Theorem gex2abl 18254

Description: A group with exponent 2 (or 1) is abelian. (Contributed by Mario Carneiro, 24-Apr-2016.)

Hypotheses

Ref	Expression
gexex.1	|- X = ( Base ` G )
gexex.2	|- E = (gEx ` G )

Assertion

Ref	Expression
gex2abl	|- ( ( G e. Grp /\ E || 2 ) -> G e. Abel )

Proof of Theorem gex2abl

Dummy variables x y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		gexex.1	. . 3 |- X = ( Base ` G )
2	1	a1i 11	. 2 |- ( ( G e. Grp /\ E || 2 ) -> X = ( Base ` G ) )
3		eqidd 2623	. 2 |- ( ( G e. Grp /\ E || 2 ) -> ( +g ` G ) = ( +g ` G ) )
4		simpl 473	. 2 |- ( ( G e. Grp /\ E || 2 ) -> G e. Grp )
5		simp1l 1085	. . . . . . . . 9 |- ( ( ( G e. Grp /\ E || 2 ) /\ x e. X /\ y e. X ) -> G e. Grp )
6		simp2 1062	. . . . . . . . 9 |- ( ( ( G e. Grp /\ E || 2 ) /\ x e. X /\ y e. X ) -> x e. X )
7		simp3 1063	. . . . . . . . 9 |- ( ( ( G e. Grp /\ E || 2 ) /\ x e. X /\ y e. X ) -> y e. X )
8		eqid 2622	. . . . . . . . . 10 |- ( +g ` G ) = ( +g ` G )
9	1, 8	grpass 17431	. . . . . . . . 9 |- ( ( G e. Grp /\ ( x e. X /\ y e. X /\ y e. X ) ) -> ( ( x ( +g ` G ) y ) ( +g ` G ) y ) = ( x ( +g ` G ) ( y ( +g ` G ) y ) ) )
10	5, 6, 7, 7, 9	syl13anc 1328	. . . . . . . 8 |- ( ( ( G e. Grp /\ E || 2 ) /\ x e. X /\ y e. X ) -> ( ( x ( +g ` G ) y ) ( +g ` G ) y ) = ( x ( +g ` G ) ( y ( +g ` G ) y ) ) )
11		eqid 2622	. . . . . . . . . . . 12 |- (.g ` G ) = (.g ` G )
12	1, 11, 8	mulg2 17550	. . . . . . . . . . 11 |- ( y e. X -> ( 2 (.g ` G ) y ) = ( y ( +g ` G ) y ) )
13	7, 12	syl 17	. . . . . . . . . 10 |- ( ( ( G e. Grp /\ E || 2 ) /\ x e. X /\ y e. X ) -> ( 2 (.g ` G ) y ) = ( y ( +g ` G ) y ) )
14		simp1r 1086	. . . . . . . . . . 11 |- ( ( ( G e. Grp /\ E || 2 ) /\ x e. X /\ y e. X ) -> E || 2 )
15		gexex.2	. . . . . . . . . . . 12 |- E = (gEx ` G )
16		eqid 2622	. . . . . . . . . . . 12 |- ( 0g ` G ) = ( 0g ` G )
17	1, 15, 11, 16	gexdvdsi 17998	. . . . . . . . . . 11 |- ( ( G e. Grp /\ y e. X /\ E || 2 ) -> ( 2 (.g ` G ) y ) = ( 0g ` G ) )
18	5, 7, 14, 17	syl3anc 1326	. . . . . . . . . 10 |- ( ( ( G e. Grp /\ E || 2 ) /\ x e. X /\ y e. X ) -> ( 2 (.g ` G ) y ) = ( 0g ` G ) )
19	13, 18	eqtr3d 2658	. . . . . . . . 9 |- ( ( ( G e. Grp /\ E || 2 ) /\ x e. X /\ y e. X ) -> ( y ( +g ` G ) y ) = ( 0g ` G ) )
