Users' Mathboxes Mathbox for Stefan O'Rear < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  hashgcdeq Unicode version

Theorem hashgcdeq 27517
Description: Number of initial natural numbers with specified divisors. (Contributed by Stefan O'Rear, 12-Sep-2015.)
Assertion
Ref Expression
hashgcdeq  |-  ( ( M  e.  NN  /\  N  e.  NN )  ->  ( # `  {
x  e.  ( 0..^ M )  |  ( x  gcd  M )  =  N } )  =  if ( N 
||  M ,  ( phi `  ( M  /  N ) ) ,  0 ) )
Distinct variable groups:    x, M    x, N

Proof of Theorem hashgcdeq
Dummy variables  z 
y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqeq2 2292 . 2  |-  ( ( phi `  ( M  /  N ) )  =  if ( N 
||  M ,  ( phi `  ( M  /  N ) ) ,  0 )  -> 
( ( # `  {
x  e.  ( 0..^ M )  |  ( x  gcd  M )  =  N } )  =  ( phi `  ( M  /  N
) )  <->  ( # `  {
x  e.  ( 0..^ M )  |  ( x  gcd  M )  =  N } )  =  if ( N 
||  M ,  ( phi `  ( M  /  N ) ) ,  0 ) ) )
2 eqeq2 2292 . 2  |-  ( 0  =  if ( N 
||  M ,  ( phi `  ( M  /  N ) ) ,  0 )  -> 
( ( # `  {
x  e.  ( 0..^ M )  |  ( x  gcd  M )  =  N } )  =  0  <->  ( # `  {
x  e.  ( 0..^ M )  |  ( x  gcd  M )  =  N } )  =  if ( N 
||  M ,  ( phi `  ( M  /  N ) ) ,  0 ) ) )
3 nndivdvds 12537 . . . . 5  |-  ( ( M  e.  NN  /\  N  e.  NN )  ->  ( N  ||  M  <->  ( M  /  N )  e.  NN ) )
43biimpa 470 . . . 4  |-  ( ( ( M  e.  NN  /\  N  e.  NN )  /\  N  ||  M
)  ->  ( M  /  N )  e.  NN )
5 dfphi2 12842 . . . 4  |-  ( ( M  /  N )  e.  NN  ->  ( phi `  ( M  /  N ) )  =  ( # `  {
y  e.  ( 0..^ ( M  /  N
) )  |  ( y  gcd  ( M  /  N ) )  =  1 } ) )
64, 5syl 15 . . 3  |-  ( ( ( M  e.  NN  /\  N  e.  NN )  /\  N  ||  M
)  ->  ( phi `  ( M  /  N
) )  =  (
# `  { y  e.  ( 0..^ ( M  /  N ) )  |  ( y  gcd  ( M  /  N
) )  =  1 } ) )
7 eqid 2283 . . . . . 6  |-  { y  e.  ( 0..^ ( M  /  N ) )  |  ( y  gcd  ( M  /  N ) )  =  1 }  =  {
y  e.  ( 0..^ ( M  /  N
) )  |  ( y  gcd  ( M  /  N ) )  =  1 }
8 eqid 2283 . . . . . 6  |-  { x  e.  ( 0..^ M )  |  ( x  gcd  M )  =  N }  =  { x  e.  ( 0..^ M )  |  ( x  gcd  M
)  =  N }
9 eqid 2283 . . . . . 6  |-  ( z  e.  { y  e.  ( 0..^ ( M  /  N ) )  |  ( y  gcd  ( M  /  N
) )  =  1 }  |->  ( z  x.  N ) )  =  ( z  e.  {
y  e.  ( 0..^ ( M  /  N
) )  |  ( y  gcd  ( M  /  N ) )  =  1 }  |->  ( z  x.  N ) )
107, 8, 9hashgcdlem 27516 . . . . 5  |-  ( ( M  e.  NN  /\  N  e.  NN  /\  N  ||  M )  ->  (
z  e.  { y  e.  ( 0..^ ( M  /  N ) )  |  ( y  gcd  ( M  /  N ) )  =  1 }  |->  ( z  x.  N ) ) : { y  e.  ( 0..^ ( M  /  N ) )  |  ( y  gcd  ( M  /  N
) )  =  1 } -1-1-onto-> { x  e.  ( 0..^ M )  |  ( x  gcd  M
)  =  N }
)
