odzdvds - Intuitionistic Logic Explorer

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Theorem odzdvds 12199

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 9232	. . . . . . . . . 10
2		zq 9585	. . . . . . . . . 10
3	1, 2	syl 14	. . . . . . . . 9
4	3	adantl 275	. . . . . . . 8 $/\$ $/\$ $/\$
5		odzcl 12197	. . . . . . . . . 10 $/\$ $/\$
6	5	adantr 274	. . . . . . . . 9 $/\$ $/\$ $/\$
7		nnq 9592	. . . . . . . . 9
8	6, 7	syl 14	. . . . . . . 8 $/\$ $/\$ $/\$
9	6	nngt0d 8922	. . . . . . . 8 $/\$ $/\$ $/\$
10		modqlt 10289	. . . . . . . 8 $/\$ $/\$
11	4, 8, 9, 10	syl3anc 1233	. . . . . . 7 $/\$ $/\$ $/\$
12	1	adantl 275	. . . . . . . . . 10 $/\$ $/\$ $/\$
13	12, 6	zmodcld 10301	. . . . . . . . 9 $/\$ $/\$ $/\$
14	13	nn0zd 9332	. . . . . . . 8 $/\$ $/\$ $/\$
15	6	nnzd 9333	. . . . . . . 8 $/\$ $/\$ $/\$
16		zltnle 9258	. . . . . . . 8 $/\$
17	14, 15, 16	syl2anc 409	. . . . . . 7 $/\$ $/\$ $/\$
18	11, 17	mpbid 146	. . . . . 6 $/\$ $/\$ $/\$
19		1zzd 9239	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ $/\$
20		nnuz 9522	. . . . . . . . . . 11
21	20	rabeqi 2723	. . . . . . . . . 10
22		oveq2 5861	. . . . . . . . . . . . . . 15
23	22	oveq1d 5868	. . . . . . . . . . . . . 14
24	23	breq2d 4001	. . . . . . . . . . . . 13
25	24	elrab 2886	. . . . . . . . . . . 12 $/\$
26	25	biimpri 132	. . . . . . . . . . 11 $/\$
27	26	adantl 275	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ $/\$
28		simp1 992	. . . . . . . . . . . 12 $/\$ $/\$
29	28	ad3antrrr 489	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
30		simp2 993	. . . . . . . . . . . . . 14 $/\$ $/\$
31	30	ad3antrrr 489	. . . . . . . . . . . . 13 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
32		elfznn 10010	. . . . . . . . . . . . . . 15
33	32	nnnn0d 9188	. . . . . . . . . . . . . 14
34	33	adantl 275	. . . . . . . . . . . . 13 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
35		zexpcl 10491	. . . . . . . . . . . . 13 $/\$
36	31, 34, 35	syl2anc 409	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
37		peano2zm 9250	. . . . . . . . . . . 12
38	36, 37	syl 14	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
39		dvdsdc 11760	. . . . . . . . . . 11 $/\$ DECID
40	29, 38, 39	syl2anc 409	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ DECID
41	19, 21, 27, 40	infssuzledc 11905	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$ inf
