Theorem dvdsppwf1o 27037

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) ↦ (𝑃↑𝑛))

Assertion

Ref	Expression
dvdsppwf1o	⊢ ((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℕ0) → 𝐹:(0...𝐴)–1-1-onto→{𝑥 ∈ ℕ ∣ 𝑥 ∥ (𝑃↑𝐴)})

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.

Step	Hyp	Ref	Expression
1		dvdsppwf1o.f	. 2 ⊢ 𝐹 = (𝑛 ∈ (0...𝐴) ↦ (𝑃↑𝑛))
2		breq1 5142	. . 3 ⊢ (𝑥 = (𝑃↑𝑛) → (𝑥 ∥ (𝑃↑𝐴) ↔ (𝑃↑𝑛) ∥ (𝑃↑𝐴)))
3		prmnn 16610	. . . . 5 ⊢ (𝑃 ∈ ℙ → 𝑃 ∈ ℕ)
4	3	adantr 480	. . . 4 ⊢ ((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℕ0) → 𝑃 ∈ ℕ)
5		elfznn0 13592	. . . 4 ⊢ (𝑛 ∈ (0...𝐴) → 𝑛 ∈ ℕ0)
6		nnexpcl 14038	. . . 4 ⊢ ((𝑃 ∈ ℕ ∧ 𝑛 ∈ ℕ0) → (𝑃↑𝑛) ∈ ℕ)
7	4, 5, 6	syl2an 595	. . 3 ⊢ (((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℕ0) ∧ 𝑛 ∈ (0...𝐴)) → (𝑃↑𝑛) ∈ ℕ)
8		prmz 16611	. . . . 5 ⊢ (𝑃 ∈ ℙ → 𝑃 ∈ ℤ)
9	8	ad2antrr 723	. . . 4 ⊢ (((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℕ0) ∧ 𝑛 ∈ (0...𝐴)) → 𝑃 ∈ ℤ)
10	5	adantl 481	. . . 4 ⊢ (((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℕ0) ∧ 𝑛 ∈ (0...𝐴)) → 𝑛 ∈ ℕ0)
11		elfzuz3 13496	. . . . 5 ⊢ (𝑛 ∈ (0...𝐴) → 𝐴 ∈ (ℤ≥‘𝑛))
12	11	adantl 481	. . . 4 ⊢ (((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℕ0) ∧ 𝑛 ∈ (0...𝐴)) → 𝐴 ∈ (ℤ≥‘𝑛))
13		dvdsexp 16270	. . . 4 ⊢ ((𝑃 ∈ ℤ ∧ 𝑛 ∈ ℕ0 ∧ 𝐴 ∈ (ℤ≥‘𝑛)) → (𝑃↑𝑛) ∥ (𝑃↑𝐴))
14	9, 10, 12, 13	syl3anc 1368	. . 3 ⊢ (((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℕ0) ∧ 𝑛 ∈ (0...𝐴)) → (𝑃↑𝑛) ∥ (𝑃↑𝐴))
15	2, 7, 14	elrabd 3678	. 2 ⊢ (((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℕ0) ∧ 𝑛 ∈ (0...𝐴)) → (𝑃↑𝑛) ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ (𝑃↑𝐴)})
16		simpl 482	. . . 4 ⊢ ((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℕ0) → 𝑃 ∈ ℙ)
17		elrabi 3670	. . . 4 ⊢ (𝑚 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ (𝑃↑𝐴)} → 𝑚 ∈ ℕ)
18		pccl 16783	. . . 4 ⊢ ((𝑃 ∈ ℙ ∧ 𝑚 ∈ ℕ) → (𝑃 pCnt 𝑚) ∈ ℕ0)
19	16, 17, 18	syl2an 595	. . 3 ⊢ (((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℕ0) ∧ 𝑚 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ (𝑃↑𝐴)}) → (𝑃 pCnt 𝑚) ∈ ℕ0)
20	16	adantr 480	. . . . 5 ⊢ (((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℕ0) ∧ 𝑚 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ (𝑃↑𝐴)}) → 𝑃 ∈ ℙ)
21	17	adantl 481	. . . . . 6 ⊢ (((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℕ0) ∧ 𝑚 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ (𝑃↑𝐴)}) → 𝑚 ∈ ℕ)
22	21	nnzd 12583	. . . . 5 ⊢ (((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℕ0) ∧ 𝑚 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ (𝑃↑𝐴)}) → 𝑚 ∈ ℤ)
