Theorem odzcllem 12940

Description: - Lemma for odzcl 12941, 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 12939	. . 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 9604	. . . 4 |- ( ( N e. NN /\ A e. ZZ /\ ( A gcd N ) = 1 ) -> 1 e. ZZ )
3		nnuz 9890	. . . . 5 |- NN = ( ZZ>= ` 1 )
4	3	rabeqi 2806	. . . 4 |- { n e. NN | N || ( ( A ^ n ) - 1 ) } = { n e. ( ZZ>= ` 1 ) | N || ( ( A ^ n ) - 1 ) }
5		oveq2 6058	. . . . . . 7 |- ( n = ( phi ` N ) -> ( A ^ n ) = ( A ^ ( phi ` N ) ) )
6	5	oveq1d 6065	. . . . . 6 |- ( n = ( phi ` N ) -> ( ( A ^ n ) - 1 ) = ( ( A ^ ( phi ` N ) ) - 1 ) )
7	6	breq2d 4121	. . . . 5 |- ( n = ( phi ` N ) -> ( N || ( ( A ^ n ) - 1 ) <-> N || ( ( A ^ ( phi ` N ) ) - 1 ) ) )
8		phicl 12912	. . . . . 6 |- ( N e. NN -> ( phi ` N ) e. NN )
9	8	3ad2ant1 1045	. . . . 5 |- ( ( N e. NN /\ A e. ZZ /\ ( A gcd N ) = 1 ) -> ( phi ` N ) e. NN )
10		eulerth 12930	. . . . . 6 |- ( ( N e. NN /\ A e. ZZ /\ ( A gcd N ) = 1 ) -> ( ( A ^ ( phi ` N ) ) mod N ) = ( 1 mod N ) )
11		simp1 1024	. . . . . . 7 |- ( ( N e. NN /\ A e. ZZ /\ ( A gcd N ) = 1 ) -> N e. NN )
12		simp2 1025	. . . . . . . 8 |- ( ( N e. NN /\ A e. ZZ /\ ( A gcd N ) = 1 ) -> A e. ZZ )
13	9	nnnn0d 9553	. . . . . . . 8 |- ( ( N e. NN /\ A e. ZZ /\ ( A gcd N ) = 1 ) -> ( phi ` N ) e. NN0 )
14		zexpcl 10916	. . . . . . . 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 9603	. . . . . . . 8 |- 1 e. ZZ
17		moddvds 12485	. . . . . . . 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 1363	. . . . . . 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 2975	. . . 4 |- ( ( N e. NN /\ A e. ZZ /\ ( A gcd N ) = 1 ) -> ( phi ` N ) e. { n e. NN | N || ( ( A ^ n ) - 1 ) } )
22		elfznn 10388	. . . . . . . . 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 9553	. . . . . . 7 |- ( ( ( N e. NN /\ A e. ZZ /\ ( A gcd N ) = 1 ) /\ n e. ( 1 ... ( phi ` N ) ) ) -> n e. NN0 )
25		zexpcl 10916	. . . . . . 7 |- ( ( A e. ZZ /\ n e. NN0 ) -> ( A ^ n ) e. ZZ )
26	12, 24, 25	syl2an2r 599	. . . . . 6 |- ( ( ( N e. NN /\ A e. ZZ /\ ( A gcd N ) = 1 ) /\ n e. ( 1 ... ( phi ` N ) ) ) -> ( A ^ n ) e. ZZ )
27		peano2zm 9615	. . . . . 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 12484	. . . . 5 |- ( ( N e. NN /\ ( ( A ^ n ) - 1 ) e. ZZ ) -> DECID N || ( ( A ^ n ) - 1 ) )
30	11, 28, 29	syl2an2r 599	. . . 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 10595	. . 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 2309	. 2 |- ( ( N e. NN /\ A e. ZZ /\ ( A gcd N ) = 1 ) -> ( ( odZ ` N ) ` A ) e. { n e. NN | N || ( ( A ^ n ) - 1 ) } )
33		oveq2 6058	. . . . 5 |- ( n = ( ( odZ ` N ) ` A ) -> ( A ^ n ) = ( A ^ ( ( odZ ` N ) ` A ) ) )
34	33	oveq1d 6065	. . . 4 |- ( n = ( ( odZ ` N ) ` A ) -> ( ( A ^ n ) - 1 ) = ( ( A ^ ( ( odZ ` N ) ` A ) ) - 1 ) )
35	34	breq2d 4121	. . 3 |- ( n = ( ( odZ ` N ) ` A ) -> ( N || ( ( A ^ n ) - 1 ) <-> N || ( ( A ^ ( ( odZ ` N ) ` A ) ) - 1 ) ) )
36	35	elrab 2973	. 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 842   /\ w3a 1005   = wceq 1398   e. wcel 2203   {crab 2524   class class class wbr 4109   ` cfv 5352  (class class class)co 6050  infcinf 7274   RRcr 8126   1c1 8128   < clt 8308   - cmin 8444   NNcn 9237   NN0cn0 9496   ZZcz 9577   ZZ>=cuz 9853   ...cfz 10342   mod cmo 10684   ^cexp 10900   || cdvds 12473   gcd cgcd 12649   odZcodz 12905   phicphi 12906

