Theorem dvdsppwf1o 27331

Description: A bijection between the divisors of a prime power and the integers less than or equal to the exponent. (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 5113	. . 3 ⊢ (𝑥 = (𝑃↑𝑛) → (𝑥 ∥ (𝑃↑𝐴) ↔ (𝑃↑𝑛) ∥ (𝑃↑𝐴)))
3		prmnn 16733	. . . . 5 ⊢ (𝑃 ∈ ℙ → 𝑃 ∈ ℕ)
4	3	adantr 485	. . . 4 ⊢ ((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℕ0) → 𝑃 ∈ ℕ)
5		elfznn0 13650	. . . 4 ⊢ (𝑛 ∈ (0...𝐴) → 𝑛 ∈ ℕ0)
6		nnexpcl 14112	. . . 4 ⊢ ((𝑃 ∈ ℕ ∧ 𝑛 ∈ ℕ0) → (𝑃↑𝑛) ∈ ℕ)
7	4, 5, 6	syl2an 607	. . 3 ⊢ (((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℕ0) ∧ 𝑛 ∈ (0...𝐴)) → (𝑃↑𝑛) ∈ ℕ)
8		prmz 16734	. . . . 5 ⊢ (𝑃 ∈ ℙ → 𝑃 ∈ ℤ)
9	8	ad2antrr 738	. . . 4 ⊢ (((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℕ0) ∧ 𝑛 ∈ (0...𝐴)) → 𝑃 ∈ ℤ)
10	5	adantl 486	. . . 4 ⊢ (((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℕ0) ∧ 𝑛 ∈ (0...𝐴)) → 𝑛 ∈ ℕ0)
11		elfzuz3 13550	. . . . 5 ⊢ (𝑛 ∈ (0...𝐴) → 𝐴 ∈ (ℤ≥‘𝑛))
12	11	adantl 486	. . . 4 ⊢ (((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℕ0) ∧ 𝑛 ∈ (0...𝐴)) → 𝐴 ∈ (ℤ≥‘𝑛))
13		dvdsexp 16387	. . . 4 ⊢ ((𝑃 ∈ ℤ ∧ 𝑛 ∈ ℕ0 ∧ 𝐴 ∈ (ℤ≥‘𝑛)) → (𝑃↑𝑛) ∥ (𝑃↑𝐴))
14	9, 10, 12, 13	syl3anc 1398	. . 3 ⊢ (((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℕ0) ∧ 𝑛 ∈ (0...𝐴)) → (𝑃↑𝑛) ∥ (𝑃↑𝐴))
15	2, 7, 14	elrabd 3653	. 2 ⊢ (((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℕ0) ∧ 𝑛 ∈ (0...𝐴)) → (𝑃↑𝑛) ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ (𝑃↑𝐴)})
16		simpl 487	. . . 4 ⊢ ((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℕ0) → 𝑃 ∈ ℙ)
17		elrabi 3647	. . . 4 ⊢ (𝑚 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ (𝑃↑𝐴)} → 𝑚 ∈ ℕ)
18		pccl 16910	. . . 4 ⊢ ((𝑃 ∈ ℙ ∧ 𝑚 ∈ ℕ) → (𝑃 pCnt 𝑚) ∈ ℕ0)
19	16, 17, 18	syl2an 607	. . 3 ⊢ (((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℕ0) ∧ 𝑚 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ (𝑃↑𝐴)}) → (𝑃 pCnt 𝑚) ∈ ℕ0)
20	16	adantr 485	. . . . 5 ⊢ (((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℕ0) ∧ 𝑚 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ (𝑃↑𝐴)}) → 𝑃 ∈ ℙ)
21	17	adantl 486	. . . . . 6 ⊢ (((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℕ0) ∧ 𝑚 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ (𝑃↑𝐴)}) → 𝑚 ∈ ℕ)
22	21	nnzd 12618	. . . . 5 ⊢ (((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℕ0) ∧ 𝑚 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ (𝑃↑𝐴)}) → 𝑚 ∈ ℤ)
23	8	ad2antrr 738	. . . . . 6 ⊢ (((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℕ0) ∧ 𝑚 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ (𝑃↑𝐴)}) → 𝑃 ∈ ℤ)
24		simplr 780	. . . . . 6 ⊢ (((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℕ0) ∧ 𝑚 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ (𝑃↑𝐴)}) → 𝐴 ∈ ℕ0)
25		zexpcl 14114	. . . . . 6 ⊢ ((𝑃 ∈ ℤ ∧ 𝐴 ∈ ℕ0) → (𝑃↑𝐴) ∈ ℤ)
26	23, 24, 25	syl2anc 595	. . . . 5 ⊢ (((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℕ0) ∧ 𝑚 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ (𝑃↑𝐴)}) → (𝑃↑𝐴) ∈ ℤ)
27		breq1 5113	. . . . . . . 8 ⊢ (𝑥 = 𝑚 → (𝑥 ∥ (𝑃↑𝐴) ↔ 𝑚 ∥ (𝑃↑𝐴)))
28	27	elrab 3651	. . . . . . 7 ⊢ (𝑚 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ (𝑃↑𝐴)} ↔ (𝑚 ∈ ℕ ∧ 𝑚 ∥ (𝑃↑𝐴)))
29	28	simprbi 502	. . . . . 6 ⊢ (𝑚 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ (𝑃↑𝐴)} → 𝑚 ∥ (𝑃↑𝐴))
30	29	adantl 486	. . . . 5 ⊢ (((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℕ0) ∧ 𝑚 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ (𝑃↑𝐴)}) → 𝑚 ∥ (𝑃↑𝐴))
