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Theorem dvdsppwf1o 20924
Description: A bijection from the divisors of a prime power to the integers less than the prime count. (Contributed by Mario Carneiro, 5-May-2016.)
Hypothesis
Ref Expression
dvdsppwf1o.f  |-  F  =  ( n  e.  ( 0 ... A ) 
|->  ( P ^ n
) )
Assertion
Ref Expression
dvdsppwf1o  |-  ( ( P  e.  Prime  /\  A  e.  NN0 )  ->  F : ( 0 ... A ) -1-1-onto-> { x  e.  NN  |  x  ||  ( P ^ A ) } )
Distinct variable groups:    x, n, A    P, n, x
Allowed substitution hints:    F( x, n)

Proof of Theorem dvdsppwf1o
Dummy variable  m is distinct from all other variables.
StepHypRef Expression
1 dvdsppwf1o.f . 2  |-  F  =  ( n  e.  ( 0 ... A ) 
|->  ( P ^ n
) )
2 prmnn 13037 . . . . 5  |-  ( P  e.  Prime  ->  P  e.  NN )
32adantr 452 . . . 4  |-  ( ( P  e.  Prime  /\  A  e.  NN0 )  ->  P  e.  NN )
4 elfznn0 11039 . . . 4  |-  ( n  e.  ( 0 ... A )  ->  n  e.  NN0 )
5 nnexpcl 11349 . . . 4  |-  ( ( P  e.  NN  /\  n  e.  NN0 )  -> 
( P ^ n
)  e.  NN )
63, 4, 5syl2an 464 . . 3  |-  ( ( ( P  e.  Prime  /\  A  e.  NN0 )  /\  n  e.  (
0 ... A ) )  ->  ( P ^
n )  e.  NN )
7 prmz 13038 . . . . 5  |-  ( P  e.  Prime  ->  P  e.  ZZ )
87ad2antrr 707 . . . 4  |-  ( ( ( P  e.  Prime  /\  A  e.  NN0 )  /\  n  e.  (
0 ... A ) )  ->  P  e.  ZZ )
94adantl 453 . . . 4  |-  ( ( ( P  e.  Prime  /\  A  e.  NN0 )  /\  n  e.  (
0 ... A ) )  ->  n  e.  NN0 )
10 elfzuz3 11012 . . . . 5  |-  ( n  e.  ( 0 ... A )  ->  A  e.  ( ZZ>= `  n )
)
1110adantl 453 . . . 4  |-  ( ( ( P  e.  Prime  /\  A  e.  NN0 )  /\  n  e.  (
0 ... A ) )  ->  A  e.  (
ZZ>= `  n ) )
12 dvdsexp 12860 . . . 4  |-  ( ( P  e.  ZZ  /\  n  e.  NN0  /\  A  e.  ( ZZ>= `  n )
)  ->  ( P ^ n )  ||  ( P ^ A ) )
138, 9, 11, 12syl3anc 1184 . . 3  |-  ( ( ( P  e.  Prime  /\  A  e.  NN0 )  /\  n  e.  (
0 ... A ) )  ->  ( P ^
n )  ||  ( P ^ A ) )
14 breq1 4175 . . . 4  |-  ( x  =  ( P ^
n )  ->  (
x  ||  ( P ^ A )  <->  ( P ^ n )  ||  ( P ^ A ) ) )
1514elrab 3052 . . 3  |-  ( ( P ^ n )  e.  { x  e.  NN  |  x  ||  ( P ^ A ) }  <->  ( ( P ^ n )  e.  NN  /\  ( P ^ n )  ||  ( P ^ A ) ) )
166, 13, 15sylanbrc 646 . 2  |-  ( ( ( P  e.  Prime  /\  A  e.  NN0 )  /\  n  e.  (
0 ... A ) )  ->  ( P ^
n )  e.  {
x  e.  NN  |  x  ||  ( P ^ A ) } )
17 simpl 444 . . . 4  |-  ( ( P  e.  Prime  /\  A  e.  NN0 )  ->  P  e.  Prime )
18 elrabi 3050 . . . 4  |-  ( m  e.  { x  e.  NN  |  x  ||  ( P ^ A ) }  ->  m  e.  NN )
19 pccl 13178 . . . 4  |-  ( ( P  e.  Prime  /\  m  e.  NN )  ->  ( P  pCnt  m )  e. 
