Theorem odzcllem 12598

Description: - Lemma for odzcl 12599, 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 12597	. . 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 9401	. . . 4 |- ( ( N e. NN /\ A e. ZZ /\ ( A gcd N ) = 1 ) -> 1 e. ZZ )
3		nnuz 9686	. . . . 5 |- NN = ( ZZ>= ` 1 )
4	3	rabeqi 2765	. . . 4 |- { n e. NN | N || ( ( A ^ n ) - 1 ) } = { n e. ( ZZ>= ` 1 ) | N || ( ( A ^ n ) - 1 ) }
5		oveq2 5954	. . . . . . 7 |- ( n = ( phi ` N ) -> ( A ^ n ) = ( A ^ ( phi ` N ) ) )
6	5	oveq1d 5961	. . . . . 6 |- ( n = ( phi ` N ) -> ( ( A ^ n ) - 1 ) = ( ( A ^ ( phi ` N ) ) - 1 ) )
7	6	breq2d 4057	. . . . 5 |- ( n = ( phi ` N ) -> ( N || ( ( A ^ n ) - 1 ) <-> N || ( ( A ^ ( phi ` N ) ) - 1 ) ) )
8		phicl 12570	. . . . . 6 |- ( N e. NN -> ( phi ` N ) e. NN )
9	8	3ad2ant1 1021	. . . . 5 |- ( ( N e. NN /\ A e. ZZ /\ ( A gcd N ) = 1 ) -> ( phi ` N ) e. NN )
10		eulerth 12588	. . . . . 6 |- ( ( N e. NN /\ A e. ZZ /\ ( A gcd N ) = 1 ) -> ( ( A ^ ( phi ` N ) ) mod N ) = ( 1 mod N ) )
11		simp1 1000	. . . . . . 7 |- ( ( N e. NN /\ A e. ZZ /\ ( A gcd N ) = 1 ) -> N e. NN )
12		simp2 1001	. . . . . . . 8 |- ( ( N e. NN /\ A e. ZZ /\ ( A gcd N ) = 1 ) -> A e. ZZ )
13	9	nnnn0d 9350	. . . . . . . 8 |- ( ( N e. NN /\ A e. ZZ /\ ( A gcd N ) = 1 ) -> ( phi ` N ) e. NN0 )
14		zexpcl 10701	. . . . . . . 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 9400	. . . . . . . 8 |- 1 e. ZZ
17		moddvds 12143	. . . . . . . 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 1339	. . . . . . 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 2931	. . . 4 |- ( ( N e. NN /\ A e. ZZ /\ ( A gcd N ) = 1 ) -> ( phi ` N ) e. { n e. NN | N || ( ( A ^ n ) - 1 ) } )
22		elfznn 10178	. . . . . . . . 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 9350	. . . . . . 7 |- ( ( ( N e. NN /\ A e. ZZ /\ ( A gcd N ) = 1 ) /\ n e. ( 1 ... ( phi ` N ) ) ) -> n e. NN0 )
25		zexpcl 10701	. . . . . . 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 9412	. . . . . 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 12142	. . . . 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 10380	. . 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 2282	. 2 |- ( ( N e. NN /\ A e. ZZ /\ ( A gcd N ) = 1 ) -> ( ( odZ ` N ) ` A ) e. { n e. NN | N || ( ( A ^ n ) - 1 ) } )
33		oveq2 5954	. . . . 5 |- ( n = ( ( odZ ` N ) ` A ) -> ( A ^ n ) = ( A ^ ( ( odZ ` N ) ` A ) ) )
34	33	oveq1d 5961	. . . 4 |- ( n = ( ( odZ ` N ) ` A ) -> ( ( A ^ n ) - 1 ) = ( ( A ^ ( ( odZ ` N ) ` A ) ) - 1 ) )
35	34	breq2d 4057	. . 3 |- ( n = ( ( odZ ` N ) ` A ) -> ( N || ( ( A ^ n ) - 1 ) <-> N || ( ( A ^ ( ( odZ ` N ) ` A ) ) - 1 ) ) )
36	35	elrab 2929	. 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 836   /\ w3a 981   = wceq 1373   e. wcel 2176   {crab 2488   class class class wbr 4045   ` cfv 5272  (class class class)co 5946  infcinf 7087   RRcr 7926   1c1 7928   < clt 8109   - cmin 8245   NNcn 9038   NN0cn0 9297   ZZcz 9374   ZZ>=cuz 9650   ...cfz 10132   mod cmo 10469   ^cexp 10685   || cdvds 12131   gcd cgcd 12307   odZcodz 12563   phicphi 12564

