odzdvds - Intuitionistic Logic Explorer

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Theorem odzdvds 12947

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 9599	. . . . . . . . . 10
2		zq 9961	. . . . . . . . . 10
3	1, 2	syl 14	. . . . . . . . 9
4	3	adantl 277	. . . . . . . 8 $/\$ $/\$ $/\$
5		odzcl 12945	. . . . . . . . . 10 $/\$ $/\$
6	5	adantr 276	. . . . . . . . 9 $/\$ $/\$ $/\$
7		nnq 9968	. . . . . . . . 9
8	6, 7	syl 14	. . . . . . . 8 $/\$ $/\$ $/\$
9	6	nngt0d 9283	. . . . . . . 8 $/\$ $/\$ $/\$
10		modqlt 10699	. . . . . . . 8 $/\$ $/\$
11	4, 8, 9, 10	syl3anc 1274	. . . . . . 7 $/\$ $/\$ $/\$
12	1	adantl 277	. . . . . . . . . 10 $/\$ $/\$ $/\$
13	12, 6	zmodcld 10711	. . . . . . . . 9 $/\$ $/\$ $/\$
14	13	nn0zd 9701	. . . . . . . 8 $/\$ $/\$ $/\$
15	6	nnzd 9702	. . . . . . . 8 $/\$ $/\$ $/\$
16		zltnle 9625	. . . . . . . 8 $/\$
17	14, 15, 16	syl2anc 411	. . . . . . 7 $/\$ $/\$ $/\$
18	11, 17	mpbid 147	. . . . . 6 $/\$ $/\$ $/\$
19		1zzd 9606	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ $/\$
20		nnuz 9893	. . . . . . . . . . 11
21	20	rabeqi 2808	. . . . . . . . . 10
22		oveq2 6060	. . . . . . . . . . . . . . 15
23	22	oveq1d 6067	. . . . . . . . . . . . . 14
24	23	breq2d 4123	. . . . . . . . . . . . 13
25	24	elrab 2975	. . . . . . . . . . . 12 $/\$
26	25	biimpri 133	. . . . . . . . . . 11 $/\$
27	26	adantl 277	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ $/\$
28		simp1 1024	. . . . . . . . . . . 12 $/\$ $/\$
29	28	ad3antrrr 492	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
30		simp2 1025	. . . . . . . . . . . . . 14 $/\$ $/\$
31	30	ad3antrrr 492	. . . . . . . . . . . . 13 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
32		elfznn 10391	. . . . . . . . . . . . . . 15
33	32	nnnn0d 9555	. . . . . . . . . . . . . 14
34	33	adantl 277	. . . . . . . . . . . . 13 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
35		zexpcl 10920	. . . . . . . . . . . . 13 $/\$
36	31, 34, 35	syl2anc 411	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
37		peano2zm 9617	. . . . . . . . . . . 12
38	36, 37	syl 14	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
39		dvdsdc 12488	. . . . . . . . . . 11 $/\$ DECID
40	29, 38, 39	syl2anc 411	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ DECID
41	19, 21, 27, 40	infssuzledc 10598	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$ inf
