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Theorem phicl2 14685
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 14684 . 2  |-  ( N  e.  NN  ->  ( phi `  N )  =  ( # `  {
x  e.  ( 1 ... N )  |  ( x  gcd  N
)  =  1 } ) )
2 fzfi 12182 . . . . . . 7  |-  ( 1 ... N )  e. 
Fin
3 ssrab2 3552 . . . . . . 7  |-  { x  e.  ( 1 ... N
)  |  ( x  gcd  N )  =  1 }  C_  (
1 ... N )
4 ssfi 7798 . . . . . . 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 676 . . . . . 6  |-  { x  e.  ( 1 ... N
)  |  ( x  gcd  N )  =  1 }  e.  Fin
6 hashcl 12535 . . . . . 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 10962 . . . 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 10967 . . . . 5  |-  1  e.  ZZ
11 hashsng 12546 . . . . 5  |-  ( 1  e.  ZZ  ->  ( # `
 { 1 } )  =  1 )
1210, 11ax-mp 5 . . . 4  |-  ( # `  { 1 } )  =  1
13 ovex 6333 . . . . . . 7  |-  ( 1 ... N )  e. 
_V
1413rabex 4576 . . . . . 6  |-  { x  e.  ( 1 ... N
)  |  ( x  gcd  N )  =  1 }  e.  _V
15 eluzfz1 11804 . . . . . . . . 9  |-  ( N  e.  ( ZZ>= `  1
)  ->  1  e.  ( 1 ... N
) )
16 nnuz 11194 . . . . . . . . 9  |-  NN  =  ( ZZ>= `  1 )
1715, 16eleq2s 2537 . . . . . . . 8  |-  ( N  e.  NN  ->  1  e.  ( 1 ... N
) )
18 nnz 10959 . . . . . . . . 9  |-  ( N  e.  NN  ->  N  e.  ZZ )
19 1gcd 14475 . . . . . . . . 9  |-  ( N  e.  ZZ  ->  (
1  gcd  N )  =  1 )
2018, 19syl 17 . . . . . . . 8  |-  ( N  e.  NN  ->  (
1  gcd  N )  =  1 )
21 oveq1 6312 . . . . . . . . . 10  |-  ( x  =  1  ->  (
x  gcd  N )  =  ( 1  gcd 
N ) )
2221eqeq1d 2431 . . . . . . . . 9  |-  ( x  =  1  ->  (
( x  gcd  N
)  =  1  <->  (
1  gcd  N )  =  1 ) )
2322elrab 3235 . . . . . . . 8  |-  ( 1  e.  { x  e.  ( 1 ... N
)  |  ( x  gcd  N )  =  1 }  <->  ( 1  e.  ( 1 ... N )  /\  (
1  gcd  N )  =  1 ) )
2417, 20, 23sylanbrc 668 . . . . . . 7  |-  ( N  e.  NN  ->  1  e.  { x  e.  ( 1 ... N )  |  ( x  gcd  N )  =  1 } )
2524snssd 4148 . . . . . 6  |-  ( N  e.  NN  ->  { 1 }  C_  { x  e.  ( 1 ... N
)  |  ( x  gcd  N )  =  1 } )
26 ssdomg 7622 . . . . . 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 65 . . . . 5  |-  ( N  e.  NN  ->  { 1 }  ~<_  { x  e.  ( 1 ... N
)  |  ( x  gcd  N )  =  1 } )
28 snfi 7657 . . . . . 6  |-  { 1 }  e.  Fin
29 hashdom 12555 . . . . . 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 676 . . . . 5  |-  ( (
# `  { 1 } )  <_  ( # `
 { x  e.  ( 1 ... N
)  |  ( x  gcd  N )  =  1 } )  <->  { 1 }  ~<_  { x  e.  ( 1 ... N
)  |  ( x  gcd  N )  =  1 } )
3127, 30sylibr 215 . . . 4  |-  ( N  e.  NN  ->  ( # `
 { 1 } )  <_  ( # `  {
x  e.  ( 1 ... N )  |  ( x  gcd  N
)  =  1 } ) )
3212, 31syl5eqbrr 4460 . . 3  |-  ( N  e.  NN  ->  1  <_  ( # `  {
x  e.  ( 1 ... N )  |  ( x  gcd  N
)  =  1 } ) )
33 ssdomg 7622 . . . . . 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 12555 . . . . . 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 676 . . . . 5  |-  ( (
# `  { x  e.  ( 1 ... N
)  |  ( x  gcd  N )  =  1 } )  <_ 
( # `  ( 1 ... N ) )  <->  { x  e.  (
1 ... N )  |  ( x  gcd  N
)  =  1 }  ~<_  ( 1 ... N
) )
3734, 36mpbir 212 . . . 4  |-  ( # `  { x  e.  ( 1 ... N )  |  ( x  gcd  N )  =  1 } )  <_  ( # `  (
