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Theorem matval 26877
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 2796 . . 3  |-  ( R  e.  V  ->  R  e.  _V )
3 id 19 . . . . . . 7  |-  ( r  =  R  ->  r  =  R )
4 id 19 . . . . . . . 8  |-  ( n  =  N  ->  n  =  N )
54, 4xpeq12d 4714 . . . . . . 7  |-  ( n  =  N  ->  (
n  X.  n )  =  ( N  X.  N ) )
63, 5oveqan12rd 5878 . . . . . 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 2333 . . . . 5  |-  ( ( n  =  N  /\  r  =  R )  ->  ( r freeLMod  ( n  X.  n ) )  =  G )
94, 4opeq12d 3804 . . . . . . . . . 10  |-  ( n  =  N  ->  <. n ,  n >.  =  <. N ,  N >. )
109, 4opeq12d 3804 . . . . . . . . 9  |-  ( n  =  N  ->  <. <. n ,  n >. ,  n >.  = 
<. <. N ,  N >. ,  N >. )
11 df-ot 3650 . . . . . . . . 9  |-  <. n ,  n ,  n >.  = 
<. <. n ,  n >. ,  n >.
12 df-ot 3650 . . . . . . . . 9  |-  <. N ,  N ,  N >.  = 
<. <. N ,  N >. ,  N >.
1310, 11, 123eqtr4g 2340 . . . . . . . 8  |-  ( n  =  N  ->  <. n ,  n ,  n >.  = 
<. N ,  N ,  N >. )
143, 13oveqan12rd 5878 . . . . . . 7  |-  ( ( n  =  N  /\  r  =  R )  ->  ( r maMul  <. n ,  n ,  n >. )  =  ( R maMul  <. N ,  N ,  N >. ) )
15 matval.t . . . . . . 7  |-  .x.  =  ( R maMul  <. N ,  N ,  N >. )
1614, 15syl6eqr 2333 . . . . . 6  |-  ( ( n  =  N  /\  r  =  R )  ->  ( r maMul  <. n ,  n ,  n >. )  =  .x.  )
1716opeq2d 3803 . . . . 5  |-  ( ( n  =  N  /\  r  =  R )  -> 
<. ( .r `  ndx ) ,  ( r maMul  <.
n ,  n ,  n >. ) >.  =  <. ( .r `  ndx ) ,  .x.  >. )
188, 17oveq12d 5876 . . . 4  |-  ( ( n  =  N  /\  r  =  R )  ->  ( ( r freeLMod  (
n  X.  n ) ) sSet  <. ( .r `  ndx ) ,  ( r maMul  <. n ,  n ,  n >. ) >. )  =  ( G sSet  <. ( .r `  ndx ) ,  .x.  >. ) )
19 df-mat 26854 . . . 4  |- Mat  =  ( n  e.  Fin , 
r  e.  _V  |->  ( ( r freeLMod  ( n  X.  n ) ) sSet  <. ( .r `  ndx ) ,  ( r maMul  <.
n ,  n ,  n >. ) >. )
)
20 ovex 5883 . . . 4  |-  ( G sSet  <. ( .r `  ndx ) ,  .x.  >. )  e.  _V
2118, 19, 20ovmpt2a 5978 . . 3  |-  ( ( N  e.  Fin  /\  R  e.  _V )  ->  ( N Mat  R )  =  ( G sSet  <. ( .r `  ndx ) ,  .x.  >. ) )
222, 21sylan2 460 . 2  |-  ( ( N  e.  Fin  /\  R  e.  V )  ->  ( N Mat  R )  =  ( G sSet  <. ( .r `  ndx ) ,  .x.  >. ) )
231, 22syl5eq 2327 1  |-  ( ( N  e.  Fin  /\  R  e.  V )  ->  A  =  ( G sSet  <. ( .r `  ndx ) ,  .x.  >. )
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1623    e. wcel 1684   _Vcvv 2788   <.cop 3643   <.cotp 3644    X. cxp 4687   ` cfv 5255  (class class class)co 5858   Fincfn 6863   ndxcnx 13145   sSet csts 13146   .rcmulr 13209   freeLMod cfrlm 26624   maMul cmmul 26851   Mat cmat 26852
This theorem is referenced by:  matmulr  26879  matbas  26880  matplusg  26881  matsca  26882  matvsca  26883
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-ot 3650  df-uni 3828  df-br 4024  df-opab 4078  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-iota 5219  df-fun 5257  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-mat 26854
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