Theorem invghm 13977

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 13937	. . 3 |- ( G e. Abel -> G e. Grp )
4		invghm.m	. . . . 5 |- I = ( invg ` G )
5	1, 4	grpinvf 13691	. . . 4 |- ( G e. Grp -> I : B --> B )
6	3, 5	syl 14	. . 3 |- ( G e. Abel -> I : B --> B )
7	1, 2, 4	ablinvadd 13958	. . . 4 |- ( ( G e. Abel /\ x e. B /\ y e. B ) -> ( I ` ( x ( +g ` G ) y ) ) = ( ( I ` x ) ( +g ` G ) ( I ` y ) ) )
8	7	3expb 1231	. . 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 13900	. 2 |- ( G e. Abel -> I e. ( G GrpHom G ) )
10		ghmgrp1 13893	. . 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 13722	. . . . . . . 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 1274	. . . . . . 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 5652	. . . . . 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 13692	. . . . . . . 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 13692	. . . . . . . 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 13896	. . . . . . 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 1274	. . . . . 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 13711	. . . . . . . 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 13711	. . . . . . . 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 6046	. . . . . 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 13652	. . . . . . 7 |- ( ( G e. Grp /\ y e. B /\ x e. B ) -> ( y ( +g ` G ) x ) e. B )
31	11, 12, 13, 30	syl3anc 1274	. . . . . 6 |- ( ( I e. ( G GrpHom G ) /\ ( x e. B /\ y e. B ) ) -> ( y ( +g ` G ) x ) e. B )
32	1, 4	grpinvinv 13711	. . . . . 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 2615	. . 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 13942	. . 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 1398   e. wcel 2202   A.wral 2511   -->wf 5329   ` cfv 5333  (class class class)co 6028   Basecbs 13143   +g cplusg 13221   Grpcgrp 13644   invgcminusg 13645   GrpHom cghm 13888   Abelcabl 13933

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2204  ax-14 2205  ax-ext 2213  ax-coll 4209  ax-sep 4212  ax-pow 4270  ax-pr 4305  ax-un 4536  ax-setind 4641  ax-cnex 8166  ax-resscn 8167  ax-1re 8169  ax-addrcl 8172

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ne 2404  df-ral 2516  df-rex 2517  df-reu 2518  df-rmo 2519  df-rab 2520  df-v 2805  df-sbc 3033  df-csb 3129  df-dif 3203  df-un 3205  df-in 3207  df-ss 3214  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-uni 3899  df-int 3934  df-iun 3977  df-br 4094  df-opab 4156  df-mpt 4157  df-id 4396  df-xp 4737  df-rel 4738  df-cnv 4739  df-co 4740  df-dm 4741  df-rn 4742  df-res 4743  df-ima 4744  df-iota 5293  df-fun 5335  df-fn 5336  df-f 5337  df-f1 5338  df-fo 5339  df-f1o 5340  df-fv 5341  df-riota 5981  df-ov 6031  df-oprab 6032  df-mpo 6033  df-inn 9187  df-2 9245  df-ndx 13146  df-slot 13147  df-base 13149  df-plusg 13234  df-0g 13402  df-mgm 13500  df-sgrp 13546  df-mnd 13561  df-grp 13647  df-minusg 13648  df-ghm 13889  df-cmn 13934  df-abl 13935

This theorem is referenced by: (None)