1 ... N ) )
38 nnnn0 10876 . . . . 5  |-  ( N  e.  NN  ->  N  e.  NN0 )
39 hashfz1 12526 . . . . 5  |-  ( N  e.  NN0  ->  ( # `  ( 1 ... N
) )  =  N )
4038, 39syl 17 . . . 4  |-  ( N  e.  NN  ->  ( # `
 ( 1 ... N ) )  =  N )
4137, 40syl5breq 4461 . . 3  |-  ( N  e.  NN  ->  ( # `
 { x  e.  ( 1 ... N
)  |  ( x  gcd  N )  =  1 } )  <_  N )
42 elfz1 11787 . . . 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 667 . . 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 1188 . 2  |-  ( N  e.  NN  ->  ( # `
 { x  e.  ( 1 ... N
)  |  ( x  gcd  N )  =  1 } )  e.  ( 1 ... N
) )
451, 44eqeltrd 2517 1  |-  ( N  e.  NN  ->  ( phi `  N )  e.  ( 1 ... N
) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 187    /\ w3a 982    = wceq 1437    e. wcel 1870   {crab 2786   _Vcvv 3087    C_ wss 3442   {csn 4002   class class class wbr 4426   ` cfv 5601  (class class class)co 6305    ~<_ cdom 7575   Fincfn 7577   1c1 9539    <_ cle 9675   NNcn 10609   NN0cn0 10869   ZZcz 10937   ZZ>=cuz 11159   ...cfz 11782   #chash 12512    gcd cgcd 14442   phicphi 14681
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-8 1872  ax-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-sep 4548  ax-nul 4556  ax-pow 4603  ax-pr 4661  ax-un 6597  ax-cnex 9594  ax-resscn 9595  ax-1cn 9596  ax-icn 9597  ax-addcl 9598  ax-addrcl 9599  ax-mulcl 9600  ax-mulrcl 9601  ax-mulcom 9602  ax-addass 9603  ax-mulass 9604  ax-distr 9605  ax-i2m1 9606  ax-1ne0 9607  ax-1rid 9608  ax-rnegex 9609  ax-rrecex 9610  ax-cnre 9611  ax-pre-lttri 9612  ax-pre-lttrn 9613  ax-pre-ltadd 9614  ax-pre-mulgt0 9615  ax-pre-sup 9616
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1790  df-eu 2270  df-mo 2271  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-nel 2628  df-ral 2787  df-rex 2788  df-reu 2789  df-rmo 2790  df-rab 2791  df-v 3089  df-sbc 3306  df-csb 3402  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-pss 3458  df-nul 3768  df-if 3916  df-pw 3987  df-sn 4003  df-pr 4005  df-tp 4007  df-op 4009  df-uni 4223  df-int 4259  df-iun 4304  df-br 4427  df-opab 4485  df-mpt 4486  df-tr 4521  df-eprel 4765  df-id 4769  df-po 4775  df-so 4776  df-fr 4813  df-we 4815  df-xp 4860  df-rel 4861  df-cnv 4862  df-co 4863  df-dm 4864  df-rn 4865  df-res 4866  df-ima 4867  df-pred 5399  df-ord 5445  df-on 5446  df-lim 5447  df-suc 5448  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-riota 6267  df-ov 6308  df-oprab 6309  df-mpt2 6310  df-om 6707  df-1st 6807  df-2nd 6808  df-wrecs 7036  df-recs 7098  df-rdg 7136  df-1o 7190  df-oadd 7194  df-er 7371  df-en 7578  df-dom 7579  df-sdom 7580  df-fin 7581  df-sup 7962  df-inf 7963  df-card 8372  df-pnf 9676  df-mnf 9677  df-xr 9678  df-ltxr 9679  df-le 9680  df-sub 9861  df-neg 9862  df-div 10269  df-nn 10610  df-2 10668  df-3 10669  df-n0 10870  df-z 10938  df-uz 11160  df-rp 11303  df-fz 11783  df-seq 12211  df-exp 12270  df-hash 12513  df-cj 13141  df-re 13142  df-im 13143  df-sqrt 13277  df-abs 13278  df-dvds 14284  df-gcd 14443  df-phi 14683
This theorem is referenced by:  phicl  14686  phi1  14690
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