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Theorem eulerthlem1 13170
Description: Lemma for eulerth 13172. (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 970 . . . . . 6  |-  ( ph  ->  A  e.  ZZ )
32adantr 452 . . . . 5  |-  ( (
ph  /\  x  e.  T )  ->  A  e.  ZZ )
4 eulerth.4 . . . . . . . . . 10  |-  ( ph  ->  F : T -1-1-onto-> S )
5 f1of 5674 . . . . . . . . . 10  |-  ( F : T -1-1-onto-> S  ->  F : T
--> S )
64, 5syl 16 . . . . . . . . 9  |-  ( ph  ->  F : T --> S )
76ffvelrnda 5870 . . . . . . . 8  |-  ( (
ph  /\  x  e.  T )  ->  ( F `  x )  e.  S )
8 oveq1 6088 . . . . . . . . . 10  |-  ( y  =  ( F `  x )  ->  (
y  gcd  N )  =  ( ( F `
 x )  gcd 
N ) )
98eqeq1d 2444 . . . . . . . . 9  |-  ( y  =  ( F `  x )  ->  (
( y  gcd  N
)  =  1  <->  (
( F `  x
)  gcd  N )  =  1 ) )
10 eulerth.2 . . . . . . . . 9  |-  S  =  { y  e.  ( 0..^ N )  |  ( y  gcd  N
)  =  1 }
119, 10elrab2 3094 . . . . . . . 8  |-  ( ( F `  x )  e.  S  <->  ( ( F `  x )  e.  ( 0..^ N )  /\  ( ( F `
 x )  gcd 
N )  =  1 ) )
127, 11sylib 189 . . . . . . 7  |-  ( (
ph  /\  x  e.  T )  ->  (
( F `  x
)  e.  ( 0..^ N )  /\  (
( F `  x
)  gcd  N )  =  1 ) )
1312simpld 446 . . . . . 6  |-  ( (
ph  /\  x  e.  T )  ->  ( F `  x )  e.  ( 0..^ N ) )
14 elfzoelz 11140 . . . . . 6  |-  ( ( F `  x )  e.  ( 0..^ N )  ->  ( F `  x )  e.  ZZ )
1513, 14syl 16 . . . . 5  |-  ( (
ph  /\  x  e.  T )  ->  ( F `  x )  e.  ZZ )
163, 15zmulcld 10381 . . . 4  |-  ( (
ph  /\  x  e.  T )  ->  ( A  x.  ( F `  x ) )  e.  ZZ )
171simp1d 969 . . . . 5  |-  ( ph  ->  N  e.  NN )
1817adantr 452 . . . 4  |-  ( (
ph  /\  x  e.  T )  ->  N  e.  NN )
19 zmodfzo 11269 . . . 4  |-  ( ( ( A  x.  ( F `  x )
)  e.  ZZ  /\  N  e.  NN )  ->  ( ( A  x.  ( F `  x ) )  mod  N )  e.  ( 0..^ N ) )
2016, 18, 19syl2anc 643 . . 3  |-  ( (
ph  /\  x  e.  T )  ->  (
( A  x.  ( F `  x )
)  mod  N )  e.  ( 0..^ N ) )
21 modgcd 13036 . . . . 5  |-  ( ( ( A  x.  ( F `  x )
)  e.  ZZ  /\  N  e.  NN )  ->  ( ( ( A  x.  ( F `  x ) )  mod 
N )  gcd  N
)  =  ( ( A  x.  ( F `
 x ) )  gcd  N ) )
2216, 18, 21syl2anc 643 . . . 4  |-  ( (
ph  /\  x  e.  T )  ->  (
( ( A  x.  ( F `  x ) )  mod  N )  gcd  N )  =  ( ( A  x.  ( F `  x ) )  gcd  N ) )
2317nnzd 10374 . . . . . 6  |-  ( ph  ->  N  e.  ZZ )
2423adantr 452 . . . . 5  |-  ( (
ph  /\  x  e.  T )  ->  N  e.  ZZ )
25 gcdcom 13020 . . . . 5  |-  ( ( ( A  x.  ( F `  x )
)  e.  ZZ  /\  N  e.  ZZ )  ->  ( ( A  x.  ( F `  x ) )  gcd  N )  =  ( N  gcd  ( A  x.  ( F `  x )
) ) )
2616, 24, 25syl2anc 643 . . . 4  |-  ( (
ph  /\  x  e.  T )  ->  (
( A  x.  ( F `  x )
)  gcd  N )  =  ( N  gcd  ( A  x.  ( F `  x )
) ) )
27 gcdcom 13020 . . . . . . . 8  |-  ( ( N  e.  ZZ  /\  A  e.  ZZ )  ->  ( N  gcd  A
)  =  ( A  gcd  N ) )
2823, 2, 27syl2anc 643 . . . . . . 7  |-  ( ph  ->  ( N  gcd  A
)  =  ( A  gcd  N ) )
291simp3d 971 . . . . . . 7  |-  ( ph  ->  ( A  gcd  N
)  =  1 )
3028, 29eqtrd 2468 . . . . . 6  |-  ( ph  ->  ( N  gcd  A
)  =  1 )
3130adantr 452 . . . . 5  |-  ( (
ph  /\  x  e.  T )  ->  ( N  gcd  A )  =  1 )
32 gcdcom 13020 . . . . . . 7  |-  ( ( N  e.  ZZ  /\  ( F `  x )  e.  ZZ )  -> 
( N  gcd  ( F `  x )
)  =  ( ( F `  x )  gcd  N ) )
3324, 15, 32syl2anc 643 . . . . . 6  |-  ( (
