Theorem matval 19039

Description: Value of the matrix algebra. (Contributed by Stefan O'Rear, 4-Sep-2015.)

Hypotheses

Ref	Expression
matval.a	|- A = ( N Mat R )
matval.g	|- G = ( R freeLMod ( N X. N ) )
matval.t	|- .x. = ( R maMul <. N , N , N >. )

Assertion

Ref	Expression
matval	|- ( ( N e. Fin /\ R e. V ) -> A = ( G sSet <. ( .r ` ndx ) , .x. >. ) )

Proof of Theorem matval

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

Step	Hyp	Ref	Expression
1		matval.a	. 2 |- A = ( N Mat R )
2		elex 3118	. . 3 |- ( R e. V -> R e. _V )
3		id 22	. . . . . . 7 |- ( r = R -> r = R )
4		id 22	. . . . . . . 8 |- ( n = N -> n = N )
5	4	sqxpeqd 5034	. . . . . . 7 |- ( n = N -> ( n X. n ) = ( N X. N ) )
6	3, 5	oveqan12rd 6316	. . . . . 6 |- ( ( n = N /\ r = R ) -> ( r freeLMod ( n X. n ) ) = ( R freeLMod ( N X. N ) ) )
7		matval.g	. . . . . 6 |- G = ( R freeLMod ( N X. N ) )
8	6, 7	syl6eqr 2516	. . . . 5 |- ( ( n = N /\ r = R ) -> ( r freeLMod ( n X. n ) ) = G )
9	4, 4, 4	oteq123d 4234	. . . . . . . 8 |- ( n = N -> <. n , n , n >. = <. N , N , N >. )
10	3, 9	oveqan12rd 6316	. . . . . . 7 |- ( ( n = N /\ r = R ) -> ( r maMul <. n , n , n >. ) = ( R maMul <. N , N , N >. ) )
11		matval.t	. . . . . . 7 |- .x. = ( R maMul <. N , N , N >. )
12	10, 11	syl6eqr 2516	. . . . . 6 |- ( ( n = N /\ r = R ) -> ( r maMul <. n , n , n >. ) = .x. )
13	12	opeq2d 4226	. . . . 5 |- ( ( n = N /\ r = R ) -> <. ( .r ` ndx ) , ( r maMul <. n , n , n >. ) >. = <. ( .r ` ndx ) , .x. >. )
14	8, 13	oveq12d 6314	. . . 4 |- ( ( n = N /\ r = R ) -> ( ( r freeLMod ( n X. n ) ) sSet <. ( .r ` ndx ) , ( r maMul <. n , n , n >. ) >. ) = ( G sSet <. ( .r ` ndx ) , .x. >. ) )
15		df-mat 19036	. . . 4 |- Mat = ( n e. Fin , r e. _V |-> ( ( r freeLMod ( n X. n ) ) sSet <. ( .r ` ndx ) , ( r maMul <. n , n , n >. ) >. ) )
16		ovex 6324	. . . 4 |- ( G sSet <. ( .r ` ndx ) , .x. >. ) e. _V
17	14, 15, 16	ovmpt2a 6432	. . 3 |- ( ( N e. Fin /\ R e. _V ) -> ( N Mat R ) = ( G sSet <. ( .r ` ndx ) , .x. >. ) )
18	2, 17	sylan2 474	. 2 |- ( ( N e. Fin /\ R e. V ) -> ( N Mat R ) = ( G sSet <. ( .r ` ndx ) , .x. >. ) )
19	1, 18	syl5eq 2510	1 |- ( ( N e. Fin /\ R e. V ) -> A = ( G sSet <. ( .r ` ndx ) , .x. >. ) )

Syntax hints:   -> wi 4   /\ wa 369   = wceq 1395   e. wcel 1819   _Vcvv 3109   <.cop 4038   <.cotp 4040   X. cxp 5006   ` cfv 5594  (class class class)co 6296   Fincfn 7535   ndxcnx 14640   sSet csts 14641   .rcmulr 14712   freeLMod cfrlm 18903   maMul cmmul 19011   Mat cmat 19035

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-sep 4578  ax-nul 4586  ax-pr 4695

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3111  df-sbc 3328  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-sn 4033  df-pr 4035  df-op 4039  df-ot 4041  df-uni 4252  df-br 4457  df-opab 4516  df-id 4804  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-iota 5557  df-fun 5596  df-fv 5602  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-mat 19036

This theorem is referenced by:  matbas  19041  matplusg  19042  matsca  19043  matvsca  19044  matmulr  19066