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2205  ax-14 2206  ax-ext 2214  ax-coll 4225  ax-sep 4228  ax-nul 4236  ax-pow 4287  ax-pr 4322  ax-un 4554  ax-setind 4659  ax-iinf 4710  ax-cnex 8218  ax-resscn 8219  ax-1cn 8220  ax-1re 8221  ax-icn 8222  ax-addcl 8223  ax-addrcl 8224  ax-mulcl 8225  ax-mulrcl 8226  ax-addcom 8227  ax-mulcom 8228  ax-addass 8229  ax-mulass 8230  ax-distr 8231  ax-i2m1 8232  ax-0lt1 8233  ax-1rid 8234  ax-0id 8235  ax-rnegex 8236  ax-precex 8237  ax-cnre 8238  ax-pre-ltirr 8239  ax-pre-ltwlin 8240  ax-pre-lttrn 8241  ax-pre-apti 8242  ax-pre-ltadd 8243  ax-pre-mulgt0 8244  ax-pre-mulext 8245  ax-arch 8246  ax-caucvg 8247

This theorem depends on definitions:  df-bi 117  df-stab 839  df-dc 843  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ne 2413  df-nel 2508  df-ral 2525  df-rex 2526  df-reu 2527  df-rmo 2528  df-rab 2529  df-v 2815  df-sbc 3043  df-csb 3139  df-dif 3213  df-un 3215  df-in 3217  df-ss 3224  df-nul 3509  df-if 3621  df-pw 3671  df-sn 3695  df-pr 3696  df-op 3698  df-uni 3915  df-int 3950  df-iun 3993  df-br 4110  df-opab 4172  df-mpt 4173  df-tr 4209  df-id 4414  df-po 4417  df-iso 4418  df-iord 4487  df-on 4489  df-ilim 4490  df-suc 4492  df-iom 4713  df-xp 4755  df-rel 4756  df-cnv 4757  df-co 4758  df-dm 4759  df-rn 4760  df-res 4761  df-ima 4762  df-iota 5312  df-fun 5354  df-fn 5355  df-f 5356  df-f1 5357  df-fo 5358  df-f1o 5359  df-fv 5360  df-isom 5361  df-riota 6003  df-ov 6053  df-oprab 6054  df-mpo 6055  df-1st 6334  df-2nd 6335  df-recs 6536  df-irdg 6601  df-frec 6622  df-1o 6647  df-oadd 6651  df-er 6767  df-en 6976  df-dom 6977  df-fin 6978  df-sup 7275  df-inf 7276  df-pnf 8310  df-mnf 8311  df-xr 8312  df-ltxr 8313  df-le 8314  df-sub 8446  df-neg 8447  df-reap 8849  df-ap 8856  df-div 8947  df-inn 9238  df-2 9296  df-3 9297  df-4 9298  df-n0 9497  df-z 9578  df-uz 9854  df-q 9952  df-rp 9987  df-fz 10343  df-fzo 10477  df-fl 10630  df-mod 10685  df-seqfrec 10810  df-exp 10901  df-ihash 11139  df-cj 11527  df-re 11528  df-im 11529  df-rsqrt 11683  df-abs 11684  df-clim 11964  df-proddc 12237  df-dvds 12474  df-gcd 12650  df-odz 12907  df-phi 12908

This theorem is referenced by:  odzcl  12941  odzid  12942