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Theorem phicl2 14174
Description: Bounds and closure for the value of the Euler  phi function. (Contributed by Mario Carneiro, 23-Feb-2014.)
Assertion
Ref Expression
phicl2  |-  ( N  e.  NN  ->  ( phi `  N )  e.  ( 1 ... N
) )

Proof of Theorem phicl2
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 phival 14173 . 2  |-  ( N  e.  NN  ->  ( phi `  N )  =  ( # `  {
x  e.  ( 1 ... N )  |  ( x  gcd  N
)  =  1 } ) )
2 fzfi 12062 . . . . . . 7  |-  ( 1 ... N )  e. 
Fin
3 ssrab2 3590 . . . . . . 7  |-  { x  e.  ( 1 ... N
)  |  ( x  gcd  N )  =  1 }  C_  (
1 ... N )
4 ssfi 7752 . . . . . . 7  |-  ( ( ( 1 ... N
)  e.  Fin  /\  { x  e.  ( 1 ... N )  |  ( x  gcd  N
)  =  1 } 
C_  ( 1 ... N ) )  ->  { x  e.  (
1 ... N )  |  ( x  gcd  N
)  =  1 }  e.  Fin )
52, 3, 4mp2an 672 . . . . . 6  |-  { x  e.  ( 1 ... N
)  |  ( x  gcd  N )  =  1 }  e.  Fin
6 hashcl 12408 . . . . . 6  |-  ( { x  e.  ( 1 ... N )  |  ( x  gcd  N
)  =  1 }  e.  Fin  ->  ( # `
 { x  e.  ( 1 ... N
)  |  ( x  gcd  N )  =  1 } )  e. 
NN0 )
75, 6ax-mp 5 . . . . 5  |-  ( # `  { x  e.  ( 1 ... N )  |  ( x  gcd  N )  =  1 } )  e.  NN0
87nn0zi 10901 . . . 4  |-  ( # `  { x  e.  ( 1 ... N )  |  ( x  gcd  N )  =  1 } )  e.  ZZ
98a1i 11 . . 3  |-  ( N  e.  NN  ->  ( # `
 { x  e.  ( 1 ... N
)  |  ( x  gcd  N )  =  1 } )  e.  ZZ )
10 1z 10906 . . . . 5  |-  1  e.  ZZ
11 hashsng 12418 . . . . 5  |-  ( 1  e.  ZZ  ->  ( # `
 { 1 } )  =  1 )
1210, 11ax-mp 5 . . . 4  |-  ( # `  { 1 } )  =  1
13 ovex 6320 . . . . . . 7  |-  ( 1 ... N )  e. 
_V
1413rabex 4604 . . . . . 6  |-  { x  e.  ( 1 ... N
)  |  ( x  gcd  N )  =  1 }  e.  _V
15 eluzfz1 11705 . . . . . . . . 9  |-  ( N  e.  ( ZZ>= `  1
)  ->  1  e.  ( 1 ... N
) )
16 nnuz 11129 . . . . . . . . 9  |-  NN  =  ( ZZ>= `  1 )
1715, 16eleq2s 2575 . . . . . . . 8  |-  ( N  e.  NN  ->  1  e.  ( 1 ... N
) )
18 nnz 10898 . . . . . . . . 9  |-  ( N  e.  NN  ->  N  e.  ZZ )
19 1gcd 14051 . . . . . . . . 9  |-  ( N  e.  ZZ  ->  (
1  gcd  N )  =  1 )
2018, 19syl 16 . . . . . . . 8  |-  ( N  e.  NN  ->  (
1  gcd  N )  =  1 )
21 oveq1 6302 . . . . . . . . . 10  |-  ( x  =  1  ->  (
x  gcd  N )  =  ( 1  gcd 
N ) )
2221eqeq1d 2469 . . . . . . . . 9  |-  ( x  =  1  ->  (
( x  gcd  N
)  =  1  <->  (
1  gcd  N )  =  1 ) )
2322elrab 3266 . . . . . . . 8  |-  ( 1  e.  { x  e.  ( 1 ... N
)  |  ( x  gcd  N )  =  1 }  <->  ( 1  e.  ( 1 ... N )  /\  (
1  gcd  N )  =  1 ) )
2417, 20, 23sylanbrc 664 . . . . . . 7  |-  ( N  e.  NN  ->  1  e.  { x  e.  ( 1 ... N )  |  ( x  gcd  N )  =  1 } )
2524snssd 4178 . . . . . 6  |-  ( N  e.  NN  ->  { 1 }  C_  { x  e.  ( 1 ... N