42	41	ex 115	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ inf
43	42	ancomsd 269	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ inf
44		odzval 12943	. . . . . . . . 9 $/\$ $/\$ inf
45	44	adantr 276	. . . . . . . 8 $/\$ $/\$ $/\$ inf
46	45	breq1d 4121	. . . . . . 7 $/\$ $/\$ $/\$ inf
47	43, 46	sylibrd 169	. . . . . 6 $/\$ $/\$ $/\$ $/\$
48	18, 47	mtod 669	. . . . 5 $/\$ $/\$ $/\$ $/\$
49		imnan 697	. . . . 5 $/\$
50	48, 49	sylibr 134	. . . 4 $/\$ $/\$ $/\$
51		elnn0 9500	. . . . . 6 $\/$
52	13, 51	sylib 122	. . . . 5 $/\$ $/\$ $/\$ $\/$
53	52	ord 732	. . . 4 $/\$ $/\$ $/\$
54	50, 53	syld 45	. . 3 $/\$ $/\$ $/\$
55		simpl1 1027	. . . . . . 7 $/\$ $/\$ $/\$
56	55	nnzd 9702	. . . . . 6 $/\$ $/\$ $/\$
57		dvds0 12496	. . . . . 6
58	56, 57	syl 14	. . . . 5 $/\$ $/\$ $/\$
59		simpl2 1028	. . . . . . . . 9 $/\$ $/\$ $/\$
60	59	zcnd 9704	. . . . . . . 8 $/\$ $/\$ $/\$
61	60	exp0d 11033	. . . . . . 7 $/\$ $/\$ $/\$
62	61	oveq1d 6067	. . . . . 6 $/\$ $/\$ $/\$
63		1m1e0 9308	. . . . . 6
64	62, 63	eqtrdi 2283	. . . . 5 $/\$ $/\$ $/\$
65	58, 64	breqtrrd 4139	. . . 4 $/\$ $/\$ $/\$
66		oveq2 6060	. . . . . 6
67	66	oveq1d 6067	. . . . 5
68	67	breq2d 4123	. . . 4
69	65, 68	syl5ibrcom 157	. . 3 $/\$ $/\$ $/\$
70	54, 69	impbid 129	. 2 $/\$ $/\$ $/\$
71	6	nnnn0d 9555	. . . . . . . . 9 $/\$ $/\$ $/\$
72		znq 9959	. . . . . . . . . . 11 $/\$
73	12, 6, 72	syl2anc 411	. . . . . . . . . 10 $/\$ $/\$ $/\$
74		nn0ge0 9523	. . . . . . . . . . . 12
75	74	adantl 277	. . . . . . . . . . 11 $/\$ $/\$ $/\$
76		nn0re 9507	. . . . . . . . . . . . 13
77	76	adantl 277	. . . . . . . . . . . 12 $/\$ $/\$ $/\$
78	6	nnred 9252	. . . . . . . . . . . 12 $/\$ $/\$ $/\$
79		ge0div 9147	. . . . . . . . . . . 12 $/\$ $/\$
80	77, 78, 9, 79	syl3anc 1274	. . . . . . . . . . 11 $/\$ $/\$ $/\$
81	75, 80	mpbid 147	. . . . . . . . . 10 $/\$ $/\$ $/\$
82		flqge0nn0 10657	. . . . . . . . . 10 $/\$
83	73, 81, 82	syl2anc 411	. . . . . . . . 9 $/\$ $/\$ $/\$
84	71, 83	nn0mulcld 9560	. . . . . . . 8 $/\$ $/\$ $/\$
85		zexpcl 10920	. . . . . . . 8 $/\$
86	59, 84, 85	syl2anc 411	. . . . . . 7 $/\$ $/\$ $/\$
87		zq 9961	. . . . . . 7
88	86, 87	syl 14	. . . . . 6 $/\$ $/\$ $/\$
89		1z 9605	. . . . . . 7
90		zq 9961	. . . . . . 7
91	89, 90	mp1i 10	. . . . . 6 $/\$ $/\$ $/\$
92		zexpcl 10920	. . . . . . 7 $/\$
