odzdvds - Intuitionistic Logic Explorer

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Theorem odzdvds 12245

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 9273	. . . . . . . . . 10
2		zq 9626	. . . . . . . . . 10
3	1, 2	syl 14	. . . . . . . . 9
4	3	adantl 277	. . . . . . . 8 $/\$ $/\$ $/\$
5		odzcl 12243	. . . . . . . . . 10 $/\$ $/\$
6	5	adantr 276	. . . . . . . . 9 $/\$ $/\$ $/\$
7		nnq 9633	. . . . . . . . 9
8	6, 7	syl 14	. . . . . . . 8 $/\$ $/\$ $/\$
9	6	nngt0d 8963	. . . . . . . 8 $/\$ $/\$ $/\$
10		modqlt 10333	. . . . . . . 8 $/\$ $/\$
11	4, 8, 9, 10	syl3anc 1238	. . . . . . 7 $/\$ $/\$ $/\$
12	1	adantl 277	. . . . . . . . . 10 $/\$ $/\$ $/\$
13	12, 6	zmodcld 10345	. . . . . . . . 9 $/\$ $/\$ $/\$
14	13	nn0zd 9373	. . . . . . . 8 $/\$ $/\$ $/\$
15	6	nnzd 9374	. . . . . . . 8 $/\$ $/\$ $/\$
16		zltnle 9299	. . . . . . . 8 $/\$
17	14, 15, 16	syl2anc 411	. . . . . . 7 $/\$ $/\$ $/\$
18	11, 17	mpbid 147	. . . . . 6 $/\$ $/\$ $/\$
19		1zzd 9280	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ $/\$
20		nnuz 9563	. . . . . . . . . . 11
21	20	rabeqi 2731	. . . . . . . . . 10
22		oveq2 5883	. . . . . . . . . . . . . . 15
23	22	oveq1d 5890	. . . . . . . . . . . . . 14
24	23	breq2d 4016	. . . . . . . . . . . . 13
25	24	elrab 2894	. . . . . . . . . . . 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 10054	. . . . . . . . . . . . . . 15
33	32	nnnn0d 9229	. . . . . . . . . . . . . 14
34	33	adantl 277	. . . . . . . . . . . . 13 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
35		zexpcl 10535	. . . . . . . . . . . . 13 $/\$
36	31, 34, 35	syl2anc 411	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
37		peano2zm 9291	. . . . . . . . . . . 12
38	36, 37	syl 14	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
39		dvdsdc 11805	. . . . . . . . . . 11 $/\$ DECID
40	29, 38, 39	syl2anc 411	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ DECID
41	19, 21, 27, 40	infssuzledc 11951	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$ inf
42	41	ex 115	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ inf
43	42	ancomsd 269	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ inf
44		odzval 12241	. . . . . . . . 9 $/\$ $/\$ inf
45	44	adantr 276	. . . . . . . 8 $/\$ $/\$ $/\$ inf
46	45	breq1d 4014	. . . . . . 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 9178	. . . . . 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 9374	. . . . . 6 $/\$ $/\$ $/\$
57		dvds0 11813	. . . . . 6
58	56, 57	syl 14	. . . . 5 $/\$ $/\$ $/\$
59		simpl2 1001	. . . . . . . . 9 $/\$ $/\$ $/\$
60	59	zcnd 9376	. . . . . . . 8 $/\$ $/\$ $/\$
61	60	exp0d 10648	. . . . . . 7 $/\$ $/\$ $/\$
62	61	oveq1d 5890	. . . . . 6 $/\$ $/\$ $/\$
63		1m1e0 8988	. . . . . 6
64	62, 63	eqtrdi 2226	. . . . 5 $/\$ $/\$ $/\$
65	58, 64	breqtrrd 4032	. . . 4 $/\$ $/\$ $/\$
66		oveq2 5883	. . . . . 6
67	66	oveq1d 5890	. . . . 5
68	67	breq2d 4016	. . . 4
69	65, 68	syl5ibrcom 157	. . 3 $/\$ $/\$ $/\$
70	54, 69	impbid 129	. 2 $/\$ $/\$ $/\$
71	6	nnnn0d 9229	. . . . . . . . 9 $/\$ $/\$ $/\$
72		znq 9624	. . . . . . . . . . 11 $/\$
73	12, 6, 72	syl2anc 411	. . . . . . . . . 10 $/\$ $/\$ $/\$
74		nn0ge0 9201	. . . . . . . . . . . 12
