MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  eulerthlem1 Unicode version

Theorem eulerthlem1 12849
Description: Lemma for eulerth 12851. (Contributed by Mario Carneiro, 8-May-2015.)
Hypotheses
Ref Expression
eulerth.1  |-  ( ph  ->  ( N  e.  NN  /\  A  e.  ZZ  /\  ( A  gcd  N )  =  1 ) )
eulerth.2  |-  S  =  { y  e.  ( 0..^ N )  |  ( y  gcd  N
)  =  1 }
eulerth.3  |-  T  =  ( 1 ... ( phi `  N ) )
eulerth.4  |-  ( ph  ->  F : T -1-1-onto-> S )
eulerth.5  |-  G  =  ( x  e.  T  |->  ( ( A  x.  ( F `  x ) )  mod  N ) )
Assertion
Ref Expression
eulerthlem1  |-  ( ph  ->  G : T --> S )
Distinct variable groups:    x, y, A    x, F, y    x, G, y    x, N, y   
x, S    ph, x, y   
x, T, y
Allowed substitution hint:    S( y)

Proof of Theorem eulerthlem1
StepHypRef Expression
1 eulerth.1 . . . . . . 7  |-  ( ph  ->  ( N  e.  NN  /\  A  e.  ZZ  /\  ( A  gcd  N )  =  1 ) )
21simp2d 968 . . . . . 6  |-  ( ph  ->  A  e.  ZZ )
32adantr 451 . . . . 5  |-  ( (
ph  /\  x  e.  T )  ->  A  e.  ZZ )
4 eulerth.4 . . . . . . . . . 10  |-  ( ph  ->  F : T -1-1-onto-> S )
5 f1of 5472 . . . . . . . . . 10  |-  ( F : T -1-1-onto-> S  ->  F : T
--> S )
64, 5syl 15 . . . . . . . . 9  |-  ( ph  ->  F : T --> S )
7 ffvelrn 5663 . . . . . . . . 9  |-  ( ( F : T --> S  /\  x  e.  T )  ->  ( F `  x
)  e.  S )
86, 7sylan 457 . . . . . . . 8  |-  ( (
ph  /\  x  e.  T )  ->  ( F `  x )  e.  S )
9 oveq1 5865 . . . . . . . . . 10  |-  ( y  =  ( F `  x )  ->  (
y  gcd  N )  =  ( ( F `
 x )  gcd 
N ) )
109eqeq1d 2291 . . . . . . . . 9  |-  ( y  =  ( F `  x )  ->  (
( y  gcd  N
)  =  1  <->  (
( F `  x
)  gcd  N )  =  1 ) )
11 eulerth.2 . . . . . . . . 9  |-  S  =  { y  e.  ( 0..^ N )  |  ( y  gcd  N
)  =  1 }
1210, 11elrab2 2925 . . . . . . . 8  |-  ( ( F `  x )  e.  S  <->  ( ( F `  x )  e.  ( 0..^ N )  /\  ( ( F `
 x )  gcd 
N )  =  1 ) )
138, 12sylib 188 . . . . . . 7  |-  ( (
ph  /\  x  e.  T )  ->  (
( F `  x
)  e.  ( 0..^ N )  /\  (
( F `  x
)  gcd  N )  =  1 ) )
1413simpld 445 . . . . . 6  |-  ( (
ph  /\  x  e.  T )  ->  ( F `  x )  e.  ( 0..^ N ) )
15 elfzoelz 10875 . . . . . 6  |-  ( ( F `  x )  e.  ( 0..^ N )  ->  ( F `  x )  e.  ZZ )
1614, 15syl 15 . . . . 5  |-  ( (
ph  /\  x  e.  T )  ->  ( F `  x )  e.  ZZ )
173, 16zmulcld 10123 . . . 4  |-  ( (
ph  /\  x  e.  T )  ->  ( A  x.  ( F `  x ) )  e.  ZZ )
181simp1d 967 . . . . 5  |-  ( ph  ->  N  e.  NN )
1918adantr 451 . . . 4  |-  ( (
ph  /\  x  e.  T )  ->  N  e.  NN )
20 zmodfzo 10992 . . . 4  |-  ( ( ( A  x.  ( F `  x )
)  e.  ZZ  /\  N  e.  NN )  ->  ( ( A  x.  ( F `  x ) )  mod  N )  e.  ( 0..^ N ) )
2117, 19, 20syl2anc 642 . . 3  |-  ( (
ph  /\  x  e.  T )  ->  (
( A  x.  ( F `  x )
)  mod  N )  e.  ( 0..^ N ) )
22 modgcd 12715 . . . . 5  |-  ( ( ( A  x.  ( F `  x )
)  e.  ZZ  /\  N  e.  NN )  ->  ( ( ( A  x.  ( F `  x ) )  mod 
N )  gcd  N
)  =  ( ( A  x.  ( F `
 x ) )  gcd  N ) )
2317, 19, 22syl2anc 642 . . . 4  |-  ( (
ph  /\  x  e.  T )  ->  (
( ( A  x.  ( F `  x ) )  mod  N )  gcd  N )  =  ( ( A  x.  ( F `  x ) )  gcd  N ) )
2418nnzd 10116 . . . . . 6  |-  ( ph  ->  N  e.  ZZ )
2524adantr 451 . . . . 5  |-  ( (
ph  /\  x  e.  T )  ->  N  e.  ZZ )
26 gcdcom 12699 . . . . 5  |-  ( ( ( A  x.  ( F `  x )
)  e.  ZZ  /\  N  e.  ZZ )  ->  ( ( A  x.  ( F `  x ) )  gcd  N )  =  ( N  gcd  ( A  x.  ( F `  x )
