Theorem odzcllem 12273

Description: - Lemma for odzcl 12274, 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 12272	. . 3 |- ( ( N e. NN /\ A e. ZZ /\ ( A gcd N ) = 1 ) -> ( ( odZ ` N ) ` A ) = inf ( { n e. NN | N || ( ( A ^ n ) - 1 ) } , RR , < ) )
2		1zzd 9309	. . . 4 |- ( ( N e. NN /\ A e. ZZ /\ ( A gcd N ) = 1 ) -> 1 e. ZZ )
3		nnuz 9592	. . . . 5 |- NN = ( ZZ>= ` 1 )
4	3	rabeqi 2745	. . . 4 |- { n e. NN | N || ( ( A ^ n ) - 1 ) } = { n e. ( ZZ>= ` 1 ) | N || ( ( A ^ n ) - 1 ) }
5		oveq2 5903	. . . . . . 7 |- ( n = ( phi ` N ) -> ( A ^ n ) = ( A ^ ( phi ` N ) ) )
6	5	oveq1d 5910	. . . . . 6 |- ( n = ( phi ` N ) -> ( ( A ^ n ) - 1 ) = ( ( A ^ ( phi ` N ) ) - 1 ) )
7	6	breq2d 4030	. . . . 5 |- ( n = ( phi ` N ) -> ( N || ( ( A ^ n ) - 1 ) <-> N || ( ( A ^ ( phi ` N ) ) - 1 ) ) )
8		phicl 12246	. . . . . 6 |- ( N e. NN -> ( phi ` N ) e. NN )
9	8	3ad2ant1 1020	. . . . 5 |- ( ( N e. NN /\ A e. ZZ /\ ( A gcd N ) = 1 ) -> ( phi ` N ) e. NN )
10		eulerth 12264	. . . . . 6 |- ( ( N e. NN /\ A e. ZZ /\ ( A gcd N ) = 1 ) -> ( ( A ^ ( phi ` N ) ) mod N ) = ( 1 mod N ) )
11		simp1 999	. . . . . . 7 |- ( ( N e. NN /\ A e. ZZ /\ ( A gcd N ) = 1 ) -> N e. NN )
12		simp2 1000	. . . . . . . 8 |- ( ( N e. NN /\ A e. ZZ /\ ( A gcd N ) = 1 ) -> A e. ZZ )
13	9	nnnn0d 9258	. . . . . . . 8 |- ( ( N e. NN /\ A e. ZZ /\ ( A gcd N ) = 1 ) -> ( phi ` N ) e. NN0 )
14		zexpcl 10565	. . . . . . . 8 |- ( ( A e. ZZ /\ ( phi ` N ) e. NN0 ) -> ( A ^ ( phi ` N ) ) e. ZZ )
15	12, 13, 14	syl2anc 411	. . . . . . 7 |- ( ( N e. NN /\ A e. ZZ /\ ( A gcd N ) = 1 ) -> ( A ^ ( phi ` N ) ) e. ZZ )
16		1z 9308	. . . . . . . 8 |- 1 e. ZZ
17		moddvds 11837	. . . . . . . 8 |- ( ( N e. NN /\ ( A ^ ( phi ` N ) ) e. ZZ /\ 1 e. ZZ ) -> ( ( ( A ^ ( phi ` N ) ) mod N ) = ( 1 mod N ) <-> N || ( ( A ^ ( phi ` N ) ) - 1 ) ) )
18	16, 17	mp3an3 1337	. . . . . . 7 |- ( ( N e. NN /\ ( A ^ ( phi ` N ) ) e. ZZ ) -> ( ( ( A ^ ( phi ` N ) ) mod N ) = ( 1 mod N ) <-> N || ( ( A ^ ( phi ` N ) ) - 1 ) ) )
19	11, 15, 18	syl2anc 411	. . . . . 6 |- ( ( N e. NN /\ A e. ZZ /\ ( A gcd N ) = 1 ) -> ( ( ( A ^ ( phi ` N ) ) mod N ) = ( 1 mod N ) <-> N || ( ( A ^ ( phi ` N ) ) - 1 ) ) )
20	10, 19	mpbid 147	. . . . 5 |- ( ( N e. NN /\ A e. ZZ /\ ( A gcd N ) = 1 ) -> N || ( ( A ^ ( phi ` N ) ) - 1 ) )
