odzdvds - Metamath Proof Explorer

	Metamath Proof Explorer		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > odzdvds		Structured version Visualization version Unicode version

Theorem odzdvds 15500

Description: The only powers of

that are congruent to

are the multiples of the order of

. (Contributed by Mario Carneiro, 28-Feb-2014.) (Proof shortened by AV, 26-Sep-2020.)

**Assertion**
Ref	Expression
odzdvds	$/\$ $/\$ $/\$

Proof of Theorem odzdvds Dummy variable

is distinct from all other variables.

Step	Hyp	Ref	Expression
1		nn0re 11301	. . . . . . . . 9
2	1	adantl 482	. . . . . . . 8 $/\$ $/\$ $/\$
3		odzcl 15498	. . . . . . . . . 10 $/\$ $/\$
4	3	adantr 481	. . . . . . . . 9 $/\$ $/\$ $/\$
5	4	nnrpd 11870	. . . . . . . 8 $/\$ $/\$ $/\$
6		modlt 12679	. . . . . . . 8 $/\$
7	2, 5, 6	syl2anc 693	. . . . . . 7 $/\$ $/\$ $/\$
8		nn0z 11400	. . . . . . . . . . 11
9	8	adantl 482	. . . . . . . . . 10 $/\$ $/\$ $/\$
10	9, 4	zmodcld 12691	. . . . . . . . 9 $/\$ $/\$ $/\$
11	10	nn0red 11352	. . . . . . . 8 $/\$ $/\$ $/\$
12	4	nnred 11035	. . . . . . . 8 $/\$ $/\$ $/\$
13	11, 12	ltnled 10184	. . . . . . 7 $/\$ $/\$ $/\$
14	7, 13	mpbid 222	. . . . . 6 $/\$ $/\$ $/\$
15		oveq2 6658	. . . . . . . . . . . 12
16	15	oveq1d 6665	. . . . . . . . . . 11
17	16	breq2d 4665	. . . . . . . . . 10
18	17	elrab 3363	. . . . . . . . 9 $/\$
19		ssrab2 3687	. . . . . . . . . . 11
20		nnuz 11723	. . . . . . . . . . 11
21	19, 20	sseqtri 3637	. . . . . . . . . 10
22		infssuzle 11771	. . . . . . . . . 10 $/\$ inf
23	21, 22	mpan 706	. . . . . . . . 9 inf
24	18, 23	sylbir 225	. . . . . . . 8 $/\$ inf
25	24	ancoms 469	. . . . . . 7 $/\$ inf
26		odzval 15496	. . . . . . . . 9 $/\$ $/\$ inf
27	26	adantr 481	. . . . . . . 8 $/\$ $/\$ $/\$ inf
28	27	breq1d 4663	. . . . . . 7 $/\$ $/\$ $/\$ inf
29	25, 28	syl5ibr 236	. . . . . 6 $/\$ $/\$ $/\$ $/\$
30	14, 29	mtod 189	. . . . 5 $/\$ $/\$ $/\$ $/\$
31		imnan 438	. . . . 5 $/\$
32	30, 31	sylibr 224	. . . 4 $/\$ $/\$ $/\$
33		elnn0 11294	. . . . . 6 $\/$
34	10, 33	sylib 208	. . . . 5 $/\$ $/\$ $/\$ $\/$
35	34	ord 392	. . . 4 $/\$ $/\$ $/\$
36	32, 35	syld 47	. . 3 $/\$ $/\$ $/\$
37		simpl1 1064	. . . . . . 7 $/\$ $/\$ $/\$
38	37	nnzd 11481	. . . . . 6 $/\$ $/\$ $/\$
39		dvds0 14997	. . . . . 6
40	38, 39	syl 17	. . . . 5 $/\$ $/\$ $/\$
41		simpl2 1065	. . . . . . . . 9 $/\$ $/\$ $/\$
42	41	zcnd 11483	. . . . . . . 8 $/\$ $/\$ $/\$
43	42	exp0d 13002	. . . . . . 7 $/\$ $/\$ $/\$
44	43	oveq1d 6665	. . . . . 6 $/\$ $/\$ $/\$
45		1m1e0 11089	. . . . . 6
46	44, 45	syl6eq 2672	. . . . 5 $/\$ $/\$ $/\$
47	40, 46	breqtrrd 4681	. . . 4 $/\$ $/\$ $/\$
48		oveq2 6658	. . . . . 6
49	48	oveq1d 6665	. . . . 5
50	49	breq2d 4665	. . . 4
51	47, 50	syl5ibrcom 237	. . 3 $/\$ $/\$ $/\$
52	36, 51	impbid 202	. 2 $/\$ $/\$ $/\$
53	4	nnnn0d 11351	. . . . . . . . 9 $/\$ $/\$ $/\$
54	2, 4	nndivred 11069	. . . . . . . . . 10 $/\$ $/\$ $/\$
55		nn0ge0 11318	. . . . . . . . . . . 12
56	55	adantl 482	. . . . . . . . . . 11 $/\$ $/\$ $/\$
