odzdvds - Intuitionistic Logic Explorer

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Theorem odzdvds 12439

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 9363	. . . . . . . . . 10
2		zq 9717	. . . . . . . . . 10
3	1, 2	syl 14	. . . . . . . . 9
4	3	adantl 277	. . . . . . . 8 $/\$ $/\$ $/\$
5		odzcl 12437	. . . . . . . . . 10 $/\$ $/\$
6	5	adantr 276	. . . . . . . . 9 $/\$ $/\$ $/\$
7		nnq 9724	. . . . . . . . 9
8	6, 7	syl 14	. . . . . . . 8 $/\$ $/\$ $/\$
9	6	nngt0d 9051	. . . . . . . 8 $/\$ $/\$ $/\$
10		modqlt 10442	. . . . . . . 8 $/\$ $/\$
11	4, 8, 9, 10	syl3anc 1249	. . . . . . 7 $/\$ $/\$ $/\$
12	1	adantl 277	. . . . . . . . . 10 $/\$ $/\$ $/\$
13	12, 6	zmodcld 10454	. . . . . . . . 9 $/\$ $/\$ $/\$
14	13	nn0zd 9463	. . . . . . . 8 $/\$ $/\$ $/\$
15	6	nnzd 9464	. . . . . . . 8 $/\$ $/\$ $/\$
16		zltnle 9389	. . . . . . . 8 $/\$
17	14, 15, 16	syl2anc 411	. . . . . . 7 $/\$ $/\$ $/\$
18	11, 17	mpbid 147	. . . . . 6 $/\$ $/\$ $/\$
19		1zzd 9370	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ $/\$
20		nnuz 9654	. . . . . . . . . . 11
21	20	rabeqi 2756	. . . . . . . . . 10
22		oveq2 5933	. . . . . . . . . . . . . . 15
23	22	oveq1d 5940	. . . . . . . . . . . . . 14
24	23	breq2d 4046	. . . . . . . . . . . . 13
25	24	elrab 2920	. . . . . . . . . . . 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 10146	. . . . . . . . . . . . . . 15
33	32	nnnn0d 9319	. . . . . . . . . . . . . 14
34	33	adantl 277	. . . . . . . . . . . . 13 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
35		zexpcl 10663	. . . . . . . . . . . . 13 $/\$
36	31, 34, 35	syl2anc 411	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
37		peano2zm 9381	. . . . . . . . . . . 12
38	36, 37	syl 14	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
39		dvdsdc 11980	. . . . . . . . . . 11 $/\$ DECID
40	29, 38, 39	syl2anc 411	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ DECID
41	19, 21, 27, 40	infssuzledc 10341	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$ inf
42	41	ex 115	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ inf
43	42	ancomsd 269	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ inf
44		odzval 12435	. . . . . . . . 9 $/\$ $/\$ inf
45	44	adantr 276	. . . . . . . 8 $/\$ $/\$ $/\$ inf
46	45	breq1d 4044	. . . . . . 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 9268	. . . . . 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 9464	. . . . . 6 $/\$ $/\$ $/\$
57		dvds0 11988	. . . . . 6
58	56, 57	syl 14	. . . . 5 $/\$ $/\$ $/\$
59		simpl2 1003	. . . . . . . . 9 $/\$ $/\$ $/\$
60	59	zcnd 9466	. . . . . . . 8 $/\$ $/\$ $/\$
61	60	exp0d 10776	. . . . . . 7 $/\$ $/\$ $/\$
62	61	oveq1d 5940	. . . . . 6 $/\$ $/\$ $/\$
63		1m1e0 9076	. . . . . 6
64	62, 63	eqtrdi 2245	. . . . 5 $/\$ $/\$ $/\$
65	58, 64	breqtrrd 4062	. . . 4 $/\$ $/\$ $/\$
66		oveq2 5933	. . . . . 6
67	66	oveq1d 5940	. . . . 5
68	67	breq2d 4046	. . . 4
69	65, 68	syl5ibrcom 157	. . 3 $/\$ $/\$ $/\$
70	54, 69	impbid 129	. 2 $/\$ $/\$ $/\$
71	6	nnnn0d 9319	. . . . . . . . 9 $/\$ $/\$ $/\$
72		znq 9715	. . . . . . . . . . 11 $/\$
73	12, 6, 72	syl2anc 411	. . . . . . . . . 10 $/\$ $/\$ $/\$
74		nn0ge0 9291	. . . . . . . . . . . 12