21	7, 9, 20	elrabd 2910	. . . 4 |- ( ( N e. NN /\ A e. ZZ /\ ( A gcd N ) = 1 ) -> ( phi ` N ) e. { n e. NN | N || ( ( A ^ n ) - 1 ) } )
22		elfznn 10083	. . . . . . . . 9 |- ( n e. ( 1 ... ( phi ` N ) ) -> n e. NN )
23	22	adantl 277	. . . . . . . 8 |- ( ( ( N e. NN /\ A e. ZZ /\ ( A gcd N ) = 1 ) /\ n e. ( 1 ... ( phi ` N ) ) ) -> n e. NN )
24	23	nnnn0d 9258	. . . . . . 7 |- ( ( ( N e. NN /\ A e. ZZ /\ ( A gcd N ) = 1 ) /\ n e. ( 1 ... ( phi ` N ) ) ) -> n e. NN0 )
25		zexpcl 10565	. . . . . . 7 |- ( ( A e. ZZ /\ n e. NN0 ) -> ( A ^ n ) e. ZZ )
26	12, 24, 25	syl2an2r 595	. . . . . 6 |- ( ( ( N e. NN /\ A e. ZZ /\ ( A gcd N ) = 1 ) /\ n e. ( 1 ... ( phi ` N ) ) ) -> ( A ^ n ) e. ZZ )
27		peano2zm 9320	. . . . . 6 |- ( ( A ^ n ) e. ZZ -> ( ( A ^ n ) - 1 ) e. ZZ )
28	26, 27	syl 14	. . . . 5 |- ( ( ( N e. NN /\ A e. ZZ /\ ( A gcd N ) = 1 ) /\ n e. ( 1 ... ( phi ` N ) ) ) -> ( ( A ^ n ) - 1 ) e. ZZ )
29		dvdsdc 11836	. . . . 5 |- ( ( N e. NN /\ ( ( A ^ n ) - 1 ) e. ZZ ) -> DECID N || ( ( A ^ n ) - 1 ) )
30	11, 28, 29	syl2an2r 595	. . . 4 |- ( ( ( N e. NN /\ A e. ZZ /\ ( A gcd N ) = 1 ) /\ n e. ( 1 ... ( phi ` N ) ) ) -> DECID N || ( ( A ^ n ) - 1 ) )
31	2, 4, 21, 30	infssuzcldc 11983	. . 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 ) } )
32	1, 31	eqeltrd 2266	. 2 |- ( ( N e. NN /\ A e. ZZ /\ ( A gcd N ) = 1 ) -> ( ( odZ ` N ) ` A ) e. { n e. NN | N || ( ( A ^ n ) - 1 ) } )
33		oveq2 5903	. . . . 5 |- ( n = ( ( odZ ` N ) ` A ) -> ( A ^ n ) = ( A ^ ( ( odZ ` N ) ` A ) ) )
34	33	oveq1d 5910	. . . 4 |- ( n = ( ( odZ ` N ) ` A ) -> ( ( A ^ n ) - 1 ) = ( ( A ^ ( ( odZ ` N ) ` A ) ) - 1 ) )
35	34	breq2d 4030	. . 3 |- ( n = ( ( odZ ` N ) ` A ) -> ( N || ( ( A ^ n ) - 1 ) <-> N || ( ( A ^ ( ( odZ ` N ) ` A ) ) - 1 ) ) )
36	35	elrab 2908	. 2 |- ( ( ( odZ ` N ) ` A ) e. { n e. NN | N || ( ( A ^ n ) - 1 ) } <-> ( ( ( odZ ` N ) ` A ) e. NN /\ N || ( ( A ^ ( ( odZ ` N ) ` A ) ) - 1 ) ) )
37	32, 36	sylib 122	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   /\ wa 104   <-> wb 105  DECID wdc 835   /\ w3a 980   = wceq 1364   e. wcel 2160   {crab 2472   class class class wbr 4018   ` cfv 5235  (class class class)co 5895  infcinf 7011   RRcr 7839   1c1 7841   < clt 8021   - cmin 8157   NNcn 8948   NN0cn0 9205   ZZcz 9282   ZZ>=cuz 9557   ...cfz 10037   mod cmo 10352   ^cexp 10549   || cdvds 11825   gcd cgcd 11974   odZcodz 12239   phicphi 12240

