odzdvds - Intuitionistic Logic Explorer

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Theorem odzdvds 12386

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 9340	. . . . . . . . . 10
2		zq 9694	. . . . . . . . . 10
3	1, 2	syl 14	. . . . . . . . 9
4	3	adantl 277	. . . . . . . 8 $/\$ $/\$ $/\$
5		odzcl 12384	. . . . . . . . . 10 $/\$ $/\$
6	5	adantr 276	. . . . . . . . 9 $/\$ $/\$ $/\$
7		nnq 9701	. . . . . . . . 9
8	6, 7	syl 14	. . . . . . . 8 $/\$ $/\$ $/\$
9	6	nngt0d 9028	. . . . . . . 8 $/\$ $/\$ $/\$
10		modqlt 10407	. . . . . . . 8 $/\$ $/\$
11	4, 8, 9, 10	syl3anc 1249	. . . . . . 7 $/\$ $/\$ $/\$
12	1	adantl 277	. . . . . . . . . 10 $/\$ $/\$ $/\$
13	12, 6	zmodcld 10419	. . . . . . . . 9 $/\$ $/\$ $/\$
14	13	nn0zd 9440	. . . . . . . 8 $/\$ $/\$ $/\$
15	6	nnzd 9441	. . . . . . . 8 $/\$ $/\$ $/\$
16		zltnle 9366	. . . . . . . 8 $/\$
17	14, 15, 16	syl2anc 411	. . . . . . 7 $/\$ $/\$ $/\$
18	11, 17	mpbid 147	. . . . . 6 $/\$ $/\$ $/\$
19		1zzd 9347	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ $/\$
20		nnuz 9631	. . . . . . . . . . 11
21	20	rabeqi 2753	. . . . . . . . . 10
22		oveq2 5927	. . . . . . . . . . . . . . 15
23	22	oveq1d 5934	. . . . . . . . . . . . . 14
24	23	breq2d 4042	. . . . . . . . . . . . 13
25	24	elrab 2917	. . . . . . . . . . . 12 $/\$
26	25	biimpri 133	. . . . . . . . . . 11 $/\$
27	26	adantl 277	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ $/\$
28		simp1 999	. . . . . . . . . . . 12 $/\$ $/\$
29	28	ad3antrrr 492	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
30		simp2 1000	. . . . . . . . . . . . . 14 $/\$ $/\$
31	30	ad3antrrr 492	. . . . . . . . . . . . 13 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
32		elfznn 10123	. . . . . . . . . . . . . . 15
33	32	nnnn0d 9296	. . . . . . . . . . . . . 14
34	33	adantl 277	. . . . . . . . . . . . 13 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
35		zexpcl 10628	. . . . . . . . . . . . 13 $/\$
36	31, 34, 35	syl2anc 411	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
37		peano2zm 9358	. . . . . . . . . . . 12
38	36, 37	syl 14	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
39		dvdsdc 11944	. . . . . . . . . . 11 $/\$ DECID
40	29, 38, 39	syl2anc 411	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ DECID
41	19, 21, 27, 40	infssuzledc 12090	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$ inf
