odzdvds - Intuitionistic Logic Explorer

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Theorem odzdvds 12247

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		nn0z 9275	. . . . . . . . . 10
2		zq 9628	. . . . . . . . . 10
3	1, 2	syl 14	. . . . . . . . 9
4	3	adantl 277	. . . . . . . 8 $/\$ $/\$ $/\$
5		odzcl 12245	. . . . . . . . . 10 $/\$ $/\$
6	5	adantr 276	. . . . . . . . 9 $/\$ $/\$ $/\$
7		nnq 9635	. . . . . . . . 9
8	6, 7	syl 14	. . . . . . . 8 $/\$ $/\$ $/\$
9	6	nngt0d 8965	. . . . . . . 8 $/\$ $/\$ $/\$
10		modqlt 10335	. . . . . . . 8 $/\$ $/\$
11	4, 8, 9, 10	syl3anc 1238	. . . . . . 7 $/\$ $/\$ $/\$
12	1	adantl 277	. . . . . . . . . 10 $/\$ $/\$ $/\$
13	12, 6	zmodcld 10347	. . . . . . . . 9 $/\$ $/\$ $/\$
14	13	nn0zd 9375	. . . . . . . 8 $/\$ $/\$ $/\$
15	6	nnzd 9376	. . . . . . . 8 $/\$ $/\$ $/\$
16		zltnle 9301	. . . . . . . 8 $/\$
17	14, 15, 16	syl2anc 411	. . . . . . 7 $/\$ $/\$ $/\$
18	11, 17	mpbid 147	. . . . . 6 $/\$ $/\$ $/\$
19		1zzd 9282	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ $/\$
20		nnuz 9565	. . . . . . . . . . 11
21	20	rabeqi 2732	. . . . . . . . . 10
22		oveq2 5885	. . . . . . . . . . . . . . 15
23	22	oveq1d 5892	. . . . . . . . . . . . . 14
24	23	breq2d 4017	. . . . . . . . . . . . 13
25	24	elrab 2895	. . . . . . . . . . . 12 $/\$
26	25	biimpri 133	. . . . . . . . . . 11 $/\$
27	26	adantl 277	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ $/\$
28		simp1 997	. . . . . . . . . . . 12 $/\$ $/\$
29	28	ad3antrrr 492	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
30		simp2 998	. . . . . . . . . . . . . 14 $/\$ $/\$
31	30	ad3antrrr 492	. . . . . . . . . . . . 13 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
32		elfznn 10056	. . . . . . . . . . . . . . 15
33	32	nnnn0d 9231	. . . . . . . . . . . . . 14
34	33	adantl 277	. . . . . . . . . . . . 13 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
35		zexpcl 10537	. . . . . . . . . . . . 13 $/\$
36	31, 34, 35	syl2anc 411	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
37		peano2zm 9293	. . . . . . . . . . . 12
38	36, 37	syl 14	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
39		dvdsdc 11807	. . . . . . . . . . 11 $/\$ DECID
40	29, 38, 39	syl2anc 411	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ DECID
41	19, 21, 27, 40	infssuzledc 11953	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$ inf