57	4	nngt0d 11064	. . . . . . . . . . . 12 $/\$ $/\$ $/\$
58		ge0div 10890	. . . . . . . . . . . 12 $/\$ $/\$
59	2, 12, 57, 58	syl3anc 1326	. . . . . . . . . . 11 $/\$ $/\$ $/\$
60	56, 59	mpbid 222	. . . . . . . . . 10 $/\$ $/\$ $/\$
61		flge0nn0 12621	. . . . . . . . . 10 $/\$
62	54, 60, 61	syl2anc 693	. . . . . . . . 9 $/\$ $/\$ $/\$
63	53, 62	nn0mulcld 11356	. . . . . . . 8 $/\$ $/\$ $/\$
64		zexpcl 12875	. . . . . . . 8 $/\$
65	41, 63, 64	syl2anc 693	. . . . . . 7 $/\$ $/\$ $/\$
66	65	zred 11482	. . . . . 6 $/\$ $/\$ $/\$
67		1red 10055	. . . . . 6 $/\$ $/\$ $/\$
68		zexpcl 12875	. . . . . . 7 $/\$
69	41, 10, 68	syl2anc 693	. . . . . 6 $/\$ $/\$ $/\$
70	37	nnrpd 11870	. . . . . 6 $/\$ $/\$ $/\$
71	42, 62, 53	expmuld 13011	. . . . . . . 8 $/\$ $/\$ $/\$
72	71	oveq1d 6665	. . . . . . 7 $/\$ $/\$ $/\$
73		zexpcl 12875	. . . . . . . . 9 $/\$
74	41, 53, 73	syl2anc 693	. . . . . . . 8 $/\$ $/\$ $/\$
75		1zzd 11408	. . . . . . . 8 $/\$ $/\$ $/\$
76		odzid 15499	. . . . . . . . . 10 $/\$ $/\$
77	76	adantr 481	. . . . . . . . 9 $/\$ $/\$ $/\$
78		moddvds 14991	. . . . . . . . . 10 $/\$ $/\$
79	37, 74, 75, 78	syl3anc 1326	. . . . . . . . 9 $/\$ $/\$ $/\$
80	77, 79	mpbird 247	. . . . . . . 8 $/\$ $/\$ $/\$
81		modexp 12999	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$
82	74, 75, 62, 70, 80, 81	syl221anc 1337	. . . . . . 7 $/\$ $/\$ $/\$
83	54	flcld 12599	. . . . . . . . 9 $/\$ $/\$ $/\$
84		1exp 12889	. . . . . . . . 9
85	83, 84	syl 17	. . . . . . . 8 $/\$ $/\$ $/\$
86	85	oveq1d 6665	. . . . . . 7 $/\$ $/\$ $/\$
87	72, 82, 86	3eqtrd 2660	. . . . . 6 $/\$ $/\$ $/\$
88		modmul1 12723	. . . . . 6 $/\$ $/\$ $/\$ $/\$
89	66, 67, 69, 70, 87, 88	syl221anc 1337	. . . . 5 $/\$ $/\$ $/\$
90	42, 10, 63	expaddd 13010	. . . . . . 7 $/\$ $/\$ $/\$
91		modval 12670	. . . . . . . . . . 11 $/\$
92	2, 5, 91	syl2anc 693	. . . . . . . . . 10 $/\$ $/\$ $/\$
93	92	oveq2d 6666	. . . . . . . . 9 $/\$ $/\$ $/\$
94	63	nn0cnd 11353	. . . . . . . . . 10 $/\$ $/\$ $/\$
95	2	recnd 10068	. . . . . . . . . 10 $/\$ $/\$ $/\$
96	94, 95	pncan3d 10395	. . . . . . . . 9 $/\$ $/\$ $/\$
97	93, 96	eqtrd 2656	. . . . . . . 8 $/\$ $/\$ $/\$
98	97	oveq2d 6666	. . . . . . 7 $/\$ $/\$ $/\$
99	90, 98	eqtr3d 2658	. . . . . 6 $/\$ $/\$ $/\$
100	99	oveq1d 6665	. . . . 5 $/\$ $/\$ $/\$
101	69	zcnd 11483	. . . . . . 7 $/\$ $/\$ $/\$
102	101	mulid2d 10058	. . . . . 6 $/\$ $/\$ $/\$
103	102	oveq1d 6665	. . . . 5 $/\$ $/\$ $/\$
104	89, 100, 103	3eqtr3d 2664	. . . 4 $/\$ $/\$ $/\$
105	104	eqeq1d 2624	. . 3 $/\$ $/\$ $/\$
106		zexpcl 12875	. . . . 5 $/\$
107	41, 106	sylancom 701	. . . 4 $/\$ $/\$ $/\$
108		moddvds 14991	. . . 4 $/\$ $/\$
109	37, 107, 75, 108	syl3anc 1326	. . 3 $/\$ $/\$ $/\$
110		moddvds 14991	. . . 4 $/\$ $/\$
111	37, 69, 75, 110	syl3anc 1326	. . 3 $/\$ $/\$ $/\$
112	105, 109, 111	3bitr3d 298	. 2 $/\$ $/\$ $/\$
113		dvdsval3 14987	. . 3 $/\$
114	4, 9, 113	syl2anc 693	. 2 $/\$ $/\$ $/\$
115	52, 112, 114	3bitr4d 300	1 $/\$ $/\$ $/\$

