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Theorem eulerthlem1 12865
Description: Lemma for eulerth 12867. (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 5488 . . . . . . . . . 10  |-  ( F : T -1-1-onto-> S  ->  F : T
--> S )
64, 5syl 15 . . . . . . . . 9  |-  ( ph  ->  F : T --> S )
7 ffvelrn 5679 . . . . . . . . 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 5881 . . . . . . . . . 10  |-  ( y  =  ( F `  x )  ->  (
y  gcd  N )  =  ( ( F `
 x )  gcd 
N ) )
109eqeq1d 2304 . . . . . . . . 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 2938 . . . . . . . 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 10891 . . . . . 6  |-  ( ( F `  x )  e.  ( 0..^ N )  ->  ( F `  x )  e.  ZZ )
1614, 15syl 15 . . . . 5  |-  ( (
ph  /\  x  e.  T )  ->  ( F `  x )  e.  ZZ )
173, 16zmulcld 10139 . . . 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 11008 . . . 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 12731 . . . . 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 10132 . . . . . 6  |-  ( ph  ->  N  e.  ZZ )
2524adantr 451 . . . . 5  |-  ( (
ph  /\  x  e.  T )  ->  N  e.  ZZ )
26 gcdcom 12715 . . . . 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 12715 . . . . . . . 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 2328 . . . . . 6  |-  ( ph  ->  ( N  gcd  A
)  =  1 )
3231adantr 451 . . . . 5  |-  ( (
ph  /\  x  e.  T )  ->  ( N  gcd  A )  =  1 )
33 gcdcom 12715 . . . . . . 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 2328 . . . . 5  |-  ( (
ph  /\  x  e.  T )  ->  ( N  gcd  ( F `  x ) )  =  1 )
37 rpmul 12818 . . . . . 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 2332 . . 3  |-  ( (
ph  /\  x  e.  T )  ->  (
( ( A  x.  ( F `  x ) )  mod  N )  gcd  N )  =  1 )
41 oveq1 5881 . . . . 5  |-  ( y  =  ( ( A  x.  ( F `  x ) )  mod 
N )  ->  (
y  gcd  N )  =  ( ( ( A  x.  ( F `
 x ) )  mod  N )  gcd 
N ) )
4241eqeq1d 2304 . . . 4  |-  ( y  =  ( ( A  x.  ( F `  x ) )  mod 
N )  ->  (
( y  gcd  N
)  =  1  <->  (
( ( A  x.  ( F `  x ) )  mod  N )  gcd  N )  =  1 ) )
4342, 11elrab2 2938 . . 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 5700 1  |-  ( ph  ->  G : T --> S )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    /\ w3a 934    = wceq 1632    e. wcel 1696   {crab 2560    e. cmpt 4093   -->wf 5267   -1-1-onto->wf1o 5270   ` cfv 5271  (class class class)co 5874   0cc0 8753   1c1 8754    x. cmul 8758   NNcn 9762   ZZcz 10040   ...cfz 10798  ..^cfzo 10886    mod cmo 10989    gcd cgcd 12701   phicphi 12848
This theorem is referenced by:  eulerthlem2  12866
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830  ax-pre-sup 8831
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 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-riota 6320  df-recs 6404  df-rdg 6439  df-er 6676  df-en 6880  df-dom 6881  df-sdom 6882  df-sup 7210  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-div 9440  df-nn 9763  df-2 9820  df-3 9821  df-n0 9982  df-z 10041  df-uz 10247  df-rp 10371  df-fz 10799  df-fzo 10887  df-fl 10941  df-mod 10990  df-seq 11063  df-exp 11121  df-cj 11600  df-re 11601  df-im 11602  df-sqr 11736  df-abs 11737  df-dvds 12548  df-gcd 12702
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