42	41	ex 115	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ inf
43	42	ancomsd 269	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ inf
44		odzval 12243	. . . . . . . . 9 $/\$ $/\$ inf
45	44	adantr 276	. . . . . . . 8 $/\$ $/\$ $/\$ inf
46	45	breq1d 4015	. . . . . . 7 $/\$ $/\$ $/\$ inf
47	43, 46	sylibrd 169	. . . . . 6 $/\$ $/\$ $/\$ $/\$
48	18, 47	mtod 663	. . . . 5 $/\$ $/\$ $/\$ $/\$
49		imnan 690	. . . . 5 $/\$
50	48, 49	sylibr 134	. . . 4 $/\$ $/\$ $/\$
51		elnn0 9180	. . . . . 6 $\/$
52	13, 51	sylib 122	. . . . 5 $/\$ $/\$ $/\$ $\/$
53	52	ord 724	. . . 4 $/\$ $/\$ $/\$
54	50, 53	syld 45	. . 3 $/\$ $/\$ $/\$
55		simpl1 1000	. . . . . . 7 $/\$ $/\$ $/\$
56	55	nnzd 9376	. . . . . 6 $/\$ $/\$ $/\$
57		dvds0 11815	. . . . . 6
58	56, 57	syl 14	. . . . 5 $/\$ $/\$ $/\$
59		simpl2 1001	. . . . . . . . 9 $/\$ $/\$ $/\$
60	59	zcnd 9378	. . . . . . . 8 $/\$ $/\$ $/\$
61	60	exp0d 10650	. . . . . . 7 $/\$ $/\$ $/\$
62	61	oveq1d 5892	. . . . . 6 $/\$ $/\$ $/\$
63		1m1e0 8990	. . . . . 6
64	62, 63	eqtrdi 2226	. . . . 5 $/\$ $/\$ $/\$
65	58, 64	breqtrrd 4033	. . . 4 $/\$ $/\$ $/\$
66		oveq2 5885	. . . . . 6
67	66	oveq1d 5892	. . . . 5
68	67	breq2d 4017	. . . 4
69	65, 68	syl5ibrcom 157	. . 3 $/\$ $/\$ $/\$
70	54, 69	impbid 129	. 2 $/\$ $/\$ $/\$
71	6	nnnn0d 9231	. . . . . . . . 9 $/\$ $/\$ $/\$
72		znq 9626	. . . . . . . . . . 11 $/\$
73	12, 6, 72	syl2anc 411	. . . . . . . . . 10 $/\$ $/\$ $/\$
74		nn0ge0 9203	. . . . . . . . . . . 12
75	74	adantl 277	. . . . . . . . . . 11 $/\$ $/\$ $/\$
76		nn0re 9187	. . . . . . . . . . . . 13
77	76	adantl 277	. . . . . . . . . . . 12 $/\$ $/\$ $/\$
78	6	nnred 8934	. . . . . . . . . . . 12 $/\$ $/\$ $/\$
79		ge0div 8830	. . . . . . . . . . . 12 $/\$ $/\$
80	77, 78, 9, 79	syl3anc 1238	. . . . . . . . . . 11 $/\$ $/\$ $/\$
81	75, 80	mpbid 147	. . . . . . . . . 10 $/\$ $/\$ $/\$
82		flqge0nn0 10295	. . . . . . . . . 10 $/\$
83	73, 81, 82	syl2anc 411	. . . . . . . . 9 $/\$ $/\$ $/\$
84	71, 83	nn0mulcld 9236	. . . . . . . 8 $/\$ $/\$ $/\$
85		zexpcl 10537	. . . . . . . 8 $/\$
86	59, 84, 85	syl2anc 411	. . . . . . 7 $/\$ $/\$ $/\$
87		zq 9628	. . . . . . 7
88	86, 87	syl 14	. . . . . 6 $/\$ $/\$ $/\$
89		1z 9281	. . . . . . 7
90		zq 9628	. . . . . . 7
91	89, 90	mp1i 10	. . . . . 6 $/\$ $/\$ $/\$
92		zexpcl 10537	. . . . . . 7 $/\$