11103expa 1151 . . . 4  |-  ( ( ( M  e.  NN  /\  N  e.  NN )  /\  N  ||  M
)  ->  ( z  e.  { y  e.  ( 0..^ ( M  /  N ) )  |  ( y  gcd  ( M  /  N ) )  =  1 }  |->  ( z  x.  N ) ) : { y  e.  ( 0..^ ( M  /  N ) )  |  ( y  gcd  ( M  /  N ) )  =  1 } -1-1-onto-> { x  e.  ( 0..^ M )  |  ( x  gcd  M
)  =  N }
)
12 ovex 5883 . . . . . 6  |-  ( 0..^ ( M  /  N
) )  e.  _V
1312rabex 4165 . . . . 5  |-  { y  e.  ( 0..^ ( M  /  N ) )  |  ( y  gcd  ( M  /  N ) )  =  1 }  e.  _V
1413f1oen 6882 . . . 4  |-  ( ( z  e.  { y  e.  ( 0..^ ( M  /  N ) )  |  ( y  gcd  ( M  /  N ) )  =  1 }  |->  ( z  x.  N ) ) : { y  e.  ( 0..^ ( M  /  N ) )  |  ( y  gcd  ( M  /  N
) )  =  1 } -1-1-onto-> { x  e.  ( 0..^ M )  |  ( x  gcd  M
)  =  N }  ->  { y  e.  ( 0..^ ( M  /  N ) )  |  ( y  gcd  ( M  /  N ) )  =  1 }  ~~  { x  e.  ( 0..^ M )  |  ( x  gcd  M )  =  N } )
15 hasheni 11347 . . . 4  |-  ( { y  e.  ( 0..^ ( M  /  N
) )  |  ( y  gcd  ( M  /  N ) )  =  1 }  ~~  { x  e.  ( 0..^ M )  |  ( x  gcd  M )  =  N }  ->  (
# `  { y  e.  ( 0..^ ( M  /  N ) )  |  ( y  gcd  ( M  /  N
) )  =  1 } )  =  (
# `  { x  e.  ( 0..^ M )  |  ( x  gcd  M )  =  N }
) )
1611, 14, 153syl 18 . . 3  |-  ( ( ( M  e.  NN  /\  N  e.  NN )  /\  N  ||  M
)  ->  ( # `  {
y  e.  ( 0..^ ( M  /  N
) )  |  ( y  gcd  ( M  /  N ) )  =  1 } )  =  ( # `  {
x  e.  ( 0..^ M )  |  ( x  gcd  M )  =  N } ) )
176, 16eqtr2d 2316 . 2  |-  ( ( ( M  e.  NN  /\  N  e.  NN )  /\  N  ||  M
)  ->  ( # `  {
x  e.  ( 0..^ M )  |  ( x  gcd  M )  =  N } )  =  ( phi `  ( M  /  N
) ) )
18 simprr 733 . . . . . . . . . 10  |-  ( ( ( M  e.  NN  /\  N  e.  NN )  /\  ( x  e.  ( 0..^ M )  /\  ( x  gcd  M )  =  N ) )  ->  ( x  gcd  M )  =  N )
19 elfzoelz 10875 . . . . . . . . . . . . 13  |-  ( x  e.  ( 0..^ M )  ->  x  e.  ZZ )
2019ad2antrl 708 . . . . . . . . . . . 12  |-  ( ( ( M  e.  NN  /\  N  e.  NN )  /\  ( x  e.  ( 0..^ M )  /\  ( x  gcd  M )  =  N ) )  ->  x  e.  ZZ )
21 nnz 10045 . . . . . . . . . . . . 13  |-  ( M  e.  NN  ->  M  e.  ZZ )
2221ad2antrr 706 . . . . . . . . . . . 12  |-  ( ( ( M  e.  NN  /\  N  e.  NN )  /\  ( x  e.  ( 0..^ M )  /\  ( x  gcd  M )  =  N ) )  ->  M  e.  ZZ )
23 gcddvds 12694 . . . . . . . . . . . 12  |-  ( ( x  e.  ZZ  /\  M  e.  ZZ )  ->  ( ( x  gcd  M )  ||  x  /\  ( x  gcd  M ) 
||  M ) )
2420, 22, 23syl2anc 642 . . . . . . . . . . 11  |-  ( ( ( M  e.  NN  /\  N  e.  NN )  /\  ( x  e.  ( 0..^ M )  /\  ( x  gcd  M )  =  N ) )  ->  ( (
x  gcd  M )  ||  x  /\  (
x  gcd  M )  ||  M ) )
2524simprd 449 . . . . . . . . . 10  |-  ( ( ( M  e.  NN  /\  N  e.  NN )  /\  ( x  e.  ( 0..^ M )  /\  ( x  gcd  M )  =  N ) )  ->  ( x  gcd  M )  ||  M
