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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-onto→wf1o 6353  ‘cfv 6354  (class class class)co 7155  0cc0 10536   ≤ cle 10675  ℕcn 11637  ℕ0cn0 11896  ℤcz 11980  ℤ≥cuz 12242  ...cfz 12891  ↑cexp 13428   ∥ cdvds 15606  ℙcprime 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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