42	41	ex 115	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ inf
43	42	ancomsd 269	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ inf
44		odzval 12382	. . . . . . . . 9 $/\$ $/\$ inf
45	44	adantr 276	. . . . . . . 8 $/\$ $/\$ $/\$ inf
46	45	breq1d 4040	. . . . . . 7 $/\$ $/\$ $/\$ inf
47	43, 46	sylibrd 169	. . . . . 6 $/\$ $/\$ $/\$ $/\$
48	18, 47	mtod 664	. . . . 5 $/\$ $/\$ $/\$ $/\$
49		imnan 691	. . . . 5 $/\$
50	48, 49	sylibr 134	. . . 4 $/\$ $/\$ $/\$
51		elnn0 9245	. . . . . 6 $\/$
52	13, 51	sylib 122	. . . . 5 $/\$ $/\$ $/\$ $\/$
53	52	ord 725	. . . 4 $/\$ $/\$ $/\$
54	50, 53	syld 45	. . 3 $/\$ $/\$ $/\$
55		simpl1 1002	. . . . . . 7 $/\$ $/\$ $/\$
56	55	nnzd 9441	. . . . . 6 $/\$ $/\$ $/\$
57		dvds0 11952	. . . . . 6
58	56, 57	syl 14	. . . . 5 $/\$ $/\$ $/\$
59		simpl2 1003	. . . . . . . . 9 $/\$ $/\$ $/\$
60	59	zcnd 9443	. . . . . . . 8 $/\$ $/\$ $/\$
61	60	exp0d 10741	. . . . . . 7 $/\$ $/\$ $/\$
62	61	oveq1d 5934	. . . . . 6 $/\$ $/\$ $/\$
63		1m1e0 9053	. . . . . 6
64	62, 63	eqtrdi 2242	. . . . 5 $/\$ $/\$ $/\$
65	58, 64	breqtrrd 4058	. . . 4 $/\$ $/\$ $/\$
66		oveq2 5927	. . . . . 6
67	66	oveq1d 5934	. . . . 5
68	67	breq2d 4042	. . . 4
69	65, 68	syl5ibrcom 157	. . 3 $/\$ $/\$ $/\$
70	54, 69	impbid 129	. 2 $/\$ $/\$ $/\$
71	6	nnnn0d 9296	. . . . . . . . 9 $/\$ $/\$ $/\$
72		znq 9692	. . . . . . . . . . 11 $/\$
73	12, 6, 72	syl2anc 411	. . . . . . . . . 10 $/\$ $/\$ $/\$
74		nn0ge0 9268	. . . . . . . . . . . 12
75	74	adantl 277	. . . . . . . . . . 11 $/\$ $/\$ $/\$
76		nn0re 9252	. . . . . . . . . . . . 13
77	76	adantl 277	. . . . . . . . . . . 12 $/\$ $/\$ $/\$
78	6	nnred 8997	. . . . . . . . . . . 12 $/\$ $/\$ $/\$
79		ge0div 8892	. . . . . . . . . . . 12 $/\$ $/\$
80	77, 78, 9, 79	syl3anc 1249	. . . . . . . . . . 11 $/\$ $/\$ $/\$
81	75, 80	mpbid 147	. . . . . . . . . 10 $/\$ $/\$ $/\$
82		flqge0nn0 10365	. . . . . . . . . 10 $/\$
83	73, 81, 82	syl2anc 411	. . . . . . . . 9 $/\$ $/\$ $/\$
84	71, 83	nn0mulcld 9301	. . . . . . . 8 $/\$ $/\$ $/\$
85		zexpcl 10628	. . . . . . . 8 $/\$
86	59, 84, 85	syl2anc 411	. . . . . . 7 $/\$ $/\$ $/\$
87		zq 9694	. . . . . . 7
88	86, 87	syl 14	. . . . . 6 $/\$ $/\$ $/\$
89		1z 9346	. . . . . . 7
90		zq 9694	. . . . . . 7
91	89, 90	mp1i 10	. . . . . 6 $/\$ $/\$ $/\$
92		zexpcl 10628	. . . . . . 7 $/\$
