Theorem odzcllem 12781

Description: - Lemma for odzcl 12782, 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 12780	. . 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 9484	. . . 4 |- ( ( N e. NN /\ A e. ZZ /\ ( A gcd N ) = 1 ) -> 1 e. ZZ )
3		nnuz 9770	. . . . 5 |- NN = ( ZZ>= ` 1 )
4	3	rabeqi 2792	. . . 4 |- { n e. NN | N || ( ( A ^ n ) - 1 ) } = { n e. ( ZZ>= ` 1 ) | N || ( ( A ^ n ) - 1 ) }
5		oveq2 6015	. . . . . . 7 |- ( n = ( phi ` N ) -> ( A ^ n ) = ( A ^ ( phi ` N ) ) )
6	5	oveq1d 6022	. . . . . 6 |- ( n = ( phi ` N ) -> ( ( A ^ n ) - 1 ) = ( ( A ^ ( phi ` N ) ) - 1 ) )
7	6	breq2d 4095	. . . . 5 |- ( n = ( phi ` N ) -> ( N || ( ( A ^ n ) - 1 ) <-> N || ( ( A ^ ( phi ` N ) ) - 1 ) ) )
8		phicl 12753	. . . . . 6 |- ( N e. NN -> ( phi ` N ) e. NN )
9	8	3ad2ant1 1042	. . . . 5 |- ( ( N e. NN /\ A e. ZZ /\ ( A gcd N ) = 1 ) -> ( phi ` N ) e. NN )
10		eulerth 12771	. . . . . 6 |- ( ( N e. NN /\ A e. ZZ /\ ( A gcd N ) = 1 ) -> ( ( A ^ ( phi ` N ) ) mod N ) = ( 1 mod N ) )
11		simp1 1021	. . . . . . 7 |- ( ( N e. NN /\ A e. ZZ /\ ( A gcd N ) = 1 ) -> N e. NN )
12		simp2 1022	. . . . . . . 8 |- ( ( N e. NN /\ A e. ZZ /\ ( A gcd N ) = 1 ) -> A e. ZZ )
13	9	nnnn0d 9433	. . . . . . . 8 |- ( ( N e. NN /\ A e. ZZ /\ ( A gcd N ) = 1 ) -> ( phi ` N ) e. NN0 )
14		zexpcl 10788	. . . . . . . 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 9483	. . . . . . . 8 |- 1 e. ZZ
17		moddvds 12326	. . . . . . . 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 1360	. . . . . . 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 2961	. . . 4 |- ( ( N e. NN /\ A e. ZZ /\ ( A gcd N ) = 1 ) -> ( phi ` N ) e. { n e. NN | N || ( ( A ^ n ) - 1 ) } )
22		elfznn 10262	. . . . . . . . 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 9433	. . . . . . 7 |- ( ( ( N e. NN /\ A e. ZZ /\ ( A gcd N ) = 1 ) /\ n e. ( 1 ... ( phi ` N ) ) ) -> n e. NN0 )
25		zexpcl 10788	. . . . . . 7 |- ( ( A e. ZZ /\ n e. NN0 ) -> ( A ^ n ) e. ZZ )
26	12, 24, 25	syl2an2r 597	. . . . . 6 |- ( ( ( N e. NN /\ A e. ZZ /\ ( A gcd N ) = 1 ) /\ n e. ( 1 ... ( phi ` N ) ) ) -> ( A ^ n ) e. ZZ )
27		peano2zm 9495	. . . . . 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 12325	. . . . 5 |- ( ( N e. NN /\ ( ( A ^ n ) - 1 ) e. ZZ ) -> DECID N || ( ( A ^ n ) - 1 ) )
30	11, 28, 29	syl2an2r 597	. . . 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 10467	. . 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 2306	. 2 |- ( ( N e. NN /\ A e. ZZ /\ ( A gcd N ) = 1 ) -> ( ( odZ ` N ) ` A ) e. { n e. NN | N || ( ( A ^ n ) - 1 ) } )
33		oveq2 6015	. . . . 5 |- ( n = ( ( odZ ` N ) ` A ) -> ( A ^ n ) = ( A ^ ( ( odZ ` N ) ` A ) ) )
34	33	oveq1d 6022	. . . 4 |- ( n = ( ( odZ ` N ) ` A ) -> ( ( A ^ n ) - 1 ) = ( ( A ^ ( ( odZ ` N ) ` A ) ) - 1 ) )
35	34	breq2d 4095	. . 3 |- ( n = ( ( odZ ` N ) ` A ) -> ( N || ( ( A ^ n ) - 1 ) <-> N || ( ( A ^ ( ( odZ ` N ) ` A ) ) - 1 ) ) )
36	35	elrab 2959	. 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 839   /\ w3a 1002   = wceq 1395   e. wcel 2200   {crab 2512   class class class wbr 4083   ` cfv 5318  (class class class)co 6007  infcinf 7161   RRcr 8009   1c1 8011   < clt 8192   - cmin 8328   NNcn 9121   NN0cn0 9380   ZZcz 9457   ZZ>=cuz 9733   ...cfz 10216   mod cmo 10556   ^cexp 10772   || cdvds 12314   gcd cgcd 12490   odZcodz 12746   phicphi 12747