Colors of variables: wff setvar class

Syntax hints:

wn 3

wi 4

wb 196 $\/$ wo 383 $/\$ wa 384 $/\$ w3a 1037

wceq 1483

wcel 1990

crab 2916

wss 3574 class class class wbr 4653

cfv 5888 (class class class)co 6650 infcinf 8347

cr 9935

cc0 9936

c1 9937

caddc 9939

cmul 9941

clt 10074

cle 10075

cmin 10266

cdiv 10684

cn 11020

cn0 11292

cz 11377

cuz 11687

crp 11832

cfl 12591

cmo 12668

cexp 12860

cdvds 14983

cgcd 15216

codz 15468

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-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-int 4476 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-1o 7560 df-oadd 7564 df-er 7742 df-map 7859 df-en 7956 df-dom 7957 df-sdom 7958 df-fin 7959 df-sup 8348 df-inf 8349 df-card 8765 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-2 11079 df-3 11080 df-n0 11293 df-xnn0 11364 df-z 11378 df-uz 11688 df-rp 11833 df-fz 12327 df-fzo 12466 df-fl 12593 df-mod 12669 df-seq 12802 df-exp 12861 df-hash 13118 df-cj 13839 df-re 13840 df-im 13841 df-sqrt 13975 df-abs 13976 df-dvds 14984 df-gcd 15217 df-odz 15470 df-phi 15471

This theorem is referenced by: odzphi 15501 pockthlem 15609 odz2prm2pw 41475 fmtnoprmfac2 41479

W3C validator