odzdvds - Intuitionistic Logic Explorer

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Theorem odzdvds 12173

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 9207	. . . . . . . . . 10
2		zq 9560	. . . . . . . . . 10
3	1, 2	syl 14	. . . . . . . . 9
4	3	adantl 275	. . . . . . . 8 $/\$ $/\$ $/\$
5		odzcl 12171	. . . . . . . . . 10 $/\$ $/\$
6	5	adantr 274	. . . . . . . . 9 $/\$ $/\$ $/\$
7		nnq 9567	. . . . . . . . 9
8	6, 7	syl 14	. . . . . . . 8 $/\$ $/\$ $/\$
9	6	nngt0d 8897	. . . . . . . 8 $/\$ $/\$ $/\$
10		modqlt 10264	. . . . . . . 8 $/\$ $/\$
11	4, 8, 9, 10	syl3anc 1228	. . . . . . 7 $/\$ $/\$ $/\$
12	1	adantl 275	. . . . . . . . . 10 $/\$ $/\$ $/\$
13	12, 6	zmodcld 10276	. . . . . . . . 9 $/\$ $/\$ $/\$
14	13	nn0zd 9307	. . . . . . . 8 $/\$ $/\$ $/\$
15	6	nnzd 9308	. . . . . . . 8 $/\$ $/\$ $/\$
16		zltnle 9233	. . . . . . . 8 $/\$
17	14, 15, 16	syl2anc 409	. . . . . . 7 $/\$ $/\$ $/\$
18	11, 17	mpbid 146	. . . . . 6 $/\$ $/\$ $/\$
19		1zzd 9214	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ $/\$
20		nnuz 9497	. . . . . . . . . . 11
21	20	rabeqi 2718	. . . . . . . . . 10
22		oveq2 5849	. . . . . . . . . . . . . . 15
23	22	oveq1d 5856	. . . . . . . . . . . . . 14
24	23	breq2d 3993	. . . . . . . . . . . . 13
25	24	elrab 2881	. . . . . . . . . . . 12 $/\$
26	25	biimpri 132	. . . . . . . . . . 11 $/\$
27	26	adantl 275	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ $/\$
28		simp1 987	. . . . . . . . . . . 12 $/\$ $/\$
29	28	ad3antrrr 484	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
30		simp2 988	. . . . . . . . . . . . . 14 $/\$ $/\$
31	30	ad3antrrr 484	. . . . . . . . . . . . 13 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
32		elfznn 9985	. . . . . . . . . . . . . . 15
33	32	nnnn0d 9163	. . . . . . . . . . . . . 14
34	33	adantl 275	. . . . . . . . . . . . 13 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
35		zexpcl 10466	. . . . . . . . . . . . 13 $/\$
36	31, 34, 35	syl2anc 409	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
37		peano2zm 9225	. . . . . . . . . . . 12
38	36, 37	syl 14	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
39		dvdsdc 11734	. . . . . . . . . . 11 $/\$ DECID
40	29, 38, 39	syl2anc 409	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ DECID
41	19, 21, 27, 40	infssuzledc 11879	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$ inf
