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Theorem dvdsppwf1o 22501
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 13758 . . . . 5  |-  ( P  e.  Prime  ->  P  e.  NN )
32adantr 465 . . . 4  |-  ( ( P  e.  Prime  /\  A  e.  NN0 )  ->  P  e.  NN )
4 elfznn0 11473 . . . 4  |-  ( n  e.  ( 0 ... A )  ->  n  e.  NN0 )
5 nnexpcl 11870 . . . 4  |-  ( ( P  e.  NN  /\  n  e.  NN0 )  -> 
( P ^ n
)  e.  NN )
63, 4, 5syl2an 477 . . 3  |-  ( ( ( P  e.  Prime  /\  A  e.  NN0 )  /\  n  e.  (
0 ... A ) )  ->  ( P ^
n )  e.  NN )
7 prmz 13759 . . . . 5  |-  ( P  e.  Prime  ->  P  e.  ZZ )
87ad2antrr 725 . . . 4  |-  ( ( ( P  e.  Prime  /\  A  e.  NN0 )  /\  n  e.  (
0 ... A ) )  ->  P  e.  ZZ )
94adantl 466 . . . 4  |-  ( ( ( P  e.  Prime  /\  A  e.  NN0 )  /\  n  e.  (
0 ... A ) )  ->  n  e.  NN0 )
10 elfzuz3 11442 . . . . 5  |-  ( n  e.  ( 0 ... A )  ->  A  e.  ( ZZ>= `  n )
)
1110adantl 466 . . . 4  |-  ( ( ( P  e.  Prime  /\  A  e.  NN0 )  /\  n  e.  (
0 ... A ) )  ->  A  e.  (
ZZ>= `  n ) )
12 dvdsexp 13581 . . . 4  |-  ( ( P  e.  ZZ  /\  n  e.  NN0  /\  A  e.  ( ZZ>= `  n )
)  ->  ( P ^ n )  ||  ( P ^ A ) )
138, 9, 11, 12syl3anc 1218 . . 3  |-  ( ( ( P  e.  Prime  /\  A  e.  NN0 )  /\  n  e.  (
0 ... A ) )  ->  ( P ^
n )  ||  ( P ^ A ) )
14 breq1 4290 . . . 4  |-  ( x  =  ( P ^
n )  ->  (
x  ||  ( P ^ A )  <->  ( P ^ n )  ||  ( P ^ A ) ) )
1514elrab 3112 . . 3  |-  ( ( P ^ n )  e.  { x  e.  NN  |  x  ||  ( P ^ A ) }  <->  ( ( P ^ n )  e.  NN  /\  ( P ^ n )  ||  ( P ^ A ) ) )
166, 13, 15sylanbrc 664 . 2  |-  ( ( ( P  e.  Prime  /\  A  e.  NN0 )  /\  n  e.  (
0 ... A ) )  ->  ( P ^
n )  e.  {
x  e.  NN  |  x  ||  ( P ^ A ) } )
17 simpl 457 . . . 4  |-  ( ( P  e.  Prime  /\  A  e.  NN0 )  ->  P  e.  Prime )
18 elrabi 3109 . . . 4  |-  ( m  e.  { x  e.  NN  |  x  ||  ( P ^ A ) }  ->  m  e.  NN )
19 pccl 13908 . . . 4  |-  ( ( P  e.  Prime  /\  m  e.  NN )  ->  ( P  pCnt  m )  e. 
NN0 )
2017, 18, 19syl2an 477 . . 3  |-  ( ( ( P  e.  Prime  /\  A  e.  NN0 )  /\  m  e.  { x  e.  NN  |  x  ||  ( P ^ A ) } )  ->  ( P  pCnt  m )  e. 
