Theorem matval 18311

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 2981	. . 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, 4	xpeq12d 4865	. . . . . . 7 |- ( n = N -> ( n X. n ) = ( N X. N ) )
6	3, 5	oveqan12rd 6111	. . . . . 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 2493	. . . . 5 |- ( ( n = N /\ r = R ) -> ( r freeLMod ( n X. n ) ) = G )
9	4, 4, 4	oteq123d 4074	. . . . . . . 8 |- ( n = N -> <. n , n , n >. = <. N , N , N >. )
10	3, 9	oveqan12rd 6111	. . . . . . 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 2493	. . . . . 6 |- ( ( n = N /\ r = R ) -> ( r maMul <. n , n , n >. ) = .x. )
13	12	opeq2d 4066	. . . . 5 |- ( ( n = N /\ r = R ) -> <. ( .r ` ndx ) , ( r maMul <. n , n , n >. ) >. = <. ( .r ` ndx ) , .x. >. )
14	8, 13	oveq12d 6109	. . . 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 18282	. . . 4 |- Mat = ( n e. Fin , r e. _V |-> ( ( r freeLMod ( n X. n ) ) sSet <. ( .r ` ndx ) , ( r maMul <. n , n , n >. ) >. ) )
16		ovex 6116	. . . 4 |- ( G sSet <. ( .r ` ndx ) , .x. >. ) e. _V
17	14, 15, 16	ovmpt2a 6221	. . 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 2487	1 |- ( ( N e. Fin /\ R e. V ) -> A = ( G sSet <. ( .r ` ndx ) , .x. >. ) )

Syntax hints:   -> wi 4   /\ wa 369   = wceq 1369   e. wcel 1756   _Vcvv 2972   <.cop 3883   <.cotp 3885   X. cxp 4838   ` cfv 5418  (class class class)co 6091   Fincfn 7310   ndxcnx 14171   sSet csts 14172   .rcmulr 14239   freeLMod cfrlm 18171   maMul cmmul 18279   Mat cmat 18280

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4413  ax-nul 4421  ax-pr 4531

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-ne 2608  df-ral 2720  df-rex 2721  df-rab 2724  df-v 2974  df-sbc 3187  df-dif 3331  df-un 3333  df-in 3335  df-ss 3342  df-nul 3638  df-if 3792  df-sn 3878  df-pr 3880  df-op 3884  df-ot 3886  df-uni 4092  df-br 4293  df-opab 4351  df-id 4636  df-xp 4846  df-rel 4847  df-cnv 4848  df-co 4849  df-dm 4850  df-iota 5381  df-fun 5420  df-fv 5426  df-ov 6094  df-oprab 6095  df-mpt2 6096  df-mat 18282

This theorem is referenced by:  matmulr  18313  matbas  18314  matplusg  18315  matsca  18316  matvsca  18317