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Theorem grpinvfvi 14848
Description: The group inverse function is compatible with identity-function protection. (Contributed by Stefan O'Rear, 21-Mar-2015.)
Hypothesis
Ref Expression
grpinvfvi.t  |-  N  =  ( inv g `  G )
Assertion
Ref Expression
grpinvfvi  |-  N  =  ( inv g `  (  _I  `  G ) )

Proof of Theorem grpinvfvi
StepHypRef Expression
1 grpinvfvi.t . 2  |-  N  =  ( inv g `  G )
2 fvi 5785 . . . 4  |-  ( G  e.  _V  ->  (  _I  `  G )  =  G )
32fveq2d 5734 . . 3  |-  ( G  e.  _V  ->  ( inv g `  (  _I 
`  G ) )  =  ( inv g `  G ) )
4 base0 13508 . . . . . 6  |-  (/)  =  (
Base `  (/) )
5 eqid 2438 . . . . . 6  |-  ( inv g `  (/) )  =  ( inv g `  (/) )
64, 5grpinvfn 14847 . . . . 5  |-  ( inv g `  (/) )  Fn  (/)
7 fn0 5566 . . . . 5  |-  ( ( inv g `  (/) )  Fn  (/) 
<->  ( inv g `  (/) )  =  (/) )
86, 7mpbi 201 . . . 4  |-  ( inv g `  (/) )  =  (/)
9 fvprc 5724 . . . . 5  |-  ( -.  G  e.  _V  ->  (  _I  `  G )  =  (/) )
109fveq2d 5734 . . . 4  |-  ( -.  G  e.  _V  ->  ( inv g `  (  _I  `  G ) )  =  ( inv g `  (/) ) )
11 fvprc 5724 . . . 4  |-  ( -.  G  e.  _V  ->  ( inv g `  G
)  =  (/) )
128, 10, 113eqtr4a 2496 . . 3  |-  ( -.  G  e.  _V  ->  ( inv g `  (  _I  `  G ) )  =  ( inv g `  G ) )
133, 12pm2.61i 159 . 2  |-  ( inv g `  (  _I 
`  G ) )  =  ( inv g `  G )
141, 13eqtr4i 2461 1  |-  N  =  ( inv g `  (  _I  `  G ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    = wceq 1653    e. wcel 1726   _Vcvv 2958   (/)c0 3630    _I cid 4495    Fn wfn 5451   ` cfv 5456   inv gcminusg 14688
This theorem is referenced by:  deg1invg  20031
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-rep 4322  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-reu 2714  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  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-iun 4097  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-rn 4891  df-res 4892  df-ima 4893  df-iota 5420  df-fun 5458  df-fn 5459  df-f 5460  df-f1 5461  df-fo 5462  df-f1o 5463  df-fv 5464  df-ov 6086  df-riota 6551  df-slot 13475  df-base 13476  df-minusg 14815
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