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Theorem hashgcdeq 12180
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 2180 . 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 2180 . 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 11745 . . . . 5  |-  ( ( M  e.  NN  /\  N  e.  NN )  ->  ( N  ||  M  <->  ( M  /  N )  e.  NN ) )
43biimpa 294 . . . 4  |-  ( ( ( M  e.  NN  /\  N  e.  NN )  /\  N  ||  M
)  ->  ( M  /  N )  e.  NN )
5 dfphi2 12161 . . . 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 9210 . . . . . 6  |-  0  e.  ZZ
84nnzd 9320 . . . . . 6  |-  ( ( ( M  e.  NN  /\  N  e.  NN )  /\  N  ||  M
)  ->  ( M  /  N )  e.  ZZ )
9 fzofig 10375 . . . . . 6  |-  ( ( 0  e.  ZZ  /\  ( M  /  N
)  e.  ZZ )  ->  ( 0..^ ( M  /  N ) )  e.  Fin )
107, 8, 9sylancr 412 . . . . 5  |-  ( ( ( M  e.  NN  /\  N  e.  NN )  /\  N  ||  M
)  ->  ( 0..^ ( M  /  N
) )  e.  Fin )
11 elfzoelz 10090 . . . . . . . . . 10  |-  ( y  e.  ( 0..^ ( M  /  N ) )  ->  y  e.  ZZ )
1211adantl 275 . . . . . . . . 9  |-  ( ( ( ( M  e.  NN  /\  N  e.  NN )  /\  N  ||  M )  /\  y  e.  ( 0..^ ( M  /  N ) ) )  ->  y  e.  ZZ )
138adantr 274 . . . . . . . . 9  |-  ( ( ( ( M  e.  NN  /\  N  e.  NN )  /\  N  ||  M )  /\  y  e.  ( 0..^ ( M  /  N ) ) )  ->  ( M  /  N )  e.  ZZ )
1412, 13gcdcld 11910 . . . . . . . 8  |-  ( ( ( ( M  e.  NN  /\  N  e.  NN )  /\  N  ||  M )  /\  y  e.  ( 0..^ ( M  /  N ) ) )  ->  ( y  gcd  ( M  /  N
) )  e.  NN0 )
1514nn0zd 9319 . . . . . . 7  |-  ( ( ( ( M  e.  NN  /\  N  e.  NN )  /\  N  ||  M )  /\  y  e.  ( 0..^ ( M  /  N ) ) )  ->  ( y  gcd  ( M  /  N
) )  e.  ZZ )
16 1zzd 9226 . . . . . . 7  |-  ( ( ( ( M  e.  NN  /\  N  e.  NN )  /\  N  ||  M )  /\  y  e.  ( 0..^ ( M  /  N ) ) )  ->  1  e.  ZZ )
17 zdceq 9274 . . . . . . 7  |-  ( ( ( y  gcd  ( M  /  N ) )  e.  ZZ  /\  1  e.  ZZ )  -> DECID  ( y  gcd  ( M  /  N ) )  =  1 )
1815, 16, 17syl2anc 409 . . . . . 6  |-  ( ( ( ( M  e.  NN  /\  N  e.  NN )  /\  N  ||  M )  /\  y  e.  ( 0..^ ( M  /  N ) ) )  -> DECID  ( y  gcd  ( M  /  N ) )  =  1 )
1918ralrimiva 2543 . . . . 5  |-  ( ( ( M  e.  NN  /\  N  e.  NN )  /\  N  ||  M
)  ->  A. y  e.  ( 0..^ ( M  /  N ) )DECID  ( y  gcd  ( M  /  N ) )  =  1 )
2010, 19ssfirab 6907 . . . 4  |-  ( ( ( M  e.  NN  /\  N  e.  NN )  /\  N  ||  M
)  ->  { y  e.  ( 0..^ ( M  /  N ) )  |  ( y  gcd  ( M  /  N
) )  =  1 }  e.  Fin )
21 eqid 2170 . . . . . 6  |-  { y  e.  ( 0..^ ( M  /  N ) )  |  ( y  gcd  ( M  /  N ) )  =  1 }  =  {
y  e.  ( 0..^ ( M  /  N
) )  |  ( y  gcd  ( M  /  N ) )  =  1 }
22 eqid 2170 . . . . . 6  |-  { x  e.  ( 0..^ M )  |  ( x  gcd  M )  =  N }  =  { x  e.  ( 0..^ M )  |  ( x  gcd  M