42	41	ex 114	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ inf
43	42	ancomsd 267	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ inf
44		odzval 12195	. . . . . . . . 9 $/\$ $/\$ inf
45	44	adantr 274	. . . . . . . 8 $/\$ $/\$ $/\$ inf
46	45	breq1d 3999	. . . . . . 7 $/\$ $/\$ $/\$ inf
47	43, 46	sylibrd 168	. . . . . 6 $/\$ $/\$ $/\$ $/\$
48	18, 47	mtod 658	. . . . 5 $/\$ $/\$ $/\$ $/\$
49		imnan 685	. . . . 5 $/\$
50	48, 49	sylibr 133	. . . 4 $/\$ $/\$ $/\$
51		elnn0 9137	. . . . . 6 $\/$
52	13, 51	sylib 121	. . . . 5 $/\$ $/\$ $/\$ $\/$
53	52	ord 719	. . . 4 $/\$ $/\$ $/\$
54	50, 53	syld 45	. . 3 $/\$ $/\$ $/\$
55		simpl1 995	. . . . . . 7 $/\$ $/\$ $/\$
56	55	nnzd 9333	. . . . . 6 $/\$ $/\$ $/\$
57		dvds0 11768	. . . . . 6
58	56, 57	syl 14	. . . . 5 $/\$ $/\$ $/\$
59		simpl2 996	. . . . . . . . 9 $/\$ $/\$ $/\$
60	59	zcnd 9335	. . . . . . . 8 $/\$ $/\$ $/\$
61	60	exp0d 10603	. . . . . . 7 $/\$ $/\$ $/\$
62	61	oveq1d 5868	. . . . . 6 $/\$ $/\$ $/\$
63		1m1e0 8947	. . . . . 6
64	62, 63	eqtrdi 2219	. . . . 5 $/\$ $/\$ $/\$
65	58, 64	breqtrrd 4017	. . . 4 $/\$ $/\$ $/\$
66		oveq2 5861	. . . . . 6
67	66	oveq1d 5868	. . . . 5
68	67	breq2d 4001	. . . 4
69	65, 68	syl5ibrcom 156	. . 3 $/\$ $/\$ $/\$
70	54, 69	impbid 128	. 2 $/\$ $/\$ $/\$
71	6	nnnn0d 9188	. . . . . . . . 9 $/\$ $/\$ $/\$
72		znq 9583	. . . . . . . . . . 11 $/\$
73	12, 6, 72	syl2anc 409	. . . . . . . . . 10 $/\$ $/\$ $/\$
74		nn0ge0 9160	. . . . . . . . . . . 12
75	74	adantl 275	. . . . . . . . . . 11 $/\$ $/\$ $/\$
76		nn0re 9144	. . . . . . . . . . . . 13
77	76	adantl 275	. . . . . . . . . . . 12 $/\$ $/\$ $/\$
78	6	nnred 8891	. . . . . . . . . . . 12 $/\$ $/\$ $/\$
79		ge0div 8787	. . . . . . . . . . . 12 $/\$ $/\$
80	77, 78, 9, 79	syl3anc 1233	. . . . . . . . . . 11 $/\$ $/\$ $/\$
81	75, 80	mpbid 146	. . . . . . . . . 10 $/\$ $/\$ $/\$
82		flqge0nn0 10249	. . . . . . . . . 10 $/\$
83	73, 81, 82	syl2anc 409	. . . . . . . . 9 $/\$ $/\$ $/\$
84	71, 83	nn0mulcld 9193	. . . . . . . 8 $/\$ $/\$ $/\$
85		zexpcl 10491	. . . . . . . 8 $/\$
86	59, 84, 85	syl2anc 409	. . . . . . 7 $/\$ $/\$ $/\$
87		zq 9585	. . . . . . 7
88	86, 87	syl 14	. . . . . 6 $/\$ $/\$ $/\$
89		1z 9238	. . . . . . 7
90		zq 9585	. . . . . . 7
91	89, 90	mp1i 10	. . . . . 6 $/\$ $/\$ $/\$
92		zexpcl 10491	. . . . . . 7 $/\$
