Theorem odzcllem 15497

Description: - Lemma for odzcl 15498, showing existence of a recurrent point for the exponential. (Contributed by Mario Carneiro, 28-Feb-2014.) (Proof shortened by AV, 26-Sep-2020.)

Assertion

Ref	Expression
odzcllem	|- ( ( N e. NN /\ A e. ZZ /\ ( A gcd N ) = 1 ) -> ( ( ( odZ ` N ) ` A ) e. NN /\ N || ( ( A ^ ( ( odZ ` N ) ` A ) ) - 1 ) ) )

Proof of Theorem odzcllem

Dummy variable n is distinct from all other variables.

Step	Hyp	Ref	Expression
1		odzval 15496	. . 3 |- ( ( N e. NN /\ A e. ZZ /\ ( A gcd N ) = 1 ) -> ( ( odZ ` N ) ` A ) = inf ( { n e. NN | N || ( ( A ^ n ) - 1 ) } , RR , < ) )
2		ssrab2 3687	. . . . 5 |- { n e. NN | N || ( ( A ^ n ) - 1 ) } C_ NN
3		nnuz 11723	. . . . 5 |- NN = ( ZZ>= ` 1 )
4	2, 3	sseqtri 3637	. . . 4 |- { n e. NN | N || ( ( A ^ n ) - 1 ) } C_ ( ZZ>= ` 1 )
5		phicl 15474	. . . . . . 7 |- ( N e. NN -> ( phi ` N ) e. NN )
6	5	3ad2ant1 1082	. . . . . 6 |- ( ( N e. NN /\ A e. ZZ /\ ( A gcd N ) = 1 ) -> ( phi ` N ) e. NN )
7		eulerth 15488	. . . . . . 7 |- ( ( N e. NN /\ A e. ZZ /\ ( A gcd N ) = 1 ) -> ( ( A ^ ( phi ` N ) ) mod N ) = ( 1 mod N ) )
8		simp1 1061	. . . . . . . 8 |- ( ( N e. NN /\ A e. ZZ /\ ( A gcd N ) = 1 ) -> N e. NN )
9		simp2 1062	. . . . . . . . 9 |- ( ( N e. NN /\ A e. ZZ /\ ( A gcd N ) = 1 ) -> A e. ZZ )
10	6	nnnn0d 11351	. . . . . . . . 9 |- ( ( N e. NN /\ A e. ZZ /\ ( A gcd N ) = 1 ) -> ( phi ` N ) e. NN0 )
11		zexpcl 12875	. . . . . . . . 9 |- ( ( A e. ZZ /\ ( phi ` N ) e. NN0 ) -> ( A ^ ( phi ` N ) ) e. ZZ )
12	9, 10, 11	syl2anc 693	. . . . . . . 8 |- ( ( N e. NN /\ A e. ZZ /\ ( A gcd N ) = 1 ) -> ( A ^ ( phi ` N ) ) e. ZZ )
13		1z 11407	. . . . . . . . 9 |- 1 e. ZZ
14		moddvds 14991	. . . . . . . . 9 |- ( ( N e. NN /\ ( A ^ ( phi ` N ) ) e. ZZ /\ 1 e. ZZ ) -> ( ( ( A ^ ( phi ` N ) ) mod N ) = ( 1 mod N ) <-> N || ( ( A ^ ( phi ` N ) ) - 1 ) ) )
15	13, 14	mp3an3 1413	. . . . . . . 8 |- ( ( N e. NN /\ ( A ^ ( phi ` N ) ) e. ZZ ) -> ( ( ( A ^ ( phi ` N ) ) mod N ) = ( 1 mod N ) <-> N || ( ( A ^ ( phi ` N ) ) - 1 ) ) )
16	8, 12, 15	syl2anc 693	. . . . . . 7 |- ( ( N e. NN /\ A e. ZZ /\ ( A gcd N ) = 1 ) -> ( ( ( A ^ ( phi ` N ) ) mod N ) = ( 1 mod N ) <-> N || ( ( A ^ ( phi ` N ) ) - 1 ) ) )
17	7, 16	mpbid 222	. . . . . 6 |- ( ( N e. NN /\ A e. ZZ /\ ( A gcd N ) = 1 ) -> N || ( ( A ^ ( phi ` N ) ) - 1 ) )
18		oveq2 6658	. . . . . . . . 9 |- ( n = ( phi ` N ) -> ( A ^ n ) = ( A ^ ( phi ` N ) ) )
19	18	oveq1d 6665	. . . . . . . 8 |- ( n = ( phi ` N ) -> ( ( A ^ n ) - 1 ) = ( ( A ^ ( phi ` N ) ) - 1 ) )
20	19	breq2d 4665	. . . . . . 7 |- ( n = ( phi ` N ) -> ( N || ( ( A ^ n ) - 1 ) <-> N || ( ( A ^ ( phi ` N ) ) - 1 ) ) )
21	20	rspcev 3309	. . . . . 6 |- ( ( ( phi ` N ) e. NN /\ N || ( ( A ^ ( phi ` N ) ) - 1 ) ) -> E. n e. NN N || ( ( A ^ n ) - 1 ) )
