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Theorem matval 27444
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 2966 . . 3  |-  ( R  e.  V  ->  R  e.  _V )
3 id 21 . . . . . . 7  |-  ( r  =  R  ->  r  =  R )
4 id 21 . . . . . . . 8  |-  ( n  =  N  ->  n  =  N )
54, 4xpeq12d 4905 . . . . . . 7  |-  ( n  =  N  ->  (
n  X.  n )  =  ( N  X.  N ) )
63, 5oveqan12rd 6103 . . . . . 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 2488 . . . . 5  |-  ( ( n  =  N  /\  r  =  R )  ->  ( r freeLMod  ( n  X.  n ) )  =  G )
94, 4, 4oteq123d 4001 . . . . . . . 8  |-  ( n  =  N  ->  <. n ,  n ,  n >.  = 
<. N ,  N ,  N >. )
103, 9oveqan12rd 6103 . . . . . . 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 2488 . . . . . 6  |-  ( ( n  =  N  /\  r  =  R )  ->  ( r maMul  <. n ,  n ,  n >. )  =  .x.  )
1312opeq2d 3993 . . . . 5  |-  ( ( n  =  N  /\  r  =  R )  -> 
<. ( .r `  ndx ) ,  ( r maMul  <.
n ,  n ,  n >. ) >.  =  <. ( .r `  ndx ) ,  .x.  >. )
148, 13oveq12d 6101 . . . 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 27421 . . . 4  |- Mat  =  ( n  e.  Fin , 
r  e.  _V  |->  ( ( r freeLMod  ( n  X.  n ) ) sSet  <. ( .r `  ndx ) ,  ( r maMul  <.
n ,  n ,  n >. ) >. )
)
16 ovex 6108 . . . 4  |-  ( G sSet  <. ( .r `  ndx ) ,  .x.  >. )  e.  _V
1714, 15, 16ovmpt2a 6206 . . 3  |-  ( ( N  e.  Fin  /\  R  e.  _V )  ->  ( N Mat  R )  =  ( G sSet  <. ( .r `  ndx ) ,  .x.  >. ) )
182, 17sylan2 462 . 2  |-  ( ( N  e.  Fin  /\  R  e.  V )  ->  ( N Mat  R )  =  ( G sSet  <. ( .r `  ndx ) ,  .x.  >. ) )
191, 18syl5eq 2482 1  |-  ( ( N  e.  Fin  /\  R  e.  V )  ->  A  =  ( G sSet  <. ( .r `  ndx ) ,  .x.  >. )
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 360    = wceq 1653    e. wcel 1726   _Vcvv 2958   <.cop 3819   <.cotp 3820    X. cxp 4878   ` cfv 5456  (class class class)co 6083   Fincfn 7111   ndxcnx 13468   sSet csts 13469   .rcmulr 13532   freeLMod cfrlm 27191   maMul cmmul 27418   Mat cmat 27419
This theorem is referenced by:  matmulr  27446  matbas  27447  matplusg  27448  matsca  27449  matvsca  27450
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4332  ax-nul 4340  ax-pr 4405
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-sbc 3164  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-sn 3822  df-pr 3823  df-op 3825  df-ot 3826  df-uni 4018  df-br 4215  df-opab 4269  df-id 4500  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-iota 5420  df-fun 5458  df-fv 5464  df-ov 6086  df-oprab 6087  df-mpt2 6088  df-mat 27421
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