93	59, 13, 92	syl2anc 411	. . . . . 6 $/\$ $/\$ $/\$
94		nnq 9968	. . . . . . 7
95	55, 94	syl 14	. . . . . 6 $/\$ $/\$ $/\$
96	55	nngt0d 9283	. . . . . 6 $/\$ $/\$ $/\$
97	60, 83, 71	expmuld 11042	. . . . . . . 8 $/\$ $/\$ $/\$
98	97	oveq1d 6067	. . . . . . 7 $/\$ $/\$ $/\$
99		zexpcl 10920	. . . . . . . . 9 $/\$
100	59, 71, 99	syl2anc 411	. . . . . . . 8 $/\$ $/\$ $/\$
101		1zzd 9606	. . . . . . . 8 $/\$ $/\$ $/\$
102		odzid 12946	. . . . . . . . . 10 $/\$ $/\$
103	102	adantr 276	. . . . . . . . 9 $/\$ $/\$ $/\$
104		moddvds 12489	. . . . . . . . . 10 $/\$ $/\$
105	55, 100, 101, 104	syl3anc 1274	. . . . . . . . 9 $/\$ $/\$ $/\$
106	103, 105	mpbird 167	. . . . . . . 8 $/\$ $/\$ $/\$
107	100, 101, 83, 95, 96, 106	modqexp 11032	. . . . . . 7 $/\$ $/\$ $/\$
108	73	flqcld 10641	. . . . . . . . 9 $/\$ $/\$ $/\$
109		1exp 10934	. . . . . . . . 9
110	108, 109	syl 14	. . . . . . . 8 $/\$ $/\$ $/\$
111	110	oveq1d 6067	. . . . . . 7 $/\$ $/\$ $/\$
112	98, 107, 111	3eqtrd 2271	. . . . . 6 $/\$ $/\$ $/\$
113	88, 91, 93, 95, 96, 112	modqmul1 10743	. . . . 5 $/\$ $/\$ $/\$
114	60, 13, 84	expaddd 11041	. . . . . . 7 $/\$ $/\$ $/\$
115		modqval 10690	. . . . . . . . . . 11 $/\$ $/\$
116	4, 8, 9, 115	syl3anc 1274	. . . . . . . . . 10 $/\$ $/\$ $/\$
117	116	oveq2d 6068	. . . . . . . . 9 $/\$ $/\$ $/\$
118	84	nn0cnd 9557	. . . . . . . . . 10 $/\$ $/\$ $/\$
119	77	recnd 8304	. . . . . . . . . 10 $/\$ $/\$ $/\$
120	118, 119	pncan3d 8589	. . . . . . . . 9 $/\$ $/\$ $/\$
121	117, 120	eqtrd 2267	. . . . . . . 8 $/\$ $/\$ $/\$
122	121	oveq2d 6068	. . . . . . 7 $/\$ $/\$ $/\$
123	114, 122	eqtr3d 2269	. . . . . 6 $/\$ $/\$ $/\$
124	123	oveq1d 6067	. . . . 5 $/\$ $/\$ $/\$
125	93	zcnd 9704	. . . . . . 7 $/\$ $/\$ $/\$
126	125	mullidd 8294	. . . . . 6 $/\$ $/\$ $/\$
127	126	oveq1d 6067	. . . . 5 $/\$ $/\$ $/\$
128	113, 124, 127	3eqtr3d 2275	. . . 4 $/\$ $/\$ $/\$
129	128	eqeq1d 2243	. . 3 $/\$ $/\$ $/\$
130		zexpcl 10920	. . . . 5 $/\$
131	59, 130	sylancom 420	. . . 4 $/\$ $/\$ $/\$
132		moddvds 12489	. . . 4 $/\$ $/\$
133	55, 131, 101, 132	syl3anc 1274	. . 3 $/\$ $/\$ $/\$
134		moddvds 12489	. . . 4 $/\$ $/\$
135	55, 93, 101, 134	syl3anc 1274	. . 3 $/\$ $/\$ $/\$
136	129, 133, 135	3bitr3d 218	. 2 $/\$ $/\$ $/\$
137		dvdsval3 12481	. . 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 716 DECID wdc 842 $/\$ w3a 1005

