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Theorem dvdsppwf1o 24829
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 15323 . . . . 5 (𝑃 ∈ ℙ → 𝑃 ∈ ℕ)
32adantr 481 . . . 4 ((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℕ0) → 𝑃 ∈ ℕ)
4 elfznn0 12382 . . . 4 (𝑛 ∈ (0...𝐴) → 𝑛 ∈ ℕ0)
5 nnexpcl 12821 . . . 4 ((𝑃 ∈ ℕ ∧ 𝑛 ∈ ℕ0) → (𝑃𝑛) ∈ ℕ)
63, 4, 5syl2an 494 . . 3 (((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℕ0) ∧ 𝑛 ∈ (0...𝐴)) → (𝑃𝑛) ∈ ℕ)
7 prmz 15324 . . . . 5 (𝑃 ∈ ℙ → 𝑃 ∈ ℤ)
87ad2antrr 761 . . . 4 (((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℕ0) ∧ 𝑛 ∈ (0...𝐴)) → 𝑃 ∈ ℤ)
94adantl 482 . . . 4 (((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℕ0) ∧ 𝑛 ∈ (0...𝐴)) → 𝑛 ∈ ℕ0)
10 elfzuz3 12289 . . . . 5 (𝑛 ∈ (0...𝐴) → 𝐴 ∈ (ℤ𝑛))
1110adantl 482 . . . 4 (((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℕ0) ∧ 𝑛 ∈ (0...𝐴)) → 𝐴 ∈ (ℤ𝑛))
12 dvdsexp 14984 . . . 4 ((𝑃 ∈ ℤ ∧ 𝑛 ∈ ℕ0𝐴 ∈ (ℤ𝑛)) → (𝑃𝑛) ∥ (𝑃𝐴))
138, 9, 11, 12syl3anc 1323 . . 3 (((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℕ0) ∧ 𝑛 ∈ (0...𝐴)) → (𝑃𝑛) ∥ (𝑃𝐴))
14 breq1 4621 . . . 4 (𝑥 = (𝑃𝑛) → (𝑥 ∥ (𝑃𝐴) ↔ (𝑃𝑛) ∥ (𝑃𝐴)))
1514elrab 3350 . . 3 ((𝑃𝑛) ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ (𝑃𝐴)} ↔ ((𝑃𝑛) ∈ ℕ ∧ (𝑃𝑛) ∥ (𝑃𝐴)))
166, 13, 15sylanbrc 697 . 2 (((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℕ0) ∧ 𝑛 ∈ (0...𝐴)) → (𝑃𝑛) ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ (𝑃𝐴)})
17 simpl 473 . . . 4 ((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℕ0) → 𝑃 ∈ ℙ)
18 elrabi 3346 . . . 4 (𝑚 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ (𝑃𝐴)} → 𝑚 ∈ ℕ)
19 pccl 15489 . . . 4 ((𝑃 ∈ ℙ ∧ 𝑚 ∈ ℕ) → (𝑃 pCnt 𝑚) ∈ ℕ0)
2017, 18, 19syl2an 494 . . 3 (((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℕ0) ∧ 𝑚 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ (𝑃𝐴)}) → (𝑃 pCnt 𝑚) ∈ ℕ0)
2117adantr 481 . . . . 5 (((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℕ0) ∧ 𝑚 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ (𝑃𝐴)}) → 𝑃 ∈ ℙ)
2218adantl 482 . . . . . 6 (((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℕ0) ∧ 𝑚 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ (𝑃𝐴)}) → 𝑚 ∈ ℕ)
2322nnzd 11433 . . . . 5 (((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℕ0) ∧ 𝑚 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ (𝑃𝐴)}) → 𝑚 ∈ ℤ)
247ad2antrr 761 . . . . . 6 (((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℕ0) ∧ 𝑚 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ (𝑃𝐴)}) → 𝑃 ∈ ℤ)
25 simplr 791 . . . . . 6 (((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℕ0) ∧ 𝑚 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ (𝑃𝐴)}) → 𝐴 ∈ ℕ0)
26 zexpcl 12823 . . . . . 6 ((𝑃 ∈ ℤ ∧ 𝐴 ∈ ℕ0) → (𝑃𝐴) ∈ ℤ)
2724, 25, 26syl2anc 692 . . . . 5 (((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℕ0) ∧ 𝑚 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ (𝑃𝐴)}) → (𝑃𝐴) ∈ ℤ)
28 breq1 4621 . . . . . . . 8 (𝑥 = 𝑚 → (𝑥 ∥ (𝑃𝐴) ↔ 𝑚 ∥ (𝑃𝐴)))
2928elrab 3350 . . . . . . 7 (𝑚 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ (𝑃𝐴)} ↔ (𝑚 ∈ ℕ ∧ 𝑚 ∥ (𝑃𝐴)))
