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Theorem dvdsppwf1o 25762
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 ∈ (0...𝐴) ↦ (𝑃𝑛))
Assertion
Ref Expression
dvdsppwf1o ((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℕ0) → 𝐹:(0...𝐴)–1-1-onto→{𝑥 ∈ ℕ ∣ 𝑥 ∥ (𝑃𝐴)})
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 ∈ (0...𝐴) ↦ (𝑃𝑛))
2 breq1 5068 . . 3 (𝑥 = (𝑃𝑛) → (𝑥 ∥ (𝑃𝐴) ↔ (𝑃𝑛) ∥ (𝑃𝐴)))
3 prmnn 16017 . . . . 5 (𝑃 ∈ ℙ → 𝑃 ∈ ℕ)
43adantr 483 . . . 4 ((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℕ0) → 𝑃 ∈ ℕ)
5 elfznn0 12999 . . . 4 (𝑛 ∈ (0...𝐴) → 𝑛 ∈ ℕ0)
6 nnexpcl 13441 . . . 4 ((𝑃 ∈ ℕ ∧ 𝑛 ∈ ℕ0) → (𝑃𝑛) ∈ ℕ)
74, 5, 6syl2an 597 . . 3 (((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℕ0) ∧ 𝑛 ∈ (0...𝐴)) → (𝑃𝑛) ∈ ℕ)
8 prmz 16018 . . . . 5 (𝑃 ∈ ℙ → 𝑃 ∈ ℤ)
98ad2antrr 724 . . . 4 (((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℕ0) ∧ 𝑛 ∈ (0...𝐴)) → 𝑃 ∈ ℤ)
105adantl 484 . . . 4 (((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℕ0) ∧ 𝑛 ∈ (0...𝐴)) → 𝑛 ∈ ℕ0)
11 elfzuz3 12904 . . . . 5 (𝑛 ∈ (0...𝐴) → 𝐴 ∈ (ℤ𝑛))
1211adantl 484 . . . 4 (((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℕ0) ∧ 𝑛 ∈ (0...𝐴)) → 𝐴 ∈ (ℤ𝑛))
13 dvdsexp 15676 . . . 4 ((𝑃 ∈ ℤ ∧ 𝑛 ∈ ℕ0𝐴 ∈ (ℤ𝑛)) → (𝑃𝑛) ∥ (𝑃𝐴))
149, 10, 12, 13syl3anc 1367 . . 3 (((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℕ0) ∧ 𝑛 ∈ (0...𝐴)) → (𝑃𝑛) ∥ (𝑃𝐴))
152, 7, 14elrabd 3681 . 2 (((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℕ0) ∧ 𝑛 ∈ (0...𝐴)) → (𝑃𝑛) ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ (𝑃𝐴)})
16 simpl 485 . . . 4 ((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℕ0) → 𝑃 ∈ ℙ)
17 elrabi 3674 . . . 4 (𝑚 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ (𝑃𝐴)} → 𝑚 ∈ ℕ)
18 pccl 16185 . . . 4 ((𝑃 ∈ ℙ ∧ 𝑚 ∈ ℕ) → (𝑃 pCnt 𝑚) ∈ ℕ0)
1916, 17, 18syl2an 597 . . 3 (((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℕ0) ∧ 𝑚 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ (𝑃𝐴)}) → (𝑃 pCnt 𝑚) ∈ ℕ0)
2016adantr 483 . . . . 5 (((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℕ0) ∧ 𝑚 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ (𝑃𝐴)}) → 𝑃 ∈ ℙ)
2117adantl 484 . . . . . 6 (((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℕ0) ∧ 𝑚 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ (𝑃𝐴)}) → 𝑚 ∈ ℕ)
2221nnzd 12085 . . . . 5 (((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℕ0) ∧ 𝑚 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ (𝑃𝐴)}) → 𝑚 ∈ ℤ)
