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Theorem dvdsppwf1o 20963
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 13074 . . . . 5  |-  ( P  e.  Prime  ->  P  e.  NN )
32adantr 452 . . . 4  |-  ( ( P  e.  Prime  /\  A  e.  NN0 )  ->  P  e.  NN )
4 elfznn0 11075 . . . 4  |-  ( n  e.  ( 0 ... A )  ->  n  e.  NN0 )
5 nnexpcl 11386 . . . 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 13075 . . . . 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 11048 . . . . 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 12897 . . . 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 4207 . . . 4  |-  ( x  =  ( P ^
n )  ->  (
x  ||  ( P ^ A )  <->  ( P ^ n )  ||  ( P ^ A ) ) )
1514elrab 3084 . . 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 3082 . . . 4  |-  ( m  e.  { x  e.  NN  |  x  ||  ( P ^ A ) }  ->  m  e.  NN )
19 pccl 13215 . . . 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 10366 . . . . 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 11388 . . . . . 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 4207 . . . . . . . 8  |-  ( x  =  m  ->  (
x  ||  ( P ^ A )  <->  m  ||  ( P ^ A ) ) )
2928elrab 3084 . . . . . . 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 13241 . . . . 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 13237 . . . . 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 4228 . . 3  |-  ( ( ( P  e.  Prime  /\  A  e.  NN0 )  /\  m  e.  { x  e.  NN  |  x  ||  ( P ^ A ) } )  ->  ( P  pCnt  m )  <_  A )
37 fznn0 11105 . . . 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 6081 . . . . . . . . 9  |-  ( n  =  A  ->  ( P ^ n )  =  ( P ^ A
) )
4140breq2d 4216 . . . . . . . 8  |-  ( n  =  A  ->  (
m  ||  ( P ^ n )  <->  m  ||  ( P ^ A ) ) )
4241rspcev 3044 . . . . . . 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 13247 . . . . . . 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 6081 . . . . 5  |-  ( n  =  ( P  pCnt  m )  ->  ( P ^ n )  =  ( P ^ ( P  pCnt  m ) ) )
4948eqeq2d 2446 . . . 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 11051 . . . . . . 7  |-  ( n  e.  ( 0 ... A )  ->  n  e.  ZZ )
52 pcid 13238 . . . . . . 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 2440 . . . . 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 6081 . . . . 5  |-  ( m  =  ( P ^
n )  ->  ( P  pCnt  m )  =  ( P  pCnt  ( P ^ n ) ) )
5756eqeq2d 2446 . . . 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 6288 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 1652    e. wcel 1725   E.wrex 2698   {crab 2701   class class class wbr 4204    e. cmpt 4258   -1-1-onto->wf1o 5445   ` cfv 5446  (class class class)co 6073   0cc0 8982    <_ cle 9113   NNcn 9992   NN0cn0 10213   ZZcz 10274   ZZ>=cuz 10480   ...cfz 11035   ^cexp 11374    || cdivides 12844   Primecprime 13071    pCnt cpc 13202
This theorem is referenced by:  sgmppw  20973  0sgmppw  20974  dchrisum0flblem1  21194
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693  ax-cnex 9038  ax-resscn 9039  ax-1cn 9040  ax-icn 9041  ax-addcl 9042  ax-addrcl 9043  ax-mulcl 9044  ax-mulrcl 9045  ax-mulcom 9046  ax-addass 9047  ax-mulass 9048  ax-distr 9049  ax-i2m1 9050  ax-1ne0 9051  ax-1rid 9052  ax-rnegex 9053  ax-rrecex 9054  ax-cnre 9055  ax-pre-lttri 9056  ax-pre-lttrn 9057  ax-pre-ltadd 9058  ax-pre-mulgt0 9059  ax-pre-sup 9060
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-reu 2704  df-rmo 2705  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-int 4043  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579  df-om 4838  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-1st 6341  df-2nd 6342  df-riota 6541  df-recs 6625  df-rdg 6660  df-1o 6716  df-2o 6717  df-oadd 6720  df-er 6897  df-en 7102  df-dom 7103  df-sdom 7104  df-fin 7105  df-sup 7438  df-pnf 9114  df-mnf 9115  df-xr 9116  df-ltxr 9117  df-le 9118  df-sub 9285  df-neg 9286  df-div 9670  df-nn 9993  df-2 10050  df-3 10051  df-n0 10214  df-z 10275  df-uz 10481  df-q 10567  df-rp 10605  df-fz 11036  df-fl 11194  df-mod 11243  df-seq 11316  df-exp 11375  df-cj 11896  df-re 11897  df-im 11898  df-sqr 12032  df-abs 12033  df-dvds 12845  df-gcd 12999  df-prm 13072  df-pc 13203
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