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Theorem dvdsppwf1o 27244
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 5151 . . 3 (𝑥 = (𝑃𝑛) → (𝑥 ∥ (𝑃𝐴) ↔ (𝑃𝑛) ∥ (𝑃𝐴)))
3 prmnn 16708 . . . . 5 (𝑃 ∈ ℙ → 𝑃 ∈ ℕ)
43adantr 480 . . . 4 ((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℕ0) → 𝑃 ∈ ℕ)
5 elfznn0 13657 . . . 4 (𝑛 ∈ (0...𝐴) → 𝑛 ∈ ℕ0)
6 nnexpcl 14112 . . . 4 ((𝑃 ∈ ℕ ∧ 𝑛 ∈ ℕ0) → (𝑃𝑛) ∈ ℕ)
74, 5, 6syl2an 596 . . 3 (((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℕ0) ∧ 𝑛 ∈ (0...𝐴)) → (𝑃𝑛) ∈ ℕ)
8 prmz 16709 . . . . 5 (𝑃 ∈ ℙ → 𝑃 ∈ ℤ)
98ad2antrr 726 . . . 4 (((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℕ0) ∧ 𝑛 ∈ (0...𝐴)) → 𝑃 ∈ ℤ)
105adantl 481 . . . 4 (((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℕ0) ∧ 𝑛 ∈ (0...𝐴)) → 𝑛 ∈ ℕ0)
11 elfzuz3 13558 . . . . 5 (𝑛 ∈ (0...𝐴) → 𝐴 ∈ (ℤ𝑛))
1211adantl 481 . . . 4 (((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℕ0) ∧ 𝑛 ∈ (0...𝐴)) → 𝐴 ∈ (ℤ𝑛))
13 dvdsexp 16362 . . . 4 ((𝑃 ∈ ℤ ∧ 𝑛 ∈ ℕ0𝐴 ∈ (ℤ𝑛)) → (𝑃𝑛) ∥ (𝑃𝐴))
149, 10, 12, 13syl3anc 1370 . . 3 (((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℕ0) ∧ 𝑛 ∈ (0...𝐴)) → (𝑃𝑛) ∥ (𝑃𝐴))
152, 7, 14elrabd 3697 . 2 (((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℕ0) ∧ 𝑛 ∈ (0...𝐴)) → (𝑃𝑛) ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ (𝑃𝐴)})
16 simpl 482 . . . 4 ((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℕ0) → 𝑃 ∈ ℙ)
17 elrabi 3690 . . . 4 (𝑚 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ (𝑃𝐴)} → 𝑚 ∈ ℕ)
18 pccl 16883 . . . 4 ((𝑃 ∈ ℙ ∧ 𝑚 ∈ ℕ) → (𝑃 pCnt 𝑚) ∈ ℕ0)
1916, 17, 18syl2an 596 . . 3 (((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℕ0) ∧ 𝑚 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ (𝑃𝐴)}) → (𝑃 pCnt 𝑚) ∈ ℕ0)
2016adantr 480 . . . . 5 (((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℕ0) ∧ 𝑚 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ (𝑃𝐴)}) → 𝑃 ∈ ℙ)
2117adantl 481 . . . . . 6 (((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℕ0) ∧ 𝑚 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ (𝑃𝐴)}) → 𝑚 ∈ ℕ)
2221nnzd 12638 . . . . 5 (((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℕ0) ∧ 𝑚 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ (𝑃𝐴)}) → 𝑚 ∈ ℤ)
238ad2antrr 726 . . . . . 6 (((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℕ0) ∧ 𝑚 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ (𝑃𝐴)}) → 𝑃 ∈ ℤ)
24 simplr 769 . . . . . 6 (((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℕ0) ∧ 𝑚 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ (𝑃𝐴)}) → 𝐴 ∈ ℕ0)
25 zexpcl 14114 . . . . . 6 ((𝑃 ∈ ℤ ∧ 𝐴 ∈ ℕ0) → (𝑃𝐴) ∈ ℤ)
2623, 24, 25syl2anc 584 . . . . 5 (((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℕ0) ∧ 𝑚 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ (𝑃𝐴)}) → (𝑃𝐴) ∈ ℤ)
27 breq1 5151 . . . . . . . 8 (𝑥 = 𝑚 → (𝑥 ∥ (𝑃𝐴) ↔ 𝑚 ∥ (𝑃𝐴)))