3029simprbi 480 . . . . . 6 (𝑚 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ (𝑃𝐴)} → 𝑚 ∥ (𝑃𝐴))
3130adantl 482 . . . . 5 (((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℕ0) ∧ 𝑚 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ (𝑃𝐴)}) → 𝑚 ∥ (𝑃𝐴))
32 pcdvdstr 15515 . . . . 5 ((𝑃 ∈ ℙ ∧ (𝑚 ∈ ℤ ∧ (𝑃𝐴) ∈ ℤ ∧ 𝑚 ∥ (𝑃𝐴))) → (𝑃 pCnt 𝑚) ≤ (𝑃 pCnt (𝑃𝐴)))
3321, 23, 27, 31, 32syl13anc 1325 . . . 4 (((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℕ0) ∧ 𝑚 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ (𝑃𝐴)}) → (𝑃 pCnt 𝑚) ≤ (𝑃 pCnt (𝑃𝐴)))
34 pcidlem 15511 . . . . 5 ((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℕ0) → (𝑃 pCnt (𝑃𝐴)) = 𝐴)
3534adantr 481 . . . 4 (((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℕ0) ∧ 𝑚 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ (𝑃𝐴)}) → (𝑃 pCnt (𝑃𝐴)) = 𝐴)
3633, 35breqtrd 4644 . . 3 (((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℕ0) ∧ 𝑚 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ (𝑃𝐴)}) → (𝑃 pCnt 𝑚) ≤ 𝐴)
37 fznn0 12381 . . . 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 956 . 2 (((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℕ0) ∧ 𝑚 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ (𝑃𝐴)}) → (𝑃 pCnt 𝑚) ∈ (0...𝐴))
40 oveq2 6618 . . . . . . . . 9 (𝑛 = 𝐴 → (𝑃𝑛) = (𝑃𝐴))
4140breq2d 4630 . . . . . . . 8 (𝑛 = 𝐴 → (𝑚 ∥ (𝑃𝑛) ↔ 𝑚 ∥ (𝑃𝐴)))
4241rspcev 3298 . . . . . . 7 ((𝐴 ∈ ℕ0𝑚 ∥ (𝑃𝐴)) → ∃𝑛 ∈ ℕ0 𝑚 ∥ (𝑃𝑛))
4325, 31, 42syl2anc 692 . . . . . 6 (((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℕ0) ∧ 𝑚 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ (𝑃𝐴)}) → ∃𝑛 ∈ ℕ0 𝑚 ∥ (𝑃𝑛))
44 pcprmpw2 15521 . . . . . . 7 ((𝑃 ∈ ℙ ∧ 𝑚 ∈ ℕ) → (∃𝑛 ∈ ℕ0 𝑚 ∥ (𝑃𝑛) ↔ 𝑚 = (𝑃↑(𝑃 pCnt 𝑚))))
4517, 18, 44syl2an 494 . . . . . 6 (((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℕ0) ∧ 𝑚 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ (𝑃𝐴)}) → (∃𝑛 ∈ ℕ0 𝑚 ∥ (𝑃𝑛) ↔ 𝑚 = (𝑃↑(𝑃 pCnt 𝑚))))
4643, 45mpbid 222 . . . . 5 (((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℕ0) ∧ 𝑚 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ (𝑃𝐴)}) → 𝑚 = (𝑃↑(𝑃 pCnt 𝑚)))
4746adantrl 751 . . . 4 (((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℕ0) ∧ (𝑛 ∈ (0...𝐴) ∧ 𝑚 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ (𝑃𝐴)})) → 𝑚 = (𝑃↑(𝑃 pCnt 𝑚)))
48 oveq2 6618 . . . . 5 (𝑛 = (𝑃 pCnt 𝑚) → (𝑃𝑛) = (𝑃↑(𝑃 pCnt 𝑚)))
4948eqeq2d 2631 . . . 4 (𝑛 = (𝑃 pCnt 𝑚) → (𝑚 = (𝑃𝑛) ↔ 𝑚 = (𝑃↑(𝑃 pCnt 𝑚))))
5047, 49syl5ibrcom 237 . . 3 (((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℕ0) ∧ (𝑛 ∈ (0...𝐴) ∧ 𝑚 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ (𝑃𝐴)})) → (𝑛 = (𝑃 pCnt 𝑚) → 𝑚 = (𝑃𝑛)))
51 elfzelz 12292 . . . . . . 7 (𝑛 ∈ (0...𝐴) → 𝑛 ∈ ℤ)
52 pcid 15512 . . . . . . 7 ((𝑃 ∈ ℙ ∧ 𝑛 ∈ ℤ) → (𝑃 pCnt (𝑃𝑛)) = 𝑛)
5317, 51, 52syl2an 494 . . . . . 6 (((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℕ0) ∧ 𝑛 ∈ (0...𝐴)) → (𝑃 pCnt (𝑃𝑛)) = 𝑛)
5453eqcomd 2627 . . . . 5 (((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℕ0) ∧ 𝑛 ∈ (0...𝐴)) → 𝑛 = (𝑃 pCnt (𝑃𝑛)))
