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Theorem grpinvfvi 16218
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  =  ( invg `  G )
Assertion
Ref Expression
grpinvfvi  |-  N  =  ( invg `  (  _I  `  G ) )

Proof of Theorem grpinvfvi
StepHypRef Expression
1 grpinvfvi.t . 2  |-  N  =  ( invg `  G )
2 fvi 5930 . . . 4  |-  ( G  e.  _V  ->  (  _I  `  G )  =  G )
32fveq2d 5876 . . 3  |-  ( G  e.  _V  ->  ( invg `  (  _I 
`  G ) )  =  ( invg `  G ) )
4 base0 14685 . . . . . 6  |-  (/)  =  (
Base `  (/) )
5 eqid 2457 . . . . . 6  |-  ( invg `  (/) )  =  ( invg `  (/) )
64, 5grpinvfn 16217 . . . . 5  |-  ( invg `  (/) )  Fn  (/)
7 fn0 5706 . . . . 5  |-  ( ( invg `  (/) )  Fn  (/) 
<->  ( invg `  (/) )  =  (/) )
86, 7mpbi 208 . . . 4  |-  ( invg `  (/) )  =  (/)
9 fvprc 5866 . . . . 5  |-  ( -.  G  e.  _V  ->  (  _I  `  G )  =  (/) )
109fveq2d 5876 . . . 4  |-  ( -.  G  e.  _V  ->  ( invg `  (  _I  `  G ) )  =  ( invg `  (/) ) )
11 fvprc 5866 . . . 4  |-  ( -.  G  e.  _V  ->  ( invg `  G
)  =  (/) )
128, 10, 113eqtr4a 2524 . . 3  |-  ( -.  G  e.  _V  ->  ( invg `  (  _I  `  G ) )  =  ( invg `  G ) )
133, 12pm2.61i 164 . 2  |-  ( invg `  (  _I 
`  G ) )  =  ( invg `  G )
141, 13eqtr4i 2489 1  |-  N  =  ( invg `  (  _I  `  G ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    = wceq 1395    e. wcel 1819   _Vcvv 3109   (/)c0 3793    _I cid 4799    Fn wfn 5589   ` cfv 5594   invgcminusg 16181
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-rep 4568  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-reu 2814  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4252  df-iun 4334  df-br 4457  df-opab 4516  df-mpt 4517  df-id 4804  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-riota 6258  df-ov 6299  df-slot 14648  df-base 14649  df-minusg 16185
This theorem is referenced by:  deg1invg  22633
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