NN0 )
2017, 18, 19syl2an 464 . . 3  |-  ( ( ( P  e.  Prime  /\  A  e.  NN0 )  /\  m  e.  { x  e.  NN  |  x  ||  ( P ^ A ) } )  ->  ( P  pCnt  m )  e. 
NN0 )
2117adantr 452 . . . . 5  |-  ( ( ( P  e.  Prime  /\  A  e.  NN0 )  /\  m  e.  { x  e.  NN  |  x  ||  ( P ^ A ) } )  ->  P  e.  Prime )
2218adantl 453 . . . . . 6  |-  ( ( ( P  e.  Prime  /\  A  e.  NN0 )  /\  m  e.  { x  e.  NN  |  x  ||  ( P ^ A ) } )  ->  m  e.  NN )
2322nnzd 10330 . . . . 5  |-  ( ( ( P  e.  Prime  /\  A  e.  NN0 )  /\  m  e.  { x  e.  NN  |  x  ||  ( P ^ A ) } )  ->  m  e.  ZZ )
247ad2antrr 707 . . . . . 6  |-  ( ( ( P  e.  Prime  /\  A  e.  NN0 )  /\  m  e.  { x  e.  NN  |  x  ||  ( P ^ A ) } )  ->  P  e.  ZZ )
25 simplr 732 . . . . . 6  |-  ( ( ( P  e.  Prime  /\  A  e.  NN0 )  /\  m  e.  { x  e.  NN  |  x  ||  ( P ^ A ) } )  ->  A  e.  NN0 )
26 zexpcl 11351 . . . . . 6  |-  ( ( P  e.  ZZ  /\  A  e.  NN0 )  -> 
( P ^ A
)  e.  ZZ )
2724, 25, 26syl2anc 643 . . . . 5  |-  ( ( ( P  e.  Prime  /\  A  e.  NN0 )  /\  m  e.  { x  e.  NN  |  x  ||  ( P ^ A ) } )  ->  ( P ^ A )  e.  ZZ )
28 breq1 4175 . . . . . . . 8  |-  ( x  =  m  ->  (
x  ||  ( P ^ A )  <->  m  ||  ( P ^ A ) ) )
2928elrab 3052 . . . . . . 7  |-  ( m  e.  { x  e.  NN  |  x  ||  ( P ^ A ) }  <->  ( m  e.  NN  /\  m  ||  ( P ^ A ) ) )
3029simprbi 451 . . . . . 6  |-  ( m  e.  { x  e.  NN  |  x  ||  ( P ^ A ) }  ->  m  ||  ( P ^ A ) )
3130adantl 453 . . . . 5  |-  ( ( ( P  e.  Prime  /\  A  e.  NN0 )  /\  m  e.  { x  e.  NN  |  x  ||  ( P ^ A ) } )  ->  m  ||  ( P ^ A
) )
32 pcdvdstr 13204 . . . . 5  |-  ( ( P  e.  Prime  /\  (
m  e.  ZZ  /\  ( P ^ A )  e.  ZZ  /\  m  ||  ( P ^ A
) ) )  -> 
( P  pCnt  m
)  <_  ( P  pCnt  ( P ^ A
) ) )
3321, 23, 27, 31, 32syl13anc 1186 . . . 4  |-  ( ( ( P  e.  Prime  /\  A  e.  NN0 )  /\  m  e.  { x  e.  NN  |  x  ||  ( P ^ A ) } )  ->  ( P  pCnt  m )  <_ 
( P  pCnt  ( P ^ A ) ) )
34 pcidlem 13200 . . . . 5  |-  ( ( P  e.  Prime  /\  A  e.  NN0 )  ->  ( P  pCnt  ( P ^ A ) )  =  A )
3534adantr 452 . . . 4  |-  ( ( ( P  e.  Prime  /\  A  e.  NN0 )  /\  m  e.  { x  e.  NN  |  x  ||  ( P ^ A ) } )  ->  ( P  pCnt  ( P ^ A ) )  =  A )
3633, 35breqtrd 4196 . . 3  |-  ( ( ( P  e.  Prime  /\  A  e.  NN0 )  /\  m  e.  { x  e.  NN  |  x  ||  ( P ^ A ) } )  ->  ( P  pCnt  m )  <_  A )
37 fznn0 11069 . . . 4  |-  ( A  e.  NN0  ->  ( ( P  pCnt  m )  e.  ( 0 ... A
)  <->  ( ( P 
pCnt  m )  e.  NN0  /\  ( P  pCnt  m
)  <_  A )
) )
3825, 37syl 16 . . 3  |-  ( ( ( P  e.  Prime  /\  A  e.  NN0 )  /\  m  e.  { x  e.  NN  |  x  ||  ( P ^ A ) } )  ->  (
( P  pCnt  m
)  e.  ( 0 ... A )  <->  ( ( P  pCnt  m )  e. 
