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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.
StepHypRef 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 )
54sqxpeqd 5034 . . . . . . 7  |-  ( n  =  N  ->  (
n  X.  n )  =  ( N  X.  N ) )
63, 5oveqan12rd 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 ) )
86, 7syl6eqr 2516 . . . . 5  |-  ( ( n  =  N  /\  r  =  R )  ->  ( r freeLMod  ( n  X.  n ) )  =  G )
94, 4, 4oteq123d 4234 . . . . . . . 8  |-  ( n  =  N  ->  <. n ,  n ,  n >.  = 
<. N ,  N ,  N >. )
103, 9oveqan12rd 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 >. )
1210, 11syl6eqr 2516 . . . . . 6  |-  ( ( n  =  N  /\  r  =  R )  ->  ( r maMul  <. n ,  n ,  n >. )  =  .x.  )
1312opeq2d 4226 . . . . 5  |-  ( ( n  =  N  /\  r  =  R )  -> 
<. ( .r `  ndx ) ,  ( r maMul  <.
n ,  n ,  n >. ) >.  =  <. ( .r `  ndx ) ,  .x.  >. )
148, 13oveq12d 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
1714, 15, 16ovmpt2a 6432 . . 3  |-  ( ( N  e.  Fin  /\  R  e.  _V )  ->  ( N Mat  R )  =  ( G sSet  <. ( .r `  ndx ) ,  .x.  >. ) )
182, 17sylan2 474 . 2  |-  ( ( N  e.  Fin  /\  R  e.  V )  ->  ( N Mat  R )  =  ( G sSet  <. ( .r `  ndx ) ,  .x.  >. ) )
191, 18syl5eq 2510 1  |-  ( ( N  e.  Fin  /\  R  e.  V )  ->  A  =  ( G sSet  <. ( .r `  ndx ) ,  .x.  >. )
)
Colors of variables: wff setvar class
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
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