odzdvds - Intuitionistic Logic Explorer

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Theorem odzdvds 12383

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 9337	. . . . . . . . . 10
2		zq 9691	. . . . . . . . . 10
3	1, 2	syl 14	. . . . . . . . 9
4	3	adantl 277	. . . . . . . 8 $/\$ $/\$ $/\$
5		odzcl 12381	. . . . . . . . . 10 $/\$ $/\$
6	5	adantr 276	. . . . . . . . 9 $/\$ $/\$ $/\$
7		nnq 9698	. . . . . . . . 9
8	6, 7	syl 14	. . . . . . . 8 $/\$ $/\$ $/\$
9	6	nngt0d 9026	. . . . . . . 8 $/\$ $/\$ $/\$
10		modqlt 10404	. . . . . . . 8 $/\$ $/\$
11	4, 8, 9, 10	syl3anc 1249	. . . . . . 7 $/\$ $/\$ $/\$
12	1	adantl 277	. . . . . . . . . 10 $/\$ $/\$ $/\$
13	12, 6	zmodcld 10416	. . . . . . . . 9 $/\$ $/\$ $/\$
14	13	nn0zd 9437	. . . . . . . 8 $/\$ $/\$ $/\$
15	6	nnzd 9438	. . . . . . . 8 $/\$ $/\$ $/\$
16		zltnle 9363	. . . . . . . 8 $/\$
17	14, 15, 16	syl2anc 411	. . . . . . 7 $/\$ $/\$ $/\$
18	11, 17	mpbid 147	. . . . . 6 $/\$ $/\$ $/\$
19		1zzd 9344	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ $/\$
20		nnuz 9628	. . . . . . . . . . 11
21	20	rabeqi 2753	. . . . . . . . . 10
22		oveq2 5926	. . . . . . . . . . . . . . 15
23	22	oveq1d 5933	. . . . . . . . . . . . . 14
24	23	breq2d 4041	. . . . . . . . . . . . 13
25	24	elrab 2916	. . . . . . . . . . . 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 10120	. . . . . . . . . . . . . . 15
33	32	nnnn0d 9293	. . . . . . . . . . . . . 14
34	33	adantl 277	. . . . . . . . . . . . 13 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
35		zexpcl 10625	. . . . . . . . . . . . 13 $/\$
36	31, 34, 35	syl2anc 411	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
37		peano2zm 9355	. . . . . . . . . . . 12
38	36, 37	syl 14	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
39		dvdsdc 11941	. . . . . . . . . . 11 $/\$ DECID
40	29, 38, 39	syl2anc 411	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ DECID
41	19, 21, 27, 40	infssuzledc 12087	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$ inf
42	41	ex 115	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ inf
43	42	ancomsd 269	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ inf
44		odzval 12379	. . . . . . . . 9 $/\$ $/\$ inf
45	44	adantr 276	. . . . . . . 8 $/\$ $/\$ $/\$ inf
46	45	breq1d 4039	. . . . . . 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 9242	. . . . . 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 9438	. . . . . 6 $/\$ $/\$ $/\$
57		dvds0 11949	. . . . . 6
58	56, 57	syl 14	. . . . 5 $/\$ $/\$ $/\$
59		simpl2 1003	. . . . . . . . 9 $/\$ $/\$ $/\$
60	59	zcnd 9440	. . . . . . . 8 $/\$ $/\$ $/\$
61	60	exp0d 10738	. . . . . . 7 $/\$ $/\$ $/\$
62	61	oveq1d 5933	. . . . . 6 $/\$ $/\$ $/\$
63		1m1e0 9051	. . . . . 6
64	62, 63	eqtrdi 2242	. . . . 5 $/\$ $/\$ $/\$
65	58, 64	breqtrrd 4057	. . . 4 $/\$ $/\$ $/\$
66		oveq2 5926	. . . . . 6
67	66	oveq1d 5933	. . . . 5
68	67	breq2d 4041	. . . 4
69	65, 68	syl5ibrcom 157	. . 3 $/\$ $/\$ $/\$
70	54, 69	impbid 129	. 2 $/\$ $/\$ $/\$
71	6	nnnn0d 9293	. . . . . . . . 9 $/\$ $/\$ $/\$
72		znq 9689	. . . . . . . . . . 11 $/\$
73	12, 6, 72	syl2anc 411	. . . . . . . . . 10 $/\$ $/\$ $/\$
74		nn0ge0 9265	. . . . . . . . . . . 12
