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Theorem hashgcdeq 27494
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 2445 . 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 2445 . 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 12858 . . . . 5  |-  ( ( M  e.  NN  /\  N  e.  NN )  ->  ( N  ||  M  <->  ( M  /  N )  e.  NN ) )
43biimpa 471 . . . 4  |-  ( ( ( M  e.  NN  /\  N  e.  NN )  /\  N  ||  M
)  ->  ( M  /  N )  e.  NN )
5 dfphi2 13163 . . . 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 2436 . . . . . 6  |-  { y  e.  ( 0..^ ( M  /  N ) )  |  ( y  gcd  ( M  /  N ) )  =  1 }  =  {
y  e.  ( 0..^ ( M  /  N
) )  |  ( y  gcd  ( M  /  N ) )  =  1 }
8 eqid 2436 . . . . . 6  |-  { x  e.  ( 0..^ M )  |  ( x  gcd  M )  =  N }  =  { x  e.  ( 0..^ M )  |  ( x  gcd  M
)  =  N }
9 eqid 2436 . . . . . 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 27493 . . . . 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 1153 . . . 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 6106 . . . . . 6  |-  ( 0..^ ( M  /  N
) )  e.  _V
1312rabex 4354 . . . . 5  |-  { y  e.  ( 0..^ ( M  /  N ) )  |  ( y  gcd  ( M  /  N ) )  =  1 }  e.  _V
1413f1oen 7128 . . . 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 11632 . . . 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 19 . . 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 2469 . 2  |-  ( ( ( M  e.  NN  /\  N  e.  NN )  /\  N  ||  M
)  ->  ( # `  {
x  e.  ( 0..^ M )  |  ( x  gcd  M )  =  N } )  =  ( phi `  ( M  /  N
) ) )
18 simprr 734 . . . . . . . . . 10  |-  ( ( ( M  e.  NN  /\  N  e.  NN )  /\  ( x  e.  ( 0..^ M )  /\  ( x  gcd  M )  =  N ) )  ->  ( x  gcd  M )  =  N )
19 elfzoelz 11140 . . . . . . . . . . . . 13  |-  ( x  e.  ( 0..^ M )  ->  x  e.  ZZ )
2019ad2antrl 709 . . . . . . . . . . . 12  |-  ( ( ( M  e.  NN  /\  N  e.  NN )  /\  ( x  e.  ( 0..^ M )  /\  ( x  gcd  M )  =  N ) )  ->  x  e.  ZZ )
21 nnz 10303 . . . . . . . . . . . . 13  |-  ( M  e.  NN  ->  M  e.  ZZ )
2221ad2antrr 707 . . . . . . . . . . . 12  |-  ( ( ( M  e.  NN  /\  N  e.  NN )  /\  ( x  e.  ( 0..^ M )  /\  ( x  gcd  M )  =  N ) )  ->  M  e.  ZZ )
23 gcddvds 13015 . . . . . . . . . . . 12  |-  ( ( x  e.  ZZ  /\  M  e.  ZZ )  ->  ( ( x  gcd  M )  ||  x  /\  ( x  gcd  M ) 
||  M ) )
2420, 22, 23syl2anc 643 . . . . . . . . . . 11  |-  ( ( ( M  e.  NN  /\  N  e.  NN )  /\  ( x  e.  ( 0..^ M )  /\  ( x  gcd  M )  =  N ) )  ->  ( (
x  gcd  M )  ||  x  /\  (
x  gcd  M )  ||  M ) )
2524simprd 450 . . . . . . . . . 10  |-  ( ( ( M  e.  NN  /\  N  e.  NN )  /\  ( x  e.  ( 0..^ M )  /\  ( x  gcd  M )  =  N ) )  ->  ( x  gcd  M )  ||  M
)
2618, 25eqbrtrrd 4234 . . . . . . . . 9  |-  ( ( ( M  e.  NN  /\  N  e.  NN )  /\  ( x  e.  ( 0..^ M )  /\  ( x  gcd  M )  =  N ) )  ->  N  ||  M
)
2726expr 599 . . . . . . . 8  |-  ( ( ( M  e.  NN  /\  N  e.  NN )  /\  x  e.  ( 0..^ M ) )  ->  ( ( x  gcd  M )  =  N  ->  N  ||  M
) )
2827con3d 127 . . . . . . 7  |-  ( ( ( M  e.  NN  /\  N  e.  NN )  /\  x  e.  ( 0..^ M ) )  ->  ( -.  N  ||  M  ->  -.  (
x  gcd  M )  =  N ) )
