Theorem oddvdsnn0 17963

Description: The only multiples of A that are equal to the identity are the multiples of the order of A. (Contributed 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
oddvdsnn0	|- ( ( G e. Mnd /\ A e. X /\ N e. NN0 ) -> ( ( O ` A ) || N <-> ( N .x. A ) = .0. ) )

Proof of Theorem oddvdsnn0

Step	Hyp	Ref	Expression
1		0nn0 11307	. . . . 5 |- 0 e. NN0
2		odcl.1	. . . . . . 7 |- X = ( Base ` G )
3		odcl.2	. . . . . . 7 |- O = ( od ` G )
4		odid.3	. . . . . . 7 |- .x. = (.g ` G )
5		odid.4	. . . . . . 7 |- .0. = ( 0g ` G )
6	2, 3, 4, 5	mndodcong 17961	. . . . . 6 |- ( ( ( G e. Mnd /\ A e. X ) /\ ( N e. NN0 /\ 0 e. NN0 ) /\ ( O ` A ) e. NN ) -> ( ( O ` A ) || ( N - 0 ) <-> ( N .x. A ) = ( 0 .x. A ) ) )
7	6	3expia 1267	. . . . 5 |- ( ( ( G e. Mnd /\ A e. X ) /\ ( N e. NN0 /\ 0 e. NN0 ) ) -> ( ( O ` A ) e. NN -> ( ( O ` A ) || ( N - 0 ) <-> ( N .x. A ) = ( 0 .x. A ) ) ) )
8	1, 7	mpanr2 720	. . . 4 |- ( ( ( G e. Mnd /\ A e. X ) /\ N e. NN0 ) -> ( ( O ` A ) e. NN -> ( ( O ` A ) || ( N - 0 ) <-> ( N .x. A ) = ( 0 .x. A ) ) ) )
9	8	3impa 1259	. . 3 |- ( ( G e. Mnd /\ A e. X /\ N e. NN0 ) -> ( ( O ` A ) e. NN -> ( ( O ` A ) || ( N - 0 ) <-> ( N .x. A ) = ( 0 .x. A ) ) ) )
10		nn0cn 11302	. . . . . . 7 |- ( N e. NN0 -> N e. CC )
11	10	3ad2ant3 1084	. . . . . 6 |- ( ( G e. Mnd /\ A e. X /\ N e. NN0 ) -> N e. CC )
12	11	subid1d 10381	. . . . 5 |- ( ( G e. Mnd /\ A e. X /\ N e. NN0 ) -> ( N - 0 ) = N )
13	12	breq2d 4665	. . . 4 |- ( ( G e. Mnd /\ A e. X /\ N e. NN0 ) -> ( ( O ` A ) || ( N - 0 ) <-> ( O ` A ) || N ) )
14	2, 5, 4	mulg0 17546	. . . . . 6 |- ( A e. X -> ( 0 .x. A ) = .0. )
15	14	3ad2ant2 1083	. . . . 5 |- ( ( G e. Mnd /\ A e. X /\ N e. NN0 ) -> ( 0 .x. A ) = .0. )
16	15	eqeq2d 2632	. . . 4 |- ( ( G e. Mnd /\ A e. X /\ N e. NN0 ) -> ( ( N .x. A ) = ( 0 .x. A ) <-> ( N .x. A ) = .0. ) )
17	13, 16	bibi12d 335	. . 3 |- ( ( G e. Mnd /\ A e. X /\ N e. NN0 ) -> ( ( ( O ` A ) || ( N - 0 ) <-> ( N .x. A ) = ( 0 .x. A ) ) <-> ( ( O ` A ) || N <-> ( N .x. A ) = .0. ) ) )
18	9, 17	sylibd 229	. 2 |- ( ( G e. Mnd /\ A e. X /\ N e. NN0 ) -> ( ( O ` A ) e. NN -> ( ( O ` A ) || N <-> ( N .x. A ) = .0. ) ) )
19		simpr 477	. . . . 5 |- ( ( ( G e. Mnd /\ A e. X /\ N e. NN0 ) /\ ( O ` A ) = 0 ) -> ( O ` A ) = 0 )
20	19	breq1d 4663	. . . 4 |- ( ( ( G e. Mnd /\ A e. X /\ N e. NN0 ) /\ ( O ` A ) = 0 ) -> ( ( O ` A ) || N <-> 0 || N ) )
21		simpl3 1066	. . . . 5 |- ( ( ( G e. Mnd /\ A e. X /\ N e. NN0 ) /\ ( O ` A ) = 0 ) -> N e. NN0 )
22		nn0z 11400	. . . . 5 |- ( N e. NN0 -> N e. ZZ )
23		0dvds 15002	. . . . 5 |- ( N e. ZZ -> ( 0 || N <-> N = 0 ) )
24	21, 22, 23	3syl 18	. . . 4 |- ( ( ( G e. Mnd /\ A e. X /\ N e. NN0 ) /\ ( O ` A ) = 0 ) -> ( 0 || N <-> N = 0 ) )
25	15	adantr 481	. . . . . 6 |- ( ( ( G e. Mnd /\ A e. X /\ N e. NN0 ) /\ ( O ` A ) = 0 ) -> ( 0 .x. A ) = .0. )
26		oveq1 6657	. . . . . . 7 |- ( N = 0 -> ( N .x. A ) = ( 0 .x. A ) )
27	26	eqeq1d 2624	. . . . . 6 |- ( N = 0 -> ( ( N .x. A ) = .0. <-> ( 0 .x. A ) = .0. ) )
28	25, 27	syl5ibrcom 237	. . . . 5 |- ( ( ( G e. Mnd /\ A e. X /\ N e. NN0 ) /\ ( O ` A ) = 0 ) -> ( N = 0 -> ( N .x. A ) = .0. ) )
