odzdvds - Intuitionistic Logic Explorer

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Theorem odzdvds 13002

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 9643	. . . . . . . . . 10
2		zq 10005	. . . . . . . . . 10
3	1, 2	syl 14	. . . . . . . . 9
4	3	adantl 277	. . . . . . . 8 $/\$ $/\$ $/\$
5		odzcl 13000	. . . . . . . . . 10 $/\$ $/\$
6	5	adantr 276	. . . . . . . . 9 $/\$ $/\$ $/\$
7		nnq 10012	. . . . . . . . 9
8	6, 7	syl 14	. . . . . . . 8 $/\$ $/\$ $/\$
9	6	nngt0d 9327	. . . . . . . 8 $/\$ $/\$ $/\$
10		modqlt 10748	. . . . . . . 8 $/\$ $/\$
11	4, 8, 9, 10	syl3anc 1278	. . . . . . 7 $/\$ $/\$ $/\$
12	1	adantl 277	. . . . . . . . . 10 $/\$ $/\$ $/\$
13	12, 6	zmodcld 10760	. . . . . . . . 9 $/\$ $/\$ $/\$
14	13	nn0zd 9745	. . . . . . . 8 $/\$ $/\$ $/\$
15	6	nnzd 9746	. . . . . . . 8 $/\$ $/\$ $/\$
16		zltnle 9669	. . . . . . . 8 $/\$
17	14, 15, 16	syl2anc 415	. . . . . . 7 $/\$ $/\$ $/\$
18	11, 17	mpbid 147	. . . . . 6 $/\$ $/\$ $/\$
19		1zzd 9650	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ $/\$
20		nnuz 9937	. . . . . . . . . . 11
21	20	rabeqi 2814	. . . . . . . . . 10
22		oveq2 6083	. . . . . . . . . . . . . . 15
23	22	oveq1d 6090	. . . . . . . . . . . . . 14
24	23	breq2d 4137	. . . . . . . . . . . . 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 10438	. . . . . . . . . . . . . . 15
33	32	nnnn0d 9599	. . . . . . . . . . . . . 14
34	33	adantl 277	. . . . . . . . . . . . 13 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
35		zexpcl 10969	. . . . . . . . . . . . 13 $/\$
36	31, 34, 35	syl2anc 415	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
37		peano2zm 9661	. . . . . . . . . . . 12
38	36, 37	syl 14	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
39		dvdsdc 12543	. . . . . . . . . . 11 $/\$ DECID
40	29, 38, 39	syl2anc 415	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ DECID
41	19, 21, 27, 40	infssuzledc 10645	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$ inf
42	41	ex 115	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ inf
43	42	ancomsd 269	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ inf
44		odzval 12998	. . . . . . . . 9 $/\$ $/\$ inf
45	44	adantr 276	. . . . . . . 8 $/\$ $/\$ $/\$ inf
46	45	breq1d 4135	. . . . . . 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 9544	. . . . . 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 9746	. . . . . 6 $/\$ $/\$ $/\$
57		dvds0 12551	. . . . . 6
58	56, 57	syl 14	. . . . 5 $/\$ $/\$ $/\$
59		simpl2 1032	. . . . . . . . 9 $/\$ $/\$ $/\$
60	59	zcnd 9748	. . . . . . . 8 $/\$ $/\$ $/\$
61	60	exp0d 11083	. . . . . . 7 $/\$ $/\$ $/\$
62	61	oveq1d 6090	. . . . . 6 $/\$ $/\$ $/\$
63		1m1e0 9352	. . . . . 6
64	62, 63	eqtrdi 2287	. . . . 5 $/\$ $/\$ $/\$
65	58, 64	breqtrrd 4153	. . . 4 $/\$ $/\$ $/\$
66		oveq2 6083	. . . . . 6
67	66	oveq1d 6090	. . . . 5
68	67	breq2d 4137	. . . 4
69	65, 68	syl5ibrcom 157	. . 3 $/\$ $/\$ $/\$
70	54, 69	impbid 129	. 2 $/\$ $/\$ $/\$
71	6	nnnn0d 9599	. . . . . . . . 9 $/\$ $/\$ $/\$
72		znq 10003	. . . . . . . . . . 11 $/\$
73	12, 6, 72	syl2anc 415	. . . . . . . . . 10 $/\$ $/\$ $/\$
74		nn0ge0 9567	. . . . . . . . . . . 12
75	74	adantl 277	. . . . . . . . . . 11 $/\$ $/\$ $/\$
