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Theorem invghm 16311
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 2441 . . 3  |-  ( +g  `  G )  =  ( +g  `  G )
3 ablgrp 16275 . . 3  |-  ( G  e.  Abel  ->  G  e. 
Grp )
4 invghm.m . . . . 5  |-  I  =  ( invg `  G )
51, 4grpinvf 15575 . . . 4  |-  ( G  e.  Grp  ->  I : B --> B )
63, 5syl 16 . . 3  |-  ( G  e.  Abel  ->  I : B --> B )
71, 2, 4ablinvadd 16292 . . . 4  |-  ( ( G  e.  Abel  /\  x  e.  B  /\  y  e.  B )  ->  (
I `  ( x
( +g  `  G ) y ) )  =  ( ( I `  x ) ( +g  `  G ) ( I `
 y ) ) )
873expb 1183 . . 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 15749 . 2  |-  ( G  e.  Abel  ->  I  e.  ( G  GrpHom  G ) )
10 ghmgrp1 15742 . . 3  |-  ( I  e.  ( G  GrpHom  G )  ->  G  e.  Grp )
1110adantr 462 . . . . . . . 8  |-  ( ( I  e.  ( G 
GrpHom  G )  /\  (
x  e.  B  /\  y  e.  B )
)  ->  G  e.  Grp )
12 simprr 751 . . . . . . . 8  |-  ( ( I  e.  ( G 
GrpHom  G )  /\  (
x  e.  B  /\  y  e.  B )
)  ->  y  e.  B )
13 simprl 750 . . . . . . . 8  |-  ( ( I  e.  ( G 
GrpHom  G )  /\  (
x  e.  B  /\  y  e.  B )
)  ->  x  e.  B )
141, 2, 4grpinvadd 15597 . . . . . . . 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 1213 . . . . . . 7  |-  ( ( I  e.  ( G 
GrpHom  G )  /\  (
x  e.  B  /\  y  e.  B )
)  ->  ( I `  ( y ( +g  `  G ) x ) )  =  ( ( I `  x ) ( +g  `  G
) ( I `  y ) ) )
1615fveq2d 5692 . . . . . 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 454 . . . . . . 7  |-  ( ( I  e.  ( G 
GrpHom  G )  /\  (
x  e.  B  /\  y  e.  B )
)  ->  I  e.  ( G  GrpHom  G ) )
181, 4grpinvcl 15576 . . . . . . . 8  |-  ( ( G  e.  Grp  /\  x  e.  B )  ->  ( I `  x
)  e.  B )
1911, 13, 18syl2anc 656 . . . . . . 7  |-  ( ( I  e.  ( G 
GrpHom  G )  /\  (
x  e.  B  /\  y  e.  B )
)  ->  ( I `  x )  e.  B
)
201, 4grpinvcl 15576 . . . . . . . 8  |-  ( ( G  e.  Grp  /\  y  e.  B )  ->  ( I `  y
)  e.  B )
2111, 12, 20syl2anc 656 . . . . . . 7  |-  ( ( I  e.  ( G 
GrpHom  G )  /\  (
x  e.  B  /\  y  e.  B )
)  ->  ( I `  y )  e.  B
)
221, 2, 2ghmlin 15745 . . . . . . 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 1213 . . . . . 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 15586 . . . . . . . 8  |-  ( ( G  e.  Grp  /\  x  e.  B )  ->  ( I `  (
I `  x )
)  =  x )
2511, 13, 24syl2anc 656 . . . . . . 7  |-  ( ( I  e.  ( G 
GrpHom  G )  /\  (
x  e.  B  /\  y  e.  B )
)  ->  ( I `  ( I `  x
) )  =  x )
261, 4grpinvinv 15586 . . . . . . . 8  |-  ( ( G  e.  Grp  /\  y  e.  B )  ->  ( I `  (
I `  y )
)  =  y )
2711, 12, 26syl2anc 656 . . . . . . 7  |-  ( ( I  e.  ( G 
GrpHom  G )  /\  (
x  e.  B  /\  y  e.  B )
)  ->  ( I `  ( I `  y
) )  =  y )
