Theorem oddvds 17189

Description: The only multiples of A that are equal to the identity are the multiples of the order of A. (Contributed by Mario Carneiro, 14-Jan-2015.) (Revised by Mario Carneiro, 23-Sep-2015.)

Hypotheses

Ref	Expression
odcl.1	|- X = ( Base ` G )
odcl.2	|- O = ( od ` G )
odid.3	|- .x. = (.g ` G )
odid.4	|- .0. = ( 0g ` G )

Assertion

Ref	Expression
oddvds	|- ( ( G e. Grp /\ A e. X /\ N e. ZZ ) -> ( ( O ` A ) || N <-> ( N .x. A ) = .0. ) )

Proof of Theorem oddvds

Step	Hyp	Ref	Expression
1		simpr 463	. . . 4 |- ( ( ( G e. Grp /\ A e. X /\ N e. ZZ ) /\ ( O ` A ) e. NN ) -> ( O ` A ) e. NN )
2		simpl3 1012	. . . 4 |- ( ( ( G e. Grp /\ A e. X /\ N e. ZZ ) /\ ( O ` A ) e. NN ) -> N e. ZZ )
3		dvdsval3 14302	. . . 4 |- ( ( ( O ` A ) e. NN /\ N e. ZZ ) -> ( ( O ` A ) || N <-> ( N mod ( O ` A ) ) = 0 ) )
4	1, 2, 3	syl2anc 666	. . 3 |- ( ( ( G e. Grp /\ A e. X /\ N e. ZZ ) /\ ( O ` A ) e. NN ) -> ( ( O ` A ) || N <-> ( N mod ( O ` A ) ) = 0 ) )
5		simpl2 1011	. . . . . 6 |- ( ( ( G e. Grp /\ A e. X /\ N e. ZZ ) /\ ( O ` A ) e. NN ) -> A e. X )
6		odcl.1	. . . . . . 7 |- X = ( Base ` G )
7		odid.4	. . . . . . 7 |- .0. = ( 0g ` G )
8		odid.3	. . . . . . 7 |- .x. = (.g ` G )
9	6, 7, 8	mulg0 16756	. . . . . 6 |- ( A e. X -> ( 0 .x. A ) = .0. )
10	5, 9	syl 17	. . . . 5 |- ( ( ( G e. Grp /\ A e. X /\ N e. ZZ ) /\ ( O ` A ) e. NN ) -> ( 0 .x. A ) = .0. )
11		oveq1 6295	. . . . . 6 |- ( ( N mod ( O ` A ) ) = 0 -> ( ( N mod ( O ` A ) ) .x. A ) = ( 0 .x. A ) )
12	11	eqeq1d 2452	. . . . 5 |- ( ( N mod ( O ` A ) ) = 0 -> ( ( ( N mod ( O ` A ) ) .x. A ) = .0. <-> ( 0 .x. A ) = .0. ) )
13	10, 12	syl5ibrcom 226	. . . 4 |- ( ( ( G e. Grp /\ A e. X /\ N e. ZZ ) /\ ( O ` A ) e. NN ) -> ( ( N mod ( O ` A ) ) = 0 -> ( ( N mod ( O ` A ) ) .x. A ) = .0. ) )
14	2	zred 11037	. . . . . . . 8 |- ( ( ( G e. Grp /\ A e. X /\ N e. ZZ ) /\ ( O ` A ) e. NN ) -> N e. RR )
15	1	nnrpd 11336	. . . . . . . 8 |- ( ( ( G e. Grp /\ A e. X /\ N e. ZZ ) /\ ( O ` A ) e. NN ) -> ( O ` A ) e. RR+ )
16		modlt 12104	. . . . . . . 8 |- ( ( N e. RR /\ ( O ` A ) e. RR+ ) -> ( N mod ( O ` A ) ) < ( O ` A ) )
