odzdvds - Intuitionistic Logic Explorer

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Theorem odzdvds 13053

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 9675	. . . . . . . . . 10
2		zq 10042	. . . . . . . . . 10
3	1, 2	syl 14	. . . . . . . . 9
4	3	adantl 277	. . . . . . . 8 $/\$ $/\$ $/\$
5		odzcl 13051	. . . . . . . . . 10 $/\$ $/\$
6	5	adantr 276	. . . . . . . . 9 $/\$ $/\$ $/\$
7		nnq 10049	. . . . . . . . 9
8	6, 7	syl 14	. . . . . . . 8 $/\$ $/\$ $/\$
9	6	nngt0d 9357	. . . . . . . 8 $/\$ $/\$ $/\$
10		modqlt 10791	. . . . . . . 8 $/\$ $/\$
11	4, 8, 9, 10	syl3anc 1278	. . . . . . 7 $/\$ $/\$ $/\$
12	1	adantl 277	. . . . . . . . . 10 $/\$ $/\$ $/\$
13	12, 6	zmodcld 10803	. . . . . . . . 9 $/\$ $/\$ $/\$
14	13	nn0zd 9777	. . . . . . . 8 $/\$ $/\$ $/\$
15	6	nnzd 9778	. . . . . . . 8 $/\$ $/\$ $/\$
16		zltnle 9701	. . . . . . . 8 $/\$
17	14, 15, 16	syl2anc 415	. . . . . . 7 $/\$ $/\$ $/\$
18	11, 17	mpbid 147	. . . . . 6 $/\$ $/\$ $/\$
19		1zzd 9682	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ $/\$
20		nnuz 9974	. . . . . . . . . . 11
21	20	rabeqi 2814	. . . . . . . . . 10
22		oveq2 6096	. . . . . . . . . . . . . . 15
23	22	oveq1d 6103	. . . . . . . . . . . . . 14
24	23	breq2d 4143	. . . . . . . . . . . . 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 10477	. . . . . . . . . . . . . . 15
33	32	nnnn0d 9631	. . . . . . . . . . . . . 14
34	33	adantl 277	. . . . . . . . . . . . 13 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
35		zexpcl 11012	. . . . . . . . . . . . 13 $/\$
36	31, 34, 35	syl2anc 415	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
37		peano2zm 9693	. . . . . . . . . . . 12
38	36, 37	syl 14	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
39		dvdsdc 12590	. . . . . . . . . . 11 $/\$ DECID
40	29, 38, 39	syl2anc 415	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ DECID
41	19, 21, 27, 40	infssuzledc 10684	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$ inf
42	41	ex 115	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ inf
43	42	ancomsd 269	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ inf
44		odzval 13049	. . . . . . . . 9 $/\$ $/\$ inf
45	44	adantr 276	. . . . . . . 8 $/\$ $/\$ $/\$ inf
46	45	breq1d 4141	. . . . . . 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 9576	. . . . . 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 9778	. . . . . 6 $/\$ $/\$ $/\$
57		dvds0 12598	. . . . . 6
58	56, 57	syl 14	. . . . 5 $/\$ $/\$ $/\$
59		simpl2 1032	. . . . . . . . 9 $/\$ $/\$ $/\$
60	59	zcnd 9780	. . . . . . . 8 $/\$ $/\$ $/\$
61	60	exp0d 11126	. . . . . . 7 $/\$ $/\$ $/\$
62	61	oveq1d 6103	. . . . . 6 $/\$ $/\$ $/\$
63		1m1e0 9382	. . . . . 6
64	62, 63	eqtrdi 2287	. . . . 5 $/\$ $/\$ $/\$
65	58, 64	breqtrrd 4159	. . . 4 $/\$ $/\$ $/\$
66		oveq2 6096	. . . . . 6
67	66	oveq1d 6103	. . . . 5
68	67	breq2d 4143	. . . 4
69	65, 68	syl5ibrcom 157	. . 3 $/\$ $/\$ $/\$
70	54, 69	impbid 129	. 2 $/\$ $/\$ $/\$
71	6	nnnn0d 9631	. . . . . . . . 9 $/\$ $/\$ $/\$
72		znq 10040	. . . . . . . . . . 11 $/\$
73	12, 6, 72	syl2anc 415	. . . . . . . . . 10 $/\$ $/\$ $/\$
74		nn0ge0 9599	. . . . . . . . . . . 12
75	74	adantl 277	. . . . . . . . . . 11 $/\$ $/\$ $/\$
