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Theorem dvdsppwf1o 25332
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 prmnn 15767 . . . . 5 (𝑃 ∈ ℙ → 𝑃 ∈ ℕ)
32adantr 474 . . . 4 ((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℕ0) → 𝑃 ∈ ℕ)
4 elfznn0 12734 . . . 4 (𝑛 ∈ (0...𝐴) → 𝑛 ∈ ℕ0)
5 nnexpcl 13174 . . . 4 ((𝑃 ∈ ℕ ∧ 𝑛 ∈ ℕ0) → (𝑃𝑛) ∈ ℕ)
63, 4, 5syl2an 589 . . 3 (((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℕ0) ∧ 𝑛 ∈ (0...𝐴)) → (𝑃𝑛) ∈ ℕ)
7 prmz 15768 . . . . 5 (𝑃 ∈ ℙ → 𝑃 ∈ ℤ)
87ad2antrr 717 . . . 4 (((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℕ0) ∧ 𝑛 ∈ (0...𝐴)) → 𝑃 ∈ ℤ)
94adantl 475 . . . 4 (((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℕ0) ∧ 𝑛 ∈ (0...𝐴)) → 𝑛 ∈ ℕ0)
10 elfzuz3 12639 . . . . 5 (𝑛 ∈ (0...𝐴) → 𝐴 ∈ (ℤ𝑛))
1110adantl 475 . . . 4 (((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℕ0) ∧ 𝑛 ∈ (0...𝐴)) → 𝐴 ∈ (ℤ𝑛))
12 dvdsexp 15433 . . . 4 ((𝑃 ∈ ℤ ∧ 𝑛 ∈ ℕ0𝐴 ∈ (ℤ𝑛)) → (𝑃𝑛) ∥ (𝑃𝐴))
138, 9, 11, 12syl3anc 1494 . . 3 (((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℕ0) ∧ 𝑛 ∈ (0...𝐴)) → (𝑃𝑛) ∥ (𝑃𝐴))
14 breq1 4878 . . . 4 (𝑥 = (𝑃𝑛) → (𝑥 ∥ (𝑃𝐴) ↔ (𝑃𝑛) ∥ (𝑃𝐴)))
1514elrab 3585 . . 3 ((𝑃𝑛) ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ (𝑃𝐴)} ↔ ((𝑃𝑛) ∈ ℕ ∧ (𝑃𝑛) ∥ (𝑃𝐴)))
166, 13, 15sylanbrc 578 . 2 (((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℕ0) ∧ 𝑛 ∈ (0...𝐴)) → (𝑃𝑛) ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ (𝑃𝐴)})
17 simpl 476 . . . 4 ((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℕ0) → 𝑃 ∈ ℙ)
18 elrabi 3580 . . . 4 (𝑚 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ (𝑃𝐴)} → 𝑚 ∈ ℕ)
19 pccl 15932 . . . 4 ((𝑃 ∈ ℙ ∧ 𝑚 ∈ ℕ) → (𝑃 pCnt 𝑚) ∈ ℕ0)
2017, 18, 19syl2an 589 . . 3 (((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℕ0) ∧ 𝑚 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ (𝑃𝐴)}) → (𝑃 pCnt 𝑚) ∈ ℕ0)
2117adantr 474 . . . . 5 (((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℕ0) ∧ 𝑚 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ (𝑃𝐴)}) → 𝑃 ∈ ℙ)
2218adantl 475 . . . . . 6 (((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℕ0) ∧ 𝑚 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ (𝑃𝐴)}) → 𝑚 ∈ ℕ)
2322nnzd 11816 . . . . 5 (((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℕ0) ∧ 𝑚 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ (𝑃𝐴)}) → 𝑚 ∈ ℤ)
247ad2antrr 717 . . . . . 6 (((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℕ0) ∧ 𝑚 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ (𝑃𝐴)}) → 𝑃 ∈ ℤ)