93	59, 13, 92	syl2anc 411	. . . . . 6 $/\$ $/\$ $/\$
94		nnq 9635	. . . . . . 7
95	55, 94	syl 14	. . . . . 6 $/\$ $/\$ $/\$
96	55	nngt0d 8965	. . . . . 6 $/\$ $/\$ $/\$
97	60, 83, 71	expmuld 10659	. . . . . . . 8 $/\$ $/\$ $/\$
98	97	oveq1d 5892	. . . . . . 7 $/\$ $/\$ $/\$
99		zexpcl 10537	. . . . . . . . 9 $/\$
100	59, 71, 99	syl2anc 411	. . . . . . . 8 $/\$ $/\$ $/\$
101		1zzd 9282	. . . . . . . 8 $/\$ $/\$ $/\$
102		odzid 12246	. . . . . . . . . 10 $/\$ $/\$
103	102	adantr 276	. . . . . . . . 9 $/\$ $/\$ $/\$
104		moddvds 11808	. . . . . . . . . 10 $/\$ $/\$
105	55, 100, 101, 104	syl3anc 1238	. . . . . . . . 9 $/\$ $/\$ $/\$
106	103, 105	mpbird 167	. . . . . . . 8 $/\$ $/\$ $/\$
107	100, 101, 83, 95, 96, 106	modqexp 10649	. . . . . . 7 $/\$ $/\$ $/\$
108	73	flqcld 10279	. . . . . . . . 9 $/\$ $/\$ $/\$
109		1exp 10551	. . . . . . . . 9
110	108, 109	syl 14	. . . . . . . 8 $/\$ $/\$ $/\$
111	110	oveq1d 5892	. . . . . . 7 $/\$ $/\$ $/\$
112	98, 107, 111	3eqtrd 2214	. . . . . 6 $/\$ $/\$ $/\$
113	88, 91, 93, 95, 96, 112	modqmul1 10379	. . . . 5 $/\$ $/\$ $/\$
114	60, 13, 84	expaddd 10658	. . . . . . 7 $/\$ $/\$ $/\$
115		modqval 10326	. . . . . . . . . . 11 $/\$ $/\$
116	4, 8, 9, 115	syl3anc 1238	. . . . . . . . . 10 $/\$ $/\$ $/\$
117	116	oveq2d 5893	. . . . . . . . 9 $/\$ $/\$ $/\$
118	84	nn0cnd 9233	. . . . . . . . . 10 $/\$ $/\$ $/\$
119	77	recnd 7988	. . . . . . . . . 10 $/\$ $/\$ $/\$
120	118, 119	pncan3d 8273	. . . . . . . . 9 $/\$ $/\$ $/\$
121	117, 120	eqtrd 2210	. . . . . . . 8 $/\$ $/\$ $/\$
122	121	oveq2d 5893	. . . . . . 7 $/\$ $/\$ $/\$
123	114, 122	eqtr3d 2212	. . . . . 6 $/\$ $/\$ $/\$
124	123	oveq1d 5892	. . . . 5 $/\$ $/\$ $/\$
125	93	zcnd 9378	. . . . . . 7 $/\$ $/\$ $/\$
126	125	mulid2d 7978	. . . . . 6 $/\$ $/\$ $/\$
127	126	oveq1d 5892	. . . . 5 $/\$ $/\$ $/\$
128	113, 124, 127	3eqtr3d 2218	. . . 4 $/\$ $/\$ $/\$
129	128	eqeq1d 2186	. . 3 $/\$ $/\$ $/\$
130		zexpcl 10537	. . . . 5 $/\$
131	59, 130	sylancom 420	. . . 4 $/\$ $/\$ $/\$
132		moddvds 11808	. . . 4 $/\$ $/\$
133	55, 131, 101, 132	syl3anc 1238	. . 3 $/\$ $/\$ $/\$
134		moddvds 11808	. . . 4 $/\$ $/\$
135	55, 93, 101, 134	syl3anc 1238	. . 3 $/\$ $/\$ $/\$
136	129, 133, 135	3bitr3d 218	. 2 $/\$ $/\$ $/\$
137		dvdsval3 11800	. . 3 $/\$
138	6, 12, 137	syl2anc 411	. 2 $/\$ $/\$ $/\$
139	70, 136, 138	3bitr4d 220	1 $/\$ $/\$ $/\$

