odzdvds - Intuitionistic Logic Explorer

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Theorem odzdvds 13024

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 9664	. . . . . . . . . 10
2		zq 10026	. . . . . . . . . 10
3	1, 2	syl 14	. . . . . . . . 9
4	3	adantl 277	. . . . . . . 8 $/\$ $/\$ $/\$
5		odzcl 13022	. . . . . . . . . 10 $/\$ $/\$
6	5	adantr 276	. . . . . . . . 9 $/\$ $/\$ $/\$
7		nnq 10033	. . . . . . . . 9
8	6, 7	syl 14	. . . . . . . 8 $/\$ $/\$ $/\$
9	6	nngt0d 9348	. . . . . . . 8 $/\$ $/\$ $/\$
10		modqlt 10770	. . . . . . . 8 $/\$ $/\$
11	4, 8, 9, 10	syl3anc 1278	. . . . . . 7 $/\$ $/\$ $/\$
12	1	adantl 277	. . . . . . . . . 10 $/\$ $/\$ $/\$
13	12, 6	zmodcld 10782	. . . . . . . . 9 $/\$ $/\$ $/\$
14	13	nn0zd 9766	. . . . . . . 8 $/\$ $/\$ $/\$
15	6	nnzd 9767	. . . . . . . 8 $/\$ $/\$ $/\$
16		zltnle 9690	. . . . . . . 8 $/\$
17	14, 15, 16	syl2anc 415	. . . . . . 7 $/\$ $/\$ $/\$
18	11, 17	mpbid 147	. . . . . 6 $/\$ $/\$ $/\$
19		1zzd 9671	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ $/\$
20		nnuz 9958	. . . . . . . . . . 11
21	20	rabeqi 2814	. . . . . . . . . 10
22		oveq2 6093	. . . . . . . . . . . . . . 15
23	22	oveq1d 6100	. . . . . . . . . . . . . 14
24	23	breq2d 4142	. . . . . . . . . . . . 13
25	24	elrab 2982	. . . . . . . . . . . 12 $/\$
26	25	biimpri 133	. . . . . . . . . . 11 $/\$
27	26	adantl 277	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ $/\$
28		simp1 1028	. . . . . . . . . . . 12 $/\$ $/\$
29	28	ad3antrrr 496	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
30		simp2 1029	. . . . . . . . . . . . . 14 $/\$ $/\$
31	30	ad3antrrr 496	. . . . . . . . . . . . 13 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
32		elfznn 10460	. . . . . . . . . . . . . . 15
33	32	nnnn0d 9620	. . . . . . . . . . . . . 14
34	33	adantl 277	. . . . . . . . . . . . 13 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
35		zexpcl 10991	. . . . . . . . . . . . 13 $/\$
36	31, 34, 35	syl2anc 415	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
37		peano2zm 9682	. . . . . . . . . . . 12
38	36, 37	syl 14	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
39		dvdsdc 12565	. . . . . . . . . . 11 $/\$ DECID
40	29, 38, 39	syl2anc 415	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ DECID
41	19, 21, 27, 40	infssuzledc 10667	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$ inf
