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Theorem dvdsppwf1o 22413
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 13751 . . . . 5  |-  ( P  e.  Prime  ->  P  e.  NN )
32adantr 462 . . . 4  |-  ( ( P  e.  Prime  /\  A  e.  NN0 )  ->  P  e.  NN )
4 elfznn0 11470 . . . 4  |-  ( n  e.  ( 0 ... A )  ->  n  e.  NN0 )
5 nnexpcl 11864 . . . 4  |-  ( ( P  e.  NN  /\  n  e.  NN0 )  -> 
( P ^ n
)  e.  NN )
63, 4, 5syl2an 474 . . 3  |-  ( ( ( P  e.  Prime  /\  A  e.  NN0 )  /\  n  e.  (
0 ... A ) )  ->  ( P ^
n )  e.  NN )
7 prmz 13752 . . . . 5  |-  ( P  e.  Prime  ->  P  e.  ZZ )
87ad2antrr 720 . . . 4  |-  ( ( ( P  e.  Prime  /\  A  e.  NN0 )  /\  n  e.  (
0 ... A ) )  ->  P  e.  ZZ )
94adantl 463 . . . 4  |-  ( ( ( P  e.  Prime  /\  A  e.  NN0 )  /\  n  e.  (
0 ... A ) )  ->  n  e.  NN0 )
10 elfzuz3 11439 . . . . 5  |-  ( n  e.  ( 0 ... A )  ->  A  e.  ( ZZ>= `  n )
)
1110adantl 463 . . . 4  |-  ( ( ( P  e.  Prime  /\  A  e.  NN0 )  /\  n  e.  (
0 ... A ) )  ->  A  e.  (
ZZ>= `  n ) )
12 dvdsexp 13574 . . . 4  |-  ( ( P  e.  ZZ  /\  n  e.  NN0  /\  A  e.  ( ZZ>= `  n )
)  ->  ( P ^ n )  ||  ( P ^ A ) )
138, 9, 11, 12syl3anc 1213 . . 3  |-  ( ( ( P  e.  Prime  /\  A  e.  NN0 )  /\  n  e.  (
0 ... A ) )  ->  ( P ^
n )  ||  ( P ^ A ) )
14 breq1 4285 . . . 4  |-  ( x  =  ( P ^
n )  ->  (
x  ||  ( P ^ A )  <->  ( P ^ n )  ||  ( P ^ A ) ) )
1514elrab 3108 . . 3  |-  ( ( P ^ n )  e.  { x  e.  NN  |  x  ||  ( P ^ A ) }  <->  ( ( P ^ n )  e.  NN  /\  ( P ^ n )  ||  ( P ^ A ) ) )
166, 13, 15sylanbrc 659 . 2  |-  ( ( ( P  e.  Prime  /\  A  e.  NN0 )  /\  n  e.  (
0 ... A ) )  ->  ( P ^
n )  e.  {
x  e.  NN  |  x  ||  ( P ^ A ) } )
17 simpl 454 . . . 4  |-  ( ( P  e.  Prime  /\  A  e.  NN0 )  ->  P  e.  Prime )
18 elrabi 3105 . . . 4  |-  ( m  e.  { x  e.  NN  |  x  ||  ( P ^ A ) }  ->  m  e.  NN )
19 pccl 13901 . . . 4  |-  ( ( P  e.  Prime  /\  m  e.  NN )  ->  ( P  pCnt  m )  e. 
NN0 )
2017, 18, 19syl2an 474 . . 3  |-  ( ( ( P  e.  Prime  /\  A  e.  NN0 )  /\  m  e.  { x  e.  NN  |  x  ||  ( P ^ A ) } )  ->  ( P  pCnt  m )  e. 