75	74	adantl 277	. . . . . . . . . . 11 $/\$ $/\$ $/\$
76		nn0re 9275	. . . . . . . . . . . . 13
77	76	adantl 277	. . . . . . . . . . . 12 $/\$ $/\$ $/\$
78	6	nnred 9020	. . . . . . . . . . . 12 $/\$ $/\$ $/\$
79		ge0div 8915	. . . . . . . . . . . 12 $/\$ $/\$
80	77, 78, 9, 79	syl3anc 1249	. . . . . . . . . . 11 $/\$ $/\$ $/\$
81	75, 80	mpbid 147	. . . . . . . . . 10 $/\$ $/\$ $/\$
82		flqge0nn0 10400	. . . . . . . . . 10 $/\$
83	73, 81, 82	syl2anc 411	. . . . . . . . 9 $/\$ $/\$ $/\$
84	71, 83	nn0mulcld 9324	. . . . . . . 8 $/\$ $/\$ $/\$
85		zexpcl 10663	. . . . . . . 8 $/\$
86	59, 84, 85	syl2anc 411	. . . . . . 7 $/\$ $/\$ $/\$
87		zq 9717	. . . . . . 7
88	86, 87	syl 14	. . . . . 6 $/\$ $/\$ $/\$
89		1z 9369	. . . . . . 7
90		zq 9717	. . . . . . 7
91	89, 90	mp1i 10	. . . . . 6 $/\$ $/\$ $/\$
92		zexpcl 10663	. . . . . . 7 $/\$
93	59, 13, 92	syl2anc 411	. . . . . 6 $/\$ $/\$ $/\$
94		nnq 9724	. . . . . . 7
95	55, 94	syl 14	. . . . . 6 $/\$ $/\$ $/\$
96	55	nngt0d 9051	. . . . . 6 $/\$ $/\$ $/\$
97	60, 83, 71	expmuld 10785	. . . . . . . 8 $/\$ $/\$ $/\$
98	97	oveq1d 5940	. . . . . . 7 $/\$ $/\$ $/\$
99		zexpcl 10663	. . . . . . . . 9 $/\$
100	59, 71, 99	syl2anc 411	. . . . . . . 8 $/\$ $/\$ $/\$
101		1zzd 9370	. . . . . . . 8 $/\$ $/\$ $/\$
102		odzid 12438	. . . . . . . . . 10 $/\$ $/\$
103	102	adantr 276	. . . . . . . . 9 $/\$ $/\$ $/\$
104		moddvds 11981	. . . . . . . . . 10 $/\$ $/\$
105	55, 100, 101, 104	syl3anc 1249	. . . . . . . . 9 $/\$ $/\$ $/\$
106	103, 105	mpbird 167	. . . . . . . 8 $/\$ $/\$ $/\$
107	100, 101, 83, 95, 96, 106	modqexp 10775	. . . . . . 7 $/\$ $/\$ $/\$
108	73	flqcld 10384	. . . . . . . . 9 $/\$ $/\$ $/\$
109		1exp 10677	. . . . . . . . 9
110	108, 109	syl 14	. . . . . . . 8 $/\$ $/\$ $/\$
111	110	oveq1d 5940	. . . . . . 7 $/\$ $/\$ $/\$
112	98, 107, 111	3eqtrd 2233	. . . . . 6 $/\$ $/\$ $/\$
113	88, 91, 93, 95, 96, 112	modqmul1 10486	. . . . 5 $/\$ $/\$ $/\$
114	60, 13, 84	expaddd 10784	. . . . . . 7 $/\$ $/\$ $/\$
115		modqval 10433	. . . . . . . . . . 11 $/\$ $/\$
116	4, 8, 9, 115	syl3anc 1249	. . . . . . . . . 10 $/\$ $/\$ $/\$
117	116	oveq2d 5941	. . . . . . . . 9 $/\$ $/\$ $/\$
118	84	nn0cnd 9321	. . . . . . . . . 10 $/\$ $/\$ $/\$
119	77	recnd 8072	. . . . . . . . . 10 $/\$ $/\$ $/\$
120	118, 119	pncan3d 8357	. . . . . . . . 9 $/\$ $/\$ $/\$
121	117, 120	eqtrd 2229	. . . . . . . 8 $/\$ $/\$ $/\$
122	121	oveq2d 5941	. . . . . . 7 $/\$ $/\$ $/\$
123	114, 122	eqtr3d 2231	. . . . . 6 $/\$ $/\$ $/\$
124	123	oveq1d 5940	. . . . 5 $/\$ $/\$ $/\$
125	93	zcnd 9466	. . . . . . 7 $/\$ $/\$ $/\$
126	125	mulid2d 8062	. . . . . 6 $/\$ $/\$ $/\$
127	126	oveq1d 5940	. . . . 5 $/\$ $/\$ $/\$
128	113, 124, 127	3eqtr3d 2237	. . . 4 $/\$ $/\$ $/\$
129	128	eqeq1d 2205	. . 3 $/\$ $/\$ $/\$
130		zexpcl 10663	. . . . 5 $/\$
131	59, 130	sylancom 420	. . . 4 $/\$ $/\$ $/\$
132		moddvds 11981	. . . 4 $/\$ $/\$
133	55, 131, 101, 132	syl3anc 1249	. . 3 $/\$ $/\$ $/\$
134		moddvds 11981	. . . 4 $/\$ $/\$
135	55, 93, 101, 134	syl3anc 1249	. . 3 $/\$ $/\$ $/\$
136	129, 133, 135	3bitr3d 218	. 2 $/\$ $/\$ $/\$
137		dvdsval3 11973	. . 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 2167

