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Theorem grpinvfvi 15678
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 5844 . . . 4  |-  ( G  e.  _V  ->  (  _I  `  G )  =  G )
32fveq2d 5790 . . 3  |-  ( G  e.  _V  ->  ( invg `  (  _I 
`  G ) )  =  ( invg `  G ) )
4 base0 14312 . . . . . 6  |-  (/)  =  (
Base `  (/) )
5 eqid 2451 . . . . . 6  |-  ( invg `  (/) )  =  ( invg `  (/) )
64, 5grpinvfn 15677 . . . . 5  |-  ( invg `  (/) )  Fn  (/)
7 fn0 5625 . . . . 5  |-  ( ( invg `  (/) )  Fn  (/) 
<->  ( invg `  (/) )  =  (/) )
86, 7mpbi 208 . . . 4  |-  ( invg `  (/) )  =  (/)
9 fvprc 5780 . . . . 5  |-  ( -.  G  e.  _V  ->  (  _I  `  G )  =  (/) )
109fveq2d 5790 . . . 4  |-  ( -.  G  e.  _V  ->  ( invg `  (  _I  `  G ) )  =  ( invg `  (/) ) )
11 fvprc 5780 . . . 4  |-  ( -.  G  e.  _V  ->  ( invg `  G
)  =  (/) )
128, 10, 113eqtr4a 2517 . . 3  |-  ( -.  G  e.  _V  ->  ( invg `  (  _I  `  G ) )  =  ( invg `  G ) )
133, 12pm2.61i 164 . 2  |-  ( invg `  (  _I 
`  G ) )  =  ( invg `  G )
141, 13eqtr4i 2482 1  |-  N  =  ( invg `  (  _I  `  G ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    = wceq 1370    e. wcel 1758   _Vcvv 3065   (/)c0 3732    _I cid 4726    Fn wfn 5508   ` cfv 5513   invgcminusg 15510
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-rep 4498  ax-sep 4508  ax-nul 4516  ax-pow 4565  ax-pr 4626
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2599  df-ne 2644  df-ral 2798  df-rex 2799  df-reu 2800  df-rab 2802  df-v 3067  df-sbc 3282  df-csb 3384  df-dif 3426  df-un 3428  df-in 3430  df-ss 3437  df-nul 3733  df-if 3887  df-sn 3973  df-pr 3975  df-op 3979  df-uni 4187  df-iun 4268  df-br 4388  df-opab 4446  df-mpt 4447  df-id 4731  df-xp 4941  df-rel 4942  df-cnv 4943  df-co 4944  df-dm 4945  df-rn 4946  df-res 4947  df-ima 4948  df-iota 5476  df-fun 5515  df-fn 5516  df-f 5517  df-f1 5518  df-fo 5519  df-f1o 5520  df-fv 5521  df-riota 6148  df-ov 6190  df-slot 14277  df-base 14278  df-minusg 15645
This theorem is referenced by:  deg1invg  21691
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