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Theorem grpinvfvi 14572
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 5617 . . . 4  |-  ( G  e.  _V  ->  (  _I  `  G )  =  G )
32fveq2d 5567 . . 3  |-  ( G  e.  _V  ->  ( inv g `  (  _I 
`  G ) )  =  ( inv g `  G ) )
4 base0 13232 . . . . . 6  |-  (/)  =  (
Base `  (/) )
5 eqid 2316 . . . . . 6  |-  ( inv g `  (/) )  =  ( inv g `  (/) )
64, 5grpinvfn 14571 . . . . 5  |-  ( inv g `  (/) )  Fn  (/)
7 fn0 5400 . . . . 5  |-  ( ( inv g `  (/) )  Fn  (/) 
<->  ( inv g `  (/) )  =  (/) )
86, 7mpbi 199 . . . 4  |-  ( inv g `  (/) )  =  (/)
9 fvprc 5557 . . . . 5  |-  ( -.  G  e.  _V  ->  (  _I  `  G )  =  (/) )
109fveq2d 5567 . . . 4  |-  ( -.  G  e.  _V  ->  ( inv g `  (  _I  `  G ) )  =  ( inv g `  (/) ) )
11 fvprc 5557 . . . 4  |-  ( -.  G  e.  _V  ->  ( inv g `  G
)  =  (/) )
128, 10, 113eqtr4a 2374 . . 3  |-  ( -.  G  e.  _V  ->  ( inv g `  (  _I  `  G ) )  =  ( inv g `  G ) )
133, 12pm2.61i 156 . 2  |-  ( inv g `  (  _I 
`  G ) )  =  ( inv g `  G )
141, 13eqtr4i 2339 1  |-  N  =  ( inv g `  (  _I  `  G ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    = wceq 1633    e. wcel 1701   _Vcvv 2822   (/)c0 3489    _I cid 4341    Fn wfn 5287   ` cfv 5292   inv gcminusg 14412
This theorem is referenced by:  deg1invg  19545
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1537  ax-5 1548  ax-17 1607  ax-9 1645  ax-8 1666  ax-13 1703  ax-14 1705  ax-6 1720  ax-7 1725  ax-11 1732  ax-12 1897  ax-ext 2297  ax-rep 4168  ax-sep 4178  ax-nul 4186  ax-pow 4225  ax-pr 4251
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1533  df-nf 1536  df-sb 1640  df-eu 2180  df-mo 2181  df-clab 2303  df-cleq 2309  df-clel 2312  df-nfc 2441  df-ne 2481  df-ral 2582  df-rex 2583  df-reu 2584  df-rab 2586  df-v 2824  df-sbc 3026  df-csb 3116  df-dif 3189  df-un 3191  df-in 3193  df-ss 3200  df-nul 3490  df-if 3600  df-sn 3680  df-pr 3681  df-op 3683  df-uni 3865  df-iun 3944  df-br 4061  df-opab 4115  df-mpt 4116  df-id 4346  df-xp 4732  df-rel 4733  df-cnv 4734  df-co 4735  df-dm 4736  df-rn 4737  df-res 4738  df-ima 4739  df-iota 5256  df-fun 5294  df-fn 5295  df-f 5296  df-f1 5297  df-fo 5298  df-f1o 5299  df-fv 5300  df-ov 5903  df-riota 6346  df-slot 13199  df-base 13200  df-minusg 14539
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