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Theorem matrcl 27445
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 3635 . 2  |-  ( X  e.  B  ->  -.  B  =  (/) )
2 matrcl.a . . . . 5  |-  A  =  ( N Mat  R )
3 df-mat 27421 . . . . . 6  |- Mat  =  ( a  e.  Fin , 
b  e.  _V  |->  ( ( b freeLMod  ( a  X.  a ) ) sSet  <. ( .r `  ndx ) ,  ( b maMul  <.
a ,  a ,  a >. ) >. )
)
43mpt2ndm0 6475 . . . . 5  |-  ( -.  ( N  e.  Fin  /\  R  e.  _V )  ->  ( N Mat  R )  =  (/) )
52, 4syl5eq 2482 . . . 4  |-  ( -.  ( N  e.  Fin  /\  R  e.  _V )  ->  A  =  (/) )
65fveq2d 5734 . . 3  |-  ( -.  ( N  e.  Fin  /\  R  e.  _V )  ->  ( Base `  A
)  =  ( Base `  (/) ) )
7 matrcl.b . . 3  |-  B  =  ( Base `  A
)
8 base0 13508 . . 3  |-  (/)  =  (
Base `  (/) )
96, 7, 83eqtr4g 2495 . 2  |-  ( -.  ( N  e.  Fin  /\  R  e.  _V )  ->  B  =  (/) )
101, 9nsyl2 122 1  |-  ( X  e.  B  ->  ( N  e.  Fin  /\  R  e.  _V ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 360    = wceq 1653    e. wcel 1726   _Vcvv 2958   (/)c0 3630   <.cop 3819   <.cotp 3820    X. cxp 4878   ` cfv 5456  (class class class)co 6083   Fincfn 7111   ndxcnx 13468   sSet csts 13469   Basecbs 13471   .rcmulr 13532   freeLMod cfrlm 27191   maMul cmmul 27418   Mat cmat 27419
This theorem is referenced by:  matbas2i  27455  matplusg2  27456  matvsca2  27457
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-13 1728  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-pow 4379  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-uni 4018  df-br 4215  df-opab 4269  df-mpt 4270  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-slot 13475  df-base 13476  df-mat 27421
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