wceq 1398

wcel 2205

crab 2526 class class class wbr 4111

cfv 5354 (class class class)co 6052 infcinf 7276

cr 8128

cc0 8129

c1 8130

caddc 8132

cmul 8134

clt 8310

cle 8311

cmin 8446

cdiv 8948

cn 9239

cn0 9498

cz 9579

cuz 9856

cq 9954

cfz 10345

cfl 10632

cmo 10688

cexp 10904

cdvds 12477

cgcd 12653

codz 12909

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 619 ax-in2 620 ax-io 717 ax-5 1496 ax-7 1497 ax-gen 1498 ax-ie1 1542 ax-ie2 1543 ax-8 1553 ax-10 1554 ax-11 1555 ax-i12 1556 ax-bndl 1558 ax-4 1559 ax-17 1575 ax-i9 1579 ax-ial 1583 ax-i5r 1584 ax-13 2207 ax-14 2208 ax-ext 2216 ax-coll 4227 ax-sep 4230 ax-nul 4238 ax-pow 4289 ax-pr 4324 ax-un 4556 ax-setind 4661 ax-iinf 4712 ax-cnex 8220 ax-resscn 8221 ax-1cn 8222 ax-1re 8223 ax-icn 8224 ax-addcl 8225 ax-addrcl 8226 ax-mulcl 8227 ax-mulrcl 8228 ax-addcom 8229 ax-mulcom 8230 ax-addass 8231 ax-mulass 8232 ax-distr 8233 ax-i2m1 8234 ax-0lt1 8235 ax-1rid 8236 ax-0id 8237 ax-rnegex 8238 ax-precex 8239 ax-cnre 8240 ax-pre-ltirr 8241 ax-pre-ltwlin 8242 ax-pre-lttrn 8243 ax-pre-apti 8244 ax-pre-ltadd 8245 ax-pre-mulgt0 8246 ax-pre-mulext 8247 ax-arch 8248 ax-caucvg 8249

This theorem depends on definitions: df-bi 117 df-stab 839 df-dc 843 df-3or 1006 df-3an 1007 df-tru 1401 df-fal 1404 df-nf 1510 df-sb 1812 df-eu 2085 df-mo 2086 df-clab 2221 df-cleq 2227 df-clel 2230 df-nfc 2375 df-ne 2415 df-nel 2510 df-ral 2527 df-rex 2528 df-reu 2529 df-rmo 2530 df-rab 2531 df-v 2817 df-sbc 3045 df-csb 3141 df-dif 3215 df-un 3217 df-in 3219 df-ss 3226 df-nul 3511 df-if 3623 df-pw 3673 df-sn 3697 df-pr 3698 df-op 3700 df-uni 3917 df-int 3952 df-iun 3995 df-br 4112 df-opab 4174 df-mpt 4175 df-tr 4211 df-id 4416 df-po 4419 df-iso 4420 df-iord 4489 df-on 4491 df-ilim 4492 df-suc 4494 df-iom 4715 df-xp 4757 df-rel 4758 df-cnv 4759 df-co 4760 df-dm 4761 df-rn 4762 df-res 4763 df-ima 4764 df-iota 5314 df-fun 5356 df-fn 5357 df-f 5358 df-f1 5359 df-fo 5360 df-f1o 5361 df-fv 5362 df-isom 5363 df-riota 6005 df-ov 6055 df-oprab 6056 df-mpo 6057 df-1st 6336 df-2nd 6337 df-recs 6538 df-irdg 6603 df-frec 6624 df-1o 6649 df-oadd 6653 df-er 6769 df-en 6978 df-dom 6979 df-fin 6980 df-sup 7277 df-inf 7278 df-pnf 8312 df-mnf 8313 df-xr 8314 df-ltxr 8315 df-le 8316 df-sub 8448 df-neg 8449 df-reap 8851 df-ap 8858 df-div 8949 df-inn 9240 df-2 9298 df-3 9299 df-4 9300 df-n0 9499 df-z 9580 df-uz 9857 df-q 9955 df-rp 9990 df-fz 10346 df-fzo 10481 df-fl 10634 df-mod 10689 df-seqfrec 10814 df-exp 10905 df-ihash 11143 df-cj 11531 df-re 11532 df-im 11533 df-rsqrt 11687 df-abs 11688 df-clim 11968 df-proddc 12241 df-dvds 12478 df-gcd 12654 df-odz 12911 df-phi 12912

This theorem is referenced by: odzphi 12948 pockthlem 13058

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