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Theorem matval 18311
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 2981 . . 3  |-  ( R  e.  V  ->  R  e.  _V )
3 id 22 . . . . . . 7  |-  ( r  =  R  ->  r  =  R )
4 id 22 . . . . . . . 8  |-  ( n  =  N  ->  n  =  N )
54, 4xpeq12d 4865 . . . . . . 7  |-  ( n  =  N  ->  (
n  X.  n )  =  ( N  X.  N ) )
63, 5oveqan12rd 6111 . . . . . 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 2493 . . . . 5  |-  ( ( n  =  N  /\  r  =  R )  ->  ( r freeLMod  ( n  X.  n ) )  =  G )
94, 4, 4oteq123d 4074 . . . . . . . 8  |-  ( n  =  N  ->  <. n ,  n ,  n >.  = 
<. N ,  N ,  N >. )
103, 9oveqan12rd 6111 . . . . . . 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 2493 . . . . . 6  |-  ( ( n  =  N  /\  r  =  R )  ->  ( r maMul  <. n ,  n ,  n >. )  =  .x.  )
1312opeq2d 4066 . . . . 5  |-  ( ( n  =  N  /\  r  =  R )  -> 
<. ( .r `  ndx ) ,  ( r maMul  <.
n ,  n ,  n >. ) >.  =  <. ( .r `  ndx ) ,  .x.  >. )
148, 13oveq12d 6109 . . . 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 18282 . . . 4  |- Mat  =  ( n  e.  Fin , 
r  e.  _V  |->  ( ( r freeLMod  ( n  X.  n ) ) sSet  <. ( .r `  ndx ) ,  ( r maMul  <.
n ,  n ,  n >. ) >. )
)
16 ovex 6116 . . . 4  |-  ( G sSet  <. ( .r `  ndx ) ,  .x.  >. )  e.  _V
1714, 15, 16ovmpt2a 6221 . . 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 2487 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 1369    e. wcel 1756   _Vcvv 2972   <.cop 3883   <.cotp 3885    X. cxp 4838   ` cfv 5418  (class class class)co 6091   Fincfn 7310   ndxcnx 14171   sSet csts 14172   .rcmulr 14239   freeLMod cfrlm 18171   maMul cmmul 18279   Mat cmat 18280
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4413  ax-nul 4421  ax-pr 4531
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-ne 2608  df-ral 2720  df-rex 2721  df-rab 2724  df-v 2974  df-sbc 3187  df-dif 3331  df-un 3333  df-in 3335  df-ss 3342  df-nul 3638  df-if 3792  df-sn 3878  df-pr 3880  df-op 3884  df-ot 3886  df-uni 4092  df-br 4293  df-opab 4351  df-id 4636  df-xp 4846  df-rel 4847  df-cnv 4848  df-co 4849  df-dm 4850  df-iota 5381  df-fun 5420  df-fv 5426  df-ov 6094  df-oprab 6095  df-mpt2 6096  df-mat 18282
This theorem is referenced by:  matmulr  18313  matbas  18314  matplusg  18315  matsca  18316  matvsca  18317
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