42	41	ex 114	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ inf
43	42	ancomsd 267	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ inf
44		odzval 12169	. . . . . . . . 9 $/\$ $/\$ inf
45	44	adantr 274	. . . . . . . 8 $/\$ $/\$ $/\$ inf
46	45	breq1d 3991	. . . . . . 7 $/\$ $/\$ $/\$ inf
47	43, 46	sylibrd 168	. . . . . 6 $/\$ $/\$ $/\$ $/\$
48	18, 47	mtod 653	. . . . 5 $/\$ $/\$ $/\$ $/\$
49		imnan 680	. . . . 5 $/\$
50	48, 49	sylibr 133	. . . 4 $/\$ $/\$ $/\$
51		elnn0 9112	. . . . . 6 $\/$
52	13, 51	sylib 121	. . . . 5 $/\$ $/\$ $/\$ $\/$
53	52	ord 714	. . . 4 $/\$ $/\$ $/\$
54	50, 53	syld 45	. . 3 $/\$ $/\$ $/\$
55		simpl1 990	. . . . . . 7 $/\$ $/\$ $/\$
56	55	nnzd 9308	. . . . . 6 $/\$ $/\$ $/\$
57		dvds0 11742	. . . . . 6
58	56, 57	syl 14	. . . . 5 $/\$ $/\$ $/\$
59		simpl2 991	. . . . . . . . 9 $/\$ $/\$ $/\$
60	59	zcnd 9310	. . . . . . . 8 $/\$ $/\$ $/\$
61	60	exp0d 10578	. . . . . . 7 $/\$ $/\$ $/\$
62	61	oveq1d 5856	. . . . . 6 $/\$ $/\$ $/\$
63		1m1e0 8922	. . . . . 6
64	62, 63	eqtrdi 2214	. . . . 5 $/\$ $/\$ $/\$
65	58, 64	breqtrrd 4009	. . . 4 $/\$ $/\$ $/\$
66		oveq2 5849	. . . . . 6
67	66	oveq1d 5856	. . . . 5
68	67	breq2d 3993	. . . 4
69	65, 68	syl5ibrcom 156	. . 3 $/\$ $/\$ $/\$
70	54, 69	impbid 128	. 2 $/\$ $/\$ $/\$
71	6	nnnn0d 9163	. . . . . . . . 9 $/\$ $/\$ $/\$
72		znq 9558	. . . . . . . . . . 11 $/\$
73	12, 6, 72	syl2anc 409	. . . . . . . . . 10 $/\$ $/\$ $/\$
74		nn0ge0 9135	. . . . . . . . . . . 12
75	74	adantl 275	. . . . . . . . . . 11 $/\$ $/\$ $/\$
76		nn0re 9119	. . . . . . . . . . . . 13
77	76	adantl 275	. . . . . . . . . . . 12 $/\$ $/\$ $/\$
78	6	nnred 8866	. . . . . . . . . . . 12 $/\$ $/\$ $/\$
79		ge0div 8762	. . . . . . . . . . . 12 $/\$ $/\$
80	77, 78, 9, 79	syl3anc 1228	. . . . . . . . . . 11 $/\$ $/\$ $/\$
81	75, 80	mpbid 146	. . . . . . . . . 10 $/\$ $/\$ $/\$
82		flqge0nn0 10224	. . . . . . . . . 10 $/\$
83	73, 81, 82	syl2anc 409	. . . . . . . . 9 $/\$ $/\$ $/\$
84	71, 83	nn0mulcld 9168	. . . . . . . 8 $/\$ $/\$ $/\$
85		zexpcl 10466	. . . . . . . 8 $/\$
86	59, 84, 85	syl2anc 409	. . . . . . 7 $/\$ $/\$ $/\$
87		zq 9560	. . . . . . 7
88	86, 87	syl 14	. . . . . 6 $/\$ $/\$ $/\$
89		1z 9213	. . . . . . 7
90		zq 9560	. . . . . . 7
91	89, 90	mp1i 10	. . . . . 6 $/\$ $/\$ $/\$
92		zexpcl 10466	. . . . . . 7 $/\$