20	19	oveq2d 6666	. . . . . . . 8 |- ( ( ( G e. Grp /\ E || 2 ) /\ x e. X /\ y e. X ) -> ( x ( +g ` G ) ( y ( +g ` G ) y ) ) = ( x ( +g ` G ) ( 0g ` G ) ) )
21	1, 8, 16	grprid 17453	. . . . . . . . 9 |- ( ( G e. Grp /\ x e. X ) -> ( x ( +g ` G ) ( 0g ` G ) ) = x )
22	5, 6, 21	syl2anc 693	. . . . . . . 8 |- ( ( ( G e. Grp /\ E || 2 ) /\ x e. X /\ y e. X ) -> ( x ( +g ` G ) ( 0g ` G ) ) = x )
23	10, 20, 22	3eqtrd 2660	. . . . . . 7 |- ( ( ( G e. Grp /\ E || 2 ) /\ x e. X /\ y e. X ) -> ( ( x ( +g ` G ) y ) ( +g ` G ) y ) = x )
24	23	oveq1d 6665	. . . . . 6 |- ( ( ( G e. Grp /\ E || 2 ) /\ x e. X /\ y e. X ) -> ( ( ( x ( +g ` G ) y ) ( +g ` G ) y ) ( +g ` G ) x ) = ( x ( +g ` G ) x ) )
25	1, 11, 8	mulg2 17550	. . . . . . 7 |- ( x e. X -> ( 2 (.g ` G ) x ) = ( x ( +g ` G ) x ) )
26	6, 25	syl 17	. . . . . 6 |- ( ( ( G e. Grp /\ E || 2 ) /\ x e. X /\ y e. X ) -> ( 2 (.g ` G ) x ) = ( x ( +g ` G ) x ) )
27	1, 15, 11, 16	gexdvdsi 17998	. . . . . . 7 |- ( ( G e. Grp /\ x e. X /\ E || 2 ) -> ( 2 (.g ` G ) x ) = ( 0g ` G ) )
28	5, 6, 14, 27	syl3anc 1326	. . . . . 6 |- ( ( ( G e. Grp /\ E || 2 ) /\ x e. X /\ y e. X ) -> ( 2 (.g ` G ) x ) = ( 0g ` G ) )
29	24, 26, 28	3eqtr2d 2662	. . . . 5 |- ( ( ( G e. Grp /\ E || 2 ) /\ x e. X /\ y e. X ) -> ( ( ( x ( +g ` G ) y ) ( +g ` G ) y ) ( +g ` G ) x ) = ( 0g ` G ) )
30	1, 8	grpcl 17430	. . . . . . 7 |- ( ( G e. Grp /\ x e. X /\ y e. X ) -> ( x ( +g ` G ) y ) e. X )
31	5, 6, 7, 30	syl3anc 1326	. . . . . 6 |- ( ( ( G e. Grp /\ E || 2 ) /\ x e. X /\ y e. X ) -> ( x ( +g ` G ) y ) e. X )
32	1, 15, 11, 16	gexdvdsi 17998	. . . . . 6 |- ( ( G e. Grp /\ ( x ( +g ` G ) y ) e. X /\ E || 2 ) -> ( 2 (.g ` G ) ( x ( +g ` G ) y ) ) = ( 0g ` G ) )
33	5, 31, 14, 32	syl3anc 1326	. . . . 5 |- ( ( ( G e. Grp /\ E || 2 ) /\ x e. X /\ y e. X ) -> ( 2 (.g ` G ) ( x ( +g ` G ) y ) ) = ( 0g ` G ) )
34	1, 11, 8	mulg2 17550	. . . . . 6 |- ( ( x ( +g ` G ) y ) e. X -> ( 2 (.g ` G ) ( x ( +g ` G ) y ) ) = ( ( x ( +g ` G ) y ) ( +g ` G ) ( x ( +g ` G ) y ) ) )
35	31, 34	syl 17	. . . . 5 |- ( ( ( G e. Grp /\ E || 2 ) /\ x e. X /\ y e. X ) -> ( 2 (.g ` G ) ( x ( +g ` G ) y ) ) = ( ( x ( +g ` G ) y ) ( +g ` G ) ( x ( +g ` G ) y ) ) )
36	29, 33, 35	3eqtr2d 2662	. . . 4 |- ( ( ( G e. Grp /\ E || 2 ) /\ x e. X /\ y e. X ) -> ( ( ( x ( +g ` G ) y ) ( +g ` G ) y ) ( +g ` G ) x ) = ( ( x ( +g ` G ) y ) ( +g ` G ) ( x ( +g ` G ) y ) ) )
