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Theorem hashgcdeq 11647
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 2104 . 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 2104 . 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 11245 . . . . 5  |-  ( ( M  e.  NN  /\  N  e.  NN )  ->  ( N  ||  M  <->  ( M  /  N )  e.  NN ) )
43biimpa 291 . . . 4  |-  ( ( ( M  e.  NN  /\  N  e.  NN )  /\  N  ||  M
)  ->  ( M  /  N )  e.  NN )
5 dfphi2 11639 . . . 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 8859 . . . . . 6  |-  0  e.  ZZ
84nnzd 8966 . . . . . 6  |-  ( ( ( M  e.  NN  /\  N  e.  NN )  /\  N  ||  M
)  ->  ( M  /  N )  e.  ZZ )
9 fzofig 9988 . . . . . 6  |-  ( ( 0  e.  ZZ  /\  ( M  /  N
)  e.  ZZ )  ->  ( 0..^ ( M  /  N ) )  e.  Fin )
107, 8, 9sylancr 406 . . . . 5  |-  ( ( ( M  e.  NN  /\  N  e.  NN )  /\  N  ||  M
)  ->  ( 0..^ ( M  /  N
) )  e.  Fin )
11 elfzoelz 9707 . . . . . . . . . 10  |-  ( y  e.  ( 0..^ ( M  /  N ) )  ->  y  e.  ZZ )
1211adantl 272 . . . . . . . . 9  |-  ( ( ( ( M  e.  NN  /\  N  e.  NN )  /\  N  ||  M )  /\  y  e.  ( 0..^ ( M  /  N ) ) )  ->  y  e.  ZZ )
138adantr 271 . . . . . . . . 9  |-  ( ( ( ( M  e.  NN  /\  N  e.  NN )  /\  N  ||  M )  /\  y  e.  ( 0..^ ( M  /  N ) ) )  ->  ( M  /  N )  e.  ZZ )
1412, 13gcdcld 11403 . . . . . . . 8  |-  ( ( ( ( M  e.  NN  /\  N  e.  NN )  /\  N  ||  M )  /\  y  e.  ( 0..^ ( M  /  N ) ) )  ->  ( y  gcd  ( M  /  N
) )  e.  NN0 )
1514nn0zd 8965 . . . . . . 7  |-  ( ( ( ( M  e.  NN  /\  N  e.  NN )  /\  N  ||  M )  /\  y  e.  ( 0..^ ( M  /  N ) ) )  ->  ( y  gcd  ( M  /  N
) )  e.  ZZ )
16 1zzd 8875 . . . . . . 7  |-  ( ( ( ( M  e.  NN  /\  N  e.  NN )  /\  N  ||  M )  /\  y  e.  ( 0..^ ( M  /  N ) ) )  ->  1  e.  ZZ )
17 zdceq 8920 . . . . . . 7  |-  ( ( ( y  gcd  ( M  /  N ) )  e.  ZZ  /\  1  e.  ZZ )  -> DECID  ( y  gcd  ( M  /  N ) )  =  1 )
1815, 16, 17syl2anc 404 . . . . . 6  |-  ( ( ( ( M  e.  NN  /\  N  e.  NN )  /\  N  ||  M )  /\  y  e.  ( 0..^ ( M  /  N ) ) )  -> DECID  ( y  gcd  ( M  /  N ) )  =  1 )
1918ralrimiva 2458 . . . . 5  |-  ( ( ( M  e.  NN  /\  N  e.  NN )  /\  N  ||  M
)  ->  A. y  e.  ( 0..^ ( M  /  N ) )DECID  ( y  gcd  ( M  /  N ) )  =  1 )
2010, 19ssfirab 6723 . . . 4  |-  ( ( ( M  e.  NN  /\  N  e.  NN )  /\  N  ||  M
)  ->  { y  e.  ( 0..^ ( M  /  N ) )  |  ( y  gcd  ( M  /  N
) )  =  1 }  e.  Fin )
21 eqid 2095 . . . . . 6  |-  { y  e.  ( 0..^ ( M  /  N ) )  |  ( y  gcd  ( M  /  N ) )  =  1 }  =  {
y  e.  ( 0..^ ( M  /  N
) )  |  ( y  gcd  ( M  /  N ) )  =  1 }
22 eqid 2095 . . . . . 6  |-  { x  e.  ( 0..^ M )  |  ( x  gcd  M )  =  N }  =  { x  e.  ( 0..^ M )  |  ( x  gcd  M
)  =  N }
23 eqid 2095 . . . . . 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 11646 . . . . 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 1146 . . . 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 10329 . . 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 2128 . 2  |-  ( ( ( M  e.  NN  /\  N  e.  NN )  /\  N  ||  M