5554adantrr 752 . . . 4 (((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℕ0) ∧ (𝑛 ∈ (0...𝐴) ∧ 𝑚 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ (𝑃𝐴)})) → 𝑛 = (𝑃 pCnt (𝑃𝑛)))
56 oveq2 6618 . . . . 5 (𝑚 = (𝑃𝑛) → (𝑃 pCnt 𝑚) = (𝑃 pCnt (𝑃𝑛)))
5756eqeq2d 2631 . . . 4 (𝑚 = (𝑃𝑛) → (𝑛 = (𝑃 pCnt 𝑚) ↔ 𝑛 = (𝑃 pCnt (𝑃𝑛))))
5855, 57syl5ibrcom 237 . . 3 (((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℕ0) ∧ (𝑛 ∈ (0...𝐴) ∧ 𝑚 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ (𝑃𝐴)})) → (𝑚 = (𝑃𝑛) → 𝑛 = (𝑃 pCnt 𝑚)))
5950, 58impbid 202 . 2 (((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℕ0) ∧ (𝑛 ∈ (0...𝐴) ∧ 𝑚 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ (𝑃𝐴)})) → (𝑛 = (𝑃 pCnt 𝑚) ↔ 𝑚 = (𝑃𝑛)))
601, 16, 39, 59f1o2d 6847 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 196  wa 384   = wceq 1480  wcel 1987  wrex 2908  {crab 2911   class class class wbr 4618  cmpt 4678  1-1-ontowf1o 5851  cfv 5852  (class class class)co 6610  0cc0 9888  cle 10027  cn 10972  0cn0 11244  cz 11329  cuz 11639  ...cfz 12276  cexp 12808  cdvds 14918  cprime 15320   pCnt cpc 15476
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6909  ax-cnex 9944  ax-resscn 9945  ax-1cn 9946  ax-icn 9947  ax-addcl 9948  ax-addrcl 9949  ax-mulcl 9950  ax-mulrcl 9951  ax-mulcom 9952  ax-addass 9953  ax-mulass 9954  ax-distr 9955  ax-i2m1 9956  ax-1ne0 9957  ax-1rid 9958  ax-rnegex 9959  ax-rrecex 9960  ax-cnre 9961  ax-pre-lttri 9962  ax-pre-lttrn 9963  ax-pre-ltadd 9964  ax-pre-mulgt0 9965  ax-pre-sup 9966
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-nel 2894  df-ral 2912  df-rex 2913  df-reu 2914  df-rmo 2915  df-rab 2916  df-v 3191  df-sbc 3422  df-csb 3519  df-dif 3562  df-un 3564  df-in 3566  df-ss 3573  df-pss 3575  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-tp 4158  df-op 4160  df-uni 4408  df-int 4446  df-iun 4492  df-br 4619  df-opab 4679  df-mpt 4680  df-tr 4718  df-eprel 4990  df-id 4994  df-po 5000  df-so 5001  df-fr 5038  df-we 5040  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-pred 5644  df-ord 5690  df-on 5691  df-lim 5692  df-suc 5693  df-iota 5815  df-fun 5854  df-fn 5855  df-f 5856  df-f1 5857  df-fo 5858  df-f1o 5859  df-fv 5860  df-riota 6571  df-ov 6613  df-oprab 6614  df-mpt2 6615  df-om 7020  df-1st 7120  df-2nd 7121  df-wrecs 7359  df-recs 7420  df-rdg 7458  df-1o 7512  df-2o 7513  df-oadd 7516  df-er 7694  df-en 7908  df-dom 7909  df-sdom 7910  df-fin 7911  df-sup 8300  df-inf 8301  df-pnf 10028  df-mnf 10029  df-xr 10030  df-ltxr 10031  df-le 10032  df-sub 10220  df-neg 10221  df-div 10637  df-nn 10973  df-2 11031  df-3 11032  df-n0 11245  df-z 11330  df-uz 11640  df-q 11741  df-rp 11785  df-fz 12277  df-fl 12541  df-mod 12617  df-seq 12750  df-exp 12809  df-cj 13781  df-re 13782  df-im 13783  df-sqrt 13917  df-abs 13918  df-dvds 14919  df-gcd 15152  df-prm 15321  df-pc 15477
This theorem is referenced by:  sgmppw  24839  0sgmppw  24840  dchrisum0flblem1  25114
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