42	41	ex 115	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ inf
43	42	ancomsd 269	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ inf
44		odzval 13020	. . . . . . . . 9 $/\$ $/\$ inf
45	44	adantr 276	. . . . . . . 8 $/\$ $/\$ $/\$ inf
46	45	breq1d 4140	. . . . . . 7 $/\$ $/\$ $/\$ inf
47	43, 46	sylibrd 169	. . . . . 6 $/\$ $/\$ $/\$ $/\$
48	18, 47	mtod 673	. . . . 5 $/\$ $/\$ $/\$ $/\$
49		imnan 701	. . . . 5 $/\$
50	48, 49	sylibr 134	. . . 4 $/\$ $/\$ $/\$
51		elnn0 9565	. . . . . 6 $\/$
52	13, 51	sylib 122	. . . . 5 $/\$ $/\$ $/\$ $\/$
53	52	ord 736	. . . 4 $/\$ $/\$ $/\$
54	50, 53	syld 45	. . 3 $/\$ $/\$ $/\$
55		simpl1 1031	. . . . . . 7 $/\$ $/\$ $/\$
56	55	nnzd 9767	. . . . . 6 $/\$ $/\$ $/\$
57		dvds0 12573	. . . . . 6
58	56, 57	syl 14	. . . . 5 $/\$ $/\$ $/\$
59		simpl2 1032	. . . . . . . . 9 $/\$ $/\$ $/\$
60	59	zcnd 9769	. . . . . . . 8 $/\$ $/\$ $/\$
61	60	exp0d 11105	. . . . . . 7 $/\$ $/\$ $/\$
62	61	oveq1d 6100	. . . . . 6 $/\$ $/\$ $/\$
63		1m1e0 9373	. . . . . 6
64	62, 63	eqtrdi 2287	. . . . 5 $/\$ $/\$ $/\$
65	58, 64	breqtrrd 4158	. . . 4 $/\$ $/\$ $/\$
66		oveq2 6093	. . . . . 6
67	66	oveq1d 6100	. . . . 5
68	67	breq2d 4142	. . . 4
69	65, 68	syl5ibrcom 157	. . 3 $/\$ $/\$ $/\$
70	54, 69	impbid 129	. 2 $/\$ $/\$ $/\$
71	6	nnnn0d 9620	. . . . . . . . 9 $/\$ $/\$ $/\$
72		znq 10024	. . . . . . . . . . 11 $/\$
73	12, 6, 72	syl2anc 415	. . . . . . . . . 10 $/\$ $/\$ $/\$
74		nn0ge0 9588	. . . . . . . . . . . 12
75	74	adantl 277	. . . . . . . . . . 11 $/\$ $/\$ $/\$
76		nn0re 9572	. . . . . . . . . . . . 13
77	76	adantl 277	. . . . . . . . . . . 12 $/\$ $/\$ $/\$
78	6	nnred 9317	. . . . . . . . . . . 12 $/\$ $/\$ $/\$
79		ge0div 9201	. . . . . . . . . . . 12 $/\$ $/\$
80	77, 78, 9, 79	syl3anc 1278	. . . . . . . . . . 11 $/\$ $/\$ $/\$
81	75, 80	mpbid 147	. . . . . . . . . 10 $/\$ $/\$ $/\$
82		flqge0nn0 10728	. . . . . . . . . 10 $/\$
83	73, 81, 82	syl2anc 415	. . . . . . . . 9 $/\$ $/\$ $/\$
84	71, 83	nn0mulcld 9625	. . . . . . . 8 $/\$ $/\$ $/\$
85		zexpcl 10991	. . . . . . . 8 $/\$
86	59, 84, 85	syl2anc 415	. . . . . . 7 $/\$ $/\$ $/\$
87		zq 10026	. . . . . . 7
88	86, 87	syl 14	. . . . . 6 $/\$ $/\$ $/\$
89		1z 9670	. . . . . . 7
90		zq 10026	. . . . . . 7
91	89, 90	mp1i 10	. . . . . 6 $/\$ $/\$ $/\$
92		zexpcl 10991	. . . . . . 7 $/\$
