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Theorem hashgcdeq 12256
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 2198 . 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 2198 . 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 11820 . . . . 5  |-  ( ( M  e.  NN  /\  N  e.  NN )  ->  ( N  ||  M  <->  ( M  /  N )  e.  NN ) )
43biimpa 296 . . . 4  |-  ( ( ( M  e.  NN  /\  N  e.  NN )  /\  N  ||  M
)  ->  ( M  /  N )  e.  NN )
5 dfphi2 12237 . . . 4  |-  ( ( M  /  N )  e.  NN  ->  ( phi `  ( M  /  N ) )  =  ( `  { y  e.  ( 0..^ ( M  /  N ) )  |  ( y  gcd  ( M  /  N
) )  =  1 } ) )
64, 5syl 14 . . 3  |-  ( ( ( M  e.  NN  /\  N  e.  NN )  /\  N  ||  M
)  ->  ( phi `  ( M  /  N
) )  =  ( `  { y  e.  ( 0..^ ( M  /  N ) )  |  ( y  gcd  ( M  /  N ) )  =  1 } ) )
7 0z 9281 . . . . . 6  |-  0  e.  ZZ
84nnzd 9391 . . . . . 6  |-  ( ( ( M  e.  NN  /\  N  e.  NN )  /\  N  ||  M
)  ->  ( M  /  N )  e.  ZZ )
9 fzofig 10449 . . . . . 6  |-  ( ( 0  e.  ZZ  /\  ( M  /  N
)  e.  ZZ )  ->  ( 0..^ ( M  /  N ) )  e.  Fin )
107, 8, 9sylancr 414 . . . . 5  |-  ( ( ( M  e.  NN  /\  N  e.  NN )  /\  N  ||  M
)  ->  ( 0..^ ( M  /  N
) )  e.  Fin )
11 elfzoelz 10164 . . . . . . . . . 10  |-  ( y  e.  ( 0..^ ( M  /  N ) )  ->  y  e.  ZZ )
1211adantl 277 . . . . . . . . 9  |-  ( ( ( ( M  e.  NN  /\  N  e.  NN )  /\  N  ||  M )  /\  y  e.  ( 0..^ ( M  /  N ) ) )  ->  y  e.  ZZ )
138adantr 276 . . . . . . . . 9  |-  ( ( ( ( M  e.  NN  /\  N  e.  NN )  /\  N  ||  M )  /\  y  e.  ( 0..^ ( M  /  N ) ) )  ->  ( M  /  N )  e.  ZZ )
1412, 13gcdcld 11986 . . . . . . . 8  |-  ( ( ( ( M  e.  NN  /\  N  e.  NN )  /\  N  ||  M )  /\  y  e.  ( 0..^ ( M  /  N ) ) )  ->  ( y  gcd  ( M  /  N
) )  e.  NN0 )
1514nn0zd 9390 . . . . . . 7  |-  ( ( ( ( M  e.  NN  /\  N  e.  NN )  /\  N  ||  M )  /\  y  e.  ( 0..^ ( M  /  N ) ) )  ->  ( y  gcd  ( M  /  N
) )  e.  ZZ )
16 1zzd 9297 . . . . . . 7  |-  ( ( ( ( M  e.  NN  /\  N  e.  NN )  /\  N  ||  M )  /\  y  e.  ( 0..^ ( M  /  N ) ) )  ->  1  e.  ZZ )
17 zdceq 9345 . . . . . . 7  |-  ( ( ( y  gcd  ( M  /  N ) )  e.  ZZ  /\  1  e.  ZZ )  -> DECID  ( y  gcd  ( M  /  N ) )  =  1 )
1815, 16, 17syl2anc 411 . . . . . 6  |-  ( ( ( ( M  e.  NN  /\  N  e.  NN )  /\  N  ||  M )  /\  y  e.  ( 0..^ ( M  /  N ) ) )  -> DECID  ( y  gcd  ( M  /  N ) )  =  1 )
1918ralrimiva 2562 . . . . 5  |-  ( ( ( M  e.  NN  /\  N  e.  NN )  /\  N  ||  M
)  ->  A. y  e.  ( 0..^ ( M  /  N ) )DECID  ( y  gcd  ( M  /  N ) )  =  1 )
2010, 19ssfirab 6950 . . . 4  |-  ( ( ( M  e.  NN  /\  N  e.  NN )  /\  N  ||  M
)  ->  { y  e.  ( 0..^ ( M  /  N ) )  |  ( y  gcd  ( M  /  N
) )  =  1 }  e.  Fin )
21 eqid 2188 . . . . . 6  |-  { y  e.  ( 0..^ ( M  /  N ) )  |  ( y  gcd  ( M  /  N ) )  =  1 }  =  {
y  e.  ( 0..^ ( M  /  N
) )  |  ( y  gcd  ( M  /  N ) )  =  1 }
22 eqid 2188 . . . . . 6  |-  { x  e.  ( 0..^ M )  |  ( x  gcd  M )  =  N }  =  { x  e.  ( 0..^ M )  |  ( x  gcd  M
)  =  N }