crab 2479 class class class wbr 4034

cfv 5259 (class class class)co 5925 infcinf 7058

cr 7895

cc0 7896

c1 7897

caddc 7899

cmul 7901

clt 8078

cle 8079

cmin 8214

cdiv 8716

cn 9007

cn0 9266

cz 9343

cuz 9618

cq 9710

cfz 10100

cfl 10375

cmo 10431

cexp 10647

cdvds 11969

cgcd 12145

codz 12401

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 1461 ax-7 1462 ax-gen 1463 ax-ie1 1507 ax-ie2 1508 ax-8 1518 ax-10 1519 ax-11 1520 ax-i12 1521 ax-bndl 1523 ax-4 1524 ax-17 1540 ax-i9 1544 ax-ial 1548 ax-i5r 1549 ax-13 2169 ax-14 2170 ax-ext 2178 ax-coll 4149 ax-sep 4152 ax-nul 4160 ax-pow 4208 ax-pr 4243 ax-un 4469 ax-setind 4574 ax-iinf 4625 ax-cnex 7987 ax-resscn 7988 ax-1cn 7989 ax-1re 7990 ax-icn 7991 ax-addcl 7992 ax-addrcl 7993 ax-mulcl 7994 ax-mulrcl 7995 ax-addcom 7996 ax-mulcom 7997 ax-addass 7998 ax-mulass 7999 ax-distr 8000 ax-i2m1 8001 ax-0lt1 8002 ax-1rid 8003 ax-0id 8004 ax-rnegex 8005 ax-precex 8006 ax-cnre 8007 ax-pre-ltirr 8008 ax-pre-ltwlin 8009 ax-pre-lttrn 8010 ax-pre-apti 8011 ax-pre-ltadd 8012 ax-pre-mulgt0 8013 ax-pre-mulext 8014 ax-arch 8015 ax-caucvg 8016

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 1475 df-sb 1777 df-eu 2048 df-mo 2049 df-clab 2183 df-cleq 2189 df-clel 2192 df-nfc 2328 df-ne 2368 df-nel 2463 df-ral 2480 df-rex 2481 df-reu 2482 df-rmo 2483 df-rab 2484 df-v 2765 df-sbc 2990 df-csb 3085 df-dif 3159 df-un 3161 df-in 3163 df-ss 3170 df-nul 3452 df-if 3563 df-pw 3608 df-sn 3629 df-pr 3630 df-op 3632 df-uni 3841 df-int 3876 df-iun 3919 df-br 4035 df-opab 4096 df-mpt 4097 df-tr 4133 df-id 4329 df-po 4332 df-iso 4333 df-iord 4402 df-on 4404 df-ilim 4405 df-suc 4407 df-iom 4628 df-xp 4670 df-rel 4671 df-cnv 4672 df-co 4673 df-dm 4674 df-rn 4675 df-res 4676 df-ima 4677 df-iota 5220 df-fun 5261 df-fn 5262 df-f 5263 df-f1 5264 df-fo 5265 df-f1o 5266 df-fv 5267 df-isom 5268 df-riota 5880 df-ov 5928 df-oprab 5929 df-mpo 5930 df-1st 6207 df-2nd 6208 df-recs 6372 df-irdg 6437 df-frec 6458 df-1o 6483 df-oadd 6487 df-er 6601 df-en 6809 df-dom 6810 df-fin 6811 df-sup 7059 df-inf 7060 df-pnf 8080 df-mnf 8081 df-xr 8082 df-ltxr 8083 df-le 8084 df-sub 8216 df-neg 8217 df-reap 8619 df-ap 8626 df-div 8717 df-inn 9008 df-2 9066 df-3 9067 df-4 9068 df-n0 9267 df-z 9344 df-uz 9619 df-q 9711 df-rp 9746 df-fz 10101 df-fzo 10235 df-fl 10377 df-mod 10432 df-seqfrec 10557 df-exp 10648 df-ihash 10885 df-cj 11024 df-re 11025 df-im 11026 df-rsqrt 11180 df-abs 11181 df-clim 11461 df-proddc 11733 df-dvds 11970 df-gcd 12146 df-odz 12403 df-phi 12404

This theorem is referenced by: odzphi 12440 pockthlem 12550

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