93	59, 13, 92	syl2anc 415	. . . . . 6 $/\$ $/\$ $/\$
94		nnq 10033	. . . . . . 7
95	55, 94	syl 14	. . . . . 6 $/\$ $/\$ $/\$
96	55	nngt0d 9348	. . . . . 6 $/\$ $/\$ $/\$
97	60, 83, 71	expmuld 11114	. . . . . . . 8 $/\$ $/\$ $/\$
98	97	oveq1d 6100	. . . . . . 7 $/\$ $/\$ $/\$
99		zexpcl 10991	. . . . . . . . 9 $/\$
100	59, 71, 99	syl2anc 415	. . . . . . . 8 $/\$ $/\$ $/\$
101		1zzd 9671	. . . . . . . 8 $/\$ $/\$ $/\$
102		odzid 13023	. . . . . . . . . 10 $/\$ $/\$
103	102	adantr 276	. . . . . . . . 9 $/\$ $/\$ $/\$
104		moddvds 12566	. . . . . . . . . 10 $/\$ $/\$
105	55, 100, 101, 104	syl3anc 1278	. . . . . . . . 9 $/\$ $/\$ $/\$
106	103, 105	mpbird 167	. . . . . . . 8 $/\$ $/\$ $/\$
107	100, 101, 83, 95, 96, 106	modqexp 11104	. . . . . . 7 $/\$ $/\$ $/\$
108	73	flqcld 10712	. . . . . . . . 9 $/\$ $/\$ $/\$
109		1exp 11005	. . . . . . . . 9
110	108, 109	syl 14	. . . . . . . 8 $/\$ $/\$ $/\$
111	110	oveq1d 6100	. . . . . . 7 $/\$ $/\$ $/\$
112	98, 107, 111	3eqtrd 2275	. . . . . 6 $/\$ $/\$ $/\$
113	88, 91, 93, 95, 96, 112	modqmul1 10814	. . . . 5 $/\$ $/\$ $/\$
114	60, 13, 84	expaddd 11113	. . . . . . 7 $/\$ $/\$ $/\$
115		modqval 10761	. . . . . . . . . . 11 $/\$ $/\$
116	4, 8, 9, 115	syl3anc 1278	. . . . . . . . . 10 $/\$ $/\$ $/\$
117	116	oveq2d 6101	. . . . . . . . 9 $/\$ $/\$ $/\$
118	84	nn0cnd 9622	. . . . . . . . . 10 $/\$ $/\$ $/\$
119	77	recnd 8354	. . . . . . . . . 10 $/\$ $/\$ $/\$
120	118, 119	pncan3d 8640	. . . . . . . . 9 $/\$ $/\$ $/\$
121	117, 120	eqtrd 2271	. . . . . . . 8 $/\$ $/\$ $/\$
122	121	oveq2d 6101	. . . . . . 7 $/\$ $/\$ $/\$
123	114, 122	eqtr3d 2273	. . . . . 6 $/\$ $/\$ $/\$
124	123	oveq1d 6100	. . . . 5 $/\$ $/\$ $/\$
125	93	zcnd 9769	. . . . . . 7 $/\$ $/\$ $/\$
126	125	mullidd 8344	. . . . . 6 $/\$ $/\$ $/\$
127	126	oveq1d 6100	. . . . 5 $/\$ $/\$ $/\$
128	113, 124, 127	3eqtr3d 2279	. . . 4 $/\$ $/\$ $/\$
129	128	eqeq1d 2247	. . 3 $/\$ $/\$ $/\$
130		zexpcl 10991	. . . . 5 $/\$
131	59, 130	sylancom 424	. . . 4 $/\$ $/\$ $/\$
132		moddvds 12566	. . . 4 $/\$ $/\$
133	55, 131, 101, 132	syl3anc 1278	. . 3 $/\$ $/\$ $/\$
134		moddvds 12566	. . . 4 $/\$ $/\$
135	55, 93, 101, 134	syl3anc 1278	. . 3 $/\$ $/\$ $/\$
136	129, 133, 135	3bitr3d 218	. 2 $/\$ $/\$ $/\$
137		dvdsval3 12558	. . 3 $/\$
138	6, 12, 137	syl2anc 415	. 2 $/\$ $/\$ $/\$
139	70, 136, 138	3bitr4d 220	1 $/\$ $/\$ $/\$