75	74	adantl 277	. . . . . . . . . . 11 $/\$ $/\$ $/\$
76		nn0re 9185	. . . . . . . . . . . . 13
77	76	adantl 277	. . . . . . . . . . . 12 $/\$ $/\$ $/\$
78	6	nnred 8932	. . . . . . . . . . . 12 $/\$ $/\$ $/\$
79		ge0div 8828	. . . . . . . . . . . 12 $/\$ $/\$
80	77, 78, 9, 79	syl3anc 1238	. . . . . . . . . . 11 $/\$ $/\$ $/\$
81	75, 80	mpbid 147	. . . . . . . . . 10 $/\$ $/\$ $/\$
82		flqge0nn0 10293	. . . . . . . . . 10 $/\$
83	73, 81, 82	syl2anc 411	. . . . . . . . 9 $/\$ $/\$ $/\$
84	71, 83	nn0mulcld 9234	. . . . . . . 8 $/\$ $/\$ $/\$
85		zexpcl 10535	. . . . . . . 8 $/\$
86	59, 84, 85	syl2anc 411	. . . . . . 7 $/\$ $/\$ $/\$
87		zq 9626	. . . . . . 7
88	86, 87	syl 14	. . . . . 6 $/\$ $/\$ $/\$
89		1z 9279	. . . . . . 7
90		zq 9626	. . . . . . 7
91	89, 90	mp1i 10	. . . . . 6 $/\$ $/\$ $/\$
92		zexpcl 10535	. . . . . . 7 $/\$
93	59, 13, 92	syl2anc 411	. . . . . 6 $/\$ $/\$ $/\$
94		nnq 9633	. . . . . . 7
95	55, 94	syl 14	. . . . . 6 $/\$ $/\$ $/\$
96	55	nngt0d 8963	. . . . . 6 $/\$ $/\$ $/\$
97	60, 83, 71	expmuld 10657	. . . . . . . 8 $/\$ $/\$ $/\$
98	97	oveq1d 5890	. . . . . . 7 $/\$ $/\$ $/\$
99		zexpcl 10535	. . . . . . . . 9 $/\$
100	59, 71, 99	syl2anc 411	. . . . . . . 8 $/\$ $/\$ $/\$
101		1zzd 9280	. . . . . . . 8 $/\$ $/\$ $/\$
102		odzid 12244	. . . . . . . . . 10 $/\$ $/\$
103	102	adantr 276	. . . . . . . . 9 $/\$ $/\$ $/\$
104		moddvds 11806	. . . . . . . . . 10 $/\$ $/\$
105	55, 100, 101, 104	syl3anc 1238	. . . . . . . . 9 $/\$ $/\$ $/\$
106	103, 105	mpbird 167	. . . . . . . 8 $/\$ $/\$ $/\$
107	100, 101, 83, 95, 96, 106	modqexp 10647	. . . . . . 7 $/\$ $/\$ $/\$
108	73	flqcld 10277	. . . . . . . . 9 $/\$ $/\$ $/\$
109		1exp 10549	. . . . . . . . 9
110	108, 109	syl 14	. . . . . . . 8 $/\$ $/\$ $/\$
111	110	oveq1d 5890	. . . . . . 7 $/\$ $/\$ $/\$
112	98, 107, 111	3eqtrd 2214	. . . . . 6 $/\$ $/\$ $/\$
113	88, 91, 93, 95, 96, 112	modqmul1 10377	. . . . 5 $/\$ $/\$ $/\$
114	60, 13, 84	expaddd 10656	. . . . . . 7 $/\$ $/\$ $/\$
115		modqval 10324	. . . . . . . . . . 11 $/\$ $/\$
116	4, 8, 9, 115	syl3anc 1238	. . . . . . . . . 10 $/\$ $/\$ $/\$
117	116	oveq2d 5891	. . . . . . . . 9 $/\$ $/\$ $/\$
118	84	nn0cnd 9231	. . . . . . . . . 10 $/\$ $/\$ $/\$
119	77	recnd 7986	. . . . . . . . . 10 $/\$ $/\$ $/\$
120	118, 119	pncan3d 8271	. . . . . . . . 9 $/\$ $/\$ $/\$
121	117, 120	eqtrd 2210	. . . . . . . 8 $/\$ $/\$ $/\$
122	121	oveq2d 5891	. . . . . . 7 $/\$ $/\$ $/\$
123	114, 122	eqtr3d 2212	. . . . . 6 $/\$ $/\$ $/\$
124	123	oveq1d 5890	. . . . 5 $/\$ $/\$ $/\$
125	93	zcnd 9376	. . . . . . 7 $/\$ $/\$ $/\$
126	125	mulid2d 7976	. . . . . 6 $/\$ $/\$ $/\$
127	126	oveq1d 5890	. . . . 5 $/\$ $/\$ $/\$
128	113, 124, 127	3eqtr3d 2218	. . . 4 $/\$ $/\$ $/\$
129	128	eqeq1d 2186	. . 3 $/\$ $/\$ $/\$
130		zexpcl 10535	. . . . 5 $/\$
131	59, 130	sylancom 420	. . . 4 $/\$ $/\$ $/\$
132		moddvds 11806	. . . 4 $/\$ $/\$
133	55, 131, 101, 132	syl3anc 1238	. . 3 $/\$ $/\$ $/\$
134		moddvds 11806	. . . 4 $/\$ $/\$
135	55, 93, 101, 134	syl3anc 1238	. . 3 $/\$ $/\$ $/\$
136	129, 133, 135	3bitr3d 218	. 2 $/\$ $/\$ $/\$
137		dvdsval3 11798	. . 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 4004