Colors of variables: wff set class

Syntax hints:

wn 3

wi 4 $/\$ wa 104

wb 105 $\/$ wo 708 DECID wdc 834 $/\$ w3a 978

wceq 1353

wcel 2148

crab 2459 class class class wbr 4005

cfv 5218 (class class class)co 5877 infcinf 6984

cr 7812

cc0 7813

c1 7814

caddc 7816

cmul 7818

clt 7994

cle 7995

cmin 8130

cdiv 8631

cn 8921

cn0 9178

cz 9255

cuz 9530

cq 9621

cfz 10010

cfl 10270

cmo 10324

cexp 10521

cdvds 11796

cgcd 11945

codz 12210

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 614 ax-in2 615 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-13 2150 ax-14 2151 ax-ext 2159 ax-coll 4120 ax-sep 4123 ax-nul 4131 ax-pow 4176 ax-pr 4211 ax-un 4435 ax-setind 4538 ax-iinf 4589 ax-cnex 7904 ax-resscn 7905 ax-1cn 7906 ax-1re 7907 ax-icn 7908 ax-addcl 7909 ax-addrcl 7910 ax-mulcl 7911 ax-mulrcl 7912 ax-addcom 7913 ax-mulcom 7914 ax-addass 7915 ax-mulass 7916 ax-distr 7917 ax-i2m1 7918 ax-0lt1 7919 ax-1rid 7920 ax-0id 7921 ax-rnegex 7922 ax-precex 7923 ax-cnre 7924 ax-pre-ltirr 7925 ax-pre-ltwlin 7926 ax-pre-lttrn 7927 ax-pre-apti 7928 ax-pre-ltadd 7929 ax-pre-mulgt0 7930 ax-pre-mulext 7931 ax-arch 7932 ax-caucvg 7933

This theorem depends on definitions: df-bi 117 df-stab 831 df-dc 835 df-3or 979 df-3an 980 df-tru 1356 df-fal 1359 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ne 2348 df-nel 2443 df-ral 2460 df-rex 2461 df-reu 2462 df-rmo 2463 df-rab 2464 df-v 2741 df-sbc 2965 df-csb 3060 df-dif 3133 df-un 3135 df-in 3137 df-ss 3144 df-nul 3425 df-if 3537 df-pw 3579 df-sn 3600 df-pr 3601 df-op 3603 df-uni 3812 df-int 3847 df-iun 3890 df-br 4006 df-opab 4067 df-mpt 4068 df-tr 4104 df-id 4295 df-po 4298 df-iso 4299 df-iord 4368 df-on 4370 df-ilim 4371 df-suc 4373 df-iom 4592 df-xp 4634 df-rel 4635 df-cnv 4636 df-co 4637 df-dm 4638 df-rn 4639 df-res 4640 df-ima 4641 df-iota 5180 df-fun 5220 df-fn 5221 df-f 5222 df-f1 5223 df-fo 5224 df-f1o 5225 df-fv 5226 df-isom 5227 df-riota 5833 df-ov 5880 df-oprab 5881 df-mpo 5882 df-1st 6143 df-2nd 6144 df-recs 6308 df-irdg 6373 df-frec 6394 df-1o 6419 df-oadd 6423 df-er 6537 df-en 6743 df-dom 6744 df-fin 6745 df-sup 6985 df-inf 6986 df-pnf 7996 df-mnf 7997 df-xr 7998 df-ltxr 7999 df-le 8000 df-sub 8132 df-neg 8133 df-reap 8534 df-ap 8541 df-div 8632 df-inn 8922 df-2 8980 df-3 8981 df-4 8982 df-n0 9179 df-z 9256 df-uz 9531 df-q 9622 df-rp 9656 df-fz 10011 df-fzo 10145 df-fl 10272 df-mod 10325 df-seqfrec 10448 df-exp 10522 df-ihash 10758 df-cj 10853 df-re 10854 df-im 10855 df-rsqrt 11009 df-abs 11010 df-clim 11289 df-proddc 11561 df-dvds 11797 df-gcd 11946 df-odz 12212 df-phi 12213

This theorem is referenced by: odzphi 12248 pockthlem 12356

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