ph  /\  x  e.  T )  ->  ( N  gcd  ( F `  x ) )  =  ( ( F `  x )  gcd  N
) )
3412simprd 450 . . . . . 6  |-  ( (
ph  /\  x  e.  T )  ->  (
( F `  x
)  gcd  N )  =  1 )
3533, 34eqtrd 2468 . . . . 5  |-  ( (
ph  /\  x  e.  T )  ->  ( N  gcd  ( F `  x ) )  =  1 )
36 rpmul 13123 . . . . . 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 ) )
3724, 3, 15, 36syl3anc 1184 . . . . 5  |-  ( (
ph  /\  x  e.  T )  ->  (
( ( N  gcd  A )  =  1  /\  ( N  gcd  ( F `  x )
)  =  1 )  ->  ( N  gcd  ( A  x.  ( F `  x )
) )  =  1 ) )
3831, 35, 37mp2and 661 . . . 4  |-  ( (
ph  /\  x  e.  T )  ->  ( N  gcd  ( A  x.  ( F `  x ) ) )  =  1 )
3922, 26, 383eqtrd 2472 . . 3  |-  ( (
ph  /\  x  e.  T )  ->  (
( ( A  x.  ( F `  x ) )  mod  N )  gcd  N )  =  1 )
40 oveq1 6088 . . . . 5  |-  ( y  =  ( ( A  x.  ( F `  x ) )  mod 
N )  ->  (
y  gcd  N )  =  ( ( ( A  x.  ( F `
 x ) )  mod  N )  gcd 
N ) )
4140eqeq1d 2444 . . . 4  |-  ( y  =  ( ( A  x.  ( F `  x ) )  mod 
N )  ->  (
( y  gcd  N
)  =  1  <->  (
( ( A  x.  ( F `  x ) )  mod  N )  gcd  N )  =  1 ) )
4241, 10elrab2 3094 . . 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 ) )
4320, 39, 42sylanbrc 646 . 2  |-  ( (
ph  /\  x  e.  T )  ->  (
( A  x.  ( F `  x )
)  mod  N )  e.  S )
44 eulerth.5 . 2  |-  G  =  ( x  e.  T  |->  ( ( A  x.  ( F `  x ) )  mod  N ) )
4543, 44fmptd 5893 1  |-  ( ph  ->  G : T --> S )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    /\ w3a 936    = wceq 1652    e. wcel 1725   {crab 2709    e. cmpt 4266   -->wf 5450   -1-1-onto->wf1o 5453   ` cfv 5454  (class class class)co 6081   0cc0 8990   1c1 8991    x. cmul 8995   NNcn 10000   ZZcz 10282   ...cfz 11043  ..^cfzo 11135    mod cmo 11250    gcd cgcd 13006   phicphi 13153
This theorem is referenced by:  eulerthlem2  13171
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701  ax-cnex 9046  ax-resscn 9047  ax-1cn 9048  ax-icn 9049  ax-addcl 9050  ax-addrcl 9051  ax-mulcl 9052  ax-mulrcl 9053  ax-mulcom 9054  ax-addass 9055  ax-mulass 9056  ax-distr 9057  ax-i2m1 9058  ax-1ne0 9059  ax-1rid 9060  ax-rnegex 9061  ax-rrecex 9062  ax-cnre 9063  ax-pre-lttri 9064  ax-pre-lttrn 9065  ax-pre-ltadd 9066  ax-pre-mulgt0 9067  ax-pre-sup 9068
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-nel 2602  df-ral 2710  df-rex 2711  df-reu 2712  df-rmo 2713  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-pss 3336  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-tp 3822  df-op 3823  df-uni 4016  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-tr 4303  df-eprel 4494  df-id 4498  df-po 4503  df-so 4504  df-fr 4541  df-we 4543  df-ord 4584  df-on 4585  df-lim 4586  df-suc 4587  df-om 4846  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-1st 6349  df-2nd 6350  df-riota 6549  df-recs 6633  df-rdg 6668  df-er 6905  df-en 7110  df-dom 7111  df-sdom 7112  df-sup 7446  df-pnf 9122  df-mnf 9123  df-xr 9124  df-ltxr 9125  df-le 9126  df-sub 9293  df-neg 9294  df-div 9678  df-nn 10001  df-2 10058  df-3 10059  df-n0 10222  df-z 10283  df-uz 10489  df-rp 10613  df-fz 11044  df-fzo 11136  df-fl 11202  df-mod 11251  df-seq 11324  df-exp 11383  df-cj 11904  df-re 11905  df-im 11906  df-sqr 12040  df-abs 12041  df-dvds 12853  df-gcd 13007
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