93	59, 13, 92	syl2anc 409	. . . . . 6 $/\$ $/\$ $/\$
94		nnq 9567	. . . . . . 7
95	55, 94	syl 14	. . . . . 6 $/\$ $/\$ $/\$
96	55	nngt0d 8897	. . . . . 6 $/\$ $/\$ $/\$
97	60, 83, 71	expmuld 10587	. . . . . . . 8 $/\$ $/\$ $/\$
98	97	oveq1d 5856	. . . . . . 7 $/\$ $/\$ $/\$
99		zexpcl 10466	. . . . . . . . 9 $/\$
100	59, 71, 99	syl2anc 409	. . . . . . . 8 $/\$ $/\$ $/\$
101		1zzd 9214	. . . . . . . 8 $/\$ $/\$ $/\$
102		odzid 12172	. . . . . . . . . 10 $/\$ $/\$
103	102	adantr 274	. . . . . . . . 9 $/\$ $/\$ $/\$
104		moddvds 11735	. . . . . . . . . 10 $/\$ $/\$
105	55, 100, 101, 104	syl3anc 1228	. . . . . . . . 9 $/\$ $/\$ $/\$
106	103, 105	mpbird 166	. . . . . . . 8 $/\$ $/\$ $/\$
107	100, 101, 83, 95, 96, 106	modqexp 10577	. . . . . . 7 $/\$ $/\$ $/\$
108	73	flqcld 10208	. . . . . . . . 9 $/\$ $/\$ $/\$
109		1exp 10480	. . . . . . . . 9
110	108, 109	syl 14	. . . . . . . 8 $/\$ $/\$ $/\$
111	110	oveq1d 5856	. . . . . . 7 $/\$ $/\$ $/\$
112	98, 107, 111	3eqtrd 2202	. . . . . 6 $/\$ $/\$ $/\$
113	88, 91, 93, 95, 96, 112	modqmul1 10308	. . . . 5 $/\$ $/\$ $/\$
114	60, 13, 84	expaddd 10586	. . . . . . 7 $/\$ $/\$ $/\$
115		modqval 10255	. . . . . . . . . . 11 $/\$ $/\$
116	4, 8, 9, 115	syl3anc 1228	. . . . . . . . . 10 $/\$ $/\$ $/\$
117	116	oveq2d 5857	. . . . . . . . 9 $/\$ $/\$ $/\$
118	84	nn0cnd 9165	. . . . . . . . . 10 $/\$ $/\$ $/\$
119	77	recnd 7923	. . . . . . . . . 10 $/\$ $/\$ $/\$
120	118, 119	pncan3d 8208	. . . . . . . . 9 $/\$ $/\$ $/\$
121	117, 120	eqtrd 2198	. . . . . . . 8 $/\$ $/\$ $/\$
122	121	oveq2d 5857	. . . . . . 7 $/\$ $/\$ $/\$
123	114, 122	eqtr3d 2200	. . . . . 6 $/\$ $/\$ $/\$
124	123	oveq1d 5856	. . . . 5 $/\$ $/\$ $/\$
125	93	zcnd 9310	. . . . . . 7 $/\$ $/\$ $/\$
126	125	mulid2d 7913	. . . . . 6 $/\$ $/\$ $/\$
127	126	oveq1d 5856	. . . . 5 $/\$ $/\$ $/\$
128	113, 124, 127	3eqtr3d 2206	. . . 4 $/\$ $/\$ $/\$
129	128	eqeq1d 2174	. . 3 $/\$ $/\$ $/\$
130		zexpcl 10466	. . . . 5 $/\$
131	59, 130	sylancom 417	. . . 4 $/\$ $/\$ $/\$
132		moddvds 11735	. . . 4 $/\$ $/\$
133	55, 131, 101, 132	syl3anc 1228	. . 3 $/\$ $/\$ $/\$
134		moddvds 11735	. . . 4 $/\$ $/\$
135	55, 93, 101, 134	syl3anc 1228	. . 3 $/\$ $/\$ $/\$
136	129, 133, 135	3bitr3d 217	. 2 $/\$ $/\$ $/\$
137		dvdsval3 11727	. . 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 698 DECID wdc 824 $/\$ w3a 968

