Theorem invghm 14038

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 2232	. . 3 |- ( +g ` G ) = ( +g ` G )
3		ablgrp 13998	. . 3 |- ( G e. Abel -> G e. Grp )
4		invghm.m	. . . . 5 |- I = ( invg ` G )
5	1, 4	grpinvf 13752	. . . 4 |- ( G e. Grp -> I : B --> B )
6	3, 5	syl 14	. . 3 |- ( G e. Abel -> I : B --> B )
7	1, 2, 4	ablinvadd 14019	. . . 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 13961	. 2 |- ( G e. Abel -> I e. ( G GrpHom G ) )
10		ghmgrp1 13954	. . 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 13783	. . . . . . . 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 5673	. . . . . 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 13753	. . . . . . . 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 13753	. . . . . . . 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 13957	. . . . . . 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 13772	. . . . . . . 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 13772	. . . . . . . 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 6067	. . . . . 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 2269	. . . . 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 13713	. . . . . . 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 13772	. . . . . 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 2267	. . . 4 |- ( ( I e. ( G GrpHom G ) /\ ( x e. B /\ y e. B ) ) -> ( x ( +g ` G ) y ) = ( y ( +g ` G ) x ) )
35	34	ralrimivva 2624	. . 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 14003	. . 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 2203   A.wral 2520   -->wf 5347   ` cfv 5351  (class class class)co 6049   Basecbs 13204   +g cplusg 13282   Grpcgrp 13705   invgcminusg 13706   GrpHom cghm 13949   Abelcabl 13994

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 2205  ax-14 2206  ax-ext 2214  ax-coll 4224  ax-sep 4227  ax-pow 4286  ax-pr 4321  ax-un 4553  ax-setind 4658  ax-cnex 8217  ax-resscn 8218  ax-1re 8220  ax-addrcl 8223

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ne 2413  df-ral 2525  df-rex 2526  df-reu 2527  df-rmo 2528  df-rab 2529  df-v 2814  df-sbc 3042  df-csb 3138  df-dif 3212  df-un 3214  df-in 3216  df-ss 3223  df-pw 3670  df-sn 3694  df-pr 3695  df-op 3697  df-uni 3914  df-int 3949  df-iun 3992  df-br 4109  df-opab 4171  df-mpt 4172  df-id 4413  df-xp 4754  df-rel 4755  df-cnv 4756  df-co 4757  df-dm 4758  df-rn 4759  df-res 4760  df-ima 4761  df-iota 5311  df-fun 5353  df-fn 5354  df-f 5355  df-f1 5356  df-fo 5357  df-f1o 5358  df-fv 5359  df-riota 6002  df-ov 6052  df-oprab 6053  df-mpo 6054  df-inn 9237  df-2 9295  df-ndx 13207  df-slot 13208  df-base 13210  df-plusg 13295  df-0g 13463  df-mgm 13561  df-sgrp 13607  df-mnd 13622  df-grp 13708  df-minusg 13709  df-ghm 13950  df-cmn 13995  df-abl 13996

This theorem is referenced by: (None)