ( Base ` r ) ^m ( 2nd ` o ) ) = ( B ^m N ) )
38		fveq2 6191	. . . . . . . . . . . 12 |- ( r = R -> ( .r ` r ) = ( .r ` R ) )
39	38	adantr 481	. . . . . . . . . . 11 |- ( ( r = R /\ o = <. M , N >. ) -> ( .r ` r ) = ( .r ` R ) )
40	39	adantl 482	. . . . . . . . . 10 |- ( ( ph /\ ( r = R /\ o = <. M , N >. ) ) -> ( .r ` r ) = ( .r ` R ) )
41		mvmulfval.t	. . . . . . . . . 10 |- .x. = ( .r ` R )
42	40, 41	syl6eqr 2674	. . . . . . . . 9 |- ( ( ph /\ ( r = R /\ o = <. M , N >. ) ) -> ( .r ` r ) = .x. )
43	42	oveqd 6667	. . . . . . . 8 |- ( ( ph /\ ( r = R /\ o = <. M , N >. ) ) -> ( ( i x j ) ( .r ` r ) ( y ` j ) ) = ( ( i x j ) .x. ( y ` j ) ) )
44	34, 43	mpteq12dv 4733	. . . . . . 7 |- ( ( ph /\ ( r = R /\ o = <. M , N >. ) ) -> ( j e. ( 2nd ` o ) |-> ( ( i x j ) ( .r ` r ) ( y ` j ) ) ) = ( j e. N |-> ( ( i x j ) .x. ( y ` j ) ) ) )
45	17, 44	oveq12d 6668	. . . . . 6 |- ( ( ph /\ ( r = R /\ o = <. M , N >. ) ) -> ( r gsumg ( j e. ( 2nd ` o ) |-> ( ( i x j ) ( .r ` r ) ( y ` j ) ) ) ) = ( R gsumg ( j e. N |-> ( ( i x j ) .x. ( y ` j ) ) ) ) )
46	28, 45	mpteq12dv 4733	. . . . 5 |- ( ( ph /\ ( r = R /\ o = <. M , N >. ) ) -> ( i e. ( 1st ` o ) |-> ( r gsumg ( j e. ( 2nd ` o ) |-> ( ( i x j ) ( .r ` r ) ( y ` j ) ) ) ) ) = ( i e. M |-> ( R gsumg ( j e. N |-> ( ( i x j ) .x. ( y ` j ) ) ) ) ) )
47	36, 37, 46	mpt2eq123dv 6717	. . . 4 |- ( ( ph /\ ( r = R /\ o = <. M , N >. ) ) -> ( x e. ( ( Base ` r ) ^m ( ( 1st ` o ) X. ( 2nd ` o ) ) ) , y e. ( ( Base ` r ) ^m ( 2nd ` o ) ) |-> ( i e. ( 1st ` o ) |-> ( r gsumg ( j e. ( 2nd ` o ) |-> ( ( i x j ) ( .r ` r ) ( y ` j ) ) ) ) ) ) = ( 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 ) ) ) ) ) ) )
48	16, 47	syl5eq 2668	. . 3 |- ( ( ph /\ ( r = R /\ o = <. M , N >. ) ) -> [_ ( 1st ` o ) / m ]_ [_ ( 2nd ` o ) / n ]_ ( x e. ( ( Base ` r ) ^m ( m X. n ) ) , y e. ( ( Base ` r ) ^m n ) |-> ( i e. m |-> ( r gsumg ( j e. n |-> ( ( i x j ) ( .r ` r ) ( y ` j ) ) ) ) ) ) = ( 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 ) ) ) ) ) ) )
49		mvmulfval.r	. . . 4 |- ( ph -> R e. V )
50		elex 3212	. . . 4 |- ( R e. V -> R e. _V )
51	49, 50	syl 17	. . 3 |- ( ph -> R e. _V )
52		opex 4932	. . . 4 |- <. M , N >. e. _V
53	52	a1i 11	. . 3 |- ( ph -> <. M , N >. e. _V )
54		ovex 6678	. . . . 5 |- ( B ^m ( M X. N ) ) e. _V
55		ovex 6678	. . . . 5 |- ( B ^m N ) e. _V
56	54, 55	mpt2ex 7247	. . . 4 |- ( 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 ) ) ) ) ) ) e. _V
57	56	a1i 11	. . 3 |- ( ph -> ( 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 ) ) ) ) ) ) e. _V )
58	3, 48, 51, 53, 57	ovmpt2d 6788	. 2 |- ( ph -> ( R maVecMul <. M , N >. ) = ( 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 ) ) ) ) ) ) )
59	1, 58	syl5eq 2668	1 |- ( 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 ) ) ) ) ) ) )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   _Vcvv 3200   [_csb 3533   <.cop 4183   |-> cmpt 4729   X. cxp 5112   ` cfv 5888  (class class class)co 6650   |-> cmpt2 6652   1stc1st 7166   2ndc2nd 7167   ^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:  mvmulval  20349  mavmuldm  20356  mavmul0g  20359