NN0 )
2117adantr 465 . . . . 5  |-  ( ( ( P  e.  Prime  /\  A  e.  NN0 )  /\  m  e.  { x  e.  NN  |  x  ||  ( P ^ A ) } )  ->  P  e.  Prime )
2218adantl 466 . . . . . 6  |-  ( ( ( P  e.  Prime  /\  A  e.  NN0 )  /\  m  e.  { x  e.  NN  |  x  ||  ( P ^ A ) } )  ->  m  e.  NN )
2322nnzd 10738 . . . . 5  |-  ( ( ( P  e.  Prime  /\  A  e.  NN0 )  /\  m  e.  { x  e.  NN  |  x  ||  ( P ^ A ) } )  ->  m  e.  ZZ )
247ad2antrr 725 . . . . . 6  |-  ( ( ( P  e.  Prime  /\  A  e.  NN0 )  /\  m  e.  { x  e.  NN  |  x  ||  ( P ^ A ) } )  ->  P  e.  ZZ )
25 simplr 754 . . . . . 6  |-  ( ( ( P  e.  Prime  /\  A  e.  NN0 )  /\  m  e.  { x  e.  NN  |  x  ||  ( P ^ A ) } )  ->  A  e.  NN0 )
26 zexpcl 11872 . . . . . 6  |-  ( ( P  e.  ZZ  /\  A  e.  NN0 )  -> 
( P ^ A
)  e.  ZZ )
2724, 25, 26syl2anc 661 . . . . 5  |-  ( ( ( P  e.  Prime  /\  A  e.  NN0 )  /\  m  e.  { x  e.  NN  |  x  ||  ( P ^ A ) } )  ->  ( P ^ A )  e.  ZZ )
28 breq1 4290 . . . . . . . 8  |-  ( x  =  m  ->  (
x  ||  ( P ^ A )  <->  m  ||  ( P ^ A ) ) )
2928elrab 3112 . . . . . . 7  |-  ( m  e.  { x  e.  NN  |  x  ||  ( P ^ A ) }  <->  ( m  e.  NN  /\  m  ||  ( P ^ A ) ) )
3029simprbi 464 . . . . . 6  |-  ( m  e.  { x  e.  NN  |  x  ||  ( P ^ A ) }  ->  m  ||  ( P ^ A ) )
3130adantl 466 . . . . 5  |-  ( ( ( P  e.  Prime  /\  A  e.  NN0 )  /\  m  e.  { x  e.  NN  |  x  ||  ( P ^ A ) } )  ->  m  ||  ( P ^ A
) )
32 pcdvdstr 13934 . . . . 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 1220 . . . 4  |-  ( ( ( P  e.  Prime  /\  A  e.  NN0 )  /\  m  e.  { x  e.  NN  |  x  ||  ( P ^ A ) } )  ->  ( P  pCnt  m )  <_ 
( P  pCnt  ( P ^ A ) ) )
34 pcidlem 13930 . . . . 5  |-  ( ( P  e.  Prime  /\  A  e.  NN0 )  ->  ( P  pCnt  ( P ^ A ) )  =  A )
3534adantr 465 . . . 4  |-  ( ( ( P  e.  Prime  /\  A  e.  NN0 )  /\  m  e.  { x  e.  NN  |  x  ||  ( P ^ A ) } )  ->  ( P  pCnt  ( P ^ A ) )  =  A )
3633, 35breqtrd 4311 . . 3  |-  ( ( ( P  e.  Prime  /\  A  e.  NN0 )  /\  m  e.  { x  e.  NN  |  x  ||  ( P ^ A ) } )  ->  ( P  pCnt  m )  <_  A )
37 fznn0 11516 . . . 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 913 . 2  |-  ( ( ( P  e.  Prime  /\  A  e.  NN0 )  /\  m  e.  { x  e.  NN  |  x  ||  ( P ^ A ) } )  ->  ( P  pCnt  m )  e.  ( 0 ... A
) )
40 oveq2 6094 . . . . . . . . 9  |-  ( n  =  A  ->  ( P ^ n )  =  ( P ^ A
) )
4140breq2d 4299 . . . . . . . 8  |-  ( n  =  A  ->  (
m  ||  ( P ^ n )  <->  m  ||  ( P ^ A ) ) )
4241rspcev 3068 . . . . . . 7  |-  ( ( A  e.  NN0  /\  m  ||  ( P ^ A ) )  ->  E. n  e.  NN0  m  ||  ( P ^
n ) )
4325, 31, 42syl2anc 661 . . . . . 6  |-  ( ( ( P  e.  Prime  /\  A  e.  NN0 )  /\  m  e.  { x  e.  NN  |  x  ||  ( P ^ A ) } )  ->  E. n  e.  NN0  m  ||  ( P ^ n ) )
44 pcprmpw2 13940 . . . . . . 7  |-  ( ( P  e.  Prime  /\  m  e.  NN )  ->  ( E. n  e.  NN0  m  ||  ( P ^
n )  <->  m  =  ( P ^ ( P 
pCnt  m ) ) ) )
4517, 18, 44syl2an 477 . . . . . 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 210 . . . . 5  |-  ( ( ( P  e.  Prime  /\  A  e.  NN0 )  /\  m  e.  { x  e.  NN  |  x  ||  ( P ^ A ) } )  ->  m  =  ( P ^
( P  pCnt  m
) ) )
4746adantrl 715 . . . 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 6094 . . . . 5  |-  ( n  =  ( P  pCnt  m )  ->  ( P ^ n )  =  ( P ^ ( P  pCnt  m ) ) )
4948eqeq2d 2449 . . . 4  |-  ( n  =  ( P  pCnt  m )  ->  ( m  =  ( P ^
n )  <->  m  =  ( P ^ ( P 
pCnt  m ) ) ) )