NN0  /\  ( P  pCnt  m )  <_  A
) ) )
3920, 36, 38mpbir2and 889 . 2  |-  ( ( ( P  e.  Prime  /\  A  e.  NN0 )  /\  m  e.  { x  e.  NN  |  x  ||  ( P ^ A ) } )  ->  ( P  pCnt  m )  e.  ( 0 ... A
) )
40 oveq2 6048 . . . . . . . . 9  |-  ( n  =  A  ->  ( P ^ n )  =  ( P ^ A
) )
4140breq2d 4184 . . . . . . . 8  |-  ( n  =  A  ->  (
m  ||  ( P ^ n )  <->  m  ||  ( P ^ A ) ) )
4241rspcev 3012 . . . . . . 7  |-  ( ( A  e.  NN0  /\  m  ||  ( P ^ A ) )  ->  E. n  e.  NN0  m  ||  ( P ^
n ) )
4325, 31, 42syl2anc 643 . . . . . 6  |-  ( ( ( P  e.  Prime  /\  A  e.  NN0 )  /\  m  e.  { x  e.  NN  |  x  ||  ( P ^ A ) } )  ->  E. n  e.  NN0  m  ||  ( P ^ n ) )
44 pcprmpw2 13210 . . . . . . 7  |-  ( ( P  e.  Prime  /\  m  e.  NN )  ->  ( E. n  e.  NN0  m  ||  ( P ^
n )  <->  m  =  ( P ^ ( P 
pCnt  m ) ) ) )
4517, 18, 44syl2an 464 . . . . . 6  |-  ( ( ( P  e.  Prime  /\  A  e.  NN0 )  /\  m  e.  { x  e.  NN  |  x  ||  ( P ^ A ) } )  ->  ( E. n  e.  NN0  m  ||  ( P ^
n )  <->  m  =  ( P ^ ( P 
pCnt  m ) ) ) )
4643, 45mpbid 202 . . . . 5  |-  ( ( ( P  e.  Prime  /\  A  e.  NN0 )  /\  m  e.  { x  e.  NN  |  x  ||  ( P ^ A ) } )  ->  m  =  ( P ^
( P  pCnt  m
) ) )
4746adantrl 697 . . . 4  |-  ( ( ( P  e.  Prime  /\  A  e.  NN0 )  /\  ( n  e.  ( 0 ... A )  /\  m  e.  {
x  e.  NN  |  x  ||  ( P ^ A ) } ) )  ->  m  =  ( P ^ ( P 
pCnt  m ) ) )
48 oveq2 6048 . . . . 5  |-  ( n  =  ( P  pCnt  m )  ->  ( P ^ n )  =  ( P ^ ( P  pCnt  m ) ) )
4948eqeq2d 2415 . . . 4  |-  ( n  =  ( P  pCnt  m )  ->  ( m  =  ( P ^
n )  <->  m  =  ( P ^ ( P 
pCnt  m ) ) ) )
5047, 49syl5ibrcom 214 . . 3  |-  ( ( ( P  e.  Prime  /\  A  e.  NN0 )  /\  ( n  e.  ( 0 ... A )  /\  m  e.  {
x  e.  NN  |  x  ||  ( P ^ A ) } ) )  ->  ( n  =  ( P  pCnt  m )  ->  m  =  ( P ^ n ) ) )
51 elfzelz 11015 . . . . . . 7  |-  ( n  e.  ( 0 ... A )  ->  n  e.  ZZ )
52 pcid 13201 . . . . . . 7  |-  ( ( P  e.  Prime  /\  n  e.  ZZ )  ->  ( P  pCnt  ( P ^
n ) )  =  n )
5317, 51, 52syl2an 464 . . . . . 6  |-  ( ( ( P  e.  Prime  /\  A  e.  NN0 )  /\  n  e.  (
0 ... A ) )  ->  ( P  pCnt  ( P ^ n ) )  =  n )
5453eqcomd 2409 . . . . 5  |-  ( ( ( P  e.  Prime  /\  A  e.  NN0 )  /\  n  e.  (
0 ... A ) )  ->  n  =  ( P  pCnt  ( P ^ n ) ) )
5554adantrr 698 . . . 4  |-  ( ( ( P  e.  Prime  /\  A  e.  NN0 )  /\  ( n  e.  ( 0 ... A )  /\  m  e.  {
x  e.  NN  |  x  ||  ( P ^ A ) } ) )  ->  n  =  ( P  pCnt  ( P ^ n ) ) )
56 oveq2 6048 . . . . 5  |-  ( m  =  ( P ^
n )  ->  ( P  pCnt  m )  =  ( P  pCnt  ( P ^ n ) ) )
5756eqeq2d 2415 . . . 4  |-  ( m  =  ( P ^
n )  ->  (
n  =  ( P 
pCnt  m )  <->  n  =  ( P  pCnt  ( P ^ n ) ) ) )