NN0 )
2117adantr 462 . . . . 5  |-  ( ( ( P  e.  Prime  /\  A  e.  NN0 )  /\  m  e.  { x  e.  NN  |  x  ||  ( P ^ A ) } )  ->  P  e.  Prime )
2218adantl 463 . . . . . 6  |-  ( ( ( P  e.  Prime  /\  A  e.  NN0 )  /\  m  e.  { x  e.  NN  |  x  ||  ( P ^ A ) } )  ->  m  e.  NN )
2322nnzd 10736 . . . . 5  |-  ( ( ( P  e.  Prime  /\  A  e.  NN0 )  /\  m  e.  { x  e.  NN  |  x  ||  ( P ^ A ) } )  ->  m  e.  ZZ )
247ad2antrr 720 . . . . . 6  |-  ( ( ( P  e.  Prime  /\  A  e.  NN0 )  /\  m  e.  { x  e.  NN  |  x  ||  ( P ^ A ) } )  ->  P  e.  ZZ )
25 simplr 749 . . . . . 6  |-  ( ( ( P  e.  Prime  /\  A  e.  NN0 )  /\  m  e.  { x  e.  NN  |  x  ||  ( P ^ A ) } )  ->  A  e.  NN0 )
26 zexpcl 11866 . . . . . 6  |-  ( ( P  e.  ZZ  /\  A  e.  NN0 )  -> 
( P ^ A
)  e.  ZZ )
2724, 25, 26syl2anc 656 . . . . 5  |-  ( ( ( P  e.  Prime  /\  A  e.  NN0 )  /\  m  e.  { x  e.  NN  |  x  ||  ( P ^ A ) } )  ->  ( P ^ A )  e.  ZZ )
28 breq1 4285 . . . . . . . 8  |-  ( x  =  m  ->  (
x  ||  ( P ^ A )  <->  m  ||  ( P ^ A ) ) )
2928elrab 3108 . . . . . . 7  |-  ( m  e.  { x  e.  NN  |  x  ||  ( P ^ A ) }  <->  ( m  e.  NN  /\  m  ||  ( P ^ A ) ) )
3029simprbi 461 . . . . . 6  |-  ( m  e.  { x  e.  NN  |  x  ||  ( P ^ A ) }  ->  m  ||  ( P ^ A ) )
3130adantl 463 . . . . 5  |-  ( ( ( P  e.  Prime  /\  A  e.  NN0 )  /\  m  e.  { x  e.  NN  |  x  ||  ( P ^ A ) } )  ->  m  ||  ( P ^ A
) )
32 pcdvdstr 13927 . . . . 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 1215 . . . 4  |-  ( ( ( P  e.  Prime  /\  A  e.  NN0 )  /\  m  e.  { x  e.  NN  |  x  ||  ( P ^ A ) } )  ->  ( P  pCnt  m )  <_ 
( P  pCnt  ( P ^ A ) ) )
34 pcidlem 13923 . . . . 5  |-  ( ( P  e.  Prime  /\  A  e.  NN0 )  ->  ( P  pCnt  ( P ^ A ) )  =  A )
3534adantr 462 . . . 4  |-  ( ( ( P  e.  Prime  /\  A  e.  NN0 )  /\  m  e.  { x  e.  NN  |  x  ||  ( P ^ A ) } )  ->  ( P  pCnt  ( P ^ A ) )  =  A )
3633, 35breqtrd 4306 . . 3  |-  ( ( ( P  e.  Prime  /\  A  e.  NN0 )  /\  m  e.  { x  e.  NN  |  x  ||  ( P ^ A ) } )  ->  ( P  pCnt  m )  <_  A )
37 fznn0 11510 . . . 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 908 . 2  |-  ( ( ( P  e.  Prime  /\  A  e.  NN0 )  /\  m  e.  { x  e.  NN  |  x  ||  ( P ^ A ) } )  ->  ( P  pCnt  m )  e.  ( 0 ... A
) )
40 oveq2 6090 . . . . . . . . 9  |-  ( n  =  A  ->  ( P ^ n )  =  ( P ^ A
) )
4140breq2d 4294 . . . . . . . 8  |-  ( n  =  A  ->  (
m  ||  ( P ^ n )  <->  m  ||  ( P ^ A ) ) )
4241rspcev 3064 . . . . . . 7  |-  ( ( A  e.  NN0  /\  m  ||  ( P ^ A ) )  ->  E. n  e.  NN0  m  ||  ( P ^
n ) )
4325, 31, 42syl2anc 656 . . . . . 6  |-  ( ( ( P  e.  Prime  /\  A  e.  NN0 )  /\  m  e.  { x  e.  NN  |  x  ||  ( P ^ A ) } )  ->  E. n  e.  NN0  m  ||  ( P ^ n ) )
44 pcprmpw2 13933 . . . . . . 7  |-  ( ( P  e.  Prime  /\  m  e.  NN )  ->  ( E. n  e.  NN0  m  ||  ( P ^
n )  <->  m  =  ( P ^ ( P 
pCnt  m ) ) ) )
4517, 18, 44syl2an 474 . . . . . 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 710 . . . 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 6090 . . . . 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 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 11442 . . . . . . 7  |-  ( n  e.  ( 0 ... A )  ->  n  e.  ZZ )
