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

Theorem hashgcdeq 30754
Description: Number of initial positive integers 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 2477 . 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 2477 . 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 13844 . . . . 5  |-  ( ( M  e.  NN  /\  N  e.  NN )  ->  ( N  ||  M  <->  ( M  /  N )  e.  NN ) )
43biimpa 484 . . . 4  |-  ( ( ( M  e.  NN  /\  N  e.  NN )  /\  N  ||  M
)  ->  ( M  /  N )  e.  NN )
5 dfphi2 14154 . . . 4  |-  ( ( M  /  N )  e.  NN  ->  ( phi `  ( M  /  N ) )  =  ( # `  {
y  e.  ( 0..^ ( M  /  N
) )  |  ( y  gcd  ( M  /  N ) )  =  1 } ) )
64, 5syl 16 . . 3  |-  ( ( ( M  e.  NN  /\  N  e.  NN )  /\  N  ||  M
)  ->  ( phi `  ( M  /  N
) )  =  (
# `  { y  e.  ( 0..^ ( M  /  N ) )  |  ( y  gcd  ( M  /  N
) )  =  1 } ) )
7 eqid 2462 . . . . . 6  |-  { y  e.  ( 0..^ ( M  /  N ) )  |  ( y  gcd  ( M  /  N ) )  =  1 }  =  {
y  e.  ( 0..^ ( M  /  N
) )  |  ( y  gcd  ( M  /  N ) )  =  1 }
8 eqid 2462 . . . . . 6  |-  { x  e.  ( 0..^ M )  |  ( x  gcd  M )  =  N }  =  { x  e.  ( 0..^ M )  |  ( x  gcd  M
)  =  N }
9 eqid 2462 . . . . . 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 30753 . . . . 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 1191 . . . 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 6302 . . . . . 6  |-  ( 0..^ ( M  /  N
) )  e.  _V
1312rabex 4593 . . . . 5  |-  { y  e.  ( 0..^ ( M  /  N ) )  |  ( y  gcd  ( M  /  N ) )  =  1 }  e.  _V
1413f1oen 7528 . . . 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 12378 . . . 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 20 . . 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 2504 . 2  |-  ( ( ( M  e.  NN  /\  N  e.  NN )  /\  N  ||  M
)  ->  ( # `  {
x  e.  ( 0..^ M )  |  ( x  gcd  M )  =  N } )  =  ( phi `  ( M  /  N
) ) )
18 simprr 756 . . . . . . . . . 10  |-  ( ( ( M  e.  NN  /\  N  e.  NN )  /\  ( x  e.  ( 0..^ M )  /\  ( x  gcd  M )  =  N ) )  ->  ( x  gcd  M )  =  N )
19 elfzoelz 11788 . . . . . . . . . . . . 13  |-  ( x  e.  ( 0..^ M )  ->  x  e.  ZZ )
2019ad2antrl 727 . . . . . . . . . . . 12  |-  ( ( ( M  e.  NN  /\  N  e.  NN )  /\  ( x  e.  ( 0..^ M )  /\  ( x  gcd  M )  =  N ) )  ->  x  e.  ZZ )
21 nnz 10877 . . . . . . . . . . . . 13  |-  ( M  e.  NN  ->  M  e.  ZZ )
2221ad2antrr 725 . . . . . . . . . . . 12  |-  ( ( ( M  e.  NN  /\  N  e.  NN )  /\  ( x  e.  ( 0..^ M )  /\  ( x  gcd  M )  =  N ) )  ->  M  e.  ZZ )
23 gcddvds 14003 . . . . . . . . . . . 12  |-  ( ( x  e.  ZZ  /\  M  e.  ZZ )  ->  ( ( x  gcd  M )  ||  x  /\  ( x  gcd  M ) 
||  M ) )
2420, 22, 23syl2anc 661 . . . . . . . . . . 11  |-  ( ( ( M  e.  NN  /\  N  e.  NN )  /\  ( x  e.  ( 0..^ M )  /\  ( x  gcd  M )  =  N ) )  ->  ( (
x  gcd  M )  ||  x  /\  (
x  gcd  M )  ||  M ) )
2524simprd 463 . . . . . . . . . 10  |-  ( ( ( M  e.  NN  /\  N  e.  NN )  /\  ( x  e.  ( 0..^ M )  /\  ( x  gcd  M )  =  N ) )  ->  ( x  gcd  M )  ||  M
)
2618, 25eqbrtrrd 4464 . . . . . . . . 9  |-  ( ( ( M  e.  NN  /\  N  e.  NN )  /\  ( x  e.  ( 0..^ M )  /\  ( x  gcd  M )  =  N ) )  ->  N  ||  M
)
2726expr 615 . . . . . . . 8  |-  ( ( ( M  e.  NN  /\  N  e.  NN )  /\  x  e.  ( 0..^ M ) )  ->  ( ( x  gcd  M )  =  N  ->  N  ||  M
) )
2827con3d 133 . . . . . . 7  |-  ( ( ( M  e.  NN  /\  N  e.  NN )  /\  x  e.  ( 0..^ M ) )  ->  ( -.  N  ||  M  ->  -.  (
x  gcd  M )  =  N ) )
2928impancom 440 . . . . . 6  |-  ( ( ( M  e.  NN  /\  N  e.  NN )  /\  -.  N  ||  M )  ->  (
