Theorem gexdvdsi 17998

Description: Any group element is annihilated by any multiple of the group exponent. (Contributed by Mario Carneiro, 24-Apr-2016.)

Hypotheses

Ref	Expression
gexcl.1	|- X = ( Base ` G )
gexcl.2	|- E = (gEx ` G )
gexid.3	|- .x. = (.g ` G )
gexid.4	|- .0. = ( 0g ` G )

Assertion

Ref	Expression
gexdvdsi	|- ( ( G e. Grp /\ A e. X /\ E || N ) -> ( N .x. A ) = .0. )

Proof of Theorem gexdvdsi

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		simp3 1063	. . . 4 |- ( ( G e. Grp /\ A e. X /\ E || N ) -> E || N )
2		dvdszrcl 14988	. . . . 5 |- ( E || N -> ( E e. ZZ /\ N e. ZZ ) )
3		divides 14985	. . . . 5 |- ( ( E e. ZZ /\ N e. ZZ ) -> ( E || N <-> E. x e. ZZ ( x x. E ) = N ) )
4	2, 3	biadan2 674	. . . 4 |- ( E || N <-> ( ( E e. ZZ /\ N e. ZZ ) /\ E. x e. ZZ ( x x. E ) = N ) )
5	1, 4	sylib 208	. . 3 |- ( ( G e. Grp /\ A e. X /\ E || N ) -> ( ( E e. ZZ /\ N e. ZZ ) /\ E. x e. ZZ ( x x. E ) = N ) )
6	5	simprd 479	. 2 |- ( ( G e. Grp /\ A e. X /\ E || N ) -> E. x e. ZZ ( x x. E ) = N )
7		simpl1 1064	. . . . . 6 |- ( ( ( G e. Grp /\ A e. X /\ E || N ) /\ x e. ZZ ) -> G e. Grp )
8		simpr 477	. . . . . 6 |- ( ( ( G e. Grp /\ A e. X /\ E || N ) /\ x e. ZZ ) -> x e. ZZ )
9	5	simplld 791	. . . . . . 7 |- ( ( G e. Grp /\ A e. X /\ E || N ) -> E e. ZZ )
10	9	adantr 481	. . . . . 6 |- ( ( ( G e. Grp /\ A e. X /\ E || N ) /\ x e. ZZ ) -> E e. ZZ )
11		simpl2 1065	. . . . . 6 |- ( ( ( G e. Grp /\ A e. X /\ E || N ) /\ x e. ZZ ) -> A e. X )
12		gexcl.1	. . . . . . 7 |- X = ( Base ` G )
13		gexid.3	. . . . . . 7 |- .x. = (.g ` G )
14	12, 13	mulgass 17579	. . . . . 6 |- ( ( G e. Grp /\ ( x e. ZZ /\ E e. ZZ /\ A e. X ) ) -> ( ( x x. E ) .x. A ) = ( x .x. ( E .x. A ) ) )
15	7, 8, 10, 11, 14	syl13anc 1328	. . . . 5 |- ( ( ( G e. Grp /\ A e. X /\ E || N ) /\ x e. ZZ ) -> ( ( x x. E ) .x. A ) = ( x .x. ( E .x. A ) ) )
16		gexcl.2	. . . . . . . 8 |- E = (gEx ` G )
17		gexid.4	. . . . . . . 8 |- .0. = ( 0g ` G )
18	12, 16, 13, 17	gexid 17996	. . . . . . 7 |- ( A e. X -> ( E .x. A ) = .0. )
19	11, 18	syl 17	. . . . . 6 |- ( ( ( G e. Grp /\ A e. X /\ E || N ) /\ x e. ZZ ) -> ( E .x. A ) = .0. )
20	19	oveq2d 6666	. . . . 5 |- ( ( ( G e. Grp /\ A e. X /\ E || N ) /\ x e. ZZ ) -> ( x .x. ( E .x. A ) ) = ( x .x. .0. ) )
21	12, 13, 17	mulgz 17568	. . . . . 6 |- ( ( G e. Grp /\ x e. ZZ ) -> ( x .x. .0. ) = .0. )
22	21	3ad2antl1 1223	. . . . 5 |- ( ( ( G e. Grp /\ A e. X /\ E || N ) /\ x e. ZZ ) -> ( x .x. .0. ) = .0. )
23	15, 20, 22	3eqtrd 2660	. . . 4 |- ( ( ( G e. Grp /\ A e. X /\ E || N ) /\ x e. ZZ ) -> ( ( x x. E ) .x. A ) = .0. )
24		oveq1 6657	. . . . 5 |- ( ( x x. E ) = N -> ( ( x x. E ) .x. A ) = ( N .x. A ) )
25	24	eqeq1d 2624	. . . 4 |- ( ( x x. E ) = N -> ( ( ( x x. E ) .x. A ) = .0. <-> ( N .x. A ) = .0. ) )
26	23, 25	syl5ibcom 235	. . 3 |- ( ( ( G e. Grp /\ A e. X /\ E || N ) /\ x e. ZZ ) -> ( ( x x. E ) = N -> ( N .x. A ) = .0. ) )
27	26	rexlimdva 3031	. 2 |- ( ( G e. Grp /\ A e. X /\ E || N ) -> ( E. x e. ZZ ( x x. E ) = N -> ( N .x. A ) = .0. ) )
28	6, 27	mpd 15	1 |- ( ( G e. Grp /\ A e. X /\ E || N ) -> ( N .x. A ) = .0. )

Syntax hints:   -> wi 4   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   E.wrex 2913   class class class wbr 4653   ` cfv 5888  (class class class)co 6650   x. cmul 9941   ZZcz 11377   || cdvds 14983   Basecbs 15857   0gc0g 16100   Grpcgrp 17422  ._gcmg 17540  gExcgex 17945

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-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

This theorem is referenced by:  gexdvds  17999  gex2abl  18254