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Theorem invoppggim 15077
Description: The inverse is an antiautomorphism on any group. (Contributed by Stefan O'Rear, 26-Aug-2015.)
Hypotheses
Ref Expression
invoppggim.o  |-  O  =  (oppg
`  G )
invoppggim.i  |-  I  =  ( inv g `  G )
Assertion
Ref Expression
invoppggim  |-  ( G  e.  Grp  ->  I  e.  ( G GrpIso  O ) )

Proof of Theorem invoppggim
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2381 . . 3  |-  ( Base `  G )  =  (
Base `  G )
2 invoppggim.o . . . 4  |-  O  =  (oppg
`  G )
32, 1oppgbas 15068 . . 3  |-  ( Base `  G )  =  (
Base `  O )
4 eqid 2381 . . 3  |-  ( +g  `  G )  =  ( +g  `  G )
5 eqid 2381 . . 3  |-  ( +g  `  O )  =  ( +g  `  O )
6 id 20 . . 3  |-  ( G  e.  Grp  ->  G  e.  Grp )
72oppggrp 15074 . . 3  |-  ( G  e.  Grp  ->  O  e.  Grp )
8 invoppggim.i . . . 4  |-  I  =  ( inv g `  G )
91, 8grpinvf 14770 . . 3  |-  ( G  e.  Grp  ->  I : ( Base `  G
) --> ( Base `  G
) )
101, 4, 8grpinvadd 14788 . . . . 5  |-  ( ( G  e.  Grp  /\  x  e.  ( Base `  G )  /\  y  e.  ( Base `  G
) )  ->  (
I `  ( x
( +g  `  G ) y ) )  =  ( ( I `  y ) ( +g  `  G ) ( I `
 x ) ) )
11103expb 1154 . . . 4  |-  ( ( G  e.  Grp  /\  ( x  e.  ( Base `  G )  /\  y  e.  ( Base `  G ) ) )  ->  ( I `  ( x ( +g  `  G ) y ) )  =  ( ( I `  y ) ( +g  `  G
) ( I `  x ) ) )
124, 2, 5oppgplus 15066 . . . 4  |-  ( ( I `  x ) ( +g  `  O
) ( I `  y ) )  =  ( ( I `  y ) ( +g  `  G ) ( I `
 x ) )
1311, 12syl6eqr 2431 . . 3  |-  ( ( G  e.  Grp  /\  ( x  e.  ( Base `  G )  /\  y  e.  ( Base `  G ) ) )  ->  ( I `  ( x ( +g  `  G ) y ) )  =  ( ( I `  x ) ( +g  `  O
) ( I `  y ) ) )
141, 3, 4, 5, 6, 7, 9, 13isghmd 14936 . 2  |-  ( G  e.  Grp  ->  I  e.  ( G  GrpHom  O ) )
151, 8, 6grpinvf1o 14782 . 2  |-  ( G  e.  Grp  ->  I : ( Base `  G
)
-1-1-onto-> ( Base `  G )
)
161, 3isgim 14970 . 2  |-  ( I  e.  ( G GrpIso  O
)  <->  ( I  e.  ( G  GrpHom  O )  /\  I : (
Base `  G ) -1-1-onto-> ( Base `  G ) ) )
1714, 15, 16sylanbrc 646 1  |-  ( G  e.  Grp  ->  I  e.  ( G GrpIso  O ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1649    e. wcel 1717   -1-1-onto->wf1o 5387   ` cfv 5388  (class class class)co 6014   Basecbs 13390   +g cplusg 13450   Grpcgrp 14606   inv gcminusg 14607    GrpHom cghm 14924   GrpIso cgim 14965  oppgcoppg 15062
This theorem is referenced by:  oppggic  15078  gsumzinv  15461  symgtrinv  27076
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2362  ax-rep 4255  ax-sep 4265  ax-nul 4273  ax-pow 4312  ax-pr 4338  ax-un 4635  ax-cnex 8973  ax-resscn 8974  ax-1cn 8975  ax-icn 8976  ax-addcl 8977  ax-addrcl 8978  ax-mulcl 8979  ax-mulrcl 8980  ax-mulcom 8981  ax-addass 8982  ax-mulass 8983  ax-distr 8984  ax-i2m1 8985  ax-1ne0 8986  ax-1rid 8987  ax-rnegex 8988  ax-rrecex 8989  ax-cnre 8990  ax-pre-lttri 8991  ax-pre-lttrn 8992  ax-pre-ltadd 8993  ax-pre-mulgt0 8994
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2236  df-mo 2237  df-clab 2368  df-cleq 2374  df-clel 2377  df-nfc 2506  df-ne 2546  df-nel 2547  df-ral 2648  df-rex 2649  df-reu 2650  df-rmo 2651  df-rab 2652  df-v 2895  df-sbc 3099  df-csb 3189  df-dif 3260  df-un 3262  df-in 3264  df-ss 3271  df-pss 3273  df-nul 3566  df-if 3677  df-pw 3738  df-sn 3757  df-pr 3758  df-tp 3759  df-op 3760  df-uni 3952  df-iun 4031  df-br 4148  df-opab 4202  df-mpt 4203  df-tr 4238  df-eprel 4429  df-id 4433  df-po 4438  df-so 4439  df-fr 4476  df-we 4478  df-ord 4519  df-on 4520  df-lim 4521  df-suc 4522  df-om 4780  df-xp 4818  df-rel 4819  df-cnv 4820  df-co 4821  df-dm 4822  df-rn 4823  df-res 4824  df-ima 4825  df-iota 5352  df-fun 5390  df-fn 5391  df-f 5392  df-f1 5393  df-fo 5394  df-f1o 5395  df-fv 5396  df-ov 6017  df-oprab 6018  df-mpt2 6019  df-tpos 6409  df-riota 6479  df-recs 6563  df-rdg 6598  df-er 6835  df-en 7040  df-dom 7041  df-sdom 7042  df-pnf 9049  df-mnf 9050  df-xr 9051  df-ltxr 9052  df-le 9053  df-sub 9219  df-neg 9220  df-nn 9927  df-2 9984  df-ndx 13393  df-slot 13394  df-base 13395  df-sets 13396  df-plusg 13463  df-0g 13648  df-mnd 14611  df-grp 14733  df-minusg 14734  df-ghm 14925  df-gim 14967  df-oppg 15063
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