22	6, 17, 21	syl2anc 693	. . . . 5 |- ( ( N e. NN /\ A e. ZZ /\ ( A gcd N ) = 1 ) -> E. n e. NN N || ( ( A ^ n ) - 1 ) )
23		rabn0 3958	. . . . 5 |- ( { n e. NN | N || ( ( A ^ n ) - 1 ) } =/= (/) <-> E. n e. NN N || ( ( A ^ n ) - 1 ) )
24	22, 23	sylibr 224	. . . 4 |- ( ( N e. NN /\ A e. ZZ /\ ( A gcd N ) = 1 ) -> { n e. NN | N || ( ( A ^ n ) - 1 ) } =/= (/) )
25		infssuzcl 11772	. . . 4 |- ( ( { n e. NN | N || ( ( A ^ n ) - 1 ) } C_ ( ZZ>= ` 1 ) /\ { n e. NN | N || ( ( A ^ n ) - 1 ) } =/= (/) ) -> inf ( { n e. NN | N || ( ( A ^ n ) - 1 ) } , RR , < ) e. { n e. NN | N || ( ( A ^ n ) - 1 ) } )
26	4, 24, 25	sylancr 695	. . 3 |- ( ( N e. NN /\ A e. ZZ /\ ( A gcd N ) = 1 ) -> inf ( { n e. NN | N || ( ( A ^ n ) - 1 ) } , RR , < ) e. { n e. NN | N || ( ( A ^ n ) - 1 ) } )
27	1, 26	eqeltrd 2701	. 2 |- ( ( N e. NN /\ A e. ZZ /\ ( A gcd N ) = 1 ) -> ( ( odZ ` N ) ` A ) e. { n e. NN | N || ( ( A ^ n ) - 1 ) } )
28		oveq2 6658	. . . . 5 |- ( n = ( ( odZ ` N ) ` A ) -> ( A ^ n ) = ( A ^ ( ( odZ ` N ) ` A ) ) )
29	28	oveq1d 6665	. . . 4 |- ( n = ( ( odZ ` N ) ` A ) -> ( ( A ^ n ) - 1 ) = ( ( A ^ ( ( odZ ` N ) ` A ) ) - 1 ) )
30	29	breq2d 4665	. . 3 |- ( n = ( ( odZ ` N ) ` A ) -> ( N || ( ( A ^ n ) - 1 ) <-> N || ( ( A ^ ( ( odZ ` N ) ` A ) ) - 1 ) ) )
31	30	elrab 3363	. 2 |- ( ( ( odZ ` N ) ` A ) e. { n e. NN | N || ( ( A ^ n ) - 1 ) } <-> ( ( ( odZ ` N ) ` A ) e. NN /\ N || ( ( A ^ ( ( odZ ` N ) ` A ) ) - 1 ) ) )
32	27, 31	sylib 208	1 |- ( ( N e. NN /\ A e. ZZ /\ ( A gcd N ) = 1 ) -> ( ( ( odZ ` N ) ` A ) e. NN /\ N || ( ( A ^ ( ( odZ ` N ) ` A ) ) - 1 ) ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   =/= wne 2794   E.wrex 2913   {crab 2916   C_ wss 3574   (/)c0 3915   class class class wbr 4653   ` cfv 5888  (class class class)co 6650  infcinf 8347   RRcr 9935   1c1 9937   < clt 10074   - cmin 10266   NNcn 11020   NN0cn0 11292   ZZcz 11377   ZZ>=cuz 11687   mod cmo 12668   ^cexp 12860   || cdvds 14983   gcd cgcd 15216   odZcodz 15468   phicphi 15469

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013  ax-pre-sup 10014

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-oadd 7564  df-er 7742  df-map 7859  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-sup 8348  df-inf 8349  df-card 8765  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-div 10685  df-nn 11021  df-2 11079  df-3 11080  df-n0 11293  df-xnn0 11364  df-z 11378  df-uz 11688  df-rp 11833  df-fz 12327  df-fzo 12466  df-fl 12593  df-mod 12669  df-seq 12802  df-exp 12861  df-hash 13118  df-cj 13839  df-re 13840  df-im 13841  df-sqrt 13975  df-abs 13976  df-dvds 14984  df-gcd 15217  df-odz 15470  df-phi 15471

This theorem is referenced by:  odzcl  15498  odzid  15499