37	1, 8	grpass 17431	. . . . 5 |- ( ( G e. Grp /\ ( ( x ( +g ` G ) y ) e. X /\ y e. X /\ x e. X ) ) -> ( ( ( x ( +g ` G ) y ) ( +g ` G ) y ) ( +g ` G ) x ) = ( ( x ( +g ` G ) y ) ( +g ` G ) ( y ( +g ` G ) x ) ) )
38	5, 31, 7, 6, 37	syl13anc 1328	. . . 4 |- ( ( ( G e. Grp /\ E || 2 ) /\ x e. X /\ y e. X ) -> ( ( ( x ( +g ` G ) y ) ( +g ` G ) y ) ( +g ` G ) x ) = ( ( x ( +g ` G ) y ) ( +g ` G ) ( y ( +g ` G ) x ) ) )
39	36, 38	eqtr3d 2658	. . 3 |- ( ( ( G e. Grp /\ E || 2 ) /\ x e. X /\ y e. X ) -> ( ( x ( +g ` G ) y ) ( +g ` G ) ( x ( +g ` G ) y ) ) = ( ( x ( +g ` G ) y ) ( +g ` G ) ( y ( +g ` G ) x ) ) )
40	1, 8	grpcl 17430	. . . . 5 |- ( ( G e. Grp /\ y e. X /\ x e. X ) -> ( y ( +g ` G ) x ) e. X )
41	5, 7, 6, 40	syl3anc 1326	. . . 4 |- ( ( ( G e. Grp /\ E || 2 ) /\ x e. X /\ y e. X ) -> ( y ( +g ` G ) x ) e. X )
42	1, 8	grplcan 17477	. . . 4 |- ( ( G e. Grp /\ ( ( x ( +g ` G ) y ) e. X /\ ( y ( +g ` G ) x ) e. X /\ ( x ( +g ` G ) y ) e. X ) ) -> ( ( ( x ( +g ` G ) y ) ( +g ` G ) ( x ( +g ` G ) y ) ) = ( ( x ( +g ` G ) y ) ( +g ` G ) ( y ( +g ` G ) x ) ) <-> ( x ( +g ` G ) y ) = ( y ( +g ` G ) x ) ) )
43	5, 31, 41, 31, 42	syl13anc 1328	. . 3 |- ( ( ( G e. Grp /\ E || 2 ) /\ x e. X /\ y e. X ) -> ( ( ( x ( +g ` G ) y ) ( +g ` G ) ( x ( +g ` G ) y ) ) = ( ( x ( +g ` G ) y ) ( +g ` G ) ( y ( +g ` G ) x ) ) <-> ( x ( +g ` G ) y ) = ( y ( +g ` G ) x ) ) )
44	39, 43	mpbid 222	. 2 |- ( ( ( G e. Grp /\ E || 2 ) /\ x e. X /\ y e. X ) -> ( x ( +g ` G ) y ) = ( y ( +g ` G ) x ) )
45	2, 3, 4, 44	isabld 18206	1 |- ( ( G e. Grp /\ E || 2 ) -> G e. Abel )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   class class class wbr 4653   ` cfv 5888  (class class class)co 6650   2c2 11070   || cdvds 14983   Basecbs 15857   +g cplusg 15941   0gc0g 16100   Grpcgrp 17422  ._gcmg 17540  gExcgex 17945   Abelcabl 18194

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-inf2 8538  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-sup 8348  df-inf 8349  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-nn 11021  df-2 11079  df-n0 11293  df-z 11378  df-uz 11688  df-fz 12327  df-seq 12802  df-dvds 14984  df-0g 16102  df-mgm 17242  df-sgrp 17284  df-mnd 17295  df-grp 17425  df-minusg 17426  df-mulg 17541  df-gex 17949  df-cmn 18195  df-abl 18196

This theorem is referenced by:  lt6abl  18296