Theorem invghm 13921

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.

Step	Hyp	Ref	Expression
1		invghm.b	. . 3 |- B = ( Base ` G )
2		eqid 2231	. . 3 |- ( +g ` G ) = ( +g ` G )
3		ablgrp 13881	. . 3 |- ( G e. Abel -> G e. Grp )
4		invghm.m	. . . . 5 |- I = ( invg ` G )
5	1, 4	grpinvf 13635	. . . 4 |- ( G e. Grp -> I : B --> B )
6	3, 5	syl 14	. . 3 |- ( G e. Abel -> I : B --> B )
7	1, 2, 4	ablinvadd 13902	. . . 4 |- ( ( G e. Abel /\ x e. B /\ y e. B ) -> ( I ` ( x ( +g ` G ) y ) ) = ( ( I ` x ) ( +g ` G ) ( I ` y ) ) )
8	7	3expb 1230	. . 3 |- ( ( G e. Abel /\ ( x e. B /\ y e. B ) ) -> ( I ` ( x ( +g ` G ) y ) ) = ( ( I ` x ) ( +g ` G ) ( I ` y ) ) )
9	1, 1, 2, 2, 3, 3, 6, 8	isghmd 13844	. 2 |- ( G e. Abel -> I e. ( G GrpHom G ) )
10		ghmgrp1 13837	. . 3 |- ( I e. ( G GrpHom G ) -> G e. Grp )
11	10	adantr 276	. . . . . . . 8 |- ( ( I e. ( G GrpHom G ) /\ ( x e. B /\ y e. B ) ) -> G e. Grp )
12		simprr 533	. . . . . . . 8 |- ( ( I e. ( G GrpHom G ) /\ ( x e. B /\ y e. B ) ) -> y e. B )
13		simprl 531	. . . . . . . 8 |- ( ( I e. ( G GrpHom G ) /\ ( x e. B /\ y e. B ) ) -> x e. B )
14	1, 2, 4	grpinvadd 13666	. . . . . . . 8 |- ( ( G e. Grp /\ y e. B /\ x e. B ) -> ( I ` ( y ( +g ` G ) x ) ) = ( ( I ` x ) ( +g ` G ) ( I ` y ) ) )
15	11, 12, 13, 14	syl3anc 1273	. . . . . . 7 |- ( ( I e. ( G GrpHom G ) /\ ( x e. B /\ y e. B ) ) -> ( I ` ( y ( +g ` G ) x ) ) = ( ( I ` x ) ( +g ` G ) ( I ` y ) ) )
16	15	fveq2d 5643	. . . . . 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 109	. . . . . . 7 |- ( ( I e. ( G GrpHom G ) /\ ( x e. B /\ y e. B ) ) -> I e. ( G GrpHom G ) )
18	1, 4	grpinvcl 13636	. . . . . . . 8 |- ( ( G e. Grp /\ x e. B ) -> ( I ` x ) e. B )
19	11, 13, 18	syl2anc 411	. . . . . . 7 |- ( ( I e. ( G GrpHom G ) /\ ( x e. B /\ y e. B ) ) -> ( I ` x ) e. B )
20	1, 4	grpinvcl 13636	. . . . . . . 8 |- ( ( G e. Grp /\ y e. B ) -> ( I ` y ) e. B )
21	11, 12, 20	syl2anc 411	. . . . . . 7 |- ( ( I e. ( G GrpHom G ) /\ ( x e. B /\ y e. B ) ) -> ( I ` y ) e. B )
22	1, 2, 2	ghmlin 13840	. . . . . . 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 ) ) ) )
23	17, 19, 21, 22	syl3anc 1273	. . . . . 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 ) ) ) )
24	1, 4	grpinvinv 13655	. . . . . . . 8 |- ( ( G e. Grp /\ x e. B ) -> ( I ` ( I ` x ) ) = x )
25	11, 13, 24	syl2anc 411	. . . . . . 7 |- ( ( I e. ( G GrpHom G ) /\ ( x e. B /\ y e. B ) ) -> ( I ` ( I ` x ) ) = x )
26	1, 4	grpinvinv 13655	. . . . . . . 8 |- ( ( G e. Grp /\ y e. B ) -> ( I ` ( I ` y ) ) = y )
27	11, 12, 26	syl2anc 411	. . . . . . 7 |- ( ( I e. ( G GrpHom G ) /\ ( x e. B /\ y e. B ) ) -> ( I ` ( I ` y ) ) = y )
28	25, 27	oveq12d 6036	. . . . . 6 |- ( ( I e. ( G GrpHom G ) /\ ( x e. B /\ y e. B ) ) -> ( ( I ` ( I ` x ) ) ( +g ` G ) ( I ` ( I ` y ) ) ) = ( x ( +g ` G ) y ) )
29	16, 23, 28	3eqtrd 2268	. . . . 5 |- ( ( I e. ( G GrpHom G ) /\ ( x e. B /\ y e. B ) ) -> ( I ` ( I ` ( y ( +g ` G ) x ) ) ) = ( x ( +g ` G ) y ) )
30	1, 2	grpcl 13596	. . . . . . 7 |- ( ( G e. Grp /\ y e. B /\ x e. B ) -> ( y ( +g ` G ) x ) e. B )
31	11, 12, 13, 30	syl3anc 1273	. . . . . 6 |- ( ( I e. ( G GrpHom G ) /\ ( x e. B /\ y e. B ) ) -> ( y ( +g ` G ) x ) e. B )
32	1, 4	grpinvinv 13655	. . . . . 6 |- ( ( G e. Grp /\ ( y ( +g ` G ) x ) e. B ) -> ( I ` ( I ` ( y ( +g ` G ) x ) ) ) = ( y ( +g ` G ) x ) )
33	11, 31, 32	syl2anc 411	. . . . 5 |- ( ( I e. ( G GrpHom G ) /\ ( x e. B /\ y e. B ) ) -> ( I ` ( I ` ( y ( +g ` G ) x ) ) ) = ( y ( +g ` G ) x ) )
34	29, 33	eqtr3d 2266	. . . 4 |- ( ( I e. ( G GrpHom G ) /\ ( x e. B /\ y e. B ) ) -> ( x ( +g ` G ) y ) = ( y ( +g ` G ) x ) )
35	34	ralrimivva 2614	. . 3 |- ( I e. ( G GrpHom G ) -> A. x e. B A. y e. B ( x ( +g ` G ) y ) = ( y ( +g ` G ) x ) )
36	1, 2	isabl2 13886	. . 3 |- ( G e. Abel <-> ( G e. Grp /\ A. x e. B A. y e. B ( x ( +g ` G ) y ) = ( y ( +g ` G ) x ) ) )
37	10, 35, 36	sylanbrc 417	. 2 |- ( I e. ( G GrpHom G ) -> G e. Abel )
38	9, 37	impbii 126	1 |- ( G e. Abel <-> I e. ( G GrpHom G ) )