52 pcid 13924 . . . . . . 7  |-  ( ( P  e.  Prime  /\  n  e.  ZZ )  ->  ( P  pCnt  ( P ^
n ) )  =  n )
5317, 51, 52syl2an 474 . . . . . 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 711 . . . 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 6090 . . . . 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 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 6303 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 1364    e. wcel 1757   E.wrex 2708   {crab 2711   class class class wbr 4282    e. cmpt 4340   -1-1-onto->wf1o 5407   ` cfv 5408  (class class class)co 6082   0cc0 9272    <_ cle 9409   NNcn 10312   NN0cn0 10569   ZZcz 10636   ZZ>=cuz 10851   ...cfz 11426   ^cexp 11851    || cdivides 13520   Primecprime 13748    pCnt cpc 13888
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1671  ax-6 1709  ax-7 1729  ax-8 1759  ax-9 1761  ax-10 1776  ax-11 1781  ax-12 1793  ax-13 1944  ax-ext 2416  ax-sep 4403  ax-nul 4411  ax-pow 4460  ax-pr 4521  ax-un 6363  ax-cnex 9328  ax-resscn 9329  ax-1cn 9330  ax-icn 9331  ax-addcl 9332  ax-addrcl 9333  ax-mulcl 9334  ax-mulrcl 9335  ax-mulcom 9336  ax-addass 9337  ax-mulass 9338  ax-distr 9339  ax-i2m1 9340  ax-1ne0 9341  ax-1rid 9342  ax-rnegex 9343  ax-rrecex 9344  ax-cnre 9345  ax-pre-lttri 9346  ax-pre-lttrn 9347  ax-pre-ltadd 9348  ax-pre-mulgt0 9349  ax-pre-sup 9350
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 961  df-3an 962  df-tru 1367  df-ex 1592  df-nf 1595  df-sb 1702  df-eu 2260  df-mo 2261  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2712  df-rex 2713  df-reu 2714  df-rmo 2715  df-rab 2716  df-v 2966  df-sbc 3178  df-csb 3279  df-dif 3321  df-un 3323  df-in 3325  df-ss 3332  df-pss 3334  df-nul 3628  df-if 3782  df-pw 3852  df-sn 3868  df-pr 3870  df-tp 3872  df-op 3874  df-uni 4082  df-int 4119  df-iun 4163  df-br 4283  df-opab 4341  df-mpt 4342  df-tr 4376  df-eprel 4621  df-id 4625  df-po 4630  df-so 4631  df-fr 4668  df-we 4670  df-ord 4711  df-on 4712  df-lim 4713  df-suc 4714  df-xp 4835  df-rel 4836  df-cnv 4837  df-co 4838  df-dm 4839  df-rn 4840  df-res 4841  df-ima 4842  df-iota 5371  df-fun 5410  df-fn 5411  df-f 5412  df-f1 5413  df-fo 5414  df-f1o 5415  df-fv 5416  df-riota 6041  df-ov 6085  df-oprab 6086  df-mpt2 6087  df-om 6468  df-1st 6568  df-2nd 6569  df-recs 6820  df-rdg 6854  df-1o 6910  df-2o 6911  df-oadd 6914  df-er 7091  df-en 7301  df-dom 7302  df-sdom 7303  df-fin 7304  df-sup 7681  df-pnf 9410  df-mnf 9411  df-xr 9412  df-ltxr 9413  df-le 9414  df-sub 9587  df-neg 9588  df-div 9984  df-nn 10313  df-2 10370  df-3 10371  df-n0 10570  df-z 10637  df-uz 10852  df-q 10944  df-rp 10982  df-fz 11427  df-fl 11628  df-mod 11695  df-seq 11793  df-exp 11852  df-cj 12574  df-re 12575  df-im 12576  df-sqr 12710  df-abs 12711  df-dvds 13521  df-gcd 13676  df-prm 13749  df-pc 13889
This theorem is referenced by:  sgmppw  22423  0sgmppw  22424  dchrisum0flblem1  22644
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