76		nn0re 9583	. . . . . . . . . . . . 13
77	76	adantl 277	. . . . . . . . . . . 12 $/\$ $/\$ $/\$
78	6	nnred 9326	. . . . . . . . . . . 12 $/\$ $/\$ $/\$
79		ge0div 9209	. . . . . . . . . . . 12 $/\$ $/\$
80	77, 78, 9, 79	syl3anc 1278	. . . . . . . . . . 11 $/\$ $/\$ $/\$
81	75, 80	mpbid 147	. . . . . . . . . 10 $/\$ $/\$ $/\$
82		flqge0nn0 10749	. . . . . . . . . 10 $/\$
83	73, 81, 82	syl2anc 415	. . . . . . . . 9 $/\$ $/\$ $/\$
84	71, 83	nn0mulcld 9636	. . . . . . . 8 $/\$ $/\$ $/\$
85		zexpcl 11012	. . . . . . . 8 $/\$
86	59, 84, 85	syl2anc 415	. . . . . . 7 $/\$ $/\$ $/\$
87		zq 10042	. . . . . . 7
88	86, 87	syl 14	. . . . . 6 $/\$ $/\$ $/\$
89		1z 9681	. . . . . . 7
90		zq 10042	. . . . . . 7
91	89, 90	mp1i 10	. . . . . 6 $/\$ $/\$ $/\$
92		zexpcl 11012	. . . . . . 7 $/\$
93	59, 13, 92	syl2anc 415	. . . . . 6 $/\$ $/\$ $/\$
94		nnq 10049	. . . . . . 7
95	55, 94	syl 14	. . . . . 6 $/\$ $/\$ $/\$
96	55	nngt0d 9357	. . . . . 6 $/\$ $/\$ $/\$
97	60, 83, 71	expmuld 11135	. . . . . . . 8 $/\$ $/\$ $/\$
98	97	oveq1d 6103	. . . . . . 7 $/\$ $/\$ $/\$
99		zexpcl 11012	. . . . . . . . 9 $/\$
100	59, 71, 99	syl2anc 415	. . . . . . . 8 $/\$ $/\$ $/\$
101		1zzd 9682	. . . . . . . 8 $/\$ $/\$ $/\$
102		odzid 13052	. . . . . . . . . 10 $/\$ $/\$
103	102	adantr 276	. . . . . . . . 9 $/\$ $/\$ $/\$
104		moddvds 12591	. . . . . . . . . 10 $/\$ $/\$
105	55, 100, 101, 104	syl3anc 1278	. . . . . . . . 9 $/\$ $/\$ $/\$
106	103, 105	mpbird 167	. . . . . . . 8 $/\$ $/\$ $/\$
107	100, 101, 83, 95, 96, 106	modqexp 11125	. . . . . . 7 $/\$ $/\$ $/\$
108	73	flqcld 10731	. . . . . . . . 9 $/\$ $/\$ $/\$
109		1exp 11026	. . . . . . . . 9
110	108, 109	syl 14	. . . . . . . 8 $/\$ $/\$ $/\$
111	110	oveq1d 6103	. . . . . . 7 $/\$ $/\$ $/\$
112	98, 107, 111	3eqtrd 2275	. . . . . 6 $/\$ $/\$ $/\$
113	88, 91, 93, 95, 96, 112	modqmul1 10835	. . . . 5 $/\$ $/\$ $/\$
114	60, 13, 84	expaddd 11134	. . . . . . 7 $/\$ $/\$ $/\$
115		modqval 10782	. . . . . . . . . . 11 $/\$ $/\$
116	4, 8, 9, 115	syl3anc 1278	. . . . . . . . . 10 $/\$ $/\$ $/\$
117	116	oveq2d 6104	. . . . . . . . 9 $/\$ $/\$ $/\$
118	84	nn0cnd 9633	. . . . . . . . . 10 $/\$ $/\$ $/\$
119	77	recnd 8360	. . . . . . . . . 10 $/\$ $/\$ $/\$
120	118, 119	pncan3d 8647	. . . . . . . . 9 $/\$ $/\$ $/\$
121	117, 120	eqtrd 2271	. . . . . . . 8 $/\$ $/\$ $/\$
122	121	oveq2d 6104	. . . . . . 7 $/\$ $/\$ $/\$
123	114, 122	eqtr3d 2273	. . . . . 6 $/\$ $/\$ $/\$
124	123	oveq1d 6103	. . . . 5 $/\$ $/\$ $/\$
125	93	zcnd 9780	. . . . . . 7 $/\$ $/\$ $/\$
126	125	mullidd 8350	. . . . . 6 $/\$ $/\$ $/\$
127	126	oveq1d 6103	. . . . 5 $/\$ $/\$ $/\$
128	113, 124, 127	3eqtr3d 2279	. . . 4 $/\$ $/\$ $/\$
129	128	eqeq1d 2247	. . 3 $/\$ $/\$ $/\$
130		zexpcl 11012	. . . . 5 $/\$
131	59, 130	sylancom 424	. . . 4 $/\$ $/\$ $/\$
132		moddvds 12591	. . . 4 $/\$ $/\$
133	55, 131, 101, 132	syl3anc 1278	. . 3 $/\$ $/\$ $/\$
134		moddvds 12591	. . . 4 $/\$ $/\$
135	55, 93, 101, 134	syl3anc 1278	. . 3 $/\$ $/\$ $/\$
136	129, 133, 135	3bitr3d 218	. 2 $/\$ $/\$ $/\$
137		dvdsval3 12583	. . 3 $/\$
138	6, 12, 137	syl2anc 415	. 2 $/\$ $/\$ $/\$
139	70, 136, 138	3bitr4d 220	1 $/\$ $/\$ $/\$