)  =  N }
23 eqid 2170 . . . . . 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 12179 . . . . 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 1198 . . . 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 10711 . . 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 2204 . 2  |-  ( ( ( M  e.  NN  /\  N  e.  NN )  /\  N  ||  M
)  ->  ( `  {
x  e.  ( 0..^ M )  |  ( x  gcd  M )  =  N } )  =  ( phi `  ( M  /  N
) ) )
28 simprr 527 . . . . . . . . . 10  |-  ( ( ( M  e.  NN  /\  N  e.  NN )  /\  ( x  e.  ( 0..^ M )  /\  ( x  gcd  M )  =  N ) )  ->  ( x  gcd  M )  =  N )
29 elfzoelz 10090 . . . . . . . . . . . . 13  |-  ( x  e.  ( 0..^ M )  ->  x  e.  ZZ )
3029ad2antrl 487 . . . . . . . . . . . 12  |-  ( ( ( M  e.  NN  /\  N  e.  NN )  /\  ( x  e.  ( 0..^ M )  /\  ( x  gcd  M )  =  N ) )  ->  x  e.  ZZ )
31 nnz 9218 . . . . . . . . . . . . 13  |-  ( M  e.  NN  ->  M  e.  ZZ )
3231ad2antrr 485 . . . . . . . . . . . 12  |-  ( ( ( M  e.  NN  /\  N  e.  NN )  /\  ( x  e.  ( 0..^ M )  /\  ( x  gcd  M )  =  N ) )  ->  M  e.  ZZ )
33 gcddvds 11905 . . . . . . . . . . . 12  |-  ( ( x  e.  ZZ  /\  M  e.  ZZ )  ->  ( ( x  gcd  M )  ||  x  /\  ( x  gcd  M ) 
||  M ) )
3430, 32, 33syl2anc 409 . . . . . . . . . . 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 4011 . . . . . . . . 9  |-  ( ( ( M  e.  NN  /\  N  e.  NN )  /\  ( x  e.  ( 0..^ M )  /\  ( x  gcd  M )  =  N ) )  ->  N  ||  M
)
3736expr 373 . . . . . . . 8  |-  ( ( ( M  e.  NN  /\  N  e.  NN )  /\  x  e.  ( 0..^ M ) )  ->  ( ( x  gcd  M )  =  N  ->  N  ||  M
) )
3837con3d 626 . . . . . . 7  |-  ( ( ( M  e.  NN  /\  N  e.  NN )  /\  x  e.  ( 0..^ M ) )  ->  ( -.  N  ||  M  ->  -.  (
x  gcd  M )  =  N ) )
3938impancom 258 . . . . . 6  |-  ( ( ( M  e.  NN  /\  N  e.  NN )  /\  -.  N  ||  M )  ->  (
x  e.  ( 0..^ M )  ->  -.  ( x  gcd  M )  =  N ) )
4039ralrimiv 2542 . . . . 5  |-  ( ( ( M  e.  NN  /\  N  e.  NN )  /\  -.  N  ||  M )  ->  A. x  e.  ( 0..^ M )  -.  ( x  gcd  M )  =  N )
41 rabeq0 3443 . . . . 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 5498 . . 3  |-  ( ( ( M  e.  NN  /\  N  e.  NN )  /\  -.  N  ||  M )  ->  ( `  { x  e.  ( 0..^ M )  |  ( x  gcd  M
)  =  N }
)  =  ( `  (/) ) )
44 hash0 10718 . . 3  |-  ( `  (/) )  =  0
4543, 44eqtrdi 2219 . 2  |-  ( ( ( M  e.  NN  /\  N  e.  NN )  /\  -.  N  ||  M )  ->  ( `  { x  e.  ( 0..^ M )  |  ( x  gcd  M
)  =  N }
)  =  0 )
46 dvdsdc 11747 . . . 4  |-  ( ( N  e.  NN  /\  M  e.  ZZ )  -> DECID  N 
||  M )
4731, 46sylan2 284 . . 3  |-  ( ( N  e.  NN  /\  M  e.  NN )  -> DECID  N 
||  M )
4847ancoms 266 . 2  |-  ( ( M  e.  NN  /\  N  e.  NN )  -> DECID  N 