238ad2antrr 724 . . . . . 6 (((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℕ0) ∧ 𝑚 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ (𝑃𝐴)}) → 𝑃 ∈ ℤ)
24 simplr 767 . . . . . 6 (((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℕ0) ∧ 𝑚 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ (𝑃𝐴)}) → 𝐴 ∈ ℕ0)
25 zexpcl 13443 . . . . . 6 ((𝑃 ∈ ℤ ∧ 𝐴 ∈ ℕ0) → (𝑃𝐴) ∈ ℤ)
2623, 24, 25syl2anc 586 . . . . 5 (((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℕ0) ∧ 𝑚 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ (𝑃𝐴)}) → (𝑃𝐴) ∈ ℤ)
27 breq1 5068 . . . . . . . 8 (𝑥 = 𝑚 → (𝑥 ∥ (𝑃𝐴) ↔ 𝑚 ∥ (𝑃𝐴)))
2827elrab 3679 . . . . . . 7 (𝑚 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ (𝑃𝐴)} ↔ (𝑚 ∈ ℕ ∧ 𝑚 ∥ (𝑃𝐴)))
2928simprbi 499 . . . . . 6 (𝑚 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ (𝑃𝐴)} → 𝑚 ∥ (𝑃𝐴))
3029adantl 484 . . . . 5 (((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℕ0) ∧ 𝑚 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ (𝑃𝐴)}) → 𝑚 ∥ (𝑃𝐴))
31 pcdvdstr 16211 . . . . 5 ((𝑃 ∈ ℙ ∧ (𝑚 ∈ ℤ ∧ (𝑃𝐴) ∈ ℤ ∧ 𝑚 ∥ (𝑃𝐴))) → (𝑃 pCnt 𝑚) ≤ (𝑃 pCnt (𝑃𝐴)))
3220, 22, 26, 30, 31syl13anc 1368 . . . 4 (((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℕ0) ∧ 𝑚 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ (𝑃𝐴)}) → (𝑃 pCnt 𝑚) ≤ (𝑃 pCnt (𝑃𝐴)))
33 pcidlem 16207 . . . . 5 ((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℕ0) → (𝑃 pCnt (𝑃𝐴)) = 𝐴)
3433adantr 483 . . . 4 (((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℕ0) ∧ 𝑚 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ (𝑃𝐴)}) → (𝑃 pCnt (𝑃𝐴)) = 𝐴)
3532, 34breqtrd 5091 . . 3 (((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℕ0) ∧ 𝑚 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ (𝑃𝐴)}) → (𝑃 pCnt 𝑚) ≤ 𝐴)
36 fznn0 12998 . . . 4 (𝐴 ∈ ℕ0 → ((𝑃 pCnt 𝑚) ∈ (0...𝐴) ↔ ((𝑃 pCnt 𝑚) ∈ ℕ0 ∧ (𝑃 pCnt 𝑚) ≤ 𝐴)))
3724, 36syl 17 . . 3 (((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℕ0) ∧ 𝑚 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ (𝑃𝐴)}) → ((𝑃 pCnt 𝑚) ∈ (0...𝐴) ↔ ((𝑃 pCnt 𝑚) ∈ ℕ0 ∧ (𝑃 pCnt 𝑚) ≤ 𝐴)))
3819, 35, 37mpbir2and 711 . 2 (((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℕ0) ∧ 𝑚 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ (𝑃𝐴)}) → (𝑃 pCnt 𝑚) ∈ (0...𝐴))
39 oveq2 7163 . . . . . . . . 9 (𝑛 = 𝐴 → (𝑃𝑛) = (𝑃𝐴))
4039breq2d 5077 . . . . . . . 8 (𝑛 = 𝐴 → (𝑚 ∥ (𝑃𝑛) ↔ 𝑚 ∥ (𝑃𝐴)))
4140rspcev 3622 . . . . . . 7 ((𝐴 ∈ ℕ0𝑚 ∥ (𝑃𝐴)) → ∃𝑛 ∈ ℕ0 𝑚 ∥ (𝑃𝑛))
