Theorem matval 20217

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 3212	. . 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 5141	. . . . . . 7 |- ( n = N -> ( n X. n ) = ( N X. N ) )
6	3, 5	oveqan12rd 6670	. . . . . 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 2674	. . . . 5 |- ( ( n = N /\ r = R ) -> ( r freeLMod ( n X. n ) ) = G )
9	4, 4, 4	oteq123d 4417	. . . . . . . 8 |- ( n = N -> <. n , n , n >. = <. N , N , N >. )
10	3, 9	oveqan12rd 6670	. . . . . . 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 2674	. . . . . 6 |- ( ( n = N /\ r = R ) -> ( r maMul <. n , n , n >. ) = .x. )
13	12	opeq2d 4409	. . . . 5 |- ( ( n = N /\ r = R ) -> <. ( .r ` ndx ) , ( r maMul <. n , n , n >. ) >. = <. ( .r ` ndx ) , .x. >. )
14	8, 13	oveq12d 6668	. . . 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 20214	. . . 4 |- Mat = ( n e. Fin , r e. _V |-> ( ( r freeLMod ( n X. n ) ) sSet <. ( .r ` ndx ) , ( r maMul <. n , n , n >. ) >. ) )
16		ovex 6678	. . . 4 |- ( G sSet <. ( .r ` ndx ) , .x. >. ) e. _V
17	14, 15, 16	ovmpt2a 6791	. . 3 |- ( ( N e. Fin /\ R e. _V ) -> ( N Mat R ) = ( G sSet <. ( .r ` ndx ) , .x. >. ) )
18	2, 17	sylan2 491	. 2 |- ( ( N e. Fin /\ R e. V ) -> ( N Mat R ) = ( G sSet <. ( .r ` ndx ) , .x. >. ) )
19	1, 18	syl5eq 2668	1 |- ( ( N e. Fin /\ R e. V ) -> A = ( G sSet <. ( .r ` ndx ) , .x. >. ) )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   _Vcvv 3200   <.cop 4183   <.cotp 4185   X. cxp 5112   ` cfv 5888  (class class class)co 6650   Fincfn 7955   ndxcnx 15854   sSet csts 15855   .rcmulr 15942   freeLMod cfrlm 20090   maMul cmmul 20189   Mat cmat 20213

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-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

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-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-ot 4186  df-uni 4437  df-br 4654  df-opab 4713  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-iota 5851  df-fun 5890  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-mat 20214

This theorem is referenced by:  matbas  20219  matplusg  20220  matsca  20221  matvsca  20222  matmulr  20244