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-coll 4199  ax-sep 4202  ax-nul 4210  ax-pow 4258  ax-pr 4293  ax-un 4524  ax-setind 4629  ax-iinf 4680  ax-cnex 8101  ax-resscn 8102  ax-1cn 8103  ax-1re 8104  ax-icn 8105  ax-addcl 8106  ax-addrcl 8107  ax-mulcl 8108  ax-mulrcl 8109  ax-addcom 8110  ax-mulcom 8111  ax-addass 8112  ax-mulass 8113  ax-distr 8114  ax-i2m1 8115  ax-0lt1 8116  ax-1rid 8117  ax-0id 8118  ax-rnegex 8119  ax-precex 8120  ax-cnre 8121  ax-pre-ltirr 8122  ax-pre-ltwlin 8123  ax-pre-lttrn 8124  ax-pre-apti 8125  ax-pre-ltadd 8126  ax-pre-mulgt0 8127  ax-pre-mulext 8128  ax-arch 8129  ax-caucvg 8130

This theorem depends on definitions:  df-bi 117  df-stab 836  df-dc 840  df-3or 1003  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-nel 2496  df-ral 2513  df-rex 2514  df-reu 2515  df-rmo 2516  df-rab 2517  df-v 2801  df-sbc 3029  df-csb 3125  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-nul 3492  df-if 3603  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3889  df-int 3924  df-iun 3967  df-br 4084  df-opab 4146  df-mpt 4147  df-tr 4183  df-id 4384  df-po 4387  df-iso 4388  df-iord 4457  df-on 4459  df-ilim 4460  df-suc 4462  df-iom 4683  df-xp 4725  df-rel 4726  df-cnv 4727  df-co 4728  df-dm 4729  df-rn 4730  df-res 4731  df-ima 4732  df-iota 5278  df-fun 5320  df-fn 5321  df-f 5322  df-f1 5323  df-fo 5324  df-f1o 5325  df-fv 5326  df-isom 5327  df-riota 5960  df-ov 6010  df-oprab 6011  df-mpo 6012  df-1st 6292  df-2nd 6293  df-recs 6457  df-irdg 6522  df-frec 6543  df-1o 6568  df-oadd 6572  df-er 6688  df-en 6896  df-dom 6897  df-fin 6898  df-sup 7162  df-inf 7163  df-pnf 8194  df-mnf 8195  df-xr 8196  df-ltxr 8197  df-le 8198  df-sub 8330  df-neg 8331  df-reap 8733  df-ap 8740  df-div 8831  df-inn 9122  df-2 9180  df-3 9181  df-4 9182  df-n0 9381  df-z 9458  df-uz 9734  df-q 9827  df-rp 9862  df-fz 10217  df-fzo 10351  df-fl 10502  df-mod 10557  df-seqfrec 10682  df-exp 10773  df-ihash 11010  df-cj 11369  df-re 11370  df-im 11371  df-rsqrt 11525  df-abs 11526  df-clim 11806  df-proddc 12078  df-dvds 12315  df-gcd 12491  df-odz 12748  df-phi 12749

This theorem is referenced by:  odzcl  12782  odzid  12783