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Theorem invghm 16715
Description: The inversion map is a group automorphism if and only if the group is abelian. (In general it is only a group homomorphism into the opposite group, but in an abelian group the opposite group coincides with the group itself.) (Contributed by Mario Carneiro, 4-May-2015.)
Hypotheses
Ref Expression
invghm.b  |-  B  =  ( Base `  G
)
invghm.m  |-  I  =  ( invg `  G )
Assertion
Ref Expression
invghm  |-  ( G  e.  Abel  <->  I  e.  ( G  GrpHom  G ) )

Proof of Theorem invghm
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 invghm.b . . 3  |-  B  =  ( Base `  G
)
2 eqid 2467 . . 3  |-  ( +g  `  G )  =  ( +g  `  G )
3 ablgrp 16676 . . 3  |-  ( G  e.  Abel  ->  G  e. 
Grp )
4 invghm.m . . . . 5  |-  I  =  ( invg `  G )
51, 4grpinvf 15966 . . . 4  |-  ( G  e.  Grp  ->  I : B --> B )
63, 5syl 16 . . 3  |-  ( G  e.  Abel  ->  I : B --> B )
71, 2, 4ablinvadd 16693 . . . 4  |-  ( ( G  e.  Abel  /\  x  e.  B  /\  y  e.  B )  ->  (
I `  ( x
( +g  `  G ) y ) )  =  ( ( I `  x ) ( +g  `  G ) ( I `
 y ) ) )
873expb 1197 . . 3  |-  ( ( G  e.  Abel  /\  (
x  e.  B  /\  y  e.  B )
)  ->  ( I `  ( x ( +g  `  G ) y ) )  =  ( ( I `  x ) ( +g  `  G
) ( I `  y ) ) )
91, 1, 2, 2, 3, 3, 6, 8isghmd 16148 . 2  |-  ( G  e.  Abel  ->  I  e.  ( G  GrpHom  G ) )
10 ghmgrp1 16141 . . 3  |-  ( I  e.  ( G  GrpHom  G )  ->  G  e.  Grp )
1110adantr 465 . . . . . . . 8  |-  ( ( I  e.  ( G 
GrpHom  G )  /\  (
x  e.  B  /\  y  e.  B )
)  ->  G  e.  Grp )
12 simprr 756 . . . . . . . 8  |-  ( ( I  e.  ( G 
GrpHom  G )  /\  (
x  e.  B  /\  y  e.  B )
)  ->  y  e.  B )
13 simprl 755 . . . . . . . 8  |-  ( ( I  e.  ( G 
GrpHom  G )  /\  (
x  e.  B  /\  y  e.  B )
)  ->  x  e.  B )
141, 2, 4grpinvadd 15988 . . . . . . . 8  |-  ( ( G  e.  Grp  /\  y  e.  B  /\  x  e.  B )  ->  ( I `  (
y ( +g  `  G
) x ) )  =  ( ( I `
 x ) ( +g  `  G ) ( I `  y
) ) )
1511, 12, 13, 14syl3anc 1228 . . . . . . 7  |-  ( ( I  e.  ( G 
GrpHom  G )  /\  (
x  e.  B  /\  y  e.  B )
)  ->  ( I `  ( y ( +g  `  G ) x ) )  =  ( ( I `  x ) ( +g  `  G
) ( I `  y ) ) )
1615fveq2d 5876 . . . . . 6  |-  ( ( I  e.  ( G 
GrpHom  G )  /\  (
x  e.  B  /\  y  e.  B )
)  ->  ( I `  ( I `  (
y ( +g  `  G
) x ) ) )  =  ( I `
 ( ( I `
 x ) ( +g  `  G ) ( I `  y
) ) ) )
17 simpl 457 . . . . . . 7  |-  ( ( I  e.  ( G 
GrpHom  G )  /\  (
x  e.  B  /\  y  e.  B )
)  ->  I  e.  ( G  GrpHom  G ) )
181, 4grpinvcl 15967 . . . . . . . 8  |-  ( ( G  e.  Grp  /\  x  e.  B )  ->  ( I `  x
)  e.  B )
1911, 13, 18syl2anc 661 . . . . . . 7  |-  ( ( I  e.  ( G 
GrpHom  G )  /\  (
x  e.  B  /\  y  e.  B )
)  ->  ( I `  x )  e.  B
)
201, 4grpinvcl 15967 . . . . . . . 8  |-  ( ( G  e.  Grp  /\  y  e.  B )  ->  ( I `  y
)  e.  B )
2111, 12, 20syl2anc 661 . . . . . . 7  |-  ( ( I  e.  ( G 
GrpHom  G )  /\  (
x  e.  B  /\  y  e.  B )
)  ->  ( I `  y )  e.  B
)
221, 2, 2ghmlin 16144 . . . . . . 7  |-  ( ( I  e.  ( G 
GrpHom  G )  /\  (
I `  x )  e.  B  /\  (
I `  y )  e.  B )  ->  (
I `  ( (
I `  x )
( +g  `  G ) ( I `  y
) ) )  =  ( ( I `  ( I `  x
) ) ( +g  `  G ) ( I `
 ( I `  y ) ) ) )
2317, 19, 21, 22syl3anc 1228 . . . . . 6  |-  ( ( I  e.  ( G 
GrpHom  G )  /\  (
x  e.  B  /\  y  e.  B )