5855, 57syl5ibrcom 214 . . 3  |-  ( ( ( P  e.  Prime  /\  A  e.  NN0 )  /\  ( n  e.  ( 0 ... A )  /\  m  e.  {
x  e.  NN  |  x  ||  ( P ^ A ) } ) )  ->  ( m  =  ( P ^
n )  ->  n  =  ( P  pCnt  m ) ) )
5950, 58impbid 184 . 2  |-  ( ( ( P  e.  Prime  /\  A  e.  NN0 )  /\  ( n  e.  ( 0 ... A )  /\  m  e.  {
x  e.  NN  |  x  ||  ( P ^ A ) } ) )  ->  ( n  =  ( P  pCnt  m )  <->  m  =  ( P ^ n ) ) )
601, 16, 39, 59f1o2d 6255 1  |-  ( ( P  e.  Prime  /\  A  e.  NN0 )  ->  F : ( 0 ... A ) -1-1-onto-> { x  e.  NN  |  x  ||  ( P ^ A ) } )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    = wceq 1649    e. wcel 1721   E.wrex 2667   {crab 2670   class class class wbr 4172    e. cmpt 4226   -1-1-onto->wf1o 5412   ` cfv 5413  (class class class)co 6040   0cc0 8946    <_ cle 9077   NNcn 9956   NN0cn0 10177   ZZcz 10238   ZZ>=cuz 10444   ...cfz 10999   ^cexp 11337    || cdivides 12807   Primecprime 13034    pCnt cpc 13165
This theorem is referenced by:  sgmppw  20934  0sgmppw  20935  dchrisum0flblem1  21155
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660  ax-cnex 9002  ax-resscn 9003  ax-1cn 9004  ax-icn 9005  ax-addcl 9006  ax-addrcl 9007  ax-mulcl 9008  ax-mulrcl 9009  ax-mulcom 9010  ax-addass 9011  ax-mulass 9012  ax-distr 9013  ax-i2m1 9014  ax-1ne0 9015  ax-1rid 9016  ax-rnegex 9017  ax-rrecex 9018  ax-cnre 9019  ax-pre-lttri 9020  ax-pre-lttrn 9021  ax-pre-ltadd 9022  ax-pre-mulgt0 9023  ax-pre-sup 9024
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-reu 2673  df-rmo 2674  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-pss 3296  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-tp 3782  df-op 3783  df-uni 3976  df-int 4011  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-tr 4263  df-eprel 4454  df-id 4458  df-po 4463  df-so 4464  df-fr 4501  df-we 4503  df-ord 4544  df-on 4545  df-lim 4546  df-suc 4547  df-om 4805  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-1st 6308  df-2nd 6309  df-riota 6508  df-recs 6592  df-rdg 6627  df-1o 6683  df-2o 6684  df-oadd 6687  df-er 6864  df-en 7069  df-dom 7070  df-sdom 7071  df-fin 7072  df-sup 7404  df-pnf 9078  df-mnf 9079  df-xr 9080  df-ltxr 9081  df-le 9082  df-sub 9249  df-neg 9250  df-div 9634  df-nn 9957  df-2 10014  df-3 10015  df-n0 10178  df-z 10239  df-uz 10445  df-q 10531  df-rp 10569  df-fz 11000  df-fl 11157  df-mod 11206  df-seq 11279  df-exp 11338  df-cj 11859  df-re 11860  df-im 11861  df-sqr 11995  df-abs 11996  df-dvds 12808  df-gcd 12962  df-prm 13035  df-pc 13166
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