93	59, 13, 92	syl2anc 411	. . . . . 6 $/\$ $/\$ $/\$
94		nnq 9701	. . . . . . 7
95	55, 94	syl 14	. . . . . 6 $/\$ $/\$ $/\$
96	55	nngt0d 9028	. . . . . 6 $/\$ $/\$ $/\$
97	60, 83, 71	expmuld 10750	. . . . . . . 8 $/\$ $/\$ $/\$
98	97	oveq1d 5934	. . . . . . 7 $/\$ $/\$ $/\$
99		zexpcl 10628	. . . . . . . . 9 $/\$
100	59, 71, 99	syl2anc 411	. . . . . . . 8 $/\$ $/\$ $/\$
101		1zzd 9347	. . . . . . . 8 $/\$ $/\$ $/\$
102		odzid 12385	. . . . . . . . . 10 $/\$ $/\$
103	102	adantr 276	. . . . . . . . 9 $/\$ $/\$ $/\$
104		moddvds 11945	. . . . . . . . . 10 $/\$ $/\$
105	55, 100, 101, 104	syl3anc 1249	. . . . . . . . 9 $/\$ $/\$ $/\$
106	103, 105	mpbird 167	. . . . . . . 8 $/\$ $/\$ $/\$
107	100, 101, 83, 95, 96, 106	modqexp 10740	. . . . . . 7 $/\$ $/\$ $/\$
108	73	flqcld 10349	. . . . . . . . 9 $/\$ $/\$ $/\$
109		1exp 10642	. . . . . . . . 9
110	108, 109	syl 14	. . . . . . . 8 $/\$ $/\$ $/\$
111	110	oveq1d 5934	. . . . . . 7 $/\$ $/\$ $/\$
112	98, 107, 111	3eqtrd 2230	. . . . . 6 $/\$ $/\$ $/\$
113	88, 91, 93, 95, 96, 112	modqmul1 10451	. . . . 5 $/\$ $/\$ $/\$
114	60, 13, 84	expaddd 10749	. . . . . . 7 $/\$ $/\$ $/\$
115		modqval 10398	. . . . . . . . . . 11 $/\$ $/\$
116	4, 8, 9, 115	syl3anc 1249	. . . . . . . . . 10 $/\$ $/\$ $/\$
117	116	oveq2d 5935	. . . . . . . . 9 $/\$ $/\$ $/\$
118	84	nn0cnd 9298	. . . . . . . . . 10 $/\$ $/\$ $/\$
119	77	recnd 8050	. . . . . . . . . 10 $/\$ $/\$ $/\$
120	118, 119	pncan3d 8335	. . . . . . . . 9 $/\$ $/\$ $/\$
121	117, 120	eqtrd 2226	. . . . . . . 8 $/\$ $/\$ $/\$
122	121	oveq2d 5935	. . . . . . 7 $/\$ $/\$ $/\$
123	114, 122	eqtr3d 2228	. . . . . 6 $/\$ $/\$ $/\$
124	123	oveq1d 5934	. . . . 5 $/\$ $/\$ $/\$
125	93	zcnd 9443	. . . . . . 7 $/\$ $/\$ $/\$
126	125	mulid2d 8040	. . . . . 6 $/\$ $/\$ $/\$
127	126	oveq1d 5934	. . . . 5 $/\$ $/\$ $/\$
128	113, 124, 127	3eqtr3d 2234	. . . 4 $/\$ $/\$ $/\$
129	128	eqeq1d 2202	. . 3 $/\$ $/\$ $/\$
130		zexpcl 10628	. . . . 5 $/\$
131	59, 130	sylancom 420	. . . 4 $/\$ $/\$ $/\$
132		moddvds 11945	. . . 4 $/\$ $/\$
133	55, 131, 101, 132	syl3anc 1249	. . 3 $/\$ $/\$ $/\$
134		moddvds 11945	. . . 4 $/\$ $/\$
135	55, 93, 101, 134	syl3anc 1249	. . 3 $/\$ $/\$ $/\$
136	129, 133, 135	3bitr3d 218	. 2 $/\$ $/\$ $/\$
137		dvdsval3 11937	. . 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 709 DECID wdc 835 $/\$ w3a 980