25 simplr 785 . . . . . 6 (((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℕ0) ∧ 𝑚 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ (𝑃𝐴)}) → 𝐴 ∈ ℕ0)
26 zexpcl 13176 . . . . . 6 ((𝑃 ∈ ℤ ∧ 𝐴 ∈ ℕ0) → (𝑃𝐴) ∈ ℤ)
2724, 25, 26syl2anc 579 . . . . 5 (((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℕ0) ∧ 𝑚 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ (𝑃𝐴)}) → (𝑃𝐴) ∈ ℤ)
28 breq1 4878 . . . . . . . 8 (𝑥 = 𝑚 → (𝑥 ∥ (𝑃𝐴) ↔ 𝑚 ∥ (𝑃𝐴)))
2928elrab 3585 . . . . . . 7 (𝑚 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ (𝑃𝐴)} ↔ (𝑚 ∈ ℕ ∧ 𝑚 ∥ (𝑃𝐴)))
3029simprbi 492 . . . . . 6 (𝑚 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ (𝑃𝐴)} → 𝑚 ∥ (𝑃𝐴))
3130adantl 475 . . . . 5 (((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℕ0) ∧ 𝑚 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ (𝑃𝐴)}) → 𝑚 ∥ (𝑃𝐴))
32 pcdvdstr 15958 . . . . 5 ((𝑃 ∈ ℙ ∧ (𝑚 ∈ ℤ ∧ (𝑃𝐴) ∈ ℤ ∧ 𝑚 ∥ (𝑃𝐴))) → (𝑃 pCnt 𝑚) ≤ (𝑃 pCnt (𝑃𝐴)))
3321, 23, 27, 31, 32syl13anc 1495 . . . 4 (((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℕ0) ∧ 𝑚 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ (𝑃𝐴)}) → (𝑃 pCnt 𝑚) ≤ (𝑃 pCnt (𝑃𝐴)))
34 pcidlem 15954 . . . . 5 ((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℕ0) → (𝑃 pCnt (𝑃𝐴)) = 𝐴)
3534adantr 474 . . . 4 (((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℕ0) ∧ 𝑚 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ (𝑃𝐴)}) → (𝑃 pCnt (𝑃𝐴)) = 𝐴)
3633, 35breqtrd 4901 . . 3 (((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℕ0) ∧ 𝑚 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ (𝑃𝐴)}) → (𝑃 pCnt 𝑚) ≤ 𝐴)
37 fznn0 12733 . . . 4 (𝐴 ∈ ℕ0 → ((𝑃 pCnt 𝑚) ∈ (0...𝐴) ↔ ((𝑃 pCnt 𝑚) ∈ ℕ0 ∧ (𝑃 pCnt 𝑚) ≤ 𝐴)))
3825, 37syl 17 . . 3 (((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℕ0) ∧ 𝑚 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ (𝑃𝐴)}) → ((𝑃 pCnt 𝑚) ∈ (0...𝐴) ↔ ((𝑃 pCnt 𝑚) ∈ ℕ0 ∧ (𝑃 pCnt 𝑚) ≤ 𝐴)))
3920, 36, 38mpbir2and 704 . 2 (((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℕ0) ∧ 𝑚 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ (𝑃𝐴)}) → (𝑃 pCnt 𝑚) ∈ (0...𝐴))
40 oveq2 6918 . . . . . . . . 9 (𝑛 = 𝐴 → (𝑃𝑛) = (𝑃𝐴))
4140breq2d 4887 . . . . . . . 8 (𝑛 = 𝐴 → (𝑚 ∥ (𝑃𝑛) ↔ 𝑚 ∥ (𝑃𝐴)))
4241rspcev 3526 . . . . . . 7 ((𝐴 ∈ ℕ0𝑚 ∥ (𝑃𝐴)) → ∃𝑛 ∈ ℕ0 𝑚 ∥ (𝑃𝑛))
4325, 31, 42syl2anc 579 . . . . . 6 (((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℕ0) ∧ 𝑚 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ (𝑃𝐴)}) → ∃𝑛 ∈ ℕ0 𝑚 ∥ (𝑃𝑛))