23 eqid 2188 . . . . . 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 ) )
2421, 22, 23hashgcdlem 12255 . . . . 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 }
)
25243expa 1204 . . . 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 }
)
2620, 25fihasheqf1od 10786 . . 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 } ) )
276, 26eqtr2d 2222 . 2  |-  ( ( ( M  e.  NN  /\  N  e.  NN )  /\  N  ||  M
)  ->  ( `  {
x  e.  ( 0..^ M )  |  ( x  gcd  M )  =  N } )  =  ( phi `  ( M  /  N
) ) )
28 simprr 531 . . . . . . . . . 10  |-  ( ( ( M  e.  NN  /\  N  e.  NN )  /\  ( x  e.  ( 0..^ M )  /\  ( x  gcd  M )  =  N ) )  ->  ( x  gcd  M )  =  N )
29 elfzoelz 10164 . . . . . . . . . . . . 13  |-  ( x  e.  ( 0..^ M )  ->  x  e.  ZZ )
3029ad2antrl 490 . . . . . . . . . . . 12  |-  ( ( ( M  e.  NN  /\  N  e.  NN )  /\  ( x  e.  ( 0..^ M )  /\  ( x  gcd  M )  =  N ) )  ->  x  e.  ZZ )
31 nnz 9289 . . . . . . . . . . . . 13  |-  ( M  e.  NN  ->  M  e.  ZZ )
3231ad2antrr 488 . . . . . . . . . . . 12  |-  ( ( ( M  e.  NN  /\  N  e.  NN )  /\  ( x  e.  ( 0..^ M )  /\  ( x  gcd  M )  =  N ) )  ->  M  e.  ZZ )
33 gcddvds 11981 . . . . . . . . . . . 12  |-  ( ( x  e.  ZZ  /\  M  e.  ZZ )  ->  ( ( x  gcd  M )  ||  x  /\  ( x  gcd  M ) 
||  M ) )
3430, 32, 33syl2anc 411 . . . . . . . . . . 11  |-  ( ( ( M  e.  NN  /\  N  e.  NN )  /\  ( x  e.  ( 0..^ M )  /\  ( x  gcd  M )  =  N ) )  ->  ( (
x  gcd  M )  ||  x  /\  (
x  gcd  M )  ||  M ) )
3534simprd 114 . . . . . . . . . 10  |-  ( ( ( M  e.  NN  /\  N  e.  NN )  /\  ( x  e.  ( 0..^ M )  /\  ( x  gcd  M )  =  N ) )  ->  ( x  gcd  M )  ||  M
)
3628, 35eqbrtrrd 4041 . . . . . . . . 9  |-  ( ( ( M  e.  NN  /\  N  e.  NN )  /\  ( x  e.  ( 0..^ M )  /\  ( x  gcd  M )  =  N ) )  ->  N  ||  M
)
3736expr 375 . . . . . . . 8  |-  ( ( ( M  e.  NN  /\  N  e.  NN )  /\  x  e.  ( 0..^ M ) )  ->  ( ( x  gcd  M )  =  N  ->  N  ||  M
) )
3837con3d 632 . . . . . . 7  |-  ( ( ( M  e.  NN  /\  N  e.  NN )  /\  x  e.  ( 0..^ M ) )  ->  ( -.  N  ||  M  ->  -.  (
x  gcd  M )  =  N ) )
3938impancom 260 . . . . . 6  |-  ( ( ( M  e.  NN  /\  N  e.  NN )  /\  -.  N  ||  M )  ->  (
x  e.  ( 0..^ M )  ->  -.  ( x  gcd  M )  =  N ) )
4039ralrimiv 2561 . . . . 5  |-  ( ( ( M  e.  NN  /\  N  e.  NN )  /\  -.  N  ||  M )  ->  A. x  e.  ( 0..^ M )  -.  ( x  gcd  M )  =  N )
41 rabeq0 3466 . . . . 5  |-  ( { x  e.  ( 0..^ M )  |  ( x  gcd  M )  =  N }  =  (/)  <->  A. x  e.  ( 0..^ M )  -.  (
x  gcd  M )  =  N )
4240, 41sylibr 134 . . . 4  |-  ( ( ( M  e.  NN  /\  N  e.  NN )  /\  -.  N  ||  M )  ->  { x  e.  ( 0..^ M )  |  ( x  gcd  M )  =  N }  =  (/) )
4342fveq2d 5533 . . 3  |-  ( ( ( M  e.  NN  /\  N  e.  NN )  /\  -.  N  ||  M )  ->  ( `  { x  e.  ( 0..^ M )  |  ( x  gcd  M
)  =  N }
)  =  ( `  (/) ) )
44 hash0 10793 . . 3  |-  ( `  (/) )  =  0
4543, 44eqtrdi 2237 . 2  |-  ( ( ( M  e.  NN  /\  N  e.  NN )  /\  -.  N  ||  M )  ->  ( `  { x  e.  ( 0..^ M )  |  ( x  gcd  M
)  =  N }
)  =  0 )