wceq 1343

wcel 2136

crab 2447 class class class wbr 3981

cfv 5187 (class class class)co 5841 infcinf 6944

cr 7748

cc0 7749

c1 7750

caddc 7752

cmul 7754

clt 7929

cle 7930

cmin 8065

cdiv 8564

cn 8853

cn0 9110

cz 9187

cuz 9462

cq 9553

cfz 9940

cfl 10199

cmo 10253

cexp 10450

cdvds 11723

cgcd 11871

codz 12136

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 604 ax-in2 605 ax-io 699 ax-5 1435 ax-7 1436 ax-gen 1437 ax-ie1 1481 ax-ie2 1482 ax-8 1492 ax-10 1493 ax-11 1494 ax-i12 1495 ax-bndl 1497 ax-4 1498 ax-17 1514 ax-i9 1518 ax-ial 1522 ax-i5r 1523 ax-13 2138 ax-14 2139 ax-ext 2147 ax-coll 4096 ax-sep 4099 ax-nul 4107 ax-pow 4152 ax-pr 4186 ax-un 4410 ax-setind 4513 ax-iinf 4564 ax-cnex 7840 ax-resscn 7841 ax-1cn 7842 ax-1re 7843 ax-icn 7844 ax-addcl 7845 ax-addrcl 7846 ax-mulcl 7847 ax-mulrcl 7848 ax-addcom 7849 ax-mulcom 7850 ax-addass 7851 ax-mulass 7852 ax-distr 7853 ax-i2m1 7854 ax-0lt1 7855 ax-1rid 7856 ax-0id 7857 ax-rnegex 7858 ax-precex 7859 ax-cnre 7860 ax-pre-ltirr 7861 ax-pre-ltwlin 7862 ax-pre-lttrn 7863 ax-pre-apti 7864 ax-pre-ltadd 7865 ax-pre-mulgt0 7866 ax-pre-mulext 7867 ax-arch 7868 ax-caucvg 7869

This theorem depends on definitions: df-bi 116 df-stab 821 df-dc 825 df-3or 969 df-3an 970 df-tru 1346 df-fal 1349 df-nf 1449 df-sb 1751 df-eu 2017 df-mo 2018 df-clab 2152 df-cleq 2158 df-clel 2161 df-nfc 2296 df-ne 2336 df-nel 2431 df-ral 2448 df-rex 2449 df-reu 2450 df-rmo 2451 df-rab 2452 df-v 2727 df-sbc 2951 df-csb 3045 df-dif 3117 df-un 3119 df-in 3121 df-ss 3128 df-nul 3409 df-if 3520 df-pw 3560 df-sn 3581 df-pr 3582 df-op 3584 df-uni 3789 df-int 3824 df-iun 3867 df-br 3982 df-opab 4043 df-mpt 4044 df-tr 4080 df-id 4270 df-po 4273 df-iso 4274 df-iord 4343 df-on 4345 df-ilim 4346 df-suc 4348 df-iom 4567 df-xp 4609 df-rel 4610 df-cnv 4611 df-co 4612 df-dm 4613 df-rn 4614 df-res 4615 df-ima 4616 df-iota 5152 df-fun 5189 df-fn 5190 df-f 5191 df-f1 5192 df-fo 5193 df-f1o 5194 df-fv 5195 df-isom 5196 df-riota 5797 df-ov 5844 df-oprab 5845 df-mpo 5846 df-1st 6105 df-2nd 6106 df-recs 6269 df-irdg 6334 df-frec 6355 df-1o 6380 df-oadd 6384 df-er 6497 df-en 6703 df-dom 6704 df-fin 6705 df-sup 6945 df-inf 6946 df-pnf 7931 df-mnf 7932 df-xr 7933 df-ltxr 7934 df-le 7935 df-sub 8067 df-neg 8068 df-reap 8469 df-ap 8476 df-div 8565 df-inn 8854 df-2 8912 df-3 8913 df-4 8914 df-n0 9111 df-z 9188 df-uz 9463 df-q 9554 df-rp 9586 df-fz 9941 df-fzo 10074 df-fl 10201 df-mod 10254 df-seqfrec 10377 df-exp 10451 df-ihash 10685 df-cj 10780 df-re 10781 df-im 10782 df-rsqrt 10936 df-abs 10937 df-clim 11216 df-proddc 11488 df-dvds 11724 df-gcd 11872 df-odz 12138 df-phi 12139

This theorem is referenced by: odzphi 12174 pockthlem 12282

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