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 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-13 2178  ax-14 2179  ax-ext 2187  ax-coll 4160  ax-sep 4163  ax-nul 4171  ax-pow 4219  ax-pr 4254  ax-un 4481  ax-setind 4586  ax-iinf 4637  ax-cnex 8018  ax-resscn 8019  ax-1cn 8020  ax-1re 8021  ax-icn 8022  ax-addcl 8023  ax-addrcl 8024  ax-mulcl 8025  ax-mulrcl 8026  ax-addcom 8027  ax-mulcom 8028  ax-addass 8029  ax-mulass 8030  ax-distr 8031  ax-i2m1 8032  ax-0lt1 8033  ax-1rid 8034  ax-0id 8035  ax-rnegex 8036  ax-precex 8037  ax-cnre 8038  ax-pre-ltirr 8039  ax-pre-ltwlin 8040  ax-pre-lttrn 8041  ax-pre-apti 8042  ax-pre-ltadd 8043  ax-pre-mulgt0 8044  ax-pre-mulext 8045  ax-arch 8046  ax-caucvg 8047

This theorem depends on definitions:  df-bi 117  df-stab 833  df-dc 837  df-3or 982  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1484  df-sb 1786  df-eu 2057  df-mo 2058  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ne 2377  df-nel 2472  df-ral 2489  df-rex 2490  df-reu 2491  df-rmo 2492  df-rab 2493  df-v 2774  df-sbc 2999  df-csb 3094  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3461  df-if 3572  df-pw 3618  df-sn 3639  df-pr 3640  df-op 3642  df-uni 3851  df-int 3886  df-iun 3929  df-br 4046  df-opab 4107  df-mpt 4108  df-tr 4144  df-id 4341  df-po 4344  df-iso 4345  df-iord 4414  df-on 4416  df-ilim 4417  df-suc 4419  df-iom 4640  df-xp 4682  df-rel 4683  df-cnv 4684  df-co 4685  df-dm 4686  df-rn 4687  df-res 4688  df-ima 4689  df-iota 5233  df-fun 5274  df-fn 5275  df-f 5276  df-f1 5277  df-fo 5278  df-f1o 5279  df-fv 5280  df-isom 5281  df-riota 5901  df-ov 5949  df-oprab 5950  df-mpo 5951  df-1st 6228  df-2nd 6229  df-recs 6393  df-irdg 6458  df-frec 6479  df-1o 6504  df-oadd 6508  df-er 6622  df-en 6830  df-dom 6831  df-fin 6832  df-sup 7088  df-inf 7089  df-pnf 8111  df-mnf 8112  df-xr 8113  df-ltxr 8114  df-le 8115  df-sub 8247  df-neg 8248  df-reap 8650  df-ap 8657  df-div 8748  df-inn 9039  df-2 9097  df-3 9098  df-4 9099  df-n0 9298  df-z 9375  df-uz 9651  df-q 9743  df-rp 9778  df-fz 10133  df-fzo 10267  df-fl 10415  df-mod 10470  df-seqfrec 10595  df-exp 10686  df-ihash 10923  df-cj 11186  df-re 11187  df-im 11188  df-rsqrt 11342  df-abs 11343  df-clim 11623  df-proddc 11895  df-dvds 12132  df-gcd 12308  df-odz 12565  df-phi 12566

This theorem is referenced by:  odzcl  12599  odzid  12600