93	59, 13, 92	syl2anc 409	. . . . . 6 $/\$ $/\$ $/\$
94		nnq 9592	. . . . . . 7
95	55, 94	syl 14	. . . . . 6 $/\$ $/\$ $/\$
96	55	nngt0d 8922	. . . . . 6 $/\$ $/\$ $/\$
97	60, 83, 71	expmuld 10612	. . . . . . . 8 $/\$ $/\$ $/\$
98	97	oveq1d 5868	. . . . . . 7 $/\$ $/\$ $/\$
99		zexpcl 10491	. . . . . . . . 9 $/\$
100	59, 71, 99	syl2anc 409	. . . . . . . 8 $/\$ $/\$ $/\$
101		1zzd 9239	. . . . . . . 8 $/\$ $/\$ $/\$
102		odzid 12198	. . . . . . . . . 10 $/\$ $/\$
103	102	adantr 274	. . . . . . . . 9 $/\$ $/\$ $/\$
104		moddvds 11761	. . . . . . . . . 10 $/\$ $/\$
105	55, 100, 101, 104	syl3anc 1233	. . . . . . . . 9 $/\$ $/\$ $/\$
106	103, 105	mpbird 166	. . . . . . . 8 $/\$ $/\$ $/\$
107	100, 101, 83, 95, 96, 106	modqexp 10602	. . . . . . 7 $/\$ $/\$ $/\$
108	73	flqcld 10233	. . . . . . . . 9 $/\$ $/\$ $/\$
109		1exp 10505	. . . . . . . . 9
110	108, 109	syl 14	. . . . . . . 8 $/\$ $/\$ $/\$
111	110	oveq1d 5868	. . . . . . 7 $/\$ $/\$ $/\$
112	98, 107, 111	3eqtrd 2207	. . . . . 6 $/\$ $/\$ $/\$
113	88, 91, 93, 95, 96, 112	modqmul1 10333	. . . . 5 $/\$ $/\$ $/\$
114	60, 13, 84	expaddd 10611	. . . . . . 7 $/\$ $/\$ $/\$
115		modqval 10280	. . . . . . . . . . 11 $/\$ $/\$
116	4, 8, 9, 115	syl3anc 1233	. . . . . . . . . 10 $/\$ $/\$ $/\$
117	116	oveq2d 5869	. . . . . . . . 9 $/\$ $/\$ $/\$
118	84	nn0cnd 9190	. . . . . . . . . 10 $/\$ $/\$ $/\$
119	77	recnd 7948	. . . . . . . . . 10 $/\$ $/\$ $/\$
120	118, 119	pncan3d 8233	. . . . . . . . 9 $/\$ $/\$ $/\$
121	117, 120	eqtrd 2203	. . . . . . . 8 $/\$ $/\$ $/\$
122	121	oveq2d 5869	. . . . . . 7 $/\$ $/\$ $/\$
123	114, 122	eqtr3d 2205	. . . . . 6 $/\$ $/\$ $/\$
124	123	oveq1d 5868	. . . . 5 $/\$ $/\$ $/\$
125	93	zcnd 9335	. . . . . . 7 $/\$ $/\$ $/\$
126	125	mulid2d 7938	. . . . . 6 $/\$ $/\$ $/\$
127	126	oveq1d 5868	. . . . 5 $/\$ $/\$ $/\$
128	113, 124, 127	3eqtr3d 2211	. . . 4 $/\$ $/\$ $/\$
129	128	eqeq1d 2179	. . 3 $/\$ $/\$ $/\$
130		zexpcl 10491	. . . . 5 $/\$
131	59, 130	sylancom 418	. . . 4 $/\$ $/\$ $/\$
132		moddvds 11761	. . . 4 $/\$ $/\$
133	55, 131, 101, 132	syl3anc 1233	. . 3 $/\$ $/\$ $/\$
134		moddvds 11761	. . . 4 $/\$ $/\$
135	55, 93, 101, 134	syl3anc 1233	. . 3 $/\$ $/\$ $/\$
136	129, 133, 135	3bitr3d 217	. 2 $/\$ $/\$ $/\$
137		dvdsval3 11753	. . 3 $/\$
138	6, 12, 137	syl2anc 409	. 2 $/\$ $/\$ $/\$
139	70, 136, 138	3bitr4d 219	1 $/\$ $/\$ $/\$