46 dvdsdc 11822 . . . 4  |-  ( ( N  e.  NN  /\  M  e.  ZZ )  -> DECID  N 
||  M )
4731, 46sylan2 286 . . 3  |-  ( ( N  e.  NN  /\  M  e.  NN )  -> DECID  N 
||  M )
4847ancoms 268 . 2  |-  ( ( M  e.  NN  /\  N  e.  NN )  -> DECID  N 
||  M )
491, 2, 27, 45, 48ifbothdadc 3580 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 104  DECID wdc 835    = wceq 1363    e. wcel 2159   A.wral 2467   {crab 2471   (/)c0 3436   ifcif 3548   class class class wbr 4017    |-> cmpt 4078   -1-1-onto->wf1o 5229   ` cfv 5230  (class class class)co 5890   Fincfn 6757   0cc0 7828   1c1 7829    x. cmul 7833    / cdiv 8646   NNcn 8936   ZZcz 9270  ..^cfzo 10159  ♯chash 10772    || cdvds 11811    gcd cgcd 11960   phicphi 12226
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-13 2161  ax-14 2162  ax-ext 2170  ax-coll 4132  ax-sep 4135  ax-nul 4143  ax-pow 4188  ax-pr 4223  ax-un 4447  ax-setind 4550  ax-iinf 4601  ax-cnex 7919  ax-resscn 7920  ax-1cn 7921  ax-1re 7922  ax-icn 7923  ax-addcl 7924  ax-addrcl 7925  ax-mulcl 7926  ax-mulrcl 7927  ax-addcom 7928  ax-mulcom 7929  ax-addass 7930  ax-mulass 7931  ax-distr 7932  ax-i2m1 7933  ax-0lt1 7934  ax-1rid 7935  ax-0id 7936  ax-rnegex 7937  ax-precex 7938  ax-cnre 7939  ax-pre-ltirr 7940  ax-pre-ltwlin 7941  ax-pre-lttrn 7942  ax-pre-apti 7943  ax-pre-ltadd 7944  ax-pre-mulgt0 7945  ax-pre-mulext 7946  ax-arch 7947  ax-caucvg 7948
This theorem depends on definitions:  df-bi 117  df-stab 832  df-dc 836  df-3or 980  df-3an 981  df-tru 1366  df-fal 1369  df-nf 1471  df-sb 1773  df-eu 2040  df-mo 2041  df-clab 2175  df-cleq 2181  df-clel 2184  df-nfc 2320  df-ne 2360  df-nel 2455  df-ral 2472  df-rex 2473  df-reu 2474  df-rmo 2475  df-rab 2476  df-v 2753  df-sbc 2977  df-csb 3072  df-dif 3145  df-un 3147  df-in 3149  df-ss 3156  df-nul 3437  df-if 3549  df-pw 3591  df-sn 3612  df-pr 3613  df-op 3615  df-uni 3824  df-int 3859  df-iun 3902  df-br 4018  df-opab 4079  df-mpt 4080  df-tr 4116  df-id 4307  df-po 4310  df-iso 4311  df-iord 4380  df-on 4382  df-ilim 4383  df-suc 4385  df-iom 4604  df-xp 4646  df-rel 4647  df-cnv 4648  df-co 4649  df-dm 4650  df-rn 4651  df-res 4652  df-ima 4653  df-iota 5192  df-fun 5232  df-fn 5233  df-f 5234  df-f1 5235  df-fo 5236  df-f1o 5237  df-fv 5238  df-riota 5846  df-ov 5893  df-oprab 5894  df-mpo 5895  df-1st 6158  df-2nd 6159  df-recs 6323  df-frec 6409  df-1o 6434  df-er 6552  df-en 6758  df-dom 6759  df-fin 6760  df-sup 7000  df-pnf 8011  df-mnf 8012  df-xr 8013  df-ltxr 8014  df-le 8015  df-sub 8147  df-neg 8148  df-reap 8549  df-ap 8556  df-div 8647  df-inn 8937  df-2 8995  df-3 8996  df-4 8997  df-n0 9194  df-z 9271  df-uz 9546  df-q 9637  df-rp 9671  df-fz 10026  df-fzo 10160  df-fl 10287  df-mod 10340  df-seqfrec 10463  df-exp 10537  df-ihash 10773  df-cj 10868  df-re 10869  df-im 10870  df-rsqrt 11024  df-abs 11025  df-dvds 11812  df-gcd 11961  df-phi 12228
This theorem is referenced by:  phisum  12257
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