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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.
StepHypRef 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 )
54sqxpeqd 4878 . . . . . . 7  |-  ( n  =  N  ->  (
n  X.  n )  =  ( N  X.  N ) )
63, 5oveqan12rd 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 ) )
86, 7syl6eqr 2513 . . . . 5  |-  ( ( n  =  N  /\  r  =  R )  ->  ( r freeLMod  ( n  X.  n ) )  =  G )
94, 4, 4oteq123d 4194 . . . . . . . 8  |-  ( n  =  N  ->  <. n ,  n ,  n >.  = 
<. N ,  N ,  N >. )
103, 9oveqan12rd 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 >. )
1210, 11syl6eqr 2513 . . . . . 6  |-  ( ( n  =  N  /\  r  =  R )  ->  ( r maMul  <. n ,  n ,  n >. )  =  .x.  )
1312opeq2d 4186 . . . . 5  |-  ( ( n  =  N  /\  r  =  R )  -> 
<. ( .r `  ndx ) ,  ( r maMul  <.
n ,  n ,  n >. ) >.  =  <. ( .r `  ndx ) ,  .x.  >. )
148, 13oveq12d 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
1714, 15, 16ovmpt2a 6453 . . 3  |-  ( ( N  e.  Fin  /\  R  e.  _V )  ->  ( N Mat  R )  =  ( G sSet  <. ( .r `  ndx ) ,  .x.  >. ) )
182, 17sylan2 481 . 2  |-  ( ( N  e.  Fin  /\  R  e.  V )  ->  ( N Mat  R )  =  ( G sSet  <. ( .r `  ndx ) ,  .x.  >. ) )
191, 18syl5eq 2507 1  |-  ( ( N  e.  Fin  /\  R  e.  V )  ->  A  =  ( G sSet  <. ( .r `  ndx ) ,  .x.  >. )
)
Colors of variables: wff setvar class
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
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