2928impancom 428 . . . . . 6  |-  ( ( ( M  e.  NN  /\  N  e.  NN )  /\  -.  N  ||  M )  ->  (
x  e.  ( 0..^ M )  ->  -.  ( x  gcd  M )  =  N ) )
3029ralrimiv 2788 . . . . 5  |-  ( ( ( M  e.  NN  /\  N  e.  NN )  /\  -.  N  ||  M )  ->  A. x  e.  ( 0..^ M )  -.  ( x  gcd  M )  =  N )
31 rabeq0 3649 . . . . 5  |-  ( { x  e.  ( 0..^ M )  |  ( x  gcd  M )  =  N }  =  (/)  <->  A. x  e.  ( 0..^ M )  -.  (
x  gcd  M )  =  N )
3230, 31sylibr 204 . . . 4  |-  ( ( ( M  e.  NN  /\  N  e.  NN )  /\  -.  N  ||  M )  ->  { x  e.  ( 0..^ M )  |  ( x  gcd  M )  =  N }  =  (/) )
3332fveq2d 5732 . . 3  |-  ( ( ( M  e.  NN  /\  N  e.  NN )  /\  -.  N  ||  M )  ->  ( # `
 { x  e.  ( 0..^ M )  |  ( x  gcd  M )  =  N }
)  =  ( # `  (/) ) )
34 hash0 11646 . . 3  |-  ( # `  (/) )  =  0
3533, 34syl6eq 2484 . 2  |-  ( ( ( M  e.  NN  /\  N  e.  NN )  /\  -.  N  ||  M )  ->  ( # `
 { x  e.  ( 0..^ M )  |  ( x  gcd  M )  =  N }
)  =  0 )
361, 2, 17, 35ifbothda 3769 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 359    = wceq 1652    e. wcel 1725   A.wral 2705   {crab 2709   (/)c0 3628   ifcif 3739   class class class wbr 4212    e. cmpt 4266   -1-1-onto->wf1o 5453   ` cfv 5454  (class class class)co 6081    ~~ cen 7106   0cc0 8990   1c1 8991    x. cmul 8995    / cdiv 9677   NNcn 10000   ZZcz 10282  ..^cfzo 11135   #chash 11618    || cdivides 12852    gcd cgcd 13006   phicphi 13153
This theorem is referenced by:  phisum  27495
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-rep 4320  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701  ax-cnex 9046  ax-resscn 9047  ax-1cn 9048  ax-icn 9049  ax-addcl 9050  ax-addrcl 9051  ax-mulcl 9052  ax-mulrcl 9053  ax-mulcom 9054  ax-addass 9055  ax-mulass 9056  ax-distr 9057  ax-i2m1 9058  ax-1ne0 9059  ax-1rid 9060  ax-rnegex 9061  ax-rrecex 9062  ax-cnre 9063  ax-pre-lttri 9064  ax-pre-lttrn 9065  ax-pre-ltadd 9066  ax-pre-mulgt0 9067  ax-pre-sup 9068
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-nel 2602  df-ral 2710  df-rex 2711  df-reu 2712  df-rmo 2713  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-pss 3336  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-tp 3822  df-op 3823  df-uni 4016  df-int 4051  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-tr 4303  df-eprel 4494  df-id 4498  df-po 4503  df-so 4504  df-fr 4541  df-we 4543  df-ord 4584  df-on 4585  df-lim 4586  df-suc 4587  df-om 4846  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-1st 6349  df-2nd 6350  df-riota 6549  df-recs 6633  df-rdg 6668  df-1o 6724  df-oadd 6728  df-er 6905  df-en 7110  df-dom 7111  df-sdom 7112  df-fin 7113  df-sup 7446  df-card 7826  df-pnf 9122  df-mnf 9123  df-xr 9124  df-ltxr 9125  df-le 9126  df-sub 9293  df-neg 9294  df-div 9678  df-nn 10001  df-2 10058  df-3 10059  df-n0 10222  df-z 10283  df-uz 10489  df-rp 10613  df-fz 11044  df-fzo 11136  df-fl 11202  df-mod 11251  df-seq 11324  df-exp 11383  df-hash 11619  df-cj 11904  df-re 11905  df-im 11906  df-sqr 12040  df-abs 12041  df-dvds 12853  df-gcd 13007  df-phi 13155
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