Colors of variables: wff set class

Syntax hints:

wn 3

wi 4 $/\$ wa 103

wb 104 $\/$ wo 703 DECID wdc 829 $/\$ w3a 973

wceq 1348

wcel 2141

crab 2452 class class class wbr 3989

cfv 5198 (class class class)co 5853 infcinf 6960

cr 7773

cc0 7774

c1 7775

caddc 7777

cmul 7779

clt 7954

cle 7955

cmin 8090

cdiv 8589

cn 8878

cn0 9135

cz 9212

cuz 9487

cq 9578

cfz 9965

cfl 10224

cmo 10278

cexp 10475

cdvds 11749

cgcd 11897

codz 12162

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 609 ax-in2 610 ax-io 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-13 2143 ax-14 2144 ax-ext 2152 ax-coll 4104 ax-sep 4107 ax-nul 4115 ax-pow 4160 ax-pr 4194 ax-un 4418 ax-setind 4521 ax-iinf 4572 ax-cnex 7865 ax-resscn 7866 ax-1cn 7867 ax-1re 7868 ax-icn 7869 ax-addcl 7870 ax-addrcl 7871 ax-mulcl 7872 ax-mulrcl 7873 ax-addcom 7874 ax-mulcom 7875 ax-addass 7876 ax-mulass 7877 ax-distr 7878 ax-i2m1 7879 ax-0lt1 7880 ax-1rid 7881 ax-0id 7882 ax-rnegex 7883 ax-precex 7884 ax-cnre 7885 ax-pre-ltirr 7886 ax-pre-ltwlin 7887 ax-pre-lttrn 7888 ax-pre-apti 7889 ax-pre-ltadd 7890 ax-pre-mulgt0 7891 ax-pre-mulext 7892 ax-arch 7893 ax-caucvg 7894

This theorem depends on definitions: df-bi 116 df-stab 826 df-dc 830 df-3or 974 df-3an 975 df-tru 1351 df-fal 1354 df-nf 1454 df-sb 1756 df-eu 2022 df-mo 2023 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-ne 2341 df-nel 2436 df-ral 2453 df-rex 2454 df-reu 2455 df-rmo 2456 df-rab 2457 df-v 2732 df-sbc 2956 df-csb 3050 df-dif 3123 df-un 3125 df-in 3127 df-ss 3134 df-nul 3415 df-if 3527 df-pw 3568 df-sn 3589 df-pr 3590 df-op 3592 df-uni 3797 df-int 3832 df-iun 3875 df-br 3990 df-opab 4051 df-mpt 4052 df-tr 4088 df-id 4278 df-po 4281 df-iso 4282 df-iord 4351 df-on 4353 df-ilim 4354 df-suc 4356 df-iom 4575 df-xp 4617 df-rel 4618 df-cnv 4619 df-co 4620 df-dm 4621 df-rn 4622 df-res 4623 df-ima 4624 df-iota 5160 df-fun 5200 df-fn 5201 df-f 5202 df-f1 5203 df-fo 5204 df-f1o 5205 df-fv 5206 df-isom 5207 df-riota 5809 df-ov 5856 df-oprab 5857 df-mpo 5858 df-1st 6119 df-2nd 6120 df-recs 6284 df-irdg 6349 df-frec 6370 df-1o 6395 df-oadd 6399 df-er 6513 df-en 6719 df-dom 6720 df-fin 6721 df-sup 6961 df-inf 6962 df-pnf 7956 df-mnf 7957 df-xr 7958 df-ltxr 7959 df-le 7960 df-sub 8092 df-neg 8093 df-reap 8494 df-ap 8501 df-div 8590 df-inn 8879 df-2 8937 df-3 8938 df-4 8939 df-n0 9136 df-z 9213 df-uz 9488 df-q 9579 df-rp 9611 df-fz 9966 df-fzo 10099 df-fl 10226 df-mod 10279 df-seqfrec 10402 df-exp 10476 df-ihash 10710 df-cj 10806 df-re 10807 df-im 10808 df-rsqrt 10962 df-abs 10963 df-clim 11242 df-proddc 11514 df-dvds 11750 df-gcd 11898 df-odz 12164 df-phi 12165

This theorem is referenced by: odzphi 12200 pockthlem 12308

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