75	74	adantl 277	. . . . . . . . . . 11 $/\$ $/\$ $/\$
76		nn0re 9249	. . . . . . . . . . . . 13
77	76	adantl 277	. . . . . . . . . . . 12 $/\$ $/\$ $/\$
78	6	nnred 8995	. . . . . . . . . . . 12 $/\$ $/\$ $/\$
79		ge0div 8890	. . . . . . . . . . . 12 $/\$ $/\$
80	77, 78, 9, 79	syl3anc 1249	. . . . . . . . . . 11 $/\$ $/\$ $/\$
81	75, 80	mpbid 147	. . . . . . . . . 10 $/\$ $/\$ $/\$
82		flqge0nn0 10362	. . . . . . . . . 10 $/\$
83	73, 81, 82	syl2anc 411	. . . . . . . . 9 $/\$ $/\$ $/\$
84	71, 83	nn0mulcld 9298	. . . . . . . 8 $/\$ $/\$ $/\$
85		zexpcl 10625	. . . . . . . 8 $/\$
86	59, 84, 85	syl2anc 411	. . . . . . 7 $/\$ $/\$ $/\$
87		zq 9691	. . . . . . 7
88	86, 87	syl 14	. . . . . 6 $/\$ $/\$ $/\$
89		1z 9343	. . . . . . 7
90		zq 9691	. . . . . . 7
91	89, 90	mp1i 10	. . . . . 6 $/\$ $/\$ $/\$
92		zexpcl 10625	. . . . . . 7 $/\$
93	59, 13, 92	syl2anc 411	. . . . . 6 $/\$ $/\$ $/\$
94		nnq 9698	. . . . . . 7
95	55, 94	syl 14	. . . . . 6 $/\$ $/\$ $/\$
96	55	nngt0d 9026	. . . . . 6 $/\$ $/\$ $/\$
97	60, 83, 71	expmuld 10747	. . . . . . . 8 $/\$ $/\$ $/\$
98	97	oveq1d 5933	. . . . . . 7 $/\$ $/\$ $/\$
99		zexpcl 10625	. . . . . . . . 9 $/\$
100	59, 71, 99	syl2anc 411	. . . . . . . 8 $/\$ $/\$ $/\$
101		1zzd 9344	. . . . . . . 8 $/\$ $/\$ $/\$
102		odzid 12382	. . . . . . . . . 10 $/\$ $/\$
103	102	adantr 276	. . . . . . . . 9 $/\$ $/\$ $/\$
104		moddvds 11942	. . . . . . . . . 10 $/\$ $/\$
105	55, 100, 101, 104	syl3anc 1249	. . . . . . . . 9 $/\$ $/\$ $/\$
106	103, 105	mpbird 167	. . . . . . . 8 $/\$ $/\$ $/\$
107	100, 101, 83, 95, 96, 106	modqexp 10737	. . . . . . 7 $/\$ $/\$ $/\$
108	73	flqcld 10346	. . . . . . . . 9 $/\$ $/\$ $/\$
109		1exp 10639	. . . . . . . . 9
110	108, 109	syl 14	. . . . . . . 8 $/\$ $/\$ $/\$
111	110	oveq1d 5933	. . . . . . 7 $/\$ $/\$ $/\$
112	98, 107, 111	3eqtrd 2230	. . . . . 6 $/\$ $/\$ $/\$
113	88, 91, 93, 95, 96, 112	modqmul1 10448	. . . . 5 $/\$ $/\$ $/\$
114	60, 13, 84	expaddd 10746	. . . . . . 7 $/\$ $/\$ $/\$
115		modqval 10395	. . . . . . . . . . 11 $/\$ $/\$
116	4, 8, 9, 115	syl3anc 1249	. . . . . . . . . 10 $/\$ $/\$ $/\$
117	116	oveq2d 5934	. . . . . . . . 9 $/\$ $/\$ $/\$
118	84	nn0cnd 9295	. . . . . . . . . 10 $/\$ $/\$ $/\$
119	77	recnd 8048	. . . . . . . . . 10 $/\$ $/\$ $/\$
120	118, 119	pncan3d 8333	. . . . . . . . 9 $/\$ $/\$ $/\$
121	117, 120	eqtrd 2226	. . . . . . . 8 $/\$ $/\$ $/\$
122	121	oveq2d 5934	. . . . . . 7 $/\$ $/\$ $/\$
123	114, 122	eqtr3d 2228	. . . . . 6 $/\$ $/\$ $/\$
124	123	oveq1d 5933	. . . . 5 $/\$ $/\$ $/\$
125	93	zcnd 9440	. . . . . . 7 $/\$ $/\$ $/\$
126	125	mulid2d 8038	. . . . . 6 $/\$ $/\$ $/\$
127	126	oveq1d 5933	. . . . 5 $/\$ $/\$ $/\$
128	113, 124, 127	3eqtr3d 2234	. . . 4 $/\$ $/\$ $/\$
129	128	eqeq1d 2202	. . 3 $/\$ $/\$ $/\$
130		zexpcl 10625	. . . . 5 $/\$
131	59, 130	sylancom 420	. . . 4 $/\$ $/\$ $/\$
132		moddvds 11942	. . . 4 $/\$ $/\$
133	55, 131, 101, 132	syl3anc 1249	. . 3 $/\$ $/\$ $/\$
134		moddvds 11942	. . . 4 $/\$ $/\$
135	55, 93, 101, 134	syl3anc 1249	. . 3 $/\$ $/\$ $/\$
136	129, 133, 135	3bitr3d 218	. 2 $/\$ $/\$ $/\$
137		dvdsval3 11934	. . 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 2164