)  ->  ( `  {
x  e.  ( 0..^ M )  |  ( x  gcd  M )  =  N } )  =  ( phi `  ( M  /  N
) ) )
28 simprr 500 . . . . . . . . . 10  |-  ( ( ( M  e.  NN  /\  N  e.  NN )  /\  ( x  e.  ( 0..^ M )  /\  ( x  gcd  M )  =  N ) )  ->  ( x  gcd  M )  =  N )
29 elfzoelz 9707 . . . . . . . . . . . . 13  |-  ( x  e.  ( 0..^ M )  ->  x  e.  ZZ )
3029ad2antrl 475 . . . . . . . . . . . 12  |-  ( ( ( M  e.  NN  /\  N  e.  NN )  /\  ( x  e.  ( 0..^ M )  /\  ( x  gcd  M )  =  N ) )  ->  x  e.  ZZ )
31 nnz 8867 . . . . . . . . . . . . 13  |-  ( M  e.  NN  ->  M  e.  ZZ )
3231ad2antrr 473 . . . . . . . . . . . 12  |-  ( ( ( M  e.  NN  /\  N  e.  NN )  /\  ( x  e.  ( 0..^ M )  /\  ( x  gcd  M )  =  N ) )  ->  M  e.  ZZ )
33 gcddvds 11398 . . . . . . . . . . . 12  |-  ( ( x  e.  ZZ  /\  M  e.  ZZ )  ->  ( ( x  gcd  M )  ||  x  /\  ( x  gcd  M ) 
||  M ) )
3430, 32, 33syl2anc 404 . . . . . . . . . . 11  |-  ( ( ( M  e.  NN  /\  N  e.  NN )  /\  ( x  e.  ( 0..^ M )  /\  ( x  gcd  M )  =  N ) )  ->  ( (
x  gcd  M )  ||  x  /\  (
x  gcd  M )  ||  M ) )
3534simprd 113 . . . . . . . . . 10  |-  ( ( ( M  e.  NN  /\  N  e.  NN )  /\  ( x  e.  ( 0..^ M )  /\  ( x  gcd  M )  =  N ) )  ->  ( x  gcd  M )  ||  M
)
3628, 35eqbrtrrd 3889 . . . . . . . . 9  |-  ( ( ( M  e.  NN  /\  N  e.  NN )  /\  ( x  e.  ( 0..^ M )  /\  ( x  gcd  M )  =  N ) )  ->  N  ||  M
)
3736expr 368 . . . . . . . 8  |-  ( ( ( M  e.  NN  /\  N  e.  NN )  /\  x  e.  ( 0..^ M ) )  ->  ( ( x  gcd  M )  =  N  ->  N  ||  M
) )
3837con3d 599 . . . . . . 7  |-  ( ( ( M  e.  NN  /\  N  e.  NN )  /\  x  e.  ( 0..^ M ) )  ->  ( -.  N  ||  M  ->  -.  (
x  gcd  M )  =  N ) )
3938impancom 257 . . . . . 6  |-  ( ( ( M  e.  NN  /\  N  e.  NN )  /\  -.  N  ||  M )  ->  (
x  e.  ( 0..^ M )  ->  -.  ( x  gcd  M )  =  N ) )
4039ralrimiv 2457 . . . . 5  |-  ( ( ( M  e.  NN  /\  N  e.  NN )  /\  -.  N  ||  M )  ->  A. x  e.  ( 0..^ M )  -.  ( x  gcd  M )  =  N )
41 rabeq0 3331 . . . . 5  |-  ( { x  e.  ( 0..^ M )  |  ( x  gcd  M )  =  N }  =  (/)  <->  A. x  e.  ( 0..^ M )  -.  (
x  gcd  M )  =  N )
4240, 41sylibr 133 . . . 4  |-  ( ( ( M  e.  NN  /\  N  e.  NN )  /\  -.  N  ||  M )  ->  { x  e.  ( 0..^ M )  |  ( x  gcd  M )  =  N }  =  (/) )
4342fveq2d 5344 . . 3  |-  ( ( ( M  e.  NN  /\  N  e.  NN )  /\  -.  N  ||  M )  ->  ( `  { x  e.  ( 0..^ M )  |  ( x  gcd  M
)  =  N }
)  =  ( `  (/) ) )
44 hash0 10336 . . 3  |-  ( `  (/) )  =  0
4543, 44syl6eq 2143 . 2  |-  ( ( ( M  e.  NN  /\  N  e.  NN )  /\  -.  N  ||  M )  ->  ( `  { x  e.  ( 0..^ M )  |  ( x  gcd  M
)  =  N }
)  =  0 )
46 dvdsdc 11247 . . . 4  |-  ( ( N  e.  NN  /\  M  e.  ZZ )  -> DECID  N 
||  M )
4731, 46sylan2 281 . . 3  |-  ( ( N  e.  NN  /\  M  e.  NN )  -> DECID  N 
||  M )
4847ancoms 265 . 2  |-  ( ( M  e.  NN  /\  N  e.  NN )  -> DECID  N 
||  M )