Colors of variables: wff set class

This proof depends on syntax axioms:

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 4131

cfv 5380 (class class class)co 6088 infcinf 7329

cr 8184

cc0 8185

c1 8186

caddc 8188

cmul 8190

clt 8366

cle 8367

cmin 8504

cdiv 9010

cn 9313

cn0 9574

cz 9655

cuz 9937

cq 10035

cfz 10428

cfl 10720

cmo 10780

cexp 10996

cdvds 12579

cgcd 12755

codz 13015

This proof depends on 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 4248 ax-sep 4251 ax-nul 4261 ax-pow 4313 ax-pr 4348 ax-un 4580 ax-setind 4686 ax-iinf 4737 ax-cnex 8276 ax-resscn 8277 ax-1cn 8278 ax-1re 8279 ax-icn 8280 ax-addcl 8281 ax-addrcl 8282 ax-mulcl 8283 ax-mulrcl 8284 ax-addcom 8285 ax-mulcom 8286 ax-addass 8287 ax-mulass 8288 ax-distr 8289 ax-i2m1 8290 ax-0lt1 8291 ax-1rid 8292 ax-0id 8293 ax-rnegex 8294 ax-precex 8295 ax-cnre 8296 ax-pre-ltirr 8297 ax-pre-ltwlin 8298 ax-pre-lttrn 8299 ax-pre-apti 8300 ax-pre-ltadd 8301 ax-pre-mulgt0 8302 ax-pre-mulext 8303 ax-arch 8304 ax-caucvg 8305

This proof 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 3639 df-pw 3690 df-sn 3715 df-pr 3716 df-op 3718 df-uni 3936 df-int 3971 df-iun 4014 df-br 4132 df-opab 4194 df-mpt 4195 df-tr 4232 df-id 4440 df-po 4443 df-iso 4444 df-iord 4513 df-on 4515 df-ilim 4516 df-suc 4518 df-iom 4740 df-xp 4782 df-rel 4783 df-cnv 4784 df-co 4785 df-dm 4786 df-rn 4787 df-res 4788 df-ima 4789 df-iota 5340 df-fun 5382 df-fn 5383 df-f 5384 df-f1 5385 df-fo 5386 df-f1o 5387 df-fv 5388 df-isom 5389 df-riota 6041 df-ov 6091 df-oprab 6092 df-mpo 6093 df-1st 6378 df-2nd 6379 df-recs 6580 df-irdg 6645 df-frec 6666 df-1o 6691 df-oadd 6695 df-er 6811 df-en 7027 df-dom 7028 df-fin 7029 df-sup 7330 df-inf 7331 df-pnf 8368 df-mnf 8369 df-xr 8370 df-ltxr 8371 df-le 8372 df-sub 8506 df-neg 8507 df-reap 8911 df-ap 8918 df-div 9011 df-inn 9314 df-2 9372 df-3 9373 df-4 9374 df-n0 9575 df-z 9656 df-uz 9938 df-q 10036 df-rp 10072 df-fz 10429 df-fzo 10567 df-fl 10722 df-mod 10781 df-seqfrec 10906 df-exp 10997 df-ihash 11237 df-cj 11629 df-re 11630 df-im 11631 df-rsqrt 11786 df-abs 11787 df-clim 12070 df-proddc 12343 df-dvds 12580 df-gcd 12756 df-odz 13017 df-phi 13018

This theorem is used by: odzphi 13054 pockthlem 13164

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