odzdvds - Intuitionistic Logic Explorer

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Theorem odzdvds 12414

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 9346	. . . . . . . . . 10
2		zq 9700	. . . . . . . . . 10
3	1, 2	syl 14	. . . . . . . . 9
4	3	adantl 277	. . . . . . . 8 $/\$ $/\$ $/\$
5		odzcl 12412	. . . . . . . . . 10 $/\$ $/\$
6	5	adantr 276	. . . . . . . . 9 $/\$ $/\$ $/\$
7		nnq 9707	. . . . . . . . 9
8	6, 7	syl 14	. . . . . . . 8 $/\$ $/\$ $/\$
9	6	nngt0d 9034	. . . . . . . 8 $/\$ $/\$ $/\$
10		modqlt 10425	. . . . . . . 8 $/\$ $/\$
11	4, 8, 9, 10	syl3anc 1249	. . . . . . 7 $/\$ $/\$ $/\$
12	1	adantl 277	. . . . . . . . . 10 $/\$ $/\$ $/\$
13	12, 6	zmodcld 10437	. . . . . . . . 9 $/\$ $/\$ $/\$
14	13	nn0zd 9446	. . . . . . . 8 $/\$ $/\$ $/\$
15	6	nnzd 9447	. . . . . . . 8 $/\$ $/\$ $/\$
16		zltnle 9372	. . . . . . . 8 $/\$
17	14, 15, 16	syl2anc 411	. . . . . . 7 $/\$ $/\$ $/\$
18	11, 17	mpbid 147	. . . . . 6 $/\$ $/\$ $/\$
19		1zzd 9353	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ $/\$
20		nnuz 9637	. . . . . . . . . . 11
21	20	rabeqi 2756	. . . . . . . . . 10
22		oveq2 5930	. . . . . . . . . . . . . . 15
23	22	oveq1d 5937	. . . . . . . . . . . . . 14
24	23	breq2d 4045	. . . . . . . . . . . . 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 10129	. . . . . . . . . . . . . . 15
33	32	nnnn0d 9302	. . . . . . . . . . . . . 14
34	33	adantl 277	. . . . . . . . . . . . 13 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
35		zexpcl 10646	. . . . . . . . . . . . 13 $/\$
36	31, 34, 35	syl2anc 411	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
37		peano2zm 9364	. . . . . . . . . . . 12
38	36, 37	syl 14	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
39		dvdsdc 11963	. . . . . . . . . . 11 $/\$ DECID
40	29, 38, 39	syl2anc 411	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ DECID
41	19, 21, 27, 40	infssuzledc 10324	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$ inf
42	41	ex 115	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ inf
43	42	ancomsd 269	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ inf
44		odzval 12410	. . . . . . . . 9 $/\$ $/\$ inf
45	44	adantr 276	. . . . . . . 8 $/\$ $/\$ $/\$ inf
46	45	breq1d 4043	. . . . . . 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 9251	. . . . . 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 9447	. . . . . 6 $/\$ $/\$ $/\$
57		dvds0 11971	. . . . . 6
58	56, 57	syl 14	. . . . 5 $/\$ $/\$ $/\$
59		simpl2 1003	. . . . . . . . 9 $/\$ $/\$ $/\$
60	59	zcnd 9449	. . . . . . . 8 $/\$ $/\$ $/\$
61	60	exp0d 10759	. . . . . . 7 $/\$ $/\$ $/\$
62	61	oveq1d 5937	. . . . . 6 $/\$ $/\$ $/\$
63		1m1e0 9059	. . . . . 6
64	62, 63	eqtrdi 2245	. . . . 5 $/\$ $/\$ $/\$
65	58, 64	breqtrrd 4061	. . . 4 $/\$ $/\$ $/\$
66		oveq2 5930	. . . . . 6
67	66	oveq1d 5937	. . . . 5
68	67	breq2d 4045	. . . 4
69	65, 68	syl5ibrcom 157	. . 3 $/\$ $/\$ $/\$
70	54, 69	impbid 129	. 2 $/\$ $/\$ $/\$
71	6	nnnn0d 9302	. . . . . . . . 9 $/\$ $/\$ $/\$
72		znq 9698	. . . . . . . . . . 11 $/\$
73	12, 6, 72	syl2anc 411	. . . . . . . . . 10 $/\$ $/\$ $/\$
74		nn0ge0 9274	. . . . . . . . . . . 12