17	14, 15, 16	syl2anc 666	. . . . . . 7 |- ( ( ( G e. Grp /\ A e. X /\ N e. ZZ ) /\ ( O ` A ) e. NN ) -> ( N mod ( O ` A ) ) < ( O ` A ) )
18	2, 1	zmodcld 12114	. . . . . . . . 9 |- ( ( ( G e. Grp /\ A e. X /\ N e. ZZ ) /\ ( O ` A ) e. NN ) -> ( N mod ( O ` A ) ) e. NN0 )
19	18	nn0red 10923	. . . . . . . 8 |- ( ( ( G e. Grp /\ A e. X /\ N e. ZZ ) /\ ( O ` A ) e. NN ) -> ( N mod ( O ` A ) ) e. RR )
20	1	nnred 10621	. . . . . . . 8 |- ( ( ( G e. Grp /\ A e. X /\ N e. ZZ ) /\ ( O ` A ) e. NN ) -> ( O ` A ) e. RR )
21	19, 20	ltnled 9779	. . . . . . 7 |- ( ( ( G e. Grp /\ A e. X /\ N e. ZZ ) /\ ( O ` A ) e. NN ) -> ( ( N mod ( O ` A ) ) < ( O ` A ) <-> -. ( O ` A ) <_ ( N mod ( O ` A ) ) ) )
22	17, 21	mpbid 214	. . . . . 6 |- ( ( ( G e. Grp /\ A e. X /\ N e. ZZ ) /\ ( O ` A ) e. NN ) -> -. ( O ` A ) <_ ( N mod ( O ` A ) ) )
23		odcl.2	. . . . . . . . . . . 12 |- O = ( od ` G )
24	6, 23, 8, 7	odlem2 17181	. . . . . . . . . . 11 |- ( ( A e. X /\ ( N mod ( O ` A ) ) e. NN /\ ( ( N mod ( O ` A ) ) .x. A ) = .0. ) -> ( O ` A ) e. ( 1 ... ( N mod ( O ` A ) ) ) )
25		elfzle2 11800	. . . . . . . . . . 11 |- ( ( O ` A ) e. ( 1 ... ( N mod ( O ` A ) ) ) -> ( O ` A ) <_ ( N mod ( O ` A ) ) )
26	24, 25	syl 17	. . . . . . . . . 10 |- ( ( A e. X /\ ( N mod ( O ` A ) ) e. NN /\ ( ( N mod ( O ` A ) ) .x. A ) = .0. ) -> ( O ` A ) <_ ( N mod ( O ` A ) ) )
27	26	3com23 1213	. . . . . . . . 9 |- ( ( A e. X /\ ( ( N mod ( O ` A ) ) .x. A ) = .0. /\ ( N mod ( O ` A ) ) e. NN ) -> ( O ` A ) <_ ( N mod ( O ` A ) ) )
28	27	3expia 1209	. . . . . . . 8 |- ( ( A e. X /\ ( ( N mod ( O ` A ) ) .x. A ) = .0. ) -> ( ( N mod ( O ` A ) ) e. NN -> ( O ` A ) <_ ( N mod ( O ` A ) ) ) )
29	28	con3d 139	. . . . . . 7 |- ( ( A e. X /\ ( ( N mod ( O ` A ) ) .x. A ) = .0. ) -> ( -. ( O ` A ) <_ ( N mod ( O ` A ) ) -> -. ( N mod ( O ` A ) ) e. NN ) )
30	29	impancom 442	. . . . . 6 |- ( ( A e. X /\ -. ( O ` A ) <_ ( N mod ( O ` A ) ) ) -> ( ( ( N mod ( O ` A ) ) .x. A ) = .0. -> -. ( N mod ( O ` A ) ) e. NN ) )
31	5, 22, 30	syl2anc 666	. . . . 5 |- ( ( ( G e. Grp /\ A e. X /\ N e. ZZ ) /\ ( O ` A ) e. NN ) -> ( ( ( N mod ( O ` A ) ) .x. A ) = .0. -> -. ( N mod ( O ` A ) ) e. NN ) )