Syntax hints:   /\ wa 104   <-> wb 105   = wceq 1397   e. wcel 2202   A.wral 2510   -->wf 5322   ` cfv 5326  (class class class)co 6018   Basecbs 13087   +g cplusg 13165   Grpcgrp 13588   invgcminusg 13589   GrpHom cghm 13832   Abelcabl 13877

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 619  ax-in2 620  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-coll 4204  ax-sep 4207  ax-pow 4264  ax-pr 4299  ax-un 4530  ax-setind 4635  ax-cnex 8123  ax-resscn 8124  ax-1re 8126  ax-addrcl 8129

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ne 2403  df-ral 2515  df-rex 2516  df-reu 2517  df-rmo 2518  df-rab 2519  df-v 2804  df-sbc 3032  df-csb 3128  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-int 3929  df-iun 3972  df-br 4089  df-opab 4151  df-mpt 4152  df-id 4390  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-res 4737  df-ima 4738  df-iota 5286  df-fun 5328  df-fn 5329  df-f 5330  df-f1 5331  df-fo 5332  df-f1o 5333  df-fv 5334  df-riota 5971  df-ov 6021  df-oprab 6022  df-mpo 6023  df-inn 9144  df-2 9202  df-ndx 13090  df-slot 13091  df-base 13093  df-plusg 13178  df-0g 13346  df-mgm 13444  df-sgrp 13490  df-mnd 13505  df-grp 13591  df-minusg 13592  df-ghm 13833  df-cmn 13878  df-abl 13879

This theorem is referenced by: (None)