76		nn0re 9551	. . . . . . . . . . . . 13
77	76	adantl 277	. . . . . . . . . . . 12 $/\$ $/\$ $/\$
78	6	nnred 9296	. . . . . . . . . . . 12 $/\$ $/\$ $/\$
79		ge0div 9191	. . . . . . . . . . . 12 $/\$ $/\$
80	77, 78, 9, 79	syl3anc 1278	. . . . . . . . . . 11 $/\$ $/\$ $/\$
81	75, 80	mpbid 147	. . . . . . . . . 10 $/\$ $/\$ $/\$
82		flqge0nn0 10706	. . . . . . . . . 10 $/\$
83	73, 81, 82	syl2anc 415	. . . . . . . . 9 $/\$ $/\$ $/\$
84	71, 83	nn0mulcld 9604	. . . . . . . 8 $/\$ $/\$ $/\$
85		zexpcl 10969	. . . . . . . 8 $/\$
86	59, 84, 85	syl2anc 415	. . . . . . 7 $/\$ $/\$ $/\$
87		zq 10005	. . . . . . 7
88	86, 87	syl 14	. . . . . 6 $/\$ $/\$ $/\$
89		1z 9649	. . . . . . 7
90		zq 10005	. . . . . . 7
91	89, 90	mp1i 10	. . . . . 6 $/\$ $/\$ $/\$
92		zexpcl 10969	. . . . . . 7 $/\$
93	59, 13, 92	syl2anc 415	. . . . . 6 $/\$ $/\$ $/\$
94		nnq 10012	. . . . . . 7
95	55, 94	syl 14	. . . . . 6 $/\$ $/\$ $/\$
96	55	nngt0d 9327	. . . . . 6 $/\$ $/\$ $/\$
97	60, 83, 71	expmuld 11092	. . . . . . . 8 $/\$ $/\$ $/\$
98	97	oveq1d 6090	. . . . . . 7 $/\$ $/\$ $/\$
99		zexpcl 10969	. . . . . . . . 9 $/\$
100	59, 71, 99	syl2anc 415	. . . . . . . 8 $/\$ $/\$ $/\$
101		1zzd 9650	. . . . . . . 8 $/\$ $/\$ $/\$
102		odzid 13001	. . . . . . . . . 10 $/\$ $/\$
103	102	adantr 276	. . . . . . . . 9 $/\$ $/\$ $/\$
104		moddvds 12544	. . . . . . . . . 10 $/\$ $/\$
105	55, 100, 101, 104	syl3anc 1278	. . . . . . . . 9 $/\$ $/\$ $/\$
106	103, 105	mpbird 167	. . . . . . . 8 $/\$ $/\$ $/\$
107	100, 101, 83, 95, 96, 106	modqexp 11082	. . . . . . 7 $/\$ $/\$ $/\$
108	73	flqcld 10690	. . . . . . . . 9 $/\$ $/\$ $/\$
109		1exp 10983	. . . . . . . . 9
110	108, 109	syl 14	. . . . . . . 8 $/\$ $/\$ $/\$
111	110	oveq1d 6090	. . . . . . 7 $/\$ $/\$ $/\$
112	98, 107, 111	3eqtrd 2275	. . . . . 6 $/\$ $/\$ $/\$
113	88, 91, 93, 95, 96, 112	modqmul1 10792	. . . . 5 $/\$ $/\$ $/\$
114	60, 13, 84	expaddd 11091	. . . . . . 7 $/\$ $/\$ $/\$
115		modqval 10739	. . . . . . . . . . 11 $/\$ $/\$
116	4, 8, 9, 115	syl3anc 1278	. . . . . . . . . 10 $/\$ $/\$ $/\$
117	116	oveq2d 6091	. . . . . . . . 9 $/\$ $/\$ $/\$
118	84	nn0cnd 9601	. . . . . . . . . 10 $/\$ $/\$ $/\$
119	77	recnd 8344	. . . . . . . . . 10 $/\$ $/\$ $/\$
120	118, 119	pncan3d 8630	. . . . . . . . 9 $/\$ $/\$ $/\$
121	117, 120	eqtrd 2271	. . . . . . . 8 $/\$ $/\$ $/\$
122	121	oveq2d 6091	. . . . . . 7 $/\$ $/\$ $/\$
123	114, 122	eqtr3d 2273	. . . . . 6 $/\$ $/\$ $/\$
124	123	oveq1d 6090	. . . . 5 $/\$ $/\$ $/\$
125	93	zcnd 9748	. . . . . . 7 $/\$ $/\$ $/\$
126	125	mullidd 8334	. . . . . 6 $/\$ $/\$ $/\$
127	126	oveq1d 6090	. . . . 5 $/\$ $/\$ $/\$
128	113, 124, 127	3eqtr3d 2279	. . . 4 $/\$ $/\$ $/\$
129	128	eqeq1d 2247	. . 3 $/\$ $/\$ $/\$
130		zexpcl 10969	. . . . 5 $/\$
131	59, 130	sylancom 424	. . . 4 $/\$ $/\$ $/\$
132		moddvds 12544	. . . 4 $/\$ $/\$
133	55, 131, 101, 132	syl3anc 1278	. . 3 $/\$ $/\$ $/\$
134		moddvds 12544	. . . 4 $/\$ $/\$
135	55, 93, 101, 134	syl3anc 1278	. . 3 $/\$ $/\$ $/\$
136	129, 133, 135	3bitr3d 218	. 2 $/\$ $/\$ $/\$
137		dvdsval3 12536	. . 3 $/\$
138	6, 12, 137	syl2anc 415	. 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 720 DECID wdc 846 $/\$ w3a 1009