44 pcprmpw2 15964 . . . . . . 7 ((𝑃 ∈ ℙ ∧ 𝑚 ∈ ℕ) → (∃𝑛 ∈ ℕ0 𝑚 ∥ (𝑃𝑛) ↔ 𝑚 = (𝑃↑(𝑃 pCnt 𝑚))))
4517, 18, 44syl2an 589 . . . . . 6 (((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℕ0) ∧ 𝑚 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ (𝑃𝐴)}) → (∃𝑛 ∈ ℕ0 𝑚 ∥ (𝑃𝑛) ↔ 𝑚 = (𝑃↑(𝑃 pCnt 𝑚))))
4643, 45mpbid 224 . . . . 5 (((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℕ0) ∧ 𝑚 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ (𝑃𝐴)}) → 𝑚 = (𝑃↑(𝑃 pCnt 𝑚)))
4746adantrl 707 . . . 4 (((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℕ0) ∧ (𝑛 ∈ (0...𝐴) ∧ 𝑚 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ (𝑃𝐴)})) → 𝑚 = (𝑃↑(𝑃 pCnt 𝑚)))
48 oveq2 6918 . . . . 5 (𝑛 = (𝑃 pCnt 𝑚) → (𝑃𝑛) = (𝑃↑(𝑃 pCnt 𝑚)))
4948eqeq2d 2835 . . . 4 (𝑛 = (𝑃 pCnt 𝑚) → (𝑚 = (𝑃𝑛) ↔ 𝑚 = (𝑃↑(𝑃 pCnt 𝑚))))
5047, 49syl5ibrcom 239 . . 3 (((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℕ0) ∧ (𝑛 ∈ (0...𝐴) ∧ 𝑚 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ (𝑃𝐴)})) → (𝑛 = (𝑃 pCnt 𝑚) → 𝑚 = (𝑃𝑛)))
51 elfzelz 12642 . . . . . . 7 (𝑛 ∈ (0...𝐴) → 𝑛 ∈ ℤ)
52 pcid 15955 . . . . . . 7 ((𝑃 ∈ ℙ ∧ 𝑛 ∈ ℤ) → (𝑃 pCnt (𝑃𝑛)) = 𝑛)
5317, 51, 52syl2an 589 . . . . . 6 (((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℕ0) ∧ 𝑛 ∈ (0...𝐴)) → (𝑃 pCnt (𝑃𝑛)) = 𝑛)
5453eqcomd 2831 . . . . 5 (((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℕ0) ∧ 𝑛 ∈ (0...𝐴)) → 𝑛 = (𝑃 pCnt (𝑃𝑛)))
5554adantrr 708 . . . 4 (((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℕ0) ∧ (𝑛 ∈ (0...𝐴) ∧ 𝑚 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ (𝑃𝐴)})) → 𝑛 = (𝑃 pCnt (𝑃𝑛)))
56 oveq2 6918 . . . . 5 (𝑚 = (𝑃𝑛) → (𝑃 pCnt 𝑚) = (𝑃 pCnt (𝑃𝑛)))
5756eqeq2d 2835 . . . 4 (𝑚 = (𝑃𝑛) → (𝑛 = (𝑃 pCnt 𝑚) ↔ 𝑛 = (𝑃 pCnt (𝑃𝑛))))
5855, 57syl5ibrcom 239 . . 3 (((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℕ0) ∧ (𝑛 ∈ (0...𝐴) ∧ 𝑚 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ (𝑃𝐴)})) → (𝑚 = (𝑃𝑛) → 𝑛 = (𝑃 pCnt 𝑚)))
5950, 58impbid 204 . 2 (((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℕ0) ∧ (𝑛 ∈ (0...𝐴) ∧ 𝑚 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ (𝑃𝐴)})) → (𝑛 = (𝑃 pCnt 𝑚) ↔ 𝑚 = (𝑃𝑛)))
601, 16, 39, 59f1o2d 7152 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 198  wa 386   = wceq 1656  wcel 2164  wrex 3118  {crab 3121   class class class wbr 4875  cmpt 4954  1-1-ontowf1o 6126  cfv 6127  (class class class)co 6910  0cc0 10259  cle 10399  cn 11357  0cn0 11625  cz 11711  cuz 11975  ...cfz 12626  cexp 13161  cdvds 15364  cprime 15764   pCnt cpc 15919