75	74	adantl 277	. . . . . . . . . . 11 $/\$ $/\$ $/\$
76		nn0re 9258	. . . . . . . . . . . . 13
77	76	adantl 277	. . . . . . . . . . . 12 $/\$ $/\$ $/\$
78	6	nnred 9003	. . . . . . . . . . . 12 $/\$ $/\$ $/\$
79		ge0div 8898	. . . . . . . . . . . 12 $/\$ $/\$
80	77, 78, 9, 79	syl3anc 1249	. . . . . . . . . . 11 $/\$ $/\$ $/\$
81	75, 80	mpbid 147	. . . . . . . . . 10 $/\$ $/\$ $/\$
82		flqge0nn0 10383	. . . . . . . . . 10 $/\$
83	73, 81, 82	syl2anc 411	. . . . . . . . 9 $/\$ $/\$ $/\$
84	71, 83	nn0mulcld 9307	. . . . . . . 8 $/\$ $/\$ $/\$
85		zexpcl 10646	. . . . . . . 8 $/\$
86	59, 84, 85	syl2anc 411	. . . . . . 7 $/\$ $/\$ $/\$
87		zq 9700	. . . . . . 7
88	86, 87	syl 14	. . . . . 6 $/\$ $/\$ $/\$
89		1z 9352	. . . . . . 7
90		zq 9700	. . . . . . 7
91	89, 90	mp1i 10	. . . . . 6 $/\$ $/\$ $/\$
92		zexpcl 10646	. . . . . . 7 $/\$
93	59, 13, 92	syl2anc 411	. . . . . 6 $/\$ $/\$ $/\$
94		nnq 9707	. . . . . . 7
95	55, 94	syl 14	. . . . . 6 $/\$ $/\$ $/\$
96	55	nngt0d 9034	. . . . . 6 $/\$ $/\$ $/\$
97	60, 83, 71	expmuld 10768	. . . . . . . 8 $/\$ $/\$ $/\$
98	97	oveq1d 5937	. . . . . . 7 $/\$ $/\$ $/\$
99		zexpcl 10646	. . . . . . . . 9 $/\$
100	59, 71, 99	syl2anc 411	. . . . . . . 8 $/\$ $/\$ $/\$
101		1zzd 9353	. . . . . . . 8 $/\$ $/\$ $/\$
102		odzid 12413	. . . . . . . . . 10 $/\$ $/\$
103	102	adantr 276	. . . . . . . . 9 $/\$ $/\$ $/\$
104		moddvds 11964	. . . . . . . . . 10 $/\$ $/\$
105	55, 100, 101, 104	syl3anc 1249	. . . . . . . . 9 $/\$ $/\$ $/\$
106	103, 105	mpbird 167	. . . . . . . 8 $/\$ $/\$ $/\$
107	100, 101, 83, 95, 96, 106	modqexp 10758	. . . . . . 7 $/\$ $/\$ $/\$
108	73	flqcld 10367	. . . . . . . . 9 $/\$ $/\$ $/\$
109		1exp 10660	. . . . . . . . 9
110	108, 109	syl 14	. . . . . . . 8 $/\$ $/\$ $/\$
111	110	oveq1d 5937	. . . . . . 7 $/\$ $/\$ $/\$
112	98, 107, 111	3eqtrd 2233	. . . . . 6 $/\$ $/\$ $/\$
113	88, 91, 93, 95, 96, 112	modqmul1 10469	. . . . 5 $/\$ $/\$ $/\$
114	60, 13, 84	expaddd 10767	. . . . . . 7 $/\$ $/\$ $/\$
115		modqval 10416	. . . . . . . . . . 11 $/\$ $/\$
116	4, 8, 9, 115	syl3anc 1249	. . . . . . . . . 10 $/\$ $/\$ $/\$
117	116	oveq2d 5938	. . . . . . . . 9 $/\$ $/\$ $/\$
118	84	nn0cnd 9304	. . . . . . . . . 10 $/\$ $/\$ $/\$
119	77	recnd 8055	. . . . . . . . . 10 $/\$ $/\$ $/\$
120	118, 119	pncan3d 8340	. . . . . . . . 9 $/\$ $/\$ $/\$
121	117, 120	eqtrd 2229	. . . . . . . 8 $/\$ $/\$ $/\$
122	121	oveq2d 5938	. . . . . . 7 $/\$ $/\$ $/\$
123	114, 122	eqtr3d 2231	. . . . . 6 $/\$ $/\$ $/\$
124	123	oveq1d 5937	. . . . 5 $/\$ $/\$ $/\$
125	93	zcnd 9449	. . . . . . 7 $/\$ $/\$ $/\$
126	125	mulid2d 8045	. . . . . 6 $/\$ $/\$ $/\$
127	126	oveq1d 5937	. . . . 5 $/\$ $/\$ $/\$
128	113, 124, 127	3eqtr3d 2237	. . . 4 $/\$ $/\$ $/\$
129	128	eqeq1d 2205	. . 3 $/\$ $/\$ $/\$
130		zexpcl 10646	. . . . 5 $/\$
131	59, 130	sylancom 420	. . . 4 $/\$ $/\$ $/\$
132		moddvds 11964	. . . 4 $/\$ $/\$
133	55, 131, 101, 132	syl3anc 1249	. . 3 $/\$ $/\$ $/\$
134		moddvds 11964	. . . 4 $/\$ $/\$
135	55, 93, 101, 134	syl3anc 1249	. . 3 $/\$ $/\$ $/\$
136	129, 133, 135	3bitr3d 218	. 2 $/\$ $/\$ $/\$
137		dvdsval3 11956	. . 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 4033