crab 2476 class class class wbr 4029

cfv 5254 (class class class)co 5918 infcinf 7042

cr 7871

cc0 7872

c1 7873

caddc 7875

cmul 7877

clt 8054

cle 8055

cmin 8190

cdiv 8691

cn 8982

cn0 9240

cz 9317

cuz 9592

cq 9684

cfz 10074

cfl 10337

cmo 10393

cexp 10609

cdvds 11930

cgcd 12079

codz 12346

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 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-13 2166 ax-14 2167 ax-ext 2175 ax-coll 4144 ax-sep 4147 ax-nul 4155 ax-pow 4203 ax-pr 4238 ax-un 4464 ax-setind 4569 ax-iinf 4620 ax-cnex 7963 ax-resscn 7964 ax-1cn 7965 ax-1re 7966 ax-icn 7967 ax-addcl 7968 ax-addrcl 7969 ax-mulcl 7970 ax-mulrcl 7971 ax-addcom 7972 ax-mulcom 7973 ax-addass 7974 ax-mulass 7975 ax-distr 7976 ax-i2m1 7977 ax-0lt1 7978 ax-1rid 7979 ax-0id 7980 ax-rnegex 7981 ax-precex 7982 ax-cnre 7983 ax-pre-ltirr 7984 ax-pre-ltwlin 7985 ax-pre-lttrn 7986 ax-pre-apti 7987 ax-pre-ltadd 7988 ax-pre-mulgt0 7989 ax-pre-mulext 7990 ax-arch 7991 ax-caucvg 7992

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 1472 df-sb 1774 df-eu 2045 df-mo 2046 df-clab 2180 df-cleq 2186 df-clel 2189 df-nfc 2325 df-ne 2365 df-nel 2460 df-ral 2477 df-rex 2478 df-reu 2479 df-rmo 2480 df-rab 2481 df-v 2762 df-sbc 2986 df-csb 3081 df-dif 3155 df-un 3157 df-in 3159 df-ss 3166 df-nul 3447 df-if 3558 df-pw 3603 df-sn 3624 df-pr 3625 df-op 3627 df-uni 3836 df-int 3871 df-iun 3914 df-br 4030 df-opab 4091 df-mpt 4092 df-tr 4128 df-id 4324 df-po 4327 df-iso 4328 df-iord 4397 df-on 4399 df-ilim 4400 df-suc 4402 df-iom 4623 df-xp 4665 df-rel 4666 df-cnv 4667 df-co 4668 df-dm 4669 df-rn 4670 df-res 4671 df-ima 4672 df-iota 5215 df-fun 5256 df-fn 5257 df-f 5258 df-f1 5259 df-fo 5260 df-f1o 5261 df-fv 5262 df-isom 5263 df-riota 5873 df-ov 5921 df-oprab 5922 df-mpo 5923 df-1st 6193 df-2nd 6194 df-recs 6358 df-irdg 6423 df-frec 6444 df-1o 6469 df-oadd 6473 df-er 6587 df-en 6795 df-dom 6796 df-fin 6797 df-sup 7043 df-inf 7044 df-pnf 8056 df-mnf 8057 df-xr 8058 df-ltxr 8059 df-le 8060 df-sub 8192 df-neg 8193 df-reap 8594 df-ap 8601 df-div 8692 df-inn 8983 df-2 9041 df-3 9042 df-4 9043 df-n0 9241 df-z 9318 df-uz 9593 df-q 9685 df-rp 9720 df-fz 10075 df-fzo 10209 df-fl 10339 df-mod 10394 df-seqfrec 10519 df-exp 10610 df-ihash 10847 df-cj 10986 df-re 10987 df-im 10988 df-rsqrt 11142 df-abs 11143 df-clim 11422 df-proddc 11694 df-dvds 11931 df-gcd 12080 df-odz 12348 df-phi 12349

This theorem is referenced by: odzphi 12384 pockthlem 12494

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