2825, 27oveq12d 6108 . . . . . 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 2477 . . . . 5  |-  ( ( I  e.  ( G 
GrpHom  G )  /\  (
x  e.  B  /\  y  e.  B )
)  ->  ( I `  ( I `  (
y ( +g  `  G
) x ) ) )  =  ( x ( +g  `  G
) y ) )
301, 2grpcl 15544 . . . . . . 7  |-  ( ( G  e.  Grp  /\  y  e.  B  /\  x  e.  B )  ->  ( y ( +g  `  G ) x )  e.  B )
3111, 12, 13, 30syl3anc 1213 . . . . . 6  |-  ( ( I  e.  ( G 
GrpHom  G )  /\  (
x  e.  B  /\  y  e.  B )
)  ->  ( y
( +g  `  G ) x )  e.  B
)
321, 4grpinvinv 15586 . . . . . 6  |-  ( ( G  e.  Grp  /\  ( y ( +g  `  G ) x )  e.  B )  -> 
( I `  (
I `  ( y
( +g  `  G ) x ) ) )  =  ( y ( +g  `  G ) x ) )
3311, 31, 32syl2anc 656 . . . . 5  |-  ( ( I  e.  ( G 
GrpHom  G )  /\  (
x  e.  B  /\  y  e.  B )
)  ->  ( I `  ( I `  (
y ( +g  `  G
) x ) ) )  =  ( y ( +g  `  G
) x ) )
3429, 33eqtr3d 2475 . . . 4  |-  ( ( I  e.  ( G 
GrpHom  G )  /\  (
x  e.  B  /\  y  e.  B )
)  ->  ( x
( +g  `  G ) y )  =  ( y ( +g  `  G
) x ) )
3534ralrimivva 2806 . . 3  |-  ( I  e.  ( G  GrpHom  G )  ->  A. x  e.  B  A. y  e.  B  ( x
( +g  `  G ) y )  =  ( y ( +g  `  G
) x ) )
361, 2isabl2 16278 . . 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 659 . 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 1364    e. wcel 1761   A.wral 2713   -->wf 5411   ` cfv 5415  (class class class)co 6090   Basecbs 14170   +g cplusg 14234   Grpcgrp 15406   invgcminusg 15407    GrpHom cghm 15737   Abelcabel 16271
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1713  ax-7 1733  ax-8 1763  ax-9 1765  ax-10 1780  ax-11 1785  ax-12 1797  ax-13 1948  ax-ext 2422  ax-rep 4400  ax-sep 4410  ax-nul 4418  ax-pow 4467  ax-pr 4528  ax-un 6371
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 962  df-tru 1367  df-ex 1592  df-nf 1595  df-sb 1706  df-eu 2261  df-mo 2262  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-ral 2718  df-rex 2719  df-reu 2720  df-rmo 2721  df-rab 2722  df-v 2972  df-sbc 3184  df-csb 3286  df-dif 3328  df-un 3330  df-in 3332  df-ss 3339  df-nul 3635  df-if 3789  df-pw 3859  df-sn 3875  df-pr 3877  df-op 3881  df-uni 4089  df-iun 4170  df-br 4290  df-opab 4348  df-mpt 4349  df-id 4632  df-xp 4842  df-rel 4843  df-cnv 4844  df-co 4845  df-dm 4846  df-rn 4847  df-res 4848  df-ima 4849  df-iota 5378  df-fun 5417  df-fn 5418  df-f 5419  df-f1 5420  df-fo 5421  df-f1o 5422  df-fv 5423  df-riota 6049  df-ov 6093  df-oprab 6094  df-mpt2 6095  df-0g 14376  df-mnd 15411  df-grp 15538  df-minusg 15539  df-ghm 15738  df-cmn 16272  df-abl 16273
This theorem is referenced by:  gsuminv  16436  gsuminvOLD  16438  invlmhm  17101  tsmsinv  19681
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