cfv 5258 (class class class)co 5922 infcinf 7049

cr 7878

cc0 7879

c1 7880

caddc 7882

cmul 7884

clt 8061

cle 8062

cmin 8197

cdiv 8699

cn 8990

cn0 9249

cz 9326

cuz 9601

cq 9693

cfz 10083

cfl 10358

cmo 10414

cexp 10630

cdvds 11952

cgcd 12120

codz 12376

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 4148 ax-sep 4151 ax-nul 4159 ax-pow 4207 ax-pr 4242 ax-un 4468 ax-setind 4573 ax-iinf 4624 ax-cnex 7970 ax-resscn 7971 ax-1cn 7972 ax-1re 7973 ax-icn 7974 ax-addcl 7975 ax-addrcl 7976 ax-mulcl 7977 ax-mulrcl 7978 ax-addcom 7979 ax-mulcom 7980 ax-addass 7981 ax-mulass 7982 ax-distr 7983 ax-i2m1 7984 ax-0lt1 7985 ax-1rid 7986 ax-0id 7987 ax-rnegex 7988 ax-precex 7989 ax-cnre 7990 ax-pre-ltirr 7991 ax-pre-ltwlin 7992 ax-pre-lttrn 7993 ax-pre-apti 7994 ax-pre-ltadd 7995 ax-pre-mulgt0 7996 ax-pre-mulext 7997 ax-arch 7998 ax-caucvg 7999

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 3451 df-if 3562 df-pw 3607 df-sn 3628 df-pr 3629 df-op 3631 df-uni 3840 df-int 3875 df-iun 3918 df-br 4034 df-opab 4095 df-mpt 4096 df-tr 4132 df-id 4328 df-po 4331 df-iso 4332 df-iord 4401 df-on 4403 df-ilim 4404 df-suc 4406 df-iom 4627 df-xp 4669 df-rel 4670 df-cnv 4671 df-co 4672 df-dm 4673 df-rn 4674 df-res 4675 df-ima 4676 df-iota 5219 df-fun 5260 df-fn 5261 df-f 5262 df-f1 5263 df-fo 5264 df-f1o 5265 df-fv 5266 df-isom 5267 df-riota 5877 df-ov 5925 df-oprab 5926 df-mpo 5927 df-1st 6198 df-2nd 6199 df-recs 6363 df-irdg 6428 df-frec 6449 df-1o 6474 df-oadd 6478 df-er 6592 df-en 6800 df-dom 6801 df-fin 6802 df-sup 7050 df-inf 7051 df-pnf 8063 df-mnf 8064 df-xr 8065 df-ltxr 8066 df-le 8067 df-sub 8199 df-neg 8200 df-reap 8602 df-ap 8609 df-div 8700 df-inn 8991 df-2 9049 df-3 9050 df-4 9051 df-n0 9250 df-z 9327 df-uz 9602 df-q 9694 df-rp 9729 df-fz 10084 df-fzo 10218 df-fl 10360 df-mod 10415 df-seqfrec 10540 df-exp 10631 df-ihash 10868 df-cj 11007 df-re 11008 df-im 11009 df-rsqrt 11163 df-abs 11164 df-clim 11444 df-proddc 11716 df-dvds 11953 df-gcd 12121 df-odz 12378 df-phi 12379

This theorem is referenced by: odzphi 12415 pockthlem 12525

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