29	2, 3, 4, 5	odlem2 17958	. . . . . . . . . . . 12 |- ( ( A e. X /\ N e. NN /\ ( N .x. A ) = .0. ) -> ( O ` A ) e. ( 1 ... N ) )
30	29	3com23 1271	. . . . . . . . . . 11 |- ( ( A e. X /\ ( N .x. A ) = .0. /\ N e. NN ) -> ( O ` A ) e. ( 1 ... N ) )
31		elfznn 12370	. . . . . . . . . . 11 |- ( ( O ` A ) e. ( 1 ... N ) -> ( O ` A ) e. NN )
32		nnne0 11053	. . . . . . . . . . 11 |- ( ( O ` A ) e. NN -> ( O ` A ) =/= 0 )
33	30, 31, 32	3syl 18	. . . . . . . . . 10 |- ( ( A e. X /\ ( N .x. A ) = .0. /\ N e. NN ) -> ( O ` A ) =/= 0 )
34	33	3expia 1267	. . . . . . . . 9 |- ( ( A e. X /\ ( N .x. A ) = .0. ) -> ( N e. NN -> ( O ` A ) =/= 0 ) )
35	34	3ad2antl2 1224	. . . . . . . 8 |- ( ( ( G e. Mnd /\ A e. X /\ N e. NN0 ) /\ ( N .x. A ) = .0. ) -> ( N e. NN -> ( O ` A ) =/= 0 ) )
36	35	necon2bd 2810	. . . . . . 7 |- ( ( ( G e. Mnd /\ A e. X /\ N e. NN0 ) /\ ( N .x. A ) = .0. ) -> ( ( O ` A ) = 0 -> -. N e. NN ) )
37		simpl3 1066	. . . . . . . . 9 |- ( ( ( G e. Mnd /\ A e. X /\ N e. NN0 ) /\ ( N .x. A ) = .0. ) -> N e. NN0 )
38		elnn0 11294	. . . . . . . . 9 |- ( N e. NN0 <-> ( N e. NN \/ N = 0 ) )
39	37, 38	sylib 208	. . . . . . . 8 |- ( ( ( G e. Mnd /\ A e. X /\ N e. NN0 ) /\ ( N .x. A ) = .0. ) -> ( N e. NN \/ N = 0 ) )
40	39	ord 392	. . . . . . 7 |- ( ( ( G e. Mnd /\ A e. X /\ N e. NN0 ) /\ ( N .x. A ) = .0. ) -> ( -. N e. NN -> N = 0 ) )
41	36, 40	syld 47	. . . . . 6 |- ( ( ( G e. Mnd /\ A e. X /\ N e. NN0 ) /\ ( N .x. A ) = .0. ) -> ( ( O ` A ) = 0 -> N = 0 ) )
42	41	impancom 456	. . . . 5 |- ( ( ( G e. Mnd /\ A e. X /\ N e. NN0 ) /\ ( O ` A ) = 0 ) -> ( ( N .x. A ) = .0. -> N = 0 ) )
43	28, 42	impbid 202	. . . 4 |- ( ( ( G e. Mnd /\ A e. X /\ N e. NN0 ) /\ ( O ` A ) = 0 ) -> ( N = 0 <-> ( N .x. A ) = .0. ) )
44	20, 24, 43	3bitrd 294	. . 3 |- ( ( ( G e. Mnd /\ A e. X /\ N e. NN0 ) /\ ( O ` A ) = 0 ) -> ( ( O ` A ) || N <-> ( N .x. A ) = .0. ) )
45	44	ex 450	. 2 |- ( ( G e. Mnd /\ A e. X /\ N e. NN0 ) -> ( ( O ` A ) = 0 -> ( ( O ` A ) || N <-> ( N .x. A ) = .0. ) ) )
46	2, 3	odcl 17955	. . . 4 |- ( A e. X -> ( O ` A ) e. NN0 )
47	46	3ad2ant2 1083	. . 3 |- ( ( G e. Mnd /\ A e. X /\ N e. NN0 ) -> ( O ` A ) e. NN0 )
48		elnn0 11294	. . 3 |- ( ( O ` A ) e. NN0 <-> ( ( O ` A ) e. NN \/ ( O ` A ) = 0 ) )
49	47, 48	sylib 208	. 2 |- ( ( G e. Mnd /\ A e. X /\ N e. NN0 ) -> ( ( O ` A ) e. NN \/ ( O ` A ) = 0 ) )
50	18, 45, 49	mpjaod 396	1 |- ( ( G e. Mnd /\ A e. X /\ N e. NN0 ) -> ( ( O ` A ) || N <-> ( N .x. A ) = .0. ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   \/ wo 383   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   =/= wne 2794   class class class wbr 4653   ` cfv 5888  (class class class)co 6650   CCcc 9934   0cc0 9936   1c1 9937   - cmin 10266   NNcn 11020   NN0cn0 11292   ZZcz 11377   ...cfz 12326   || cdvds 14983   Basecbs 15857   0gc0g 16100   Mndcmnd 17294  ._gcmg 17540   odcod 17944

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  ax-pre-sup 10014

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-div 10685  df-nn 11021  df-n0 11293  df-z 11378  df-uz 11688  df-rp 11833  df-fz 12327  df-fl 12593  df-mod 12669  df-seq 12802  df-dvds 14984  df-0g 16102  df-mgm 17242  df-sgrp 17284  df-mnd 17295  df-mulg 17541  df-od 17948

This theorem is referenced by: (None)