4224, 30, 41syl2anc 586 . . . . . 6 (((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℕ0) ∧ 𝑚 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ (𝑃𝐴)}) → ∃𝑛 ∈ ℕ0 𝑚 ∥ (𝑃𝑛))
43 pcprmpw2 16217 . . . . . . 7 ((𝑃 ∈ ℙ ∧ 𝑚 ∈ ℕ) → (∃𝑛 ∈ ℕ0 𝑚 ∥ (𝑃𝑛) ↔ 𝑚 = (𝑃↑(𝑃 pCnt 𝑚))))
4416, 17, 43syl2an 597 . . . . . 6 (((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℕ0) ∧ 𝑚 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ (𝑃𝐴)}) → (∃𝑛 ∈ ℕ0 𝑚 ∥ (𝑃𝑛) ↔ 𝑚 = (𝑃↑(𝑃 pCnt 𝑚))))
4542, 44mpbid 234 . . . . 5 (((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℕ0) ∧ 𝑚 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ (𝑃𝐴)}) → 𝑚 = (𝑃↑(𝑃 pCnt 𝑚)))
4645adantrl 714 . . . 4 (((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℕ0) ∧ (𝑛 ∈ (0...𝐴) ∧ 𝑚 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ (𝑃𝐴)})) → 𝑚 = (𝑃↑(𝑃 pCnt 𝑚)))
47 oveq2 7163 . . . . 5 (𝑛 = (𝑃 pCnt 𝑚) → (𝑃𝑛) = (𝑃↑(𝑃 pCnt 𝑚)))
4847eqeq2d 2832 . . . 4 (𝑛 = (𝑃 pCnt 𝑚) → (𝑚 = (𝑃𝑛) ↔ 𝑚 = (𝑃↑(𝑃 pCnt 𝑚))))
4946, 48syl5ibrcom 249 . . 3 (((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℕ0) ∧ (𝑛 ∈ (0...𝐴) ∧ 𝑚 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ (𝑃𝐴)})) → (𝑛 = (𝑃 pCnt 𝑚) → 𝑚 = (𝑃𝑛)))
50 elfzelz 12907 . . . . . . 7 (𝑛 ∈ (0...𝐴) → 𝑛 ∈ ℤ)
51 pcid 16208 . . . . . . 7 ((𝑃 ∈ ℙ ∧ 𝑛 ∈ ℤ) → (𝑃 pCnt (𝑃𝑛)) = 𝑛)
5216, 50, 51syl2an 597 . . . . . 6 (((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℕ0) ∧ 𝑛 ∈ (0...𝐴)) → (𝑃 pCnt (𝑃𝑛)) = 𝑛)
5352eqcomd 2827 . . . . 5 (((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℕ0) ∧ 𝑛 ∈ (0...𝐴)) → 𝑛 = (𝑃 pCnt (𝑃𝑛)))
5453adantrr 715 . . . 4 (((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℕ0) ∧ (𝑛 ∈ (0...𝐴) ∧ 𝑚 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ (𝑃𝐴)})) → 𝑛 = (𝑃 pCnt (𝑃𝑛)))
55 oveq2 7163 . . . . 5 (𝑚 = (𝑃𝑛) → (𝑃 pCnt 𝑚) = (𝑃 pCnt (𝑃𝑛)))
5655eqeq2d 2832 . . . 4 (𝑚 = (𝑃𝑛) → (𝑛 = (𝑃 pCnt 𝑚) ↔ 𝑛 = (𝑃 pCnt (𝑃𝑛))))
5754, 56syl5ibrcom 249 . . 3 (((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℕ0) ∧ (𝑛 ∈ (0...𝐴) ∧ 𝑚 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ (𝑃𝐴)})) → (𝑚 = (𝑃𝑛) → 𝑛 = (𝑃 pCnt 𝑚)))
5849, 57impbid 214 . 2 (((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℕ0) ∧ (𝑛 ∈ (0...𝐴) ∧ 𝑚 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ (𝑃𝐴)})) → (𝑛 = (𝑃 pCnt 𝑚) ↔ 𝑚 = (𝑃𝑛)))
