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Theorem matrcl 27569
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 3473 . 2  |-  ( X  e.  B  ->  -.  B  =  (/) )
2 matrcl.a . . . . 5  |-  A  =  ( N Mat  R )
3 df-mat 27545 . . . . . . . . 9  |- Mat  =  ( a  e.  Fin , 
b  e.  _V  |->  ( ( b freeLMod  ( a  X.  a ) ) sSet  <. ( .r `  ndx ) ,  ( b maMul  <.
a ,  a ,  a >. ) >. )
)
4 ovex 5899 . . . . . . . . 9  |-  ( ( b freeLMod  ( a  X.  a ) ) sSet  <. ( .r `  ndx ) ,  ( b maMul  <. a ,  a ,  a
>. ) >. )  e.  _V
53, 4dmmpt2 6210 . . . . . . . 8  |-  dom Mat  =  ( Fin  X.  _V )
65eleq2i 2360 . . . . . . 7  |-  ( <. N ,  R >.  e. 
dom Mat 
<-> 
<. N ,  R >.  e.  ( Fin  X.  _V ) )
7 opelxp 4735 . . . . . . 7  |-  ( <. N ,  R >.  e.  ( Fin  X.  _V ) 
<->  ( N  e.  Fin  /\  R  e.  _V )
)
86, 7bitri 240 . . . . . 6  |-  ( <. N ,  R >.  e. 
dom Mat 
<->  ( N  e.  Fin  /\  R  e.  _V )
)
9 df-ov 5877 . . . . . . 7  |-  ( N Mat 
R )  =  ( Mat  `  <. N ,  R >. )
10 ndmfv 5568 . . . . . . 7  |-  ( -. 
<. N ,  R >.  e. 
dom Mat  ->  ( Mat  `  <. N ,  R >. )  =  (/) )
119, 10syl5eq 2340 . . . . . 6  |-  ( -. 
<. N ,  R >.  e. 
dom Mat  ->  ( N Mat  R
)  =  (/) )
128, 11sylnbir 298 . . . . 5  |-  ( -.  ( N  e.  Fin  /\  R  e.  _V )  ->  ( N Mat  R )  =  (/) )
132, 12syl5eq 2340 . . . 4  |-  ( -.  ( N  e.  Fin  /\  R  e.  _V )  ->  A  =  (/) )
1413fveq2d 5545 . . 3  |-  ( -.  ( N  e.  Fin  /\  R  e.  _V )  ->  ( Base `  A
)  =  ( Base `  (/) ) )
15 matrcl.b . . 3  |-  B  =  ( Base `  A
)
16 base0 13201 . . 3  |-  (/)  =  (
Base `  (/) )
1714, 15, 163eqtr4g 2353 . 2  |-  ( -.  ( N  e.  Fin  /\  R  e.  _V )  ->  B  =  (/) )
181, 17nsyl2 119 1  |-  ( X  e.  B  ->  ( N  e.  Fin  /\  R  e.  _V ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 358    = wceq 1632    e. wcel 1696   _Vcvv 2801   (/)c0 3468   <.cop 3656   <.cotp 3657    X. cxp 4703   dom cdm 4705   ` cfv 5271  (class class class)co 5874   Fincfn 6879   ndxcnx 13161   sSet csts 13162   Basecbs 13164   .rcmulr 13225   freeLMod cfrlm 27315   maMul cmmul 27542   Mat cmat 27543
This theorem is referenced by:  matbas2i  27579  matplusg2  27580  matvsca2  27581
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-slot 13168  df-base 13169  df-mat 27545
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