wceq 1364

wcel 2164

crab 2476 class class class wbr 4030

cfv 5255 (class class class)co 5919 infcinf 7044

cr 7873

cc0 7874

c1 7875

caddc 7877

cmul 7879

clt 8056

cle 8057

cmin 8192

cdiv 8693

cn 8984

cn0 9243

cz 9320

cuz 9595

cq 9687

cfz 10077

cfl 10340

cmo 10396

cexp 10612

cdvds 11933

cgcd 12082

codz 12349

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 615 ax-in2 616 ax-io 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-13 2166 ax-14 2167 ax-ext 2175 ax-coll 4145 ax-sep 4148 ax-nul 4156 ax-pow 4204 ax-pr 4239 ax-un 4465 ax-setind 4570 ax-iinf 4621 ax-cnex 7965 ax-resscn 7966 ax-1cn 7967 ax-1re 7968 ax-icn 7969 ax-addcl 7970 ax-addrcl 7971 ax-mulcl 7972 ax-mulrcl 7973 ax-addcom 7974 ax-mulcom 7975 ax-addass 7976 ax-mulass 7977 ax-distr 7978 ax-i2m1 7979 ax-0lt1 7980 ax-1rid 7981 ax-0id 7982 ax-rnegex 7983 ax-precex 7984 ax-cnre 7985 ax-pre-ltirr 7986 ax-pre-ltwlin 7987 ax-pre-lttrn 7988 ax-pre-apti 7989 ax-pre-ltadd 7990 ax-pre-mulgt0 7991 ax-pre-mulext 7992 ax-arch 7993 ax-caucvg 7994

This theorem depends on definitions: df-bi 117 df-stab 832 df-dc 836 df-3or 981 df-3an 982 df-tru 1367 df-fal 1370 df-nf 1472 df-sb 1774 df-eu 2045 df-mo 2046 df-clab 2180 df-cleq 2186 df-clel 2189 df-nfc 2325 df-ne 2365 df-nel 2460 df-ral 2477 df-rex 2478 df-reu 2479 df-rmo 2480 df-rab 2481 df-v 2762 df-sbc 2987 df-csb 3082 df-dif 3156 df-un 3158 df-in 3160 df-ss 3167 df-nul 3448 df-if 3559 df-pw 3604 df-sn 3625 df-pr 3626 df-op 3628 df-uni 3837 df-int 3872 df-iun 3915 df-br 4031 df-opab 4092 df-mpt 4093 df-tr 4129 df-id 4325 df-po 4328 df-iso 4329 df-iord 4398 df-on 4400 df-ilim 4401 df-suc 4403 df-iom 4624 df-xp 4666 df-rel 4667 df-cnv 4668 df-co 4669 df-dm 4670 df-rn 4671 df-res 4672 df-ima 4673 df-iota 5216 df-fun 5257 df-fn 5258 df-f 5259 df-f1 5260 df-fo 5261 df-f1o 5262 df-fv 5263 df-isom 5264 df-riota 5874 df-ov 5922 df-oprab 5923 df-mpo 5924 df-1st 6195 df-2nd 6196 df-recs 6360 df-irdg 6425 df-frec 6446 df-1o 6471 df-oadd 6475 df-er 6589 df-en 6797 df-dom 6798 df-fin 6799 df-sup 7045 df-inf 7046 df-pnf 8058 df-mnf 8059 df-xr 8060 df-ltxr 8061 df-le 8062 df-sub 8194 df-neg 8195 df-reap 8596 df-ap 8603 df-div 8694 df-inn 8985 df-2 9043 df-3 9044 df-4 9045 df-n0 9244 df-z 9321 df-uz 9596 df-q 9688 df-rp 9723 df-fz 10078 df-fzo 10212 df-fl 10342 df-mod 10397 df-seqfrec 10522 df-exp 10613 df-ihash 10850 df-cj 10989 df-re 10990 df-im 10991 df-rsqrt 11145 df-abs 11146 df-clim 11425 df-proddc 11697 df-dvds 11934 df-gcd 12083 df-odz 12351 df-phi 12352

This theorem is referenced by: odzphi 12387 pockthlem 12497

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