x  e.  ( 0..^ M )  ->  -.  ( x  gcd  M )  =  N ) )
3029ralrimiv 2871 . . . . 5  |-  ( ( ( M  e.  NN  /\  N  e.  NN )  /\  -.  N  ||  M )  ->  A. x  e.  ( 0..^ M )  -.  ( x  gcd  M )  =  N )
31 rabeq0 3802 . . . . 5  |-  ( { x  e.  ( 0..^ M )  |  ( x  gcd  M )  =  N }  =  (/)  <->  A. x  e.  ( 0..^ M )  -.  (
x  gcd  M )  =  N )
3230, 31sylibr 212 . . . 4  |-  ( ( ( M  e.  NN  /\  N  e.  NN )  /\  -.  N  ||  M )  ->  { x  e.  ( 0..^ M )  |  ( x  gcd  M )  =  N }  =  (/) )
3332fveq2d 5863 . . 3  |-  ( ( ( M  e.  NN  /\  N  e.  NN )  /\  -.  N  ||  M )  ->  ( # `
 { x  e.  ( 0..^ M )  |  ( x  gcd  M )  =  N }
)  =  ( # `  (/) ) )
34 hash0 12394 . . 3  |-  ( # `  (/) )  =  0
3533, 34syl6eq 2519 . 2  |-  ( ( ( M  e.  NN  /\  N  e.  NN )  /\  -.  N  ||  M )  ->  ( # `
 { x  e.  ( 0..^ M )  |  ( x  gcd  M )  =  N }
)  =  0 )
361, 2, 17, 35ifbothda 3969 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 setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 369    = wceq 1374    e. wcel 1762   A.wral 2809   {crab 2813   (/)c0 3780   ifcif 3934   class class class wbr 4442    |-> cmpt 4500   -1-1-onto->wf1o 5580   ` cfv 5581  (class class class)co 6277    ~~ cen 7505   0cc0 9483   1c1 9484    x. cmul 9488    / cdiv 10197   NNcn 10527   ZZcz 10855  ..^cfzo 11783   #chash 12362    || cdivides 13838    gcd cgcd 13994   phicphi 14144
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1963  ax-ext 2440  ax-rep 4553  ax-sep 4563  ax-nul 4571  ax-pow 4620  ax-pr 4681  ax-un 6569  ax-cnex 9539  ax-resscn 9540  ax-1cn 9541  ax-icn 9542  ax-addcl 9543  ax-addrcl 9544  ax-mulcl 9545  ax-mulrcl 9546  ax-mulcom 9547  ax-addass 9548  ax-mulass 9549  ax-distr 9550  ax-i2m1 9551  ax-1ne0 9552  ax-1rid 9553  ax-rnegex 9554  ax-rrecex 9555  ax-cnre 9556  ax-pre-lttri 9557  ax-pre-lttrn 9558  ax-pre-ltadd 9559  ax-pre-mulgt0 9560  ax-pre-sup 9561
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 969  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2274  df-mo 2275  df-clab 2448  df-cleq 2454  df-clel 2457  df-nfc 2612  df-ne 2659  df-nel 2660  df-ral 2814  df-rex 2815  df-reu 2816  df-rmo 2817  df-rab 2818  df-v 3110  df-sbc 3327  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3781  df-if 3935  df-pw 4007  df-sn 4023  df-pr 4025  df-tp 4027  df-op 4029  df-uni 4241  df-int 4278  df-iun 4322  df-br 4443  df-opab 4501  df-mpt 4502  df-tr 4536  df-eprel 4786  df-id 4790  df-po 4795  df-so 4796  df-fr 4833  df-we 4835  df-ord 4876  df-on 4877  df-lim 4878  df-suc 4879  df-xp 5000  df-rel 5001  df-cnv 5002  df-co 5003  df-dm 5004  df-rn 5005  df-res 5006  df-ima 5007  df-iota 5544  df-fun 5583  df-fn 5584  df-f 5585  df-f1 5586  df-fo 5587  df-f1o 5588  df-fv 5589  df-riota 6238  df-ov 6280  df-oprab 6281  df-mpt2 6282  df-om 6674  df-1st 6776  df-2nd 6777  df-recs 7034  df-rdg 7068  df-1o 7122  df-oadd 7126  df-er 7303  df-en 7509  df-dom 7510  df-sdom 7511  df-fin 7512  df-sup 7892  df-card 8311  df-pnf 9621  df-mnf 9622  df-xr 9623  df-ltxr 9624  df-le 9625  df-sub 9798  df-neg 9799  df-div 10198  df-nn 10528  df-2 10585  df-3 10586  df-n0 10787  df-z 10856  df-uz 11074  df-rp 11212  df-fz 11664  df-fzo 11784  df-fl 11888  df-mod 11955  df-seq 12066  df-exp 12125  df-hash 12363  df-cj 12884  df-re 12885  df-im 12886  df-sqr 13020  df-abs 13021  df-dvds 13839  df-gcd 13995  df-phi 14146
This theorem is referenced by:  phisum  30755
  Copyright terms: Public domain W3C validator