23	8	ad2antrr 723	. . . . . 6 ⊢ (((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℕ0) ∧ 𝑚 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ (𝑃↑𝐴)}) → 𝑃 ∈ ℤ)
24		simplr 766	. . . . . 6 ⊢ (((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℕ0) ∧ 𝑚 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ (𝑃↑𝐴)}) → 𝐴 ∈ ℕ0)
25		zexpcl 14040	. . . . . 6 ⊢ ((𝑃 ∈ ℤ ∧ 𝐴 ∈ ℕ0) → (𝑃↑𝐴) ∈ ℤ)
26	23, 24, 25	syl2anc 583	. . . . 5 ⊢ (((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℕ0) ∧ 𝑚 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ (𝑃↑𝐴)}) → (𝑃↑𝐴) ∈ ℤ)
27		breq1 5142	. . . . . . . 8 ⊢ (𝑥 = 𝑚 → (𝑥 ∥ (𝑃↑𝐴) ↔ 𝑚 ∥ (𝑃↑𝐴)))
28	27	elrab 3676	. . . . . . 7 ⊢ (𝑚 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ (𝑃↑𝐴)} ↔ (𝑚 ∈ ℕ ∧ 𝑚 ∥ (𝑃↑𝐴)))
29	28	simprbi 496	. . . . . 6 ⊢ (𝑚 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ (𝑃↑𝐴)} → 𝑚 ∥ (𝑃↑𝐴))
30	29	adantl 481	. . . . 5 ⊢ (((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℕ0) ∧ 𝑚 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ (𝑃↑𝐴)}) → 𝑚 ∥ (𝑃↑𝐴))
31		pcdvdstr 16810	. . . . 5 ⊢ ((𝑃 ∈ ℙ ∧ (𝑚 ∈ ℤ ∧ (𝑃↑𝐴) ∈ ℤ ∧ 𝑚 ∥ (𝑃↑𝐴))) → (𝑃 pCnt 𝑚) ≤ (𝑃 pCnt (𝑃↑𝐴)))
32	20, 22, 26, 30, 31	syl13anc 1369	. . . 4 ⊢ (((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℕ0) ∧ 𝑚 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ (𝑃↑𝐴)}) → (𝑃 pCnt 𝑚) ≤ (𝑃 pCnt (𝑃↑𝐴)))
33		pcidlem 16806	. . . . 5 ⊢ ((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℕ0) → (𝑃 pCnt (𝑃↑𝐴)) = 𝐴)
34	33	adantr 480	. . . 4 ⊢ (((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℕ0) ∧ 𝑚 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ (𝑃↑𝐴)}) → (𝑃 pCnt (𝑃↑𝐴)) = 𝐴)
35	32, 34	breqtrd 5165	. . 3 ⊢ (((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℕ0) ∧ 𝑚 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ (𝑃↑𝐴)}) → (𝑃 pCnt 𝑚) ≤ 𝐴)
36		fznn0 13591	. . . 4 ⊢ (𝐴 ∈ ℕ0 → ((𝑃 pCnt 𝑚) ∈ (0...𝐴) ↔ ((𝑃 pCnt 𝑚) ∈ ℕ0 ∧ (𝑃 pCnt 𝑚) ≤ 𝐴)))
37	24, 36	syl 17	. . 3 ⊢ (((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℕ0) ∧ 𝑚 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ (𝑃↑𝐴)}) → ((𝑃 pCnt 𝑚) ∈ (0...𝐴) ↔ ((𝑃 pCnt 𝑚) ∈ ℕ0 ∧ (𝑃 pCnt 𝑚) ≤ 𝐴)))
38	19, 35, 37	mpbir2and 710	. 2 ⊢ (((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℕ0) ∧ 𝑚 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ (𝑃↑𝐴)}) → (𝑃 pCnt 𝑚) ∈ (0...𝐴))
39		oveq2 7410	. . . . . . . . 9 ⊢ (𝑛 = 𝐴 → (𝑃↑𝑛) = (𝑃↑𝐴))
40	39	breq2d 5151	. . . . . . . 8 ⊢ (𝑛 = 𝐴 → (𝑚 ∥ (𝑃↑𝑛) ↔ 𝑚 ∥ (𝑃↑𝐴)))
41	40	rspcev 3604	. . . . . . 7 ⊢ ((𝐴 ∈ ℕ0 ∧ 𝑚 ∥ (𝑃↑𝐴)) → ∃𝑛 ∈ ℕ0 𝑚 ∥ (𝑃↑𝑛))
