Theorem matval 19484

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 3065	. . 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 4878	. . . . . . 7 |- ( n = N -> ( n X. n ) = ( N X. N ) )
6	3, 5	oveqan12rd 6334	. . . . . 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 2513	. . . . 5 |- ( ( n = N /\ r = R ) -> ( r freeLMod ( n X. n ) ) = G )
9	4, 4, 4	oteq123d 4194	. . . . . . . 8 |- ( n = N -> <. n , n , n >. = <. N , N , N >. )
10	3, 9	oveqan12rd 6334	. . . . . . 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 2513	. . . . . 6 |- ( ( n = N /\ r = R ) -> ( r maMul <. n , n , n >. ) = .x. )
13	12	opeq2d 4186	. . . . 5 |- ( ( n = N /\ r = R ) -> <. ( .r ` ndx ) , ( r maMul <. n , n , n >. ) >. = <. ( .r ` ndx ) , .x. >. )
14	8, 13	oveq12d 6332	. . . 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 19481	. . . 4 |- Mat = ( n e. Fin , r e. _V |-> ( ( r freeLMod ( n X. n ) ) sSet <. ( .r ` ndx ) , ( r maMul <. n , n , n >. ) >. ) )
16		ovex 6342	. . . 4 |- ( G sSet <. ( .r ` ndx ) , .x. >. ) e. _V
17	14, 15, 16	ovmpt2a 6453	. . 3 |- ( ( N e. Fin /\ R e. _V ) -> ( N Mat R ) = ( G sSet <. ( .r ` ndx ) , .x. >. ) )
18	2, 17	sylan2 481	. 2 |- ( ( N e. Fin /\ R e. V ) -> ( N Mat R ) = ( G sSet <. ( .r ` ndx ) , .x. >. ) )
19	1, 18	syl5eq 2507	1 |- ( ( N e. Fin /\ R e. V ) -> A = ( G sSet <. ( .r ` ndx ) , .x. >. ) )

Syntax hints:   -> wi 4   /\ wa 375   = wceq 1454   e. wcel 1897   _Vcvv 3056   <.cop 3985   <.cotp 3987   X. cxp 4850   ` cfv 5600  (class class class)co 6314   Fincfn 7594   ndxcnx 15166   sSet csts 15167   .rcmulr 15239   freeLMod cfrlm 19357   maMul cmmul 19456   Mat cmat 19480

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1679  ax-4 1692  ax-5 1768  ax-6 1815  ax-7 1861  ax-9 1906  ax-10 1925  ax-11 1930  ax-12 1943  ax-13 2101  ax-ext 2441  ax-sep 4538  ax-nul 4547  ax-pr 4652

This theorem depends on definitions:  df-bi 190  df-or 376  df-an 377  df-3an 993  df-tru 1457  df-ex 1674  df-nf 1678  df-sb 1808  df-eu 2313  df-mo 2314  df-clab 2448  df-cleq 2454  df-clel 2457  df-nfc 2591  df-ne 2634  df-ral 2753  df-rex 2754  df-rab 2757  df-v 3058  df-sbc 3279  df-dif 3418  df-un 3420  df-in 3422  df-ss 3429  df-nul 3743  df-if 3893  df-sn 3980  df-pr 3982  df-op 3986  df-ot 3988  df-uni 4212  df-br 4416  df-opab 4475  df-id 4767  df-xp 4858  df-rel 4859  df-cnv 4860  df-co 4861  df-dm 4862  df-iota 5564  df-fun 5602  df-fv 5608  df-ov 6317  df-oprab 6318  df-mpt2 6319  df-mat 19481

This theorem is referenced by:  matbas  19486  matplusg  19487  matsca  19488  matvsca  19489  matmulr  19511