32		elnn0 10868	. . . . . . 7 |- ( ( N mod ( O ` A ) ) e. NN0 <-> ( ( N mod ( O ` A ) ) e. NN \/ ( N mod ( O ` A ) ) = 0 ) )
33	18, 32	sylib 200	. . . . . 6 |- ( ( ( G e. Grp /\ A e. X /\ N e. ZZ ) /\ ( O ` A ) e. NN ) -> ( ( N mod ( O ` A ) ) e. NN \/ ( N mod ( O ` A ) ) = 0 ) )
34	33	ord 379	. . . . 5 |- ( ( ( G e. Grp /\ A e. X /\ N e. ZZ ) /\ ( O ` A ) e. NN ) -> ( -. ( N mod ( O ` A ) ) e. NN -> ( N mod ( O ` A ) ) = 0 ) )
35	31, 34	syld 45	. . . 4 |- ( ( ( G e. Grp /\ A e. X /\ N e. ZZ ) /\ ( O ` A ) e. NN ) -> ( ( ( N mod ( O ` A ) ) .x. A ) = .0. -> ( N mod ( O ` A ) ) = 0 ) )
36	13, 35	impbid 194	. . 3 |- ( ( ( G e. Grp /\ A e. X /\ N e. ZZ ) /\ ( O ` A ) e. NN ) -> ( ( N mod ( O ` A ) ) = 0 <-> ( ( N mod ( O ` A ) ) .x. A ) = .0. ) )
37	6, 23, 8, 7	odmod 17188	. . . 4 |- ( ( ( G e. Grp /\ A e. X /\ N e. ZZ ) /\ ( O ` A ) e. NN ) -> ( ( N mod ( O ` A ) ) .x. A ) = ( N .x. A ) )
38	37	eqeq1d 2452	. . 3 |- ( ( ( G e. Grp /\ A e. X /\ N e. ZZ ) /\ ( O ` A ) e. NN ) -> ( ( ( N mod ( O ` A ) ) .x. A ) = .0. <-> ( N .x. A ) = .0. ) )
39	4, 36, 38	3bitrd 283	. 2 |- ( ( ( G e. Grp /\ A e. X /\ N e. ZZ ) /\ ( O ` A ) e. NN ) -> ( ( O ` A ) || N <-> ( N .x. A ) = .0. ) )
40		simpr 463	. . . 4 |- ( ( ( G e. Grp /\ A e. X /\ N e. ZZ ) /\ ( O ` A ) = 0 ) -> ( O ` A ) = 0 )
41	40	breq1d 4411	. . 3 |- ( ( ( G e. Grp /\ A e. X /\ N e. ZZ ) /\ ( O ` A ) = 0 ) -> ( ( O ` A ) || N <-> 0 || N ) )
42		simpl3 1012	. . . 4 |- ( ( ( G e. Grp /\ A e. X /\ N e. ZZ ) /\ ( O ` A ) = 0 ) -> N e. ZZ )
43		0dvds 14316	. . . 4 |- ( N e. ZZ -> ( 0 || N <-> N = 0 ) )
44	42, 43	syl 17	. . 3 |- ( ( ( G e. Grp /\ A e. X /\ N e. ZZ ) /\ ( O ` A ) = 0 ) -> ( 0 || N <-> N = 0 ) )
45		simpl2 1011	. . . . . 6 |- ( ( ( G e. Grp /\ A e. X /\ N e. ZZ ) /\ ( O ` A ) = 0 ) -> A e. X )
46	45, 9	syl 17	. . . . 5 |- ( ( ( G e. Grp /\ A e. X /\ N e. ZZ ) /\ ( O ` A ) = 0 ) -> ( 0 .x. A ) = .0. )
47		oveq1 6295	. . . . . 6 |- ( N = 0 -> ( N .x. A ) = ( 0 .x. A ) )
48	47	eqeq1d 2452	. . . . 5 |- ( N = 0 -> ( ( N .x. A ) = .0. <-> ( 0 .x. A ) = .0. ) )