591, 15, 38, 58f1o2d 7398 1 ((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℕ0) → 𝐹:(0...𝐴)–1-1-onto→{𝑥 ∈ ℕ ∣ 𝑥 ∥ (𝑃𝐴)})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 208  wa 398   = wceq 1533  wcel 2110  wrex 3139  {crab 3142   class class class wbr 5065  cmpt 5145  1-1-ontowf1o 6353  cfv 6354  (class class class)co 7155  0cc0 10536  cle 10675  cn 11637  0cn0 11896  cz 11980  cuz 12242  ...cfz 12891  cexp 13428  cdvds 15606  cprime 16014   pCnt cpc 16172
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793  ax-sep 5202  ax-nul 5209  ax-pow 5265  ax-pr 5329  ax-un 7460  ax-cnex 10592  ax-resscn 10593  ax-1cn 10594  ax-icn 10595  ax-addcl 10596  ax-addrcl 10597  ax-mulcl 10598  ax-mulrcl 10599  ax-mulcom 10600  ax-addass 10601  ax-mulass 10602  ax-distr 10603  ax-i2m1 10604  ax-1ne0 10605  ax-1rid 10606  ax-rnegex 10607  ax-rrecex 10608  ax-cnre 10609  ax-pre-lttri 10610  ax-pre-lttrn 10611  ax-pre-ltadd 10612  ax-pre-mulgt0 10613  ax-pre-sup 10614
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-nel 3124  df-ral 3143  df-rex 3144  df-reu 3145  df-rmo 3146  df-rab 3147  df-v 3496  df-sbc 3772  df-csb 3883  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-pss 3953  df-nul 4291  df-if 4467  df-pw 4540  df-sn 4567  df-pr 4569  df-tp 4571  df-op 4573  df-uni 4838  df-iun 4920  df-br 5066  df-opab 5128  df-mpt 5146  df-tr 5172  df-id 5459  df-eprel 5464  df-po 5473  df-so 5474  df-fr 5513  df-we 5515  df-xp 5560  df-rel 5561  df-cnv 5562  df-co 5563  df-dm 5564  df-rn 5565  df-res 5566  df-ima 5567  df-pred 6147  df-ord 6193  df-on 6194  df-lim 6195  df-suc 6196  df-iota 6313  df-fun 6356  df-fn 6357  df-f 6358  df-f1 6359  df-fo 6360  df-f1o 6361  df-fv 6362  df-riota 7113  df-ov 7158  df-oprab 7159  df-mpo 7160  df-om 7580  df-1st 7688  df-2nd 7689  df-wrecs 7946  df-recs 8007  df-rdg 8045  df-1o 8101  df-2o 8102  df-er 8288  df-en 8509  df-dom 8510  df-sdom 8511  df-fin 8512  df-sup 8905  df-inf 8906  df-pnf 10676  df-mnf 10677  df-xr 10678  df-ltxr 10679  df-le 10680  df-sub 10871  df-neg 10872  df-div 11297  df-nn 11638  df-2 11699  df-3 11700  df-n0 11897  df-z 11981  df-uz 12243  df-q 12348  df-rp 12389  df-fz 12892  df-fl 13161  df-mod 13237  df-seq 13369  df-exp 13429  df-cj 14457  df-re 14458  df-im 14459  df-sqrt 14593  df-abs 14594  df-dvds 15607  df-gcd 15843  df-prm 16015  df-pc 16173
This theorem is referenced by:  sgmppw  25772  0sgmppw  25773  dchrisum0flblem1  26083
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