MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  invoppggim Unicode version

Theorem invoppggim 14849
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 2296 . . 3  |-  ( Base `  G )  =  (
Base `  G )
2 invoppggim.o . . . 4  |-  O  =  (oppg
`  G )
32, 1oppgbas 14840 . . 3  |-  ( Base `  G )  =  (
Base `  O )
4 eqid 2296 . . 3  |-  ( +g  `  G )  =  ( +g  `  G )
5 eqid 2296 . . 3  |-  ( +g  `  O )  =  ( +g  `  O )
6 id 19 . . 3  |-  ( G  e.  Grp  ->  G  e.  Grp )
72oppggrp 14846 . . 3  |-  ( G  e.  Grp  ->  O  e.  Grp )
8 invoppggim.i . . . 4  |-  I  =  ( inv g `  G )
91, 8grpinvf 14542 . . 3  |-  ( G  e.  Grp  ->  I : ( Base `  G
) --> ( Base `  G
) )
101, 4, 8grpinvadd 14560 . . . . 5  |-  ( ( G  e.  Grp  /\  x  e.  ( Base `  G )  /\  y  e.  ( Base `  G
) )  ->  (
I `  ( x
( +g  `  G ) y ) )  =  ( ( I `  y ) ( +g  `  G ) ( I `
 x ) ) )
11103expb 1152 . . . 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 14838 . . . 4  |-  ( ( I `  x ) ( +g  `  O
) ( I `  y ) )  =  ( ( I `  y ) ( +g  `  G ) ( I `
 x ) )
1311, 12syl6eqr 2346 . . 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 14708 . 2  |-  ( G  e.  Grp  ->  I  e.  ( G  GrpHom  O ) )
151, 8, 6grpinvf1o 14554 . 2  |-  ( G  e.  Grp  ->  I : ( Base `  G
)
-1-1-onto-> ( Base `  G )
)
161, 3isgim 14742 . 2  |-  ( I  e.  ( G GrpIso  O
)  <->  ( I  e.  ( G  GrpHom  O )  /\  I : (
Base `  G ) -1-1-onto-> ( Base `  G ) ) )
1714, 15, 16sylanbrc 645 1  |-  ( G  e.  Grp  ->  I  e.  ( G GrpIso  O ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1632    e. wcel 1696   -1-1-onto->wf1o 5270   ` cfv 5271  (class class class)co 5874   Basecbs 13164   +g cplusg 13224   Grpcgrp 14378   inv gcminusg 14379    GrpHom cghm 14696   GrpIso cgim 14737  oppgcoppg 14834
This theorem is referenced by:  oppggic  14850  gsumzinv  15233  symgtrinv  27516
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-tpos 6250  df-riota 6320  df-recs 6404  df-rdg 6439  df-er 6676  df-en 6880  df-dom 6881  df-sdom 6882  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-nn 9763  df-2 9820  df-ndx 13167  df-slot 13168  df-base 13169  df-sets 13170  df-plusg 13237  df-0g 13420  df-mnd 14383  df-grp 14505  df-minusg 14506  df-ghm 14697  df-gim 14739  df-oppg 14835
  Copyright terms: Public domain W3C validator