)  ->  ( I `  ( ( I `  x ) ( +g  `  G ) ( I `
 y ) ) )  =  ( ( I `  ( I `
 x ) ) ( +g  `  G
) ( I `  ( I `  y
) ) ) )
241, 4grpinvinv 15977 . . . . . . . 8  |-  ( ( G  e.  Grp  /\  x  e.  B )  ->  ( I `  (
I `  x )
)  =  x )
2511, 13, 24syl2anc 661 . . . . . . 7  |-  ( ( I  e.  ( G 
GrpHom  G )  /\  (
x  e.  B  /\  y  e.  B )
)  ->  ( I `  ( I `  x
) )  =  x )
261, 4grpinvinv 15977 . . . . . . . 8  |-  ( ( G  e.  Grp  /\  y  e.  B )  ->  ( I `  (
I `  y )
)  =  y )
2711, 12, 26syl2anc 661 . . . . . . 7  |-  ( ( I  e.  ( G 
GrpHom  G )  /\  (
x  e.  B  /\  y  e.  B )
)  ->  ( I `  ( I `  y
) )  =  y )
2825, 27oveq12d 6313 . . . . . 6  |-  ( ( I  e.  ( G 
GrpHom  G )  /\  (
x  e.  B  /\  y  e.  B )
)  ->  ( (
I `  ( I `  x ) ) ( +g  `  G ) ( I `  (
I `  y )
) )  =  ( x ( +g  `  G
) y ) )
2916, 23, 283eqtrd 2512 . . . . 5  |-  ( ( I  e.  ( G 
GrpHom  G )  /\  (
x  e.  B  /\  y  e.  B )
)  ->  ( I `  ( I `  (
y ( +g  `  G
) x ) ) )  =  ( x ( +g  `  G
) y ) )
301, 2grpcl 15935 . . . . . . 7  |-  ( ( G  e.  Grp  /\  y  e.  B  /\  x  e.  B )  ->  ( y ( +g  `  G ) x )  e.  B )
3111, 12, 13, 30syl3anc 1228 . . . . . 6  |-  ( ( I  e.  ( G 
GrpHom  G )  /\  (
x  e.  B  /\  y  e.  B )
)  ->  ( y
( +g  `  G ) x )  e.  B
)
321, 4grpinvinv 15977 . . . . . 6  |-  ( ( G  e.  Grp  /\  ( y ( +g  `  G ) x )  e.  B )  -> 
( I `  (
I `  ( y
( +g  `  G ) x ) ) )  =  ( y ( +g  `  G ) x ) )
3311, 31, 32syl2anc 661 . . . . 5  |-  ( ( I  e.  ( G 
GrpHom  G )  /\  (
x  e.  B  /\  y  e.  B )
)  ->  ( I `  ( I `  (
y ( +g  `  G
) x ) ) )  =  ( y ( +g  `  G
) x ) )
3429, 33eqtr3d 2510 . . . 4  |-  ( ( I  e.  ( G 
GrpHom  G )  /\  (
x  e.  B  /\  y  e.  B )
)  ->  ( x
( +g  `  G ) y )  =  ( y ( +g  `  G
) x ) )
3534ralrimivva 2888 . . 3  |-  ( I  e.  ( G  GrpHom  G )  ->  A. x  e.  B  A. y  e.  B  ( x
( +g  `  G ) y )  =  ( y ( +g  `  G
) x ) )
361, 2isabl2 16679 . . 3  |-  ( G  e.  Abel  <->  ( G  e. 
Grp  /\  A. x  e.  B  A. y  e.  B  ( x
( +g  `  G ) y )  =  ( y ( +g  `  G
) x ) ) )
3710, 35, 36sylanbrc 664 . 2  |-  ( I  e.  ( G  GrpHom  G )  ->  G  e.  Abel )
389, 37impbii 188 1  |-  ( G  e.  Abel  <->  I  e.  ( G  GrpHom  G ) )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 184    /\ wa 369    = wceq 1379    e. wcel 1767   A.wral 2817   -->wf 5590   ` cfv 5594  (class class class)co 6295   Basecbs 14507   +g cplusg 14572   Grpcgrp 15925   invgcminusg 15926    GrpHom cghm 16136   Abelcabl 16672
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4564  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-un 6587
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2822  df-rex 2823  df-reu 2824  df-rmo 2825  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-op 4040  df-uni 4252  df-iun 4333  df-br 4454  df-opab 4512  df-mpt 4513  df-id 4801  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  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 6256  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-0g 14714  df-mgm 15746  df-sgrp 15785  df-mnd 15795  df-grp 15929  df-minusg 15930  df-ghm 16137  df-cmn 16673  df-abl 16674
This theorem is referenced by:  gsuminv  16844  gsuminvOLD  16846  invlmhm  17559  tsmsinv  20518
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