49	46, 48	syl5ibrcom 226	. . . 4 |- ( ( ( G e. Grp /\ A e. X /\ N e. ZZ ) /\ ( O ` A ) = 0 ) -> ( N = 0 -> ( N .x. A ) = .0. ) )
50	6, 23, 8, 7	odnncl 17187	. . . . . . . . 9 |- ( ( ( G e. Grp /\ A e. X /\ N e. ZZ ) /\ ( N =/= 0 /\ ( N .x. A ) = .0. ) ) -> ( O ` A ) e. NN )
51	50	nnne0d 10651	. . . . . . . 8 |- ( ( ( G e. Grp /\ A e. X /\ N e. ZZ ) /\ ( N =/= 0 /\ ( N .x. A ) = .0. ) ) -> ( O ` A ) =/= 0 )
52	51	expr 619	. . . . . . 7 |- ( ( ( G e. Grp /\ A e. X /\ N e. ZZ ) /\ N =/= 0 ) -> ( ( N .x. A ) = .0. -> ( O ` A ) =/= 0 ) )
53	52	impancom 442	. . . . . 6 |- ( ( ( G e. Grp /\ A e. X /\ N e. ZZ ) /\ ( N .x. A ) = .0. ) -> ( N =/= 0 -> ( O ` A ) =/= 0 ) )
54	53	necon4d 2647	. . . . 5 |- ( ( ( G e. Grp /\ A e. X /\ N e. ZZ ) /\ ( N .x. A ) = .0. ) -> ( ( O ` A ) = 0 -> N = 0 ) )
55	54	impancom 442	. . . 4 |- ( ( ( G e. Grp /\ A e. X /\ N e. ZZ ) /\ ( O ` A ) = 0 ) -> ( ( N .x. A ) = .0. -> N = 0 ) )
56	49, 55	impbid 194	. . 3 |- ( ( ( G e. Grp /\ A e. X /\ N e. ZZ ) /\ ( O ` A ) = 0 ) -> ( N = 0 <-> ( N .x. A ) = .0. ) )
57	41, 44, 56	3bitrd 283	. 2 |- ( ( ( G e. Grp /\ A e. X /\ N e. ZZ ) /\ ( O ` A ) = 0 ) -> ( ( O ` A ) || N <-> ( N .x. A ) = .0. ) )
58	6, 23	odcl 17178	. . . 4 |- ( A e. X -> ( O ` A ) e. NN0 )
59	58	3ad2ant2 1029	. . 3 |- ( ( G e. Grp /\ A e. X /\ N e. ZZ ) -> ( O ` A ) e. NN0 )
60		elnn0 10868	. . 3 |- ( ( O ` A ) e. NN0 <-> ( ( O ` A ) e. NN \/ ( O ` A ) = 0 ) )
61	59, 60	sylib 200	. 2 |- ( ( G e. Grp /\ A e. X /\ N e. ZZ ) -> ( ( O ` A ) e. NN \/ ( O ` A ) = 0 ) )
62	39, 57, 61	mpjaodan 794	1 |- ( ( G e. Grp /\ A e. X /\ N e. ZZ ) -> ( ( O ` A ) || N <-> ( N .x. A ) = .0. ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 188   \/ wo 370   /\ wa 371   /\ w3a 984   = wceq 1443   e. wcel 1886   =/= wne 2621   class class class wbr 4401   ` cfv 5581  (class class class)co 6288   RRcr 9535   0cc0 9536   1c1 9537   < clt 9672   <_ cle 9673   NNcn 10606   NN0cn0 10866   ZZcz 10934   RR+crp 11299   ...cfz 11781   mod cmo 12093   || cdvds 14298   Basecbs 15114   0gc0g 15331   Grpcgrp 16662  ._gcmg 16665   odcod 17158