||  M )
491, 2, 27, 45, 48ifbothdadc 3556 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 829    = wceq 1348    e. wcel 2141   A.wral 2448   {crab 2452   (/)c0 3414   ifcif 3525   class class class wbr 3987    |-> cmpt 4048   -1-1-onto->wf1o 5195   ` cfv 5196  (class class class)co 5850   Fincfn 6714   0cc0 7761   1c1 7762    x. cmul 7766    / cdiv 8576   NNcn 8865   ZZcz 9199  ..^cfzo 10085  ♯chash 10696    || cdvds 11736    gcd cgcd 11884   phicphi 12150
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-coll 4102  ax-sep 4105  ax-nul 4113  ax-pow 4158  ax-pr 4192  ax-un 4416  ax-setind 4519  ax-iinf 4570  ax-cnex 7852  ax-resscn 7853  ax-1cn 7854  ax-1re 7855  ax-icn 7856  ax-addcl 7857  ax-addrcl 7858  ax-mulcl 7859  ax-mulrcl 7860  ax-addcom 7861  ax-mulcom 7862  ax-addass 7863  ax-mulass 7864  ax-distr 7865  ax-i2m1 7866  ax-0lt1 7867  ax-1rid 7868  ax-0id 7869  ax-rnegex 7870  ax-precex 7871  ax-cnre 7872  ax-pre-ltirr 7873  ax-pre-ltwlin 7874  ax-pre-lttrn 7875  ax-pre-apti 7876  ax-pre-ltadd 7877  ax-pre-mulgt0 7878  ax-pre-mulext 7879  ax-arch 7880  ax-caucvg 7881
This theorem depends on definitions:  df-bi 116  df-stab 826  df-dc 830  df-3or 974  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-nel 2436  df-ral 2453  df-rex 2454  df-reu 2455  df-rmo 2456  df-rab 2457  df-v 2732  df-sbc 2956  df-csb 3050  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-nul 3415  df-if 3526  df-pw 3566  df-sn 3587  df-pr 3588  df-op 3590  df-uni 3795  df-int 3830  df-iun 3873  df-br 3988  df-opab 4049  df-mpt 4050  df-tr 4086  df-id 4276  df-po 4279  df-iso 4280  df-iord 4349  df-on 4351  df-ilim 4352  df-suc 4354  df-iom 4573  df-xp 4615  df-rel 4616  df-cnv 4617  df-co 4618  df-dm 4619  df-rn 4620  df-res 4621  df-ima 4622  df-iota 5158  df-fun 5198  df-fn 5199  df-f 5200  df-f1 5201  df-fo 5202  df-f1o 5203  df-fv 5204  df-riota 5806  df-ov 5853  df-oprab 5854  df-mpo 5855  df-1st 6116  df-2nd 6117  df-recs 6281  df-frec 6367  df-1o 6392  df-er 6509  df-en 6715  df-dom 6716  df-fin 6717  df-sup 6957  df-pnf 7943  df-mnf 7944  df-xr 7945  df-ltxr 7946  df-le 7947  df-sub 8079  df-neg 8080  df-reap 8481  df-ap 8488  df-div 8577  df-inn 8866  df-2 8924  df-3 8925  df-4 8926  df-n0 9123  df-z 9200  df-uz 9475  df-q 9566  df-rp 9598  df-fz 9953  df-fzo 10086  df-fl 10213  df-mod 10266  df-seqfrec 10389  df-exp 10463  df-ihash 10697  df-cj 10793  df-re 10794  df-im 10795  df-rsqrt 10949  df-abs 10950  df-dvds 11737  df-gcd 11885  df-phi 12152
This theorem is referenced by:  phisum  12181
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