cfv 5217 (class class class)co 5875 infcinf 6982

cr 7810

cc0 7811

c1 7812

caddc 7814

cmul 7816

clt 7992

cle 7993

cmin 8128

cdiv 8629

cn 8919

cn0 9176

cz 9253

cuz 9528

cq 9619

cfz 10008

cfl 10268

cmo 10322

cexp 10519

cdvds 11794

cgcd 11943

codz 12208

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 4119 ax-sep 4122 ax-nul 4130 ax-pow 4175 ax-pr 4210 ax-un 4434 ax-setind 4537 ax-iinf 4588 ax-cnex 7902 ax-resscn 7903 ax-1cn 7904 ax-1re 7905 ax-icn 7906 ax-addcl 7907 ax-addrcl 7908 ax-mulcl 7909 ax-mulrcl 7910 ax-addcom 7911 ax-mulcom 7912 ax-addass 7913 ax-mulass 7914 ax-distr 7915 ax-i2m1 7916 ax-0lt1 7917 ax-1rid 7918 ax-0id 7919 ax-rnegex 7920 ax-precex 7921 ax-cnre 7922 ax-pre-ltirr 7923 ax-pre-ltwlin 7924 ax-pre-lttrn 7925 ax-pre-apti 7926 ax-pre-ltadd 7927 ax-pre-mulgt0 7928 ax-pre-mulext 7929 ax-arch 7930 ax-caucvg 7931

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 2740 df-sbc 2964 df-csb 3059 df-dif 3132 df-un 3134 df-in 3136 df-ss 3143 df-nul 3424 df-if 3536 df-pw 3578 df-sn 3599 df-pr 3600 df-op 3602 df-uni 3811 df-int 3846 df-iun 3889 df-br 4005 df-opab 4066 df-mpt 4067 df-tr 4103 df-id 4294 df-po 4297 df-iso 4298 df-iord 4367 df-on 4369 df-ilim 4370 df-suc 4372 df-iom 4591 df-xp 4633 df-rel 4634 df-cnv 4635 df-co 4636 df-dm 4637 df-rn 4638 df-res 4639 df-ima 4640 df-iota 5179 df-fun 5219 df-fn 5220 df-f 5221 df-f1 5222 df-fo 5223 df-f1o 5224 df-fv 5225 df-isom 5226 df-riota 5831 df-ov 5878 df-oprab 5879 df-mpo 5880 df-1st 6141 df-2nd 6142 df-recs 6306 df-irdg 6371 df-frec 6392 df-1o 6417 df-oadd 6421 df-er 6535 df-en 6741 df-dom 6742 df-fin 6743 df-sup 6983 df-inf 6984 df-pnf 7994 df-mnf 7995 df-xr 7996 df-ltxr 7997 df-le 7998 df-sub 8130 df-neg 8131 df-reap 8532 df-ap 8539 df-div 8630 df-inn 8920 df-2 8978 df-3 8979 df-4 8980 df-n0 9177 df-z 9254 df-uz 9529 df-q 9620 df-rp 9654 df-fz 10009 df-fzo 10143 df-fl 10270 df-mod 10323 df-seqfrec 10446 df-exp 10520 df-ihash 10756 df-cj 10851 df-re 10852 df-im 10853 df-rsqrt 11007 df-abs 11008 df-clim 11287 df-proddc 11559 df-dvds 11795 df-gcd 11944 df-odz 12210 df-phi 12211

This theorem is referenced by: odzphi 12246 pockthlem 12354

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