5047, 49syl5ibrcom 222 . . 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 11445 . . . . . . 7  |-  ( n  e.  ( 0 ... A )  ->  n  e.  ZZ )
52 pcid 13931 . . . . . . 7  |-  ( ( P  e.  Prime  /\  n  e.  ZZ )  ->  ( P  pCnt  ( P ^
n ) )  =  n )
5317, 51, 52syl2an 477 . . . . . 6  |-  ( ( ( P  e.  Prime  /\  A  e.  NN0 )  /\  n  e.  (
0 ... A ) )  ->  ( P  pCnt  ( P ^ n ) )  =  n )
5453eqcomd 2443 . . . . 5  |-  ( ( ( P  e.  Prime  /\  A  e.  NN0 )  /\  n  e.  (
0 ... A ) )  ->  n  =  ( P  pCnt  ( P ^ n ) ) )
5554adantrr 716 . . . 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 6094 . . . . 5  |-  ( m  =  ( P ^
n )  ->  ( P  pCnt  m )  =  ( P  pCnt  ( P ^ n ) ) )
5756eqeq2d 2449 . . . 4  |-  ( m  =  ( P ^
n )  ->  (
n  =  ( P 
pCnt  m )  <->  n  =  ( P  pCnt  ( P ^ n ) ) ) )
5855, 57syl5ibrcom 222 . . 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 191 . 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 6307 1  |-  ( ( P  e.  Prime  /\  A  e.  NN0 )  ->  F : ( 0 ... A ) -1-1-onto-> { x  e.  NN  |  x  ||  ( P ^ A ) } )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1369    e. wcel 1756   E.wrex 2711   {crab 2714   class class class wbr 4287    e. cmpt 4345   -1-1-onto->wf1o 5412   ` cfv 5413  (class class class)co 6086   0cc0 9274    <_ cle 9411   NNcn 10314   NN0cn0 10571   ZZcz 10638   ZZ>=cuz 10853   ...cfz 11429   ^cexp 11857    || cdivides 13527   Primecprime 13755    pCnt cpc 13895
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2419  ax-sep 4408  ax-nul 4416  ax-pow 4465  ax-pr 4526  ax-un 6367  ax-cnex 9330  ax-resscn 9331  ax-1cn 9332  ax-icn 9333  ax-addcl 9334  ax-addrcl 9335  ax-mulcl 9336  ax-mulrcl 9337  ax-mulcom 9338  ax-addass 9339  ax-mulass 9340  ax-distr 9341  ax-i2m1 9342  ax-1ne0 9343  ax-1rid 9344  ax-rnegex 9345  ax-rrecex 9346  ax-cnre 9347  ax-pre-lttri 9348  ax-pre-lttrn 9349  ax-pre-ltadd 9350  ax-pre-mulgt0 9351  ax-pre-sup 9352
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2256  df-mo 2257  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2715  df-rex 2716  df-reu 2717  df-rmo 2718  df-rab 2719  df-v 2969  df-sbc 3182  df-csb 3284  df-dif 3326  df-un 3328  df-in 3330  df-ss 3337  df-pss 3339  df-nul 3633  df-if 3787  df-pw 3857  df-sn 3873  df-pr 3875  df-tp 3877  df-op 3879  df-uni 4087  df-int 4124  df-iun 4168  df-br 4288  df-opab 4346  df-mpt 4347  df-tr 4381  df-eprel 4627  df-id 4631  df-po 4636  df-so 4637  df-fr 4674  df-we 4676  df-ord 4717  df-on 4718  df-lim 4719  df-suc 4720  df-xp 4841  df-rel 4842  df-cnv 4843  df-co 4844  df-dm 4845  df-rn 4846  df-res 4847  df-ima 4848  df-iota 5376  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-riota 6047  df-ov 6089  df-oprab 6090  df-mpt2 6091  df-om 6472  df-1st 6572  df-2nd 6573  df-recs 6824  df-rdg 6858  df-1o 6912  df-2o 6913  df-oadd 6916  df-er 7093  df-en 7303  df-dom 7304  df-sdom 7305  df-fin 7306  df-sup 7683  df-pnf 9412  df-mnf 9413  df-xr 9414  df-ltxr 9415  df-le 9416  df-sub 9589  df-neg 9590  df-div 9986  df-nn 10315  df-2 10372  df-3 10373  df-n0 10572  df-z 10639  df-uz 10854  df-q 10946  df-rp 10984  df-fz 11430  df-fl 11634  df-mod 11701  df-seq 11799  df-exp 11858  df-cj 12580  df-re 12581  df-im 12582  df-sqr 12716  df-abs 12717  df-dvds 13528  df-gcd 13683  df-prm 13756  df-pc 13896
This theorem is referenced by:  sgmppw  22511  0sgmppw  22512  dchrisum0flblem1  22732
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