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1894  ax-4 1908  ax-5 2009  ax-6 2075  ax-7 2112  ax-8 2166  ax-9 2173  ax-10 2192  ax-11 2207  ax-12 2220  ax-13 2389  ax-ext 2803  ax-sep 5007  ax-nul 5015  ax-pow 5067  ax-pr 5129  ax-un 7214  ax-cnex 10315  ax-resscn 10316  ax-1cn 10317  ax-icn 10318  ax-addcl 10319  ax-addrcl 10320  ax-mulcl 10321  ax-mulrcl 10322  ax-mulcom 10323  ax-addass 10324  ax-mulass 10325  ax-distr 10326  ax-i2m1 10327  ax-1ne0 10328  ax-1rid 10329  ax-rnegex 10330  ax-rrecex 10331  ax-cnre 10332  ax-pre-lttri 10333  ax-pre-lttrn 10334  ax-pre-ltadd 10335  ax-pre-mulgt0 10336  ax-pre-sup 10337
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 879  df-3or 1112  df-3an 1113  df-tru 1660  df-ex 1879  df-nf 1883  df-sb 2068  df-mo 2605  df-eu 2640  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ne 3000  df-nel 3103  df-ral 3122  df-rex 3123  df-reu 3124  df-rmo 3125  df-rab 3126  df-v 3416  df-sbc 3663  df-csb 3758  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-pss 3814  df-nul 4147  df-if 4309  df-pw 4382  df-sn 4400  df-pr 4402  df-tp 4404  df-op 4406  df-uni 4661  df-iun 4744  df-br 4876  df-opab 4938  df-mpt 4955  df-tr 4978  df-id 5252  df-eprel 5257  df-po 5265  df-so 5266  df-fr 5305  df-we 5307  df-xp 5352  df-rel 5353  df-cnv 5354  df-co 5355  df-dm 5356  df-rn 5357  df-res 5358  df-ima 5359  df-pred 5924  df-ord 5970  df-on 5971  df-lim 5972  df-suc 5973  df-iota 6090  df-fun 6129  df-fn 6130  df-f 6131  df-f1 6132  df-fo 6133  df-f1o 6134  df-fv 6135  df-riota 6871  df-ov 6913  df-oprab 6914  df-mpt2 6915  df-om 7332  df-1st 7433  df-2nd 7434  df-wrecs 7677  df-recs 7739  df-rdg 7777  df-1o 7831  df-2o 7832  df-er 8014  df-en 8229  df-dom 8230  df-sdom 8231  df-fin 8232  df-sup 8623  df-inf 8624  df-pnf 10400  df-mnf 10401  df-xr 10402  df-ltxr 10403  df-le 10404  df-sub 10594  df-neg 10595  df-div 11017  df-nn 11358  df-2 11421  df-3 11422  df-n0 11626  df-z 11712  df-uz 11976  df-q 12079  df-rp 12120  df-fz 12627  df-fl 12895  df-mod 12971  df-seq 13103  df-exp 13162  df-cj 14223  df-re 14224  df-im 14225  df-sqrt 14359  df-abs 14360  df-dvds 15365  df-gcd 15597  df-prm 15765  df-pc 15920
This theorem is referenced by:  sgmppw  25342  0sgmppw  25343  dchrisum0flblem1  25617
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