Colors of variables: wff set class

This proof depends on syntax axioms:

wn 3

wi 4 $/\$ wa 104

wb 105 $\/$ wo 720 DECID wdc 846 $/\$ w3a 1009

wceq 1402

wcel 2209

crab 2532 class class class wbr 4130

cfv 5377 (class class class)co 6085 infcinf 7323

cr 8178

cc0 8179

c1 8180

caddc 8182

cmul 8184

clt 8360

cle 8361

cmin 8497

cdiv 9002

cn 9304

cn0 9563

cz 9644

cuz 9921

cq 10019

cfz 10411

cfl 10703

cmo 10759

cexp 10975

cdvds 12554

cgcd 12730

codz 12986

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 623 ax-in2 624 ax-io 721 ax-5 1500 ax-7 1501 ax-gen 1502 ax-ie1 1546 ax-ie2 1547 ax-8 1557 ax-10 1558 ax-11 1559 ax-i12 1560 ax-bndl 1562 ax-4 1563 ax-17 1579 ax-i9 1583 ax-ial 1587 ax-i5r 1588 ax-14 2212 ax-ext 2220 ax-coll 4246 ax-sep 4249 ax-nul 4259 ax-pow 4311 ax-pr 4346 ax-un 4578 ax-setind 4684 ax-iinf 4735 ax-cnex 8270 ax-resscn 8271 ax-1cn 8272 ax-1re 8273 ax-icn 8274 ax-addcl 8275 ax-addrcl 8276 ax-mulcl 8277 ax-mulrcl 8278 ax-addcom 8279 ax-mulcom 8280 ax-addass 8281 ax-mulass 8282 ax-distr 8283 ax-i2m1 8284 ax-0lt1 8285 ax-1rid 8286 ax-0id 8287 ax-rnegex 8288 ax-precex 8289 ax-cnre 8290 ax-pre-ltirr 8291 ax-pre-ltwlin 8292 ax-pre-lttrn 8293 ax-pre-apti 8294 ax-pre-ltadd 8295 ax-pre-mulgt0 8296 ax-pre-mulext 8297 ax-arch 8298 ax-caucvg 8299

This proof depends on definitions: df-bi 117 df-stab 843 df-dc 847 df-3or 1010 df-3an 1011 df-tru 1405 df-fal 1408 df-nf 1514 df-sb 1816 df-eu 2089 df-mo 2090 df-clab 2225 df-cleq 2231 df-clel 2234 df-nfc 2381 df-ne 2421 df-nel 2516 df-ral 2533 df-rex 2534 df-reu 2535 df-rmo 2536 df-rab 2537 df-v 2823 df-sbc 3052 df-csb 3148 df-dif 3222 df-un 3224 df-in 3226 df-ss 3233 df-nul 3521 df-if 3639 df-pw 3690 df-sn 3715 df-pr 3716 df-op 3718 df-uni 3936 df-int 3971 df-iun 4014 df-br 4131 df-opab 4193 df-mpt 4194 df-tr 4230 df-id 4438 df-po 4441 df-iso 4442 df-iord 4511 df-on 4513 df-ilim 4514 df-suc 4516 df-iom 4738 df-xp 4780 df-rel 4781 df-cnv 4782 df-co 4783 df-dm 4784 df-rn 4785 df-res 4786 df-ima 4787 df-iota 5337 df-fun 5379 df-fn 5380 df-f 5381 df-f1 5382 df-fo 5383 df-f1o 5384 df-fv 5385 df-isom 5386 df-riota 6038 df-ov 6088 df-oprab 6089 df-mpo 6090 df-1st 6374 df-2nd 6375 df-recs 6576 df-irdg 6641 df-frec 6662 df-1o 6687 df-oadd 6691 df-er 6807 df-en 7023 df-dom 7024 df-fin 7025 df-sup 7324 df-inf 7325 df-pnf 8362 df-mnf 8363 df-xr 8364 df-ltxr 8365 df-le 8366 df-sub 8499 df-neg 8500 df-reap 8903 df-ap 8910 df-div 9003 df-inn 9305 df-2 9363 df-3 9364 df-4 9365 df-n0 9564 df-z 9645 df-uz 9922 df-q 10020 df-rp 10055 df-fz 10412 df-fzo 10550 df-fl 10705 df-mod 10760 df-seqfrec 10885 df-exp 10976 df-ihash 11215 df-cj 11607 df-re 11608 df-im 11609 df-rsqrt 11764 df-abs 11765 df-clim 12045 df-proddc 12318 df-dvds 12555 df-gcd 12731 df-odz 12988 df-phi 12989

This theorem is used by: odzphi 13025 pockthlem 13135

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