wceq 1402

wcel 2209

crab 2532 class class class wbr 4125

cfv 5372 (class class class)co 6075 infcinf 7313

cr 8168

cc0 8169

c1 8170

caddc 8172

cmul 8174

clt 8350

cle 8351

cmin 8487

cdiv 8992

cn 9283

cn0 9542

cz 9623

cuz 9900

cq 9998

cfz 10390

cfl 10681

cmo 10737

cexp 10953

cdvds 12532

cgcd 12708

codz 12964

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 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 4241 ax-sep 4244 ax-nul 4254 ax-pow 4306 ax-pr 4341 ax-un 4573 ax-setind 4679 ax-iinf 4730 ax-cnex 8260 ax-resscn 8261 ax-1cn 8262 ax-1re 8263 ax-icn 8264 ax-addcl 8265 ax-addrcl 8266 ax-mulcl 8267 ax-mulrcl 8268 ax-addcom 8269 ax-mulcom 8270 ax-addass 8271 ax-mulass 8272 ax-distr 8273 ax-i2m1 8274 ax-0lt1 8275 ax-1rid 8276 ax-0id 8277 ax-rnegex 8278 ax-precex 8279 ax-cnre 8280 ax-pre-ltirr 8281 ax-pre-ltwlin 8282 ax-pre-lttrn 8283 ax-pre-apti 8284 ax-pre-ltadd 8285 ax-pre-mulgt0 8286 ax-pre-mulext 8287 ax-arch 8288 ax-caucvg 8289

This theorem 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 3636 df-pw 3687 df-sn 3711 df-pr 3712 df-op 3714 df-uni 3931 df-int 3966 df-iun 4009 df-br 4126 df-opab 4188 df-mpt 4189 df-tr 4225 df-id 4433 df-po 4436 df-iso 4437 df-iord 4506 df-on 4508 df-ilim 4509 df-suc 4511 df-iom 4733 df-xp 4775 df-rel 4776 df-cnv 4777 df-co 4778 df-dm 4779 df-rn 4780 df-res 4781 df-ima 4782 df-iota 5332 df-fun 5374 df-fn 5375 df-f 5376 df-f1 5377 df-fo 5378 df-f1o 5379 df-fv 5380 df-isom 5381 df-riota 6028 df-ov 6078 df-oprab 6079 df-mpo 6080 df-1st 6364 df-2nd 6365 df-recs 6566 df-irdg 6631 df-frec 6652 df-1o 6677 df-oadd 6681 df-er 6797 df-en 7013 df-dom 7014 df-fin 7015 df-sup 7314 df-inf 7315 df-pnf 8352 df-mnf 8353 df-xr 8354 df-ltxr 8355 df-le 8356 df-sub 8489 df-neg 8490 df-reap 8893 df-ap 8900 df-div 8993 df-inn 9284 df-2 9342 df-3 9343 df-4 9344 df-n0 9543 df-z 9624 df-uz 9901 df-q 9999 df-rp 10034 df-fz 10391 df-fzo 10528 df-fl 10683 df-mod 10738 df-seqfrec 10863 df-exp 10954 df-ihash 11193 df-cj 11585 df-re 11586 df-im 11587 df-rsqrt 11742 df-abs 11743 df-clim 12023 df-proddc 12296 df-dvds 12533 df-gcd 12709 df-odz 12966 df-phi 12967

This theorem is referenced by: odzphi 13003 pockthlem 13113

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