42	24, 30, 41	syl2anc 583	. . . . . 6 ⊢ (((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℕ0) ∧ 𝑚 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ (𝑃↑𝐴)}) → ∃𝑛 ∈ ℕ0 𝑚 ∥ (𝑃↑𝑛))
43		pcprmpw2 16816	. . . . . . 7 ⊢ ((𝑃 ∈ ℙ ∧ 𝑚 ∈ ℕ) → (∃𝑛 ∈ ℕ0 𝑚 ∥ (𝑃↑𝑛) ↔ 𝑚 = (𝑃↑(𝑃 pCnt 𝑚))))
44	16, 17, 43	syl2an 595	. . . . . 6 ⊢ (((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℕ0) ∧ 𝑚 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ (𝑃↑𝐴)}) → (∃𝑛 ∈ ℕ0 𝑚 ∥ (𝑃↑𝑛) ↔ 𝑚 = (𝑃↑(𝑃 pCnt 𝑚))))
45	42, 44	mpbid 231	. . . . 5 ⊢ (((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℕ0) ∧ 𝑚 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ (𝑃↑𝐴)}) → 𝑚 = (𝑃↑(𝑃 pCnt 𝑚)))
46	45	adantrl 713	. . . 4 ⊢ (((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℕ0) ∧ (𝑛 ∈ (0...𝐴) ∧ 𝑚 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ (𝑃↑𝐴)})) → 𝑚 = (𝑃↑(𝑃 pCnt 𝑚)))
47		oveq2 7410	. . . . 5 ⊢ (𝑛 = (𝑃 pCnt 𝑚) → (𝑃↑𝑛) = (𝑃↑(𝑃 pCnt 𝑚)))
48	47	eqeq2d 2735	. . . 4 ⊢ (𝑛 = (𝑃 pCnt 𝑚) → (𝑚 = (𝑃↑𝑛) ↔ 𝑚 = (𝑃↑(𝑃 pCnt 𝑚))))
49	46, 48	syl5ibrcom 246	. . 3 ⊢ (((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℕ0) ∧ (𝑛 ∈ (0...𝐴) ∧ 𝑚 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ (𝑃↑𝐴)})) → (𝑛 = (𝑃 pCnt 𝑚) → 𝑚 = (𝑃↑𝑛)))
50		elfzelz 13499	. . . . . . 7 ⊢ (𝑛 ∈ (0...𝐴) → 𝑛 ∈ ℤ)
51		pcid 16807	. . . . . . 7 ⊢ ((𝑃 ∈ ℙ ∧ 𝑛 ∈ ℤ) → (𝑃 pCnt (𝑃↑𝑛)) = 𝑛)
52	16, 50, 51	syl2an 595	. . . . . 6 ⊢ (((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℕ0) ∧ 𝑛 ∈ (0...𝐴)) → (𝑃 pCnt (𝑃↑𝑛)) = 𝑛)
53	52	eqcomd 2730	. . . . 5 ⊢ (((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℕ0) ∧ 𝑛 ∈ (0...𝐴)) → 𝑛 = (𝑃 pCnt (𝑃↑𝑛)))
54	53	adantrr 714	. . . 4 ⊢ (((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℕ0) ∧ (𝑛 ∈ (0...𝐴) ∧ 𝑚 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ (𝑃↑𝐴)})) → 𝑛 = (𝑃 pCnt (𝑃↑𝑛)))
55		oveq2 7410	. . . . 5 ⊢ (𝑚 = (𝑃↑𝑛) → (𝑃 pCnt 𝑚) = (𝑃 pCnt (𝑃↑𝑛)))
56	55	eqeq2d 2735	. . . 4 ⊢ (𝑚 = (𝑃↑𝑛) → (𝑛 = (𝑃 pCnt 𝑚) ↔ 𝑛 = (𝑃 pCnt (𝑃↑𝑛))))
57	54, 56	syl5ibrcom 246	. . 3 ⊢ (((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℕ0) ∧ (𝑛 ∈ (0...𝐴) ∧ 𝑚 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ (𝑃↑𝐴)})) → (𝑚 = (𝑃↑𝑛) → 𝑛 = (𝑃 pCnt 𝑚)))
58	49, 57	impbid 211	. 2 ⊢ (((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℕ0) ∧ (𝑛 ∈ (0...𝐴) ∧ 𝑚 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ (𝑃↑𝐴)})) → (𝑛 = (𝑃 pCnt 𝑚) ↔ 𝑚 = (𝑃↑𝑛)))
59	1, 15, 38, 58	f1o2d 7654	1 ⊢ ((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℕ0) → 𝐹:(0...𝐴)–1-1-onto→{𝑥 ∈ ℕ ∣ 𝑥 ∥ (𝑃↑𝐴)})