31		pcdvdstr 16937	. . . . 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 1399	. . . 4 ⊢ (((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℕ0) ∧ 𝑚 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ (𝑃↑𝐴)}) → (𝑃 pCnt 𝑚) ≤ (𝑃 pCnt (𝑃↑𝐴)))
33		pcidlem 16933	. . . . 5 ⊢ ((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℕ0) → (𝑃 pCnt (𝑃↑𝐴)) = 𝐴)
34	33	adantr 485	. . . 4 ⊢ (((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℕ0) ∧ 𝑚 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ (𝑃↑𝐴)}) → (𝑃 pCnt (𝑃↑𝐴)) = 𝐴)
35	32, 34	breqtrd 5138	. . 3 ⊢ (((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℕ0) ∧ 𝑚 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ (𝑃↑𝐴)}) → (𝑃 pCnt 𝑚) ≤ 𝐴)
36		fznn0 13649	. . . 4 ⊢ (𝐴 ∈ ℕ0 → ((𝑃 pCnt 𝑚) ∈ (0...𝐴) ↔ ((𝑃 pCnt 𝑚) ∈ ℕ0 ∧ (𝑃 pCnt 𝑚) ≤ 𝐴)))
37	24, 36	syl 18	. . 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 725	. 2 ⊢ (((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℕ0) ∧ 𝑚 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ (𝑃↑𝐴)}) → (𝑃 pCnt 𝑚) ∈ (0...𝐴))
39		oveq2 7420	. . . . . . . . 9 ⊢ (𝑛 = 𝐴 → (𝑃↑𝑛) = (𝑃↑𝐴))
40	39	breq2d 5122	. . . . . . . 8 ⊢ (𝑛 = 𝐴 → (𝑚 ∥ (𝑃↑𝑛) ↔ 𝑚 ∥ (𝑃↑𝐴)))
41	40	rspcev 3582	. . . . . . 7 ⊢ ((𝐴 ∈ ℕ0 ∧ 𝑚 ∥ (𝑃↑𝐴)) → ∃𝑛 ∈ ℕ0 𝑚 ∥ (𝑃↑𝑛))
42	24, 30, 41	syl2anc 595	. . . . . 6 ⊢ (((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℕ0) ∧ 𝑚 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ (𝑃↑𝐴)}) → ∃𝑛 ∈ ℕ0 𝑚 ∥ (𝑃↑𝑛))
43		pcprmpw2 16943	. . . . . . 7 ⊢ ((𝑃 ∈ ℙ ∧ 𝑚 ∈ ℕ) → (∃𝑛 ∈ ℕ0 𝑚 ∥ (𝑃↑𝑛) ↔ 𝑚 = (𝑃↑(𝑃 pCnt 𝑚))))
44	16, 17, 43	syl2an 607	. . . . . 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 235	. . . . 5 ⊢ (((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℕ0) ∧ 𝑚 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ (𝑃↑𝐴)}) → 𝑚 = (𝑃↑(𝑃 pCnt 𝑚)))
46	45	adantrl 728	. . . 4 ⊢ (((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℕ0) ∧ (𝑛 ∈ (0...𝐴) ∧ 𝑚 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ (𝑃↑𝐴)})) → 𝑚 = (𝑃↑(𝑃 pCnt 𝑚)))
47		oveq2 7420	. . . . 5 ⊢ (𝑛 = (𝑃 pCnt 𝑚) → (𝑃↑𝑛) = (𝑃↑(𝑃 pCnt 𝑚)))
48	47	eqeq2d 2774	. . . 4 ⊢ (𝑛 = (𝑃 pCnt 𝑚) → (𝑚 = (𝑃↑𝑛) ↔ 𝑚 = (𝑃↑(𝑃 pCnt 𝑚))))
49	46, 48	syl5ibrcom 250	. . 3 ⊢ (((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℕ0) ∧ (𝑛 ∈ (0...𝐴) ∧ 𝑚 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ (𝑃↑𝐴)})) → (𝑛 = (𝑃 pCnt 𝑚) → 𝑚 = (𝑃↑𝑛)))
50		elfzelz 13553	. . . . . . 7 ⊢ (𝑛 ∈ (0...𝐴) → 𝑛 ∈ ℤ)
51		pcid 16934	. . . . . . 7 ⊢ ((𝑃 ∈ ℙ ∧ 𝑛 ∈ ℤ) → (𝑃 pCnt (𝑃↑𝑛)) = 𝑛)
52	16, 50, 51	syl2an 607	. . . . . 6 ⊢ (((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℕ0) ∧ 𝑛 ∈ (0...𝐴)) → (𝑃 pCnt (𝑃↑𝑛)) = 𝑛)
53	52	eqcomd 2769	. . . . 5 ⊢ (((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℕ0) ∧ 𝑛 ∈ (0...𝐴)) → 𝑛 = (𝑃 pCnt (𝑃↑𝑛)))
54	53	adantrr 729	. . . 4 ⊢ (((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℕ0) ∧ (𝑛 ∈ (0...𝐴) ∧ 𝑚 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ (𝑃↑𝐴)})) → 𝑛 = (𝑃 pCnt (𝑃↑𝑛)))
55		oveq2 7420	. . . . 5 ⊢ (𝑚 = (𝑃↑𝑛) → (𝑃 pCnt 𝑚) = (𝑃 pCnt (𝑃↑𝑛)))
56	55	eqeq2d 2774	. . . 4 ⊢ (𝑚 = (𝑃↑𝑛) → (𝑛 = (𝑃 pCnt 𝑚) ↔ 𝑛 = (𝑃 pCnt (𝑃↑𝑛))))
57	54, 56	syl5ibrcom 250	. . 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 215	. 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 7666	1 ⊢ ((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℕ0) → 𝐹:(0...𝐴)–1-1-onto→{𝑥 ∈ ℕ ∣ 𝑥 ∥ (𝑃↑𝐴)})

