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Theorem dvdsppwf1o 20839
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 13010 . . . . 5  |-  ( P  e.  Prime  ->  P  e.  NN )
32adantr 452 . . . 4  |-  ( ( P  e.  Prime  /\  A  e.  NN0 )  ->  P  e.  NN )
4 elfznn0 11016 . . . 4  |-  ( n  e.  ( 0 ... A )  ->  n  e.  NN0 )
5 nnexpcl 11322 . . . 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 13011 . . . . 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 10989 . . . . 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 12833 . . . 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 4157 . . . 4  |-  ( x  =  ( P ^
n )  ->  (
x  ||  ( P ^ A )  <->  ( P ^ n )  ||  ( P ^ A ) ) )
1514elrab 3036 . . 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 3034 . . . 4  |-  ( m  e.  { x  e.  NN  |  x  ||  ( P ^ A ) }  ->  m  e.  NN )
19 pccl 13151 . . . 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 10307 . . . . 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 11324 . . . . . 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 4157 . . . . . . . 8  |-  ( x  =  m  ->  (
x  ||  ( P ^ A )  <->  m  ||  ( P ^ A ) ) )
2928elrab 3036 . . . . . . 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 13177 . . . . 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 13173 . . . . 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 4178 . . 3  |-  ( ( ( P  e.  Prime  /\  A  e.  NN0 )  /\  m  e.  { x  e.  NN  |  x  ||  ( P ^ A ) } )  ->  ( P  pCnt  m )  <_  A )
37 fznn0 11045 . . . 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 6029 . . . . . . . . 9  |-  ( n  =  A  ->  ( P ^ n )  =  ( P ^ A
) )
4140breq2d 4166 . . . . . . . 8  |-  ( n  =  A  ->  (
m  ||  ( P ^ n )  <->  m  ||  ( P ^ A ) ) )
4241rspcev 2996 . . . . . . 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 13183 . . . . . . 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 6029 . . . . 5  |-  ( n  =  ( P  pCnt  m )  ->  ( P ^ n )  =  ( P ^ ( P  pCnt  m ) ) )
4948eqeq2d 2399 . . . 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 10992 . . . . . . 7  |-  ( n  e.  ( 0 ... A )  ->  n  e.  ZZ )
52 pcid 13174 . . . . . . 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 2393 . . . . 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 6029 . . . . 5  |-  ( m  =  ( P ^
n )  ->  ( P  pCnt  m )  =  ( P  pCnt  ( P ^ n ) ) )
5756eqeq2d 2399 . . . 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 6236 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 1717   E.wrex 2651   {crab 2654   class class class wbr 4154    e. cmpt 4208   -1-1-onto->wf1o 5394   ` cfv 5395  (class class class)co 6021   0cc0 8924    <_ cle 9055   NNcn 9933   NN0cn0 10154   ZZcz 10215   ZZ>=cuz 10421   ...cfz 10976   ^cexp 11310    || cdivides 12780   Primecprime 13007    pCnt cpc 13138
This theorem is referenced by:  sgmppw  20849  0sgmppw  20850  dchrisum0flblem1  21070
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 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2369  ax-sep 4272  ax-nul 4280  ax-pow 4319  ax-pr 4345  ax-un 4642  ax-cnex 8980  ax-resscn 8981  ax-1cn 8982  ax-icn 8983  ax-addcl 8984  ax-addrcl 8985  ax-mulcl 8986  ax-mulrcl 8987  ax-mulcom 8988  ax-addass 8989  ax-mulass 8990  ax-distr 8991  ax-i2m1 8992  ax-1ne0 8993  ax-1rid 8994  ax-rnegex 8995  ax-rrecex 8996  ax-cnre 8997  ax-pre-lttri 8998  ax-pre-lttrn 8999  ax-pre-ltadd 9000  ax-pre-mulgt0 9001  ax-pre-sup 9002
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 2243  df-mo 2244  df-clab 2375  df-cleq 2381  df-clel 2384  df-nfc 2513  df-ne 2553  df-nel 2554  df-ral 2655  df-rex 2656  df-reu 2657  df-rmo 2658  df-rab 2659  df-v 2902  df-sbc 3106  df-csb 3196  df-dif 3267  df-un 3269  df-in 3271  df-ss 3278  df-pss 3280  df-nul 3573  df-if 3684  df-pw 3745  df-sn 3764  df-pr 3765  df-tp 3766  df-op 3767  df-uni 3959  df-int 3994  df-iun 4038  df-br 4155  df-opab 4209  df-mpt 4210  df-tr 4245  df-eprel 4436  df-id 4440  df-po 4445  df-so 4446  df-fr 4483  df-we 4485  df-ord 4526  df-on 4527  df-lim 4528  df-suc 4529  df-om 4787  df-xp 4825  df-rel 4826  df-cnv 4827  df-co 4828  df-dm 4829  df-rn 4830  df-res 4831  df-ima 4832  df-iota 5359  df-fun 5397  df-fn 5398  df-f 5399  df-f1 5400  df-fo 5401  df-f1o 5402  df-fv 5403  df-ov 6024  df-oprab 6025  df-mpt2 6026  df-1st 6289  df-2nd 6290  df-riota 6486  df-recs 6570  df-rdg 6605  df-1o 6661  df-2o 6662  df-oadd 6665  df-er 6842  df-en 7047  df-dom 7048  df-sdom 7049  df-fin 7050  df-sup 7382  df-pnf 9056  df-mnf 9057  df-xr 9058  df-ltxr 9059  df-le 9060  df-sub 9226  df-neg 9227  df-div 9611  df-nn 9934  df-2 9991  df-3 9992  df-n0 10155  df-z 10216  df-uz 10422  df-q 10508  df-rp 10546  df-fz 10977  df-fl 11130  df-mod 11179  df-seq 11252  df-exp 11311  df-cj 11832  df-re 11833  df-im 11834  df-sqr 11968  df-abs 11969  df-dvds 12781  df-gcd 12935  df-prm 13008  df-pc 13139
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