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Theorem invghm 16321
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 2443 . . 3  |-  ( +g  `  G )  =  ( +g  `  G )
3 ablgrp 16285 . . 3  |-  ( G  e.  Abel  ->  G  e. 
Grp )
4 invghm.m . . . . 5  |-  I  =  ( invg `  G )
51, 4grpinvf 15585 . . . 4  |-  ( G  e.  Grp  ->  I : B --> B )
63, 5syl 16 . . 3  |-  ( G  e.  Abel  ->  I : B --> B )
71, 2, 4ablinvadd 16302 . . . 4  |-  ( ( G  e.  Abel  /\  x  e.  B  /\  y  e.  B )  ->  (
I `  ( x
( +g  `  G ) y ) )  =  ( ( I `  x ) ( +g  `  G ) ( I `
 y ) ) )
873expb 1188 . . 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 15759 . 2  |-  ( G  e.  Abel  ->  I  e.  ( G  GrpHom  G ) )
10 ghmgrp1 15752 . . 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 15607 . . . . . . . 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 1218 . . . . . . 7  |-  ( ( I  e.  ( G 
GrpHom  G )  /\  (
x  e.  B  /\  y  e.  B )
)  ->  ( I `  ( y ( +g  `  G ) x ) )  =  ( ( I `  x ) ( +g  `  G
) ( I `  y ) ) )
1615fveq2d 5698 . . . . . 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 15586 . . . . . . . 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 15586 . . . . . . . 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 15755 . . . . . . 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 1218 . . . . . 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 15596 . . . . . . . 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 15596 . . . . . . . 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 6112 . . . . . 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 2479 . . . . 5  |-  ( ( I  e.  ( G 
GrpHom  G )  /\  (
x  e.  B  /\  y  e.  B )
)  ->  ( I `  ( I `  (
y ( +g  `  G
) x ) ) )  =  ( x ( +g  `  G
) y ) )
301, 2grpcl 15554 . . . . . . 7  |-  ( ( G  e.  Grp  /\  y  e.  B  /\  x  e.  B )  ->  ( y ( +g  `  G ) x )  e.  B )
3111, 12, 13, 30syl3anc 1218 . . . . . 6  |-  ( ( I  e.  ( G 
GrpHom  G )  /\  (
x  e.  B  /\  y  e.  B )
)  ->  ( y
( +g  `  G ) x )  e.  B
)
321, 4grpinvinv 15596 . . . . . 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 2477 . . . 4  |-  ( ( I  e.  ( G 
GrpHom  G )  /\  (
x  e.  B  /\  y  e.  B )
)  ->  ( x
( +g  `  G ) y )  =  ( y ( +g  `  G
) x ) )
3534ralrimivva 2811 . . 3  |-  ( I  e.  ( G  GrpHom  G )  ->  A. x  e.  B  A. y  e.  B  ( x
( +g  `  G ) y )  =  ( y ( +g  `  G
) x ) )
361, 2isabl2 16288 . . 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 1369    e. wcel 1756   A.wral 2718   -->wf 5417   ` cfv 5421  (class class class)co 6094   Basecbs 14177   +g cplusg 14241   Grpcgrp 15413   invgcminusg 15414    GrpHom cghm 15747   Abelcabel 16281
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4406  ax-sep 4416  ax-nul 4424  ax-pow 4473  ax-pr 4534  ax-un 6375
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2571  df-ne 2611  df-ral 2723  df-rex 2724  df-reu 2725  df-rmo 2726  df-rab 2727  df-v 2977  df-sbc 3190  df-csb 3292  df-dif 3334  df-un 3336  df-in 3338  df-ss 3345  df-nul 3641  df-if 3795  df-pw 3865  df-sn 3881  df-pr 3883  df-op 3887  df-uni 4095  df-iun 4176  df-br 4296  df-opab 4354  df-mpt 4355  df-id 4639  df-xp 4849  df-rel 4850  df-cnv 4851  df-co 4852  df-dm 4853  df-rn 4854  df-res 4855  df-ima 4856  df-iota 5384  df-fun 5423  df-fn 5424  df-f 5425  df-f1 5426  df-fo 5427  df-f1o 5428  df-fv 5429  df-riota 6055  df-ov 6097  df-oprab 6098  df-mpt2 6099  df-0g 14383  df-mnd 15418  df-grp 15548  df-minusg 15549  df-ghm 15748  df-cmn 16282  df-abl 16283
This theorem is referenced by:  gsuminv  16447  gsuminvOLD  16449  invlmhm  17126  tsmsinv  19725
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