491, 2, 27, 45, 48ifbothdadc 3442 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 103  DECID wdc 783    = wceq 1296    e. wcel 1445   A.wral 2370   {crab 2374   (/)c0 3302   ifcif 3413   class class class wbr 3867    |-> cmpt 3921   -1-1-onto->wf1o 5048   ` cfv 5049  (class class class)co 5690   Fincfn 6537   0cc0 7447   1c1 7448    x. cmul 7452    / cdiv 8236   NNcn 8520   ZZcz 8848  ..^cfzo 9702  ♯chash 10314    || cdvds 11239    gcd cgcd 11381   phicphi 11629
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 582  ax-in2 583  ax-io 668  ax-5 1388  ax-7 1389  ax-gen 1390  ax-ie1 1434  ax-ie2 1435  ax-8 1447  ax-10 1448  ax-11 1449  ax-i12 1450  ax-bndl 1451  ax-4 1452  ax-13 1456  ax-14 1457  ax-17 1471  ax-i9 1475  ax-ial 1479  ax-i5r 1480  ax-ext 2077  ax-coll 3975  ax-sep 3978  ax-nul 3986  ax-pow 4030  ax-pr 4060  ax-un 4284  ax-setind 4381  ax-iinf 4431  ax-cnex 7533  ax-resscn 7534  ax-1cn 7535  ax-1re 7536  ax-icn 7537  ax-addcl 7538  ax-addrcl 7539  ax-mulcl 7540  ax-mulrcl 7541  ax-addcom 7542  ax-mulcom 7543  ax-addass 7544  ax-mulass 7545  ax-distr 7546  ax-i2m1 7547  ax-0lt1 7548  ax-1rid 7549  ax-0id 7550  ax-rnegex 7551  ax-precex 7552  ax-cnre 7553  ax-pre-ltirr 7554  ax-pre-ltwlin 7555  ax-pre-lttrn 7556  ax-pre-apti 7557  ax-pre-ltadd 7558  ax-pre-mulgt0 7559  ax-pre-mulext 7560  ax-arch 7561  ax-caucvg 7562
This theorem depends on definitions:  df-bi 116  df-dc 784  df-3or 928  df-3an 929  df-tru 1299  df-fal 1302  df-nf 1402  df-sb 1700  df-eu 1958  df-mo 1959  df-clab 2082  df-cleq 2088  df-clel 2091  df-nfc 2224  df-ne 2263  df-nel 2358  df-ral 2375  df-rex 2376  df-reu 2377  df-rmo 2378  df-rab 2379  df-v 2635  df-sbc 2855  df-csb 2948  df-dif 3015  df-un 3017  df-in 3019  df-ss 3026  df-nul 3303  df-if 3414  df-pw 3451  df-sn 3472  df-pr 3473  df-op 3475  df-uni 3676  df-int 3711  df-iun 3754  df-br 3868  df-opab 3922  df-mpt 3923  df-tr 3959  df-id 4144  df-po 4147  df-iso 4148  df-iord 4217  df-on 4219  df-ilim 4220  df-suc 4222  df-iom 4434  df-xp 4473  df-rel 4474  df-cnv 4475  df-co 4476  df-dm 4477  df-rn 4478  df-res 4479  df-ima 4480  df-iota 5014  df-fun 5051  df-fn 5052  df-f 5053  df-f1 5054  df-fo 5055  df-f1o 5056  df-fv 5057  df-riota 5646  df-ov 5693  df-oprab 5694  df-mpt2 5695  df-1st 5949  df-2nd 5950  df-recs 6108  df-frec 6194  df-1o 6219  df-er 6332  df-en 6538  df-dom 6539  df-fin 6540  df-sup 6759  df-pnf 7621  df-mnf 7622  df-xr 7623  df-ltxr 7624  df-le 7625  df-sub 7752  df-neg 7753  df-reap 8149  df-ap 8156  df-div 8237  df-inn 8521  df-2 8579  df-3 8580  df-4 8581  df-n0 8772  df-z 8849  df-uz 9119  df-q 9204  df-rp 9234  df-fz 9574  df-fzo 9703  df-fl 9826  df-mod 9879  df-iseq 10002  df-seq3 10003  df-exp 10086  df-ihash 10315  df-cj 10407  df-re 10408  df-im 10409  df-rsqrt 10562  df-abs 10563  df-dvds 11240  df-gcd 11382  df-phi 11630
This theorem is referenced by: (None)
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