)  |  ( x  gcd  N )  =  1 } )
26 ssdomg 7573 . . . . . 6  |-  ( { x  e.  ( 1 ... N )  |  ( x  gcd  N
)  =  1 }  e.  _V  ->  ( { 1 }  C_  { x  e.  ( 1 ... N )  |  ( x  gcd  N
)  =  1 }  ->  { 1 }  ~<_  { x  e.  ( 1 ... N )  |  ( x  gcd  N )  =  1 } ) )
2714, 25, 26mpsyl 63 . . . . 5  |-  ( N  e.  NN  ->  { 1 }  ~<_  { x  e.  ( 1 ... N
)  |  ( x  gcd  N )  =  1 } )
28 snfi 7608 . . . . . 6  |-  { 1 }  e.  Fin
29 hashdom 12427 . . . . . 6  |-  ( ( { 1 }  e.  Fin  /\  { x  e.  ( 1 ... N
)  |  ( x  gcd  N )  =  1 }  e.  Fin )  ->  ( ( # `  { 1 } )  <_  ( # `  {
x  e.  ( 1 ... N )  |  ( x  gcd  N
)  =  1 } )  <->  { 1 }  ~<_  { x  e.  ( 1 ... N
)  |  ( x  gcd  N )  =  1 } ) )
3028, 5, 29mp2an 672 . . . . 5  |-  ( (
# `  { 1 } )  <_  ( # `
 { x  e.  ( 1 ... N
)  |  ( x  gcd  N )  =  1 } )  <->  { 1 }  ~<_  { x  e.  ( 1 ... N
)  |  ( x  gcd  N )  =  1 } )
3127, 30sylibr 212 . . . 4  |-  ( N  e.  NN  ->  ( # `
 { 1 } )  <_  ( # `  {
x  e.  ( 1 ... N )  |  ( x  gcd  N
)  =  1 } ) )
3212, 31syl5eqbrr 4487 . . 3  |-  ( N  e.  NN  ->  1  <_  ( # `  {
x  e.  ( 1 ... N )  |  ( x  gcd  N
)  =  1 } ) )
33 ssdomg 7573 . . . . . 6  |-  ( ( 1 ... N )  e.  _V  ->  ( { x  e.  (
1 ... N )  |  ( x  gcd  N
)  =  1 } 
C_  ( 1 ... N )  ->  { x  e.  ( 1 ... N
)  |  ( x  gcd  N )  =  1 }  ~<_  ( 1 ... N ) ) )
3413, 3, 33mp2 9 . . . . 5  |-  { x  e.  ( 1 ... N
)  |  ( x  gcd  N )  =  1 }  ~<_  ( 1 ... N )
35 hashdom 12427 . . . . . 6  |-  ( ( { x  e.  ( 1 ... N )  |  ( x  gcd  N )  =  1 }  e.  Fin  /\  (
1 ... N )  e. 
Fin )  ->  (
( # `  { x  e.  ( 1 ... N
)  |  ( x  gcd  N )  =  1 } )  <_ 
( # `  ( 1 ... N ) )  <->  { x  e.  (
1 ... N )  |  ( x  gcd  N
)  =  1 }  ~<_  ( 1 ... N
) ) )
365, 2, 35mp2an 672 . . . . 5  |-  ( (
# `  { x  e.  ( 1 ... N
)  |  ( x  gcd  N )  =  1 } )  <_ 
( # `  ( 1 ... N ) )  <->  { x  e.  (
1 ... N )  |  ( x  gcd  N
)  =  1 }  ~<_  ( 1 ... N
) )
3734, 36mpbir 209 . . . 4  |-  ( # `  { x  e.  ( 1 ... N )  |  ( x  gcd  N )  =  1 } )  <_  ( # `  (
1 ... N ) )
38 nnnn0 10814 . . . . 5  |-  ( N  e.  NN  ->  N  e.  NN0 )
39 hashfz1 12399 . . . . 5  |-  ( N  e.  NN0  ->  ( # `  ( 1 ... N
) )  =  N )
4038, 39syl 16 . . . 4  |-  ( N  e.  NN  ->  ( # `
 ( 1 ... N ) )  =  N )
4137, 40syl5breq 4488 . . 3  |-  ( N  e.  NN  ->  ( # `
 { x  e.  ( 1 ... N
)  |  ( x  gcd  N )  =  1 } )  <_  N )
42 elfz1 11689 . . . 4  |-  ( ( 1  e.  ZZ  /\  N  e.  ZZ )  ->  ( ( # `  {
x  e.  ( 1 ... N )  |  ( x  gcd  N
)  =  1 } )  e.  ( 1 ... N )  <->  ( ( # `
 { x  e.  ( 1 ... N
)  |  ( x  gcd  N )  =  1 } )  e.  ZZ  /\  1  <_ 
( # `  { x  e.  ( 1 ... N
)  |  ( x  gcd  N )  =  1 } )  /\  ( # `  { x  e.  ( 1 ... N