) ) )
2717, 25, 26syl2anc 642 . . . 4  |-  ( (
ph  /\  x  e.  T )  ->  (
( A  x.  ( F `  x )
)  gcd  N )  =  ( N  gcd  ( A  x.  ( F `  x )
) ) )
28 gcdcom 12699 . . . . . . . 8  |-  ( ( N  e.  ZZ  /\  A  e.  ZZ )  ->  ( N  gcd  A
)  =  ( A  gcd  N ) )
2924, 2, 28syl2anc 642 . . . . . . 7  |-  ( ph  ->  ( N  gcd  A
)  =  ( A  gcd  N ) )
301simp3d 969 . . . . . . 7  |-  ( ph  ->  ( A  gcd  N
)  =  1 )
3129, 30eqtrd 2315 . . . . . 6  |-  ( ph  ->  ( N  gcd  A
)  =  1 )
3231adantr 451 . . . . 5  |-  ( (
ph  /\  x  e.  T )  ->  ( N  gcd  A )  =  1 )
33 gcdcom 12699 . . . . . . 7  |-  ( ( N  e.  ZZ  /\  ( F `  x )  e.  ZZ )  -> 
( N  gcd  ( F `  x )
)  =  ( ( F `  x )  gcd  N ) )
3425, 16, 33syl2anc 642 . . . . . 6  |-  ( (
ph  /\  x  e.  T )  ->  ( N  gcd  ( F `  x ) )  =  ( ( F `  x )  gcd  N
) )
3513simprd 449 . . . . . 6  |-  ( (
ph  /\  x  e.  T )  ->  (
( F `  x
)  gcd  N )  =  1 )
3634, 35eqtrd 2315 . . . . 5  |-  ( (
ph  /\  x  e.  T )  ->  ( N  gcd  ( F `  x ) )  =  1 )
37 rpmul 12802 . . . . . 6  |-  ( ( N  e.  ZZ  /\  A  e.  ZZ  /\  ( F `  x )  e.  ZZ )  ->  (
( ( N  gcd  A )  =  1  /\  ( N  gcd  ( F `  x )
)  =  1 )  ->  ( N  gcd  ( A  x.  ( F `  x )
) )  =  1 ) )
3825, 3, 16, 37syl3anc 1182 . . . . 5  |-  ( (
ph  /\  x  e.  T )  ->  (
( ( N  gcd  A )  =  1  /\  ( N  gcd  ( F `  x )
)  =  1 )  ->  ( N  gcd  ( A  x.  ( F `  x )
) )  =  1 ) )
3932, 36, 38mp2and 660 . . . 4  |-  ( (
ph  /\  x  e.  T )  ->  ( N  gcd  ( A  x.  ( F `  x ) ) )  =  1 )
4023, 27, 393eqtrd 2319 . . 3  |-  ( (
ph  /\  x  e.  T )  ->  (
( ( A  x.  ( F `  x ) )  mod  N )  gcd  N )  =  1 )
41 oveq1 5865 . . . . 5  |-  ( y  =  ( ( A  x.  ( F `  x ) )  mod 
N )  ->  (
y  gcd  N )  =  ( ( ( A  x.  ( F `
 x ) )  mod  N )  gcd 
N ) )
4241eqeq1d 2291 . . . 4  |-  ( y  =  ( ( A  x.  ( F `  x ) )  mod 
N )  ->  (
( y  gcd  N
)  =  1  <->  (
( ( A  x.  ( F `  x ) )  mod  N )  gcd  N )  =  1 ) )
4342, 11elrab2 2925 . . 3  |-  ( ( ( A  x.  ( F `  x )
)  mod  N )  e.  S  <->  ( ( ( A  x.  ( F `
 x ) )  mod  N )  e.  ( 0..^ N )  /\  ( ( ( A  x.  ( F `
 x ) )  mod  N )  gcd 
N )  =  1 ) )
4421, 40, 43sylanbrc 645 . 2  |-  ( (
ph  /\  x  e.  T )  ->  (
( A  x.  ( F `  x )
)  mod  N )  e.  S )
45 eulerth.5 . 2  |-  G  =  ( x  e.  T  |->  ( ( A  x.  ( F `  x ) )  mod  N ) )
4644, 45fmptd 5684 1  |-  ( ph  ->  G : T --> S )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684   {crab 2547    e. cmpt 4077   -->wf 5251   -1-1-onto->wf1o 5254   ` cfv 5255  (class class class)co 5858   0cc0 8737   1c1 8738    x. cmul 8742   NNcn 9746   ZZcz 10024   ...cfz 10782  ..^cfzo 10870    mod cmo 10973    gcd cgcd 12685   phicphi 12832
This theorem is referenced by:  eulerthlem2  12850
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814  ax-pre-sup 8815
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-er 6660  df-en 6864  df-dom 6865  df-sdom 6866  df-sup 7194  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-div 9424  df-nn 9747  df-2 9804  df-3 9805  df-n0 9966  df-z 10025  df-uz 10231  df-rp 10355  df-fz 10783  df-fzo 10871  df-fl 10925  df-mod 10974  df-seq 11047  df-exp 11105  df-cj 11584  df-re 11585  df-im 11586  df-sqr 11720  df-abs 11721  df-dvds 12532  df-gcd 12686
  Copyright terms: Public domain W3C validator