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Theorem matrcl 27334
Description: Reverse closure for the matrix algebra. (Contributed by Stefan O'Rear, 5-Sep-2015.)
Hypotheses
Ref Expression
matrcl.a  |-  A  =  ( N Mat  R )
matrcl.b  |-  B  =  ( Base `  A
)
Assertion
Ref Expression
matrcl  |-  ( X  e.  B  ->  ( N  e.  Fin  /\  R  e.  _V ) )

Proof of Theorem matrcl
Dummy variables  a 
b are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 n0i 3593 . 2  |-  ( X  e.  B  ->  -.  B  =  (/) )
2 matrcl.a . . . . 5  |-  A  =  ( N Mat  R )
3 df-mat 27310 . . . . . 6  |- Mat  =  ( a  e.  Fin , 
b  e.  _V  |->  ( ( b freeLMod  ( a  X.  a ) ) sSet  <. ( .r `  ndx ) ,  ( b maMul  <.
a ,  a ,  a >. ) >. )
)
43mpt2ndm0 6432 . . . . 5  |-  ( -.  ( N  e.  Fin  /\  R  e.  _V )  ->  ( N Mat  R )  =  (/) )
52, 4syl5eq 2448 . . . 4  |-  ( -.  ( N  e.  Fin  /\  R  e.  _V )  ->  A  =  (/) )
65fveq2d 5691 . . 3  |-  ( -.  ( N  e.  Fin  /\  R  e.  _V )  ->  ( Base `  A
)  =  ( Base `  (/) ) )
7 matrcl.b . . 3  |-  B  =  ( Base `  A
)
8 base0 13461 . . 3  |-  (/)  =  (
Base `  (/) )
96, 7, 83eqtr4g 2461 . 2  |-  ( -.  ( N  e.  Fin  /\  R  e.  _V )  ->  B  =  (/) )
101, 9nsyl2 121 1  |-  ( X  e.  B  ->  ( N  e.  Fin  /\  R  e.  _V ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 359    = wceq 1649    e. wcel 1721   _Vcvv 2916   (/)c0 3588   <.cop 3777   <.cotp 3778    X. cxp 4835   ` cfv 5413  (class class class)co 6040   Fincfn 7068   ndxcnx 13421   sSet csts 13422   Basecbs 13424   .rcmulr 13485   freeLMod cfrlm 27080   maMul cmmul 27307   Mat cmat 27308
This theorem is referenced by:  matbas2i  27344  matplusg2  27345  matvsca2  27346
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-ral 2671  df-rex 2672  df-rab 2675  df-v 2918  df-sbc 3122  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-if 3700  df-sn 3780  df-pr 3781  df-op 3783  df-uni 3976  df-br 4173  df-opab 4227  df-mpt 4228  df-id 4458  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-iota 5377  df-fun 5415  df-fv 5421  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-slot 13428  df-base 13429  df-mat 27310
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