)  |  ( x  gcd  N )  =  1 } )  <_  N ) ) )
4310, 18, 42sylancr 663 . . 3  |-  ( N  e.  NN  ->  (
( # `  { x  e.  ( 1 ... N
)  |  ( x  gcd  N )  =  1 } )  e.  ( 1 ... N
)  <->  ( ( # `  { x  e.  ( 1 ... N )  |  ( x  gcd  N )  =  1 } )  e.  ZZ  /\  1  <_  ( # `  {
x  e.  ( 1 ... N )  |  ( x  gcd  N
)  =  1 } )  /\  ( # `  { x  e.  ( 1 ... N )  |  ( x  gcd  N )  =  1 } )  <_  N )
) )
449, 32, 41, 43mpbir3and 1179 . 2  |-  ( N  e.  NN  ->  ( # `
 { x  e.  ( 1 ... N
)  |  ( x  gcd  N )  =  1 } )  e.  ( 1 ... N
) )
451, 44eqeltrd 2555 1  |-  ( N  e.  NN  ->  ( phi `  N )  e.  ( 1 ... N
) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ w3a 973    = wceq 1379    e. wcel 1767   {crab 2821   _Vcvv 3118    C_ wss 3481   {csn 4033   class class class wbr 4453   ` cfv 5594  (class class class)co 6295    ~<_ cdom 7526   Fincfn 7528   1c1 9505    <_ cle 9641   NNcn 10548   NN0cn0 10807   ZZcz 10876   ZZ>=cuz 11094   ...cfz 11684   #chash 12385    gcd cgcd 14020   phicphi 14170
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-un 6587  ax-cnex 9560  ax-resscn 9561  ax-1cn 9562  ax-icn 9563  ax-addcl 9564  ax-addrcl 9565  ax-mulcl 9566  ax-mulrcl 9567  ax-mulcom 9568  ax-addass 9569  ax-mulass 9570  ax-distr 9571  ax-i2m1 9572  ax-1ne0 9573  ax-1rid 9574  ax-rnegex 9575  ax-rrecex 9576  ax-cnre 9577  ax-pre-lttri 9578  ax-pre-lttrn 9579  ax-pre-ltadd 9580  ax-pre-mulgt0 9581  ax-pre-sup 9582
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2822  df-rex 2823  df-reu 2824  df-rmo 2825  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-pss 3497  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-tp 4038  df-op 4040  df-uni 4252  df-int 4289  df-iun 4333  df-br 4454  df-opab 4512  df-mpt 4513  df-tr 4547  df-eprel 4797  df-id 4801  df-po 4806  df-so 4807  df-fr 4844  df-we 4846  df-ord 4887  df-on 4888  df-lim 4889  df-suc 4890  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-riota 6256  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-om 6696  df-1st 6795  df-2nd 6796  df-recs 7054  df-rdg 7088  df-1o 7142  df-oadd 7146  df-er 7323  df-en 7529  df-dom 7530  df-sdom 7531  df-fin 7532  df-sup 7913  df-card 8332  df-pnf 9642  df-mnf 9643  df-xr 9644  df-ltxr 9645  df-le 9646  df-sub 9819  df-neg 9820  df-div 10219  df-nn 10549  df-2 10606  df-3 10607  df-n0 10808  df-z 10877  df-uz 11095  df-rp 11233  df-fz 11685  df-seq 12088  df-exp 12147  df-hash 12386  df-cj 12912  df-re 12913  df-im 12914  df-sqrt 13048  df-abs 13049  df-dvds 13865  df-gcd 14021  df-phi 14172
This theorem is referenced by:  phicl  14175  phi1  14179
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