2827elrab 3695 . . . . . . 7 (𝑚 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ (𝑃𝐴)} ↔ (𝑚 ∈ ℕ ∧ 𝑚 ∥ (𝑃𝐴)))
2928simprbi 496 . . . . . 6 (𝑚 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ (𝑃𝐴)} → 𝑚 ∥ (𝑃𝐴))
3029adantl 481 . . . . 5 (((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℕ0) ∧ 𝑚 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ (𝑃𝐴)}) → 𝑚 ∥ (𝑃𝐴))
31 pcdvdstr 16910 . . . . 5 ((𝑃 ∈ ℙ ∧ (𝑚 ∈ ℤ ∧ (𝑃𝐴) ∈ ℤ ∧ 𝑚 ∥ (𝑃𝐴))) → (𝑃 pCnt 𝑚) ≤ (𝑃 pCnt (𝑃𝐴)))
3220, 22, 26, 30, 31syl13anc 1371 . . . 4 (((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℕ0) ∧ 𝑚 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ (𝑃𝐴)}) → (𝑃 pCnt 𝑚) ≤ (𝑃 pCnt (𝑃𝐴)))
33 pcidlem 16906 . . . . 5 ((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℕ0) → (𝑃 pCnt (𝑃𝐴)) = 𝐴)
3433adantr 480 . . . 4 (((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℕ0) ∧ 𝑚 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ (𝑃𝐴)}) → (𝑃 pCnt (𝑃𝐴)) = 𝐴)
3532, 34breqtrd 5174 . . 3 (((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℕ0) ∧ 𝑚 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ (𝑃𝐴)}) → (𝑃 pCnt 𝑚) ≤ 𝐴)
36 fznn0 13656 . . . 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 713 . 2 (((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℕ0) ∧ 𝑚 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ (𝑃𝐴)}) → (𝑃 pCnt 𝑚) ∈ (0...𝐴))
39 oveq2 7439 . . . . . . . . 9 (𝑛 = 𝐴 → (𝑃𝑛) = (𝑃𝐴))
4039breq2d 5160 . . . . . . . 8 (𝑛 = 𝐴 → (𝑚 ∥ (𝑃𝑛) ↔ 𝑚 ∥ (𝑃𝐴)))
4140rspcev 3622 . . . . . . 7 ((𝐴 ∈ ℕ0𝑚 ∥ (𝑃𝐴)) → ∃𝑛 ∈ ℕ0 𝑚 ∥ (𝑃𝑛))
4224, 30, 41syl2anc 584 . . . . . 6 (((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℕ0) ∧ 𝑚 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ (𝑃𝐴)}) → ∃𝑛 ∈ ℕ0 𝑚 ∥ (𝑃𝑛))
43 pcprmpw2 16916 . . . . . . 7 ((𝑃 ∈ ℙ ∧ 𝑚 ∈ ℕ) → (∃𝑛 ∈ ℕ0 𝑚 ∥ (𝑃𝑛) ↔ 𝑚 = (𝑃↑(𝑃 pCnt 𝑚))))
4416, 17, 43syl2an 596 . . . . . 6 (((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℕ0) ∧ 𝑚 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ (𝑃𝐴)}) → (∃𝑛 ∈ ℕ0 𝑚 ∥ (𝑃𝑛) ↔ 𝑚 = (𝑃↑(𝑃 pCnt 𝑚))))
4542, 44mpbid 232 . . . . 5 (((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℕ0) ∧ 𝑚 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ (𝑃𝐴)}) → 𝑚 = (𝑃↑(𝑃 pCnt 𝑚)))
4645adantrl 716 . . . 4 (((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℕ0) ∧ (𝑛 ∈ (0...𝐴) ∧ 𝑚 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ (𝑃𝐴)})) → 𝑚 = (𝑃↑(𝑃 pCnt 𝑚)))
47 oveq2 7439 . . . . 5 (𝑛 = (𝑃 pCnt 𝑚) → (𝑃𝑛) = (𝑃↑(𝑃 pCnt 𝑚)))
4847eqeq2d 2746 . . . 4 (𝑛 = (𝑃 pCnt 𝑚) → (𝑚 = (𝑃𝑛) ↔ 𝑚 = (𝑃↑(𝑃 pCnt 𝑚))))
4946, 48syl5ibrcom 247 . . 3 (((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℕ0) ∧ (𝑛 ∈ (0...𝐴) ∧ 𝑚 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ (𝑃𝐴)})) → (𝑛 = (𝑃 pCnt 𝑚) → 𝑚 = (𝑃𝑛)))
50 elfzelz 13561 . . . . . . 7 (𝑛 ∈ (0...𝐴) → 𝑛 ∈ ℤ)
51 pcid 16907 . . . . . . 7 ((𝑃 ∈ ℙ ∧ 𝑛 ∈ ℤ) → (𝑃 pCnt (𝑃𝑛)) = 𝑛)