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 395   = wceq 1533   ∈ wcel 2098  ∃wrex 3062  {crab 3424   class class class wbr 5139   ↦ cmpt 5222  –1-1-onto→wf1o 6533  ‘cfv 6534  (class class class)co 7402  0cc0 11107   ≤ cle 11247  ℕcn 12210  ℕ₀cn0 12470  ℤcz 12556  ℤ_≥cuz 12820  ...cfz 13482  ↑cexp 14025   ∥ cdvds 16196  ℙcprime 16607   pCnt cpc 16770

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2695  ax-sep 5290  ax-nul 5297  ax-pow 5354  ax-pr 5418  ax-un 7719  ax-cnex 11163  ax-resscn 11164  ax-1cn 11165  ax-icn 11166  ax-addcl 11167  ax-addrcl 11168  ax-mulcl 11169  ax-mulrcl 11170  ax-mulcom 11171  ax-addass 11172  ax-mulass 11173  ax-distr 11174  ax-i2m1 11175  ax-1ne0 11176  ax-1rid 11177  ax-rnegex 11178  ax-rrecex 11179  ax-cnre 11180  ax-pre-lttri 11181  ax-pre-lttrn 11182  ax-pre-ltadd 11183  ax-pre-mulgt0 11184  ax-pre-sup 11185

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2526  df-eu 2555  df-clab 2702  df-cleq 2716  df-clel 2802  df-nfc 2877  df-ne 2933  df-nel 3039  df-ral 3054  df-rex 3063  df-rmo 3368  df-reu 3369  df-rab 3425  df-v 3468  df-sbc 3771  df-csb 3887  df-dif 3944  df-un 3946  df-in 3948  df-ss 3958  df-pss 3960  df-nul 4316  df-if 4522  df-pw 4597  df-sn 4622  df-pr 4624  df-op 4628  df-uni 4901  df-iun 4990  df-br 5140  df-opab 5202  df-mpt 5223  df-tr 5257  df-id 5565  df-eprel 5571  df-po 5579  df-so 5580  df-fr 5622  df-we 5624  df-xp 5673  df-rel 5674  df-cnv 5675  df-co 5676  df-dm 5677  df-rn 5678  df-res 5679  df-ima 5680  df-pred 6291  df-ord 6358  df-on 6359  df-lim 6360  df-suc 6361  df-iota 6486  df-fun 6536  df-fn 6537  df-f 6538  df-f1 6539  df-fo 6540  df-f1o 6541  df-fv 6542  df-riota 7358  df-ov 7405  df-oprab 7406  df-mpo 7407  df-om 7850  df-1st 7969  df-2nd 7970  df-frecs 8262  df-wrecs 8293  df-recs 8367  df-rdg 8406  df-1o 8462  df-2o 8463  df-er 8700  df-en 8937  df-dom 8938  df-sdom 8939  df-fin 8940  df-sup 9434  df-inf 9435  df-pnf 11248  df-mnf 11249  df-xr 11250  df-ltxr 11251  df-le 11252  df-sub 11444  df-neg 11445  df-div 11870  df-nn 12211  df-2 12273  df-3 12274  df-n0 12471  df-z 12557  df-uz 12821  df-q 12931  df-rp 12973  df-fz 13483  df-fl 13755  df-mod 13833  df-seq 13965  df-exp 14026  df-cj 15044  df-re 15045  df-im 15046  df-sqrt 15180  df-abs 15181  df-dvds 16197  df-gcd 16435  df-prm 16608  df-pc 16771

This theorem is referenced by:  sgmppw  27049  0sgmppw  27050  dchrisum0flblem1  27360