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1668  ax-4 1681  ax-5 1757  ax-6 1804  ax-7 1850  ax-8 1888  ax-9 1895  ax-10 1914  ax-11 1919  ax-12 1932  ax-13 2090  ax-ext 2430  ax-rep 4514  ax-sep 4524  ax-nul 4533  ax-pow 4580  ax-pr 4638  ax-un 6580  ax-inf2 8143  ax-cnex 9592  ax-resscn 9593  ax-1cn 9594  ax-icn 9595  ax-addcl 9596  ax-addrcl 9597  ax-mulcl 9598  ax-mulrcl 9599  ax-mulcom 9600  ax-addass 9601  ax-mulass 9602  ax-distr 9603  ax-i2m1 9604  ax-1ne0 9605  ax-1rid 9606  ax-rnegex 9607  ax-rrecex 9608  ax-cnre 9609  ax-pre-lttri 9610  ax-pre-lttrn 9611  ax-pre-ltadd 9612  ax-pre-mulgt0 9613  ax-pre-sup 9614

This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 985  df-3an 986  df-tru 1446  df-ex 1663  df-nf 1667  df-sb 1797  df-eu 2302  df-mo 2303  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2580  df-ne 2623  df-nel 2624  df-ral 2741  df-rex 2742  df-reu 2743  df-rmo 2744  df-rab 2745  df-v 3046  df-sbc 3267  df-csb 3363  df-dif 3406  df-un 3408  df-in 3410  df-ss 3417  df-pss 3419  df-nul 3731  df-if 3881  df-pw 3952  df-sn 3968  df-pr 3970  df-tp 3972  df-op 3974  df-uni 4198  df-iun 4279  df-br 4402  df-opab 4461  df-mpt 4462  df-tr 4497  df-eprel 4744  df-id 4748  df-po 4754  df-so 4755  df-fr 4792  df-we 4794  df-xp 4839  df-rel 4840  df-cnv 4841  df-co 4842  df-dm 4843  df-rn 4844  df-res 4845  df-ima 4846  df-pred 5379  df-ord 5425  df-on 5426  df-lim 5427  df-suc 5428  df-iota 5545  df-fun 5583  df-fn 5584  df-f 5585  df-f1 5586  df-fo 5587  df-f1o 5588  df-fv 5589  df-riota 6250  df-ov 6291  df-oprab 6292  df-mpt2 6293  df-om 6690  df-1st 6790  df-2nd 6791  df-wrecs 7025  df-recs 7087  df-rdg 7125  df-er 7360  df-en 7567  df-dom 7568  df-sdom 7569  df-sup 7953  df-inf 7954  df-pnf 9674  df-mnf 9675  df-xr 9676  df-ltxr 9677  df-le 9678  df-sub 9859  df-neg 9860  df-div 10267  df-nn 10607  df-2 10665  df-3 10666  df-n0 10867  df-z 10935  df-uz 11157  df-rp 11300  df-fz 11782  df-fl 12025  df-mod 12094  df-seq 12211  df-exp 12270  df-cj 13155  df-re 13156  df-im 13157  df-sqrt 13291  df-abs 13292  df-dvds 14299  df-0g 15333  df-mgm 16481  df-sgrp 16520  df-mnd 16530  df-grp 16666  df-minusg 16667  df-sbg 16668  df-mulg 16669  df-od 17165

This theorem is referenced by:  oddvdsi  17190  odcong  17191  odeq  17192  odmulgid  17198  odbezout  17202  gexdvds2  17230  gexod  17231  gexcl3  17232  odadd1  17479  odadd2  17480  oddvdssubg  17486  pgpfac1lem3a  17702  chrdvds  19092  dchrfi  24176  dchrabs  24181  dchrptlem2  24186  idomodle  36064