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 2162  ax-14 2163  ax-ext 2171  ax-coll 4133  ax-sep 4136  ax-nul 4144  ax-pow 4192  ax-pr 4227  ax-un 4451  ax-setind 4554  ax-iinf 4605  ax-cnex 7931  ax-resscn 7932  ax-1cn 7933  ax-1re 7934  ax-icn 7935  ax-addcl 7936  ax-addrcl 7937  ax-mulcl 7938  ax-mulrcl 7939  ax-addcom 7940  ax-mulcom 7941  ax-addass 7942  ax-mulass 7943  ax-distr 7944  ax-i2m1 7945  ax-0lt1 7946  ax-1rid 7947  ax-0id 7948  ax-rnegex 7949  ax-precex 7950  ax-cnre 7951  ax-pre-ltirr 7952  ax-pre-ltwlin 7953  ax-pre-lttrn 7954  ax-pre-apti 7955  ax-pre-ltadd 7956  ax-pre-mulgt0 7957  ax-pre-mulext 7958  ax-arch 7959  ax-caucvg 7960

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 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ne 2361  df-nel 2456  df-ral 2473  df-rex 2474  df-reu 2475  df-rmo 2476  df-rab 2477  df-v 2754  df-sbc 2978  df-csb 3073  df-dif 3146  df-un 3148  df-in 3150  df-ss 3157  df-nul 3438  df-if 3550  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-int 3860  df-iun 3903  df-br 4019  df-opab 4080  df-mpt 4081  df-tr 4117  df-id 4311  df-po 4314  df-iso 4315  df-iord 4384  df-on 4386  df-ilim 4387  df-suc 4389  df-iom 4608  df-xp 4650  df-rel 4651  df-cnv 4652  df-co 4653  df-dm 4654  df-rn 4655  df-res 4656  df-ima 4657  df-iota 5196  df-fun 5237  df-fn 5238  df-f 5239  df-f1 5240  df-fo 5241  df-f1o 5242  df-fv 5243  df-isom 5244  df-riota 5851  df-ov 5898  df-oprab 5899  df-mpo 5900  df-1st 6164  df-2nd 6165  df-recs 6329  df-irdg 6394  df-frec 6415  df-1o 6440  df-oadd 6444  df-er 6558  df-en 6766  df-dom 6767  df-fin 6768  df-sup 7012  df-inf 7013  df-pnf 8023  df-mnf 8024  df-xr 8025  df-ltxr 8026  df-le 8027  df-sub 8159  df-neg 8160  df-reap 8561  df-ap 8568  df-div 8659  df-inn 8949  df-2 9007  df-3 9008  df-4 9009  df-n0 9206  df-z 9283  df-uz 9558  df-q 9649  df-rp 9683  df-fz 10038  df-fzo 10172  df-fl 10300  df-mod 10353  df-seqfrec 10476  df-exp 10550  df-ihash 10787  df-cj 10882  df-re 10883  df-im 10884  df-rsqrt 11038  df-abs 11039  df-clim 11318  df-proddc 11590  df-dvds 11826  df-gcd 11975  df-odz 12241  df-phi 12242

This theorem is referenced by:  odzcl  12274  odzid  12275