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400   = wceq 1570   ∈ wcel 2143  ∃wrex 3089  {crab 3416   class class class wbr 5110   ↦ cmpt 5193  –1-1-onto→wf1o 6537  ‘cfv 6538  (class class class)co 7412  0cc0 11101   ≤ cle 11245  ℕcn 12234  ℕ₀cn0 12505  ℤcz 12592  ℤ_≥cuz 12863  ...cfz 13536  ↑cexp 14099   ∥ cdvds 16311  ℙcprime 16730   pCnt cpc 16897

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-sep 5258  ax-nul 5270  ax-pow 5338  ax-pr 5406  ax-un 7734  ax-cnex 11157  ax-resscn 11158  ax-1cn 11159  ax-icn 11160  ax-addcl 11161  ax-addrcl 11162  ax-mulcl 11163  ax-mulrcl 11164  ax-mulcom 11165  ax-addass 11166  ax-mulass 11167  ax-distr 11168  ax-i2m1 11169  ax-1ne0 11170  ax-1rid 11171  ax-rnegex 11172  ax-rrecex 11173  ax-cnre 11174  ax-pre-lttri 11175  ax-pre-lttrn 11176  ax-pre-ltadd 11177  ax-pre-mulgt0 11178  ax-pre-sup 11179

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-nel 3065  df-ral 3080  df-rex 3090  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3457  df-sbc 3746  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-pss 3926  df-nul 4288  df-if 4489  df-pw 4565  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4874  df-iun 4959  df-br 5111  df-opab 5175  df-mpt 5194  df-tr 5220  df-id 5558  df-eprel 5563  df-po 5571  df-so 5572  df-fr 5616  df-we 5618  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-res 5675  df-ima 5676  df-pred 6304  df-ord 6365  df-on 6366  df-lim 6367  df-suc 6368  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545  df-fv 6546  df-riota 7369  df-ov 7415  df-oprab 7416  df-mpo 7417  df-om 7864  df-1st 7987  df-2nd 7988  df-frecs 8279  df-wrecs 8310  df-recs 8359  df-rdg 8398  df-1o 8454  df-2o 8455  df-er 8695  df-en 8945  df-dom 8946  df-sdom 8947  df-fin 8948  df-sup 9403  df-inf 9404  df-pnf 11246  df-mnf 11247  df-xr 11248  df-ltxr 11249  df-le 11250  df-sub 11444  df-neg 11445  df-div 11873  df-nn 12235  df-2 12304  df-3 12305  df-n0 12506  df-z 12593  df-uz 12864  df-q 12974  df-rp 13018  df-fz 13537  df-fl 13827  df-mod 13905  df-seq 14040  df-exp 14100  df-cj 15152  df-re 15153  df-im 15154  df-sqrt 15288  df-abs 15289  df-dvds 16312  df-gcd 16554  df-prm 16731  df-pc 16898

This theorem is referenced by:  sgmppw  27342  0sgmppw  27343  dchrisum0flblem1  27653