5216, 50, 51syl2an 596 . . . . . 6 (((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℕ0) ∧ 𝑛 ∈ (0...𝐴)) → (𝑃 pCnt (𝑃𝑛)) = 𝑛)
5352eqcomd 2741 . . . . 5 (((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℕ0) ∧ 𝑛 ∈ (0...𝐴)) → 𝑛 = (𝑃 pCnt (𝑃𝑛)))
5453adantrr 717 . . . 4 (((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℕ0) ∧ (𝑛 ∈ (0...𝐴) ∧ 𝑚 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ (𝑃𝐴)})) → 𝑛 = (𝑃 pCnt (𝑃𝑛)))
55 oveq2 7439 . . . . 5 (𝑚 = (𝑃𝑛) → (𝑃 pCnt 𝑚) = (𝑃 pCnt (𝑃𝑛)))
5655eqeq2d 2746 . . . 4 (𝑚 = (𝑃𝑛) → (𝑛 = (𝑃 pCnt 𝑚) ↔ 𝑛 = (𝑃 pCnt (𝑃𝑛))))
5754, 56syl5ibrcom 247 . . 3 (((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℕ0) ∧ (𝑛 ∈ (0...𝐴) ∧ 𝑚 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ (𝑃𝐴)})) → (𝑚 = (𝑃𝑛) → 𝑛 = (𝑃 pCnt 𝑚)))
5849, 57impbid 212 . 2 (((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℕ0) ∧ (𝑛 ∈ (0...𝐴) ∧ 𝑚 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ (𝑃𝐴)})) → (𝑛 = (𝑃 pCnt 𝑚) ↔ 𝑚 = (𝑃𝑛)))
591, 15, 38, 58f1o2d 7687 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 206  wa 395   = wceq 1537  wcel 2106  wrex 3068  {crab 3433   class class class wbr 5148  cmpt 5231  1-1-ontowf1o 6562  cfv 6563  (class class class)co 7431  0cc0 11153  cle 11294  cn 12264  0cn0 12524  cz 12611  cuz 12876  ...cfz 13544  cexp 14099  cdvds 16287  cprime 16705   pCnt cpc 16870
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 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754  ax-cnex 11209  ax-resscn 11210  ax-1cn 11211  ax-icn 11212  ax-addcl 11213  ax-addrcl 11214  ax-mulcl 11215  ax-mulrcl 11216  ax-mulcom 11217  ax-addass 11218  ax-mulass 11219  ax-distr 11220  ax-i2m1 11221  ax-1ne0 11222  ax-1rid 11223  ax-rnegex 11224  ax-rrecex 11225  ax-cnre 11226  ax-pre-lttri 11227  ax-pre-lttrn 11228  ax-pre-ltadd 11229  ax-pre-mulgt0 11230  ax-pre-sup 11231
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-nel 3045  df-ral 3060  df-rex 3069  df-rmo 3378  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-pred 6323  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8013  df-2nd 8014  df-frecs 8305  df-wrecs 8336  df-recs 8410  df-rdg 8449  df-1o 8505  df-2o 8506  df-er 8744  df-en 8985  df-dom 8986  df-sdom 8987  df-fin 8988  df-sup 9480  df-inf 9481  df-pnf 11295  df-mnf 11296  df-xr 11297  df-ltxr 11298  df-le 11299  df-sub 11492  df-neg 11493  df-div 11919  df-nn 12265  df-2 12327  df-3 12328  df-n0 12525  df-z 12612  df-uz 12877  df-q 12989  df-rp 13033  df-fz 13545  df-fl 13829  df-mod 13907  df-seq 14040  df-exp 14100  df-cj 15135  df-re 15136  df-im 15137  df-sqrt 15271  df-abs 15272  df-dvds 16288  df-gcd 16529  df-prm 16706  df-pc 16871
This theorem is referenced by:  sgmppw  27256  0sgmppw  27257  dchrisum0flblem1  27567
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