)
2618, 25eqbrtrrd 4045 . . . . . . . . 9  |-  ( ( ( M  e.  NN  /\  N  e.  NN )  /\  ( x  e.  ( 0..^ M )  /\  ( x  gcd  M )  =  N ) )  ->  N  ||  M
)
2726expr 598 . . . . . . . 8  |-  ( ( ( M  e.  NN  /\  N  e.  NN )  /\  x  e.  ( 0..^ M ) )  ->  ( ( x  gcd  M )  =  N  ->  N  ||  M
) )
2827con3d 125 . . . . . . 7  |-  ( ( ( M  e.  NN  /\  N  e.  NN )  /\  x  e.  ( 0..^ M ) )  ->  ( -.  N  ||  M  ->  -.  (
x  gcd  M )  =  N ) )
2928impancom 427 . . . . . 6  |-  ( ( ( M  e.  NN  /\  N  e.  NN )  /\  -.  N  ||  M )  ->  (
x  e.  ( 0..^ M )  ->  -.  ( x  gcd  M )  =  N ) )
3029ralrimiv 2625 . . . . 5  |-  ( ( ( M  e.  NN  /\  N  e.  NN )  /\  -.  N  ||  M )  ->  A. x  e.  ( 0..^ M )  -.  ( x  gcd  M )  =  N )
31 rabeq0 3476 . . . . 5  |-  ( { x  e.  ( 0..^ M )  |  ( x  gcd  M )  =  N }  =  (/)  <->  A. x  e.  ( 0..^ M )  -.  (
x  gcd  M )  =  N )
3230, 31sylibr 203 . . . 4  |-  ( ( ( M  e.  NN  /\  N  e.  NN )  /\  -.  N  ||  M )  ->  { x  e.  ( 0..^ M )  |  ( x  gcd  M )  =  N }  =  (/) )
3332fveq2d 5529 . . 3  |-  ( ( ( M  e.  NN  /\  N  e.  NN )  /\  -.  N  ||  M )  ->  ( # `
 { x  e.  ( 0..^ M )  |  ( x  gcd  M )  =  N }
)  =  ( # `  (/) ) )
34 hash0 11355 . . 3  |-  ( # `  (/) )  =  0
3533, 34syl6eq 2331 . 2  |-  ( ( ( M  e.  NN  /\  N  e.  NN )  /\  -.  N  ||  M )  ->  ( # `
 { x  e.  ( 0..^ M )  |  ( x  gcd  M )  =  N }
)  =  0 )
361, 2, 17, 35ifbothda 3595 1  |-  ( ( M  e.  NN  /\  N  e.  NN )  ->  ( # `  {
x  e.  ( 0..^ M )  |  ( x  gcd  M )  =  N } )  =  if ( N 
||  M ,  ( phi `  ( M  /  N ) ) ,  0 ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 358    = wceq 1623    e. wcel 1684   A.wral 2543   {crab 2547   (/)c0 3455   ifcif 3565   class class class wbr 4023    e. cmpt 4077   -1-1-onto->wf1o 5254   ` cfv 5255  (class class class)co 5858    ~~ cen 6860   0cc0 8737   1c1 8738    x. cmul 8742    / cdiv 9423   NNcn 9746   ZZcz 10024  ..^cfzo 10870   #chash 11337    || cdivides 12531    gcd cgcd 12685   phicphi 12832
This theorem is referenced by:  phisum  27518
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-rep 4131  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-int 3863  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-1o 6479  df-oadd 6483  df-er 6660  df-en 6864  df-dom 6865  df-sdom 6866  df-fin 6867  df-sup 7194  df-card 7572  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-hash 11338  df-cj 11584  df-re 11585  df-im 11586  df-sqr 11720  df-abs 11721  df-dvds 12532  df-gcd 12686  df-phi 12834
  Copyright terms: Public domain W3C validator