Theorem dvdsppwf1o 15719

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 4091	. . 3 ⊢ (𝑥 = (𝑃↑𝑛) → (𝑥 ∥ (𝑃↑𝐴) ↔ (𝑃↑𝑛) ∥ (𝑃↑𝐴)))
3		prmnn 12687	. . . . 5 ⊢ (𝑃 ∈ ℙ → 𝑃 ∈ ℕ)
4	3	adantr 276	. . . 4 ⊢ ((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℕ0) → 𝑃 ∈ ℕ)
5		elfznn0 10349	. . . 4 ⊢ (𝑛 ∈ (0...𝐴) → 𝑛 ∈ ℕ0)
6		nnexpcl 10815	. . . 4 ⊢ ((𝑃 ∈ ℕ ∧ 𝑛 ∈ ℕ0) → (𝑃↑𝑛) ∈ ℕ)
7	4, 5, 6	syl2an 289	. . 3 ⊢ (((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℕ0) ∧ 𝑛 ∈ (0...𝐴)) → (𝑃↑𝑛) ∈ ℕ)
8		prmz 12688	. . . . 5 ⊢ (𝑃 ∈ ℙ → 𝑃 ∈ ℤ)
9	8	ad2antrr 488	. . . 4 ⊢ (((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℕ0) ∧ 𝑛 ∈ (0...𝐴)) → 𝑃 ∈ ℤ)
10	5	adantl 277	. . . 4 ⊢ (((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℕ0) ∧ 𝑛 ∈ (0...𝐴)) → 𝑛 ∈ ℕ0)
11		elfzuz3 10257	. . . . 5 ⊢ (𝑛 ∈ (0...𝐴) → 𝐴 ∈ (ℤ≥‘𝑛))
12	11	adantl 277	. . . 4 ⊢ (((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℕ0) ∧ 𝑛 ∈ (0...𝐴)) → 𝐴 ∈ (ℤ≥‘𝑛))
13		dvdsexp 12427	. . . 4 ⊢ ((𝑃 ∈ ℤ ∧ 𝑛 ∈ ℕ0 ∧ 𝐴 ∈ (ℤ≥‘𝑛)) → (𝑃↑𝑛) ∥ (𝑃↑𝐴))
14	9, 10, 12, 13	syl3anc 1273	. . 3 ⊢ (((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℕ0) ∧ 𝑛 ∈ (0...𝐴)) → (𝑃↑𝑛) ∥ (𝑃↑𝐴))
15	2, 7, 14	elrabd 2964	. 2 ⊢ (((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℕ0) ∧ 𝑛 ∈ (0...𝐴)) → (𝑃↑𝑛) ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ (𝑃↑𝐴)})
16		simpl 109	. . . 4 ⊢ ((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℕ0) → 𝑃 ∈ ℙ)
17		elrabi 2959	. . . 4 ⊢ (𝑚 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ (𝑃↑𝐴)} → 𝑚 ∈ ℕ)
18		pccl 12877	. . . 4 ⊢ ((𝑃 ∈ ℙ ∧ 𝑚 ∈ ℕ) → (𝑃 pCnt 𝑚) ∈ ℕ0)
19	16, 17, 18	syl2an 289	. . 3 ⊢ (((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℕ0) ∧ 𝑚 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ (𝑃↑𝐴)}) → (𝑃 pCnt 𝑚) ∈ ℕ0)
20	16	adantr 276	. . . . 5 ⊢ (((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℕ0) ∧ 𝑚 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ (𝑃↑𝐴)}) → 𝑃 ∈ ℙ)
21	17	adantl 277	. . . . . 6 ⊢ (((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℕ0) ∧ 𝑚 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ (𝑃↑𝐴)}) → 𝑚 ∈ ℕ)
22	21	nnzd 9601	. . . . 5 ⊢ (((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℕ0) ∧ 𝑚 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ (𝑃↑𝐴)}) → 𝑚 ∈ ℤ)
23	8	ad2antrr 488	. . . . . 6 ⊢ (((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℕ0) ∧ 𝑚 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ (𝑃↑𝐴)}) → 𝑃 ∈ ℤ)
24		simplr 529	. . . . . 6 ⊢ (((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℕ0) ∧ 𝑚 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ (𝑃↑𝐴)}) → 𝐴 ∈ ℕ0)
25		zexpcl 10817	. . . . . 6 ⊢ ((𝑃 ∈ ℤ ∧ 𝐴 ∈ ℕ0) → (𝑃↑𝐴) ∈ ℤ)
26	23, 24, 25	syl2anc 411	. . . . 5 ⊢ (((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℕ0) ∧ 𝑚 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ (𝑃↑𝐴)}) → (𝑃↑𝐴) ∈ ℤ)
27		breq1 4091	. . . . . . . 8 ⊢ (𝑥 = 𝑚 → (𝑥 ∥ (𝑃↑𝐴) ↔ 𝑚 ∥ (𝑃↑𝐴)))
28	27	elrab 2962	. . . . . . 7 ⊢ (𝑚 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ (𝑃↑𝐴)} ↔ (𝑚 ∈ ℕ ∧ 𝑚 ∥ (𝑃↑𝐴)))
29	28	simprbi 275	. . . . . 6 ⊢ (𝑚 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ (𝑃↑𝐴)} → 𝑚 ∥ (𝑃↑𝐴))
30	29	adantl 277	. . . . 5 ⊢ (((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℕ0) ∧ 𝑚 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ (𝑃↑𝐴)}) → 𝑚 ∥ (𝑃↑𝐴))
31		pcdvdstr 12905	. . . . 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 1275	. . . 4 ⊢ (((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℕ0) ∧ 𝑚 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ (𝑃↑𝐴)}) → (𝑃 pCnt 𝑚) ≤ (𝑃 pCnt (𝑃↑𝐴)))
33		pcidlem 12901	. . . . 5 ⊢ ((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℕ0) → (𝑃 pCnt (𝑃↑𝐴)) = 𝐴)
34	33	adantr 276	. . . 4 ⊢ (((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℕ0) ∧ 𝑚 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ (𝑃↑𝐴)}) → (𝑃 pCnt (𝑃↑𝐴)) = 𝐴)
35	32, 34	breqtrd 4114	. . 3 ⊢ (((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℕ0) ∧ 𝑚 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ (𝑃↑𝐴)}) → (𝑃 pCnt 𝑚) ≤ 𝐴)
36		fznn0 10348	. . . 4 ⊢ (𝐴 ∈ ℕ0 → ((𝑃 pCnt 𝑚) ∈ (0...𝐴) ↔ ((𝑃 pCnt 𝑚) ∈ ℕ0 ∧ (𝑃 pCnt 𝑚) ≤ 𝐴)))
37	24, 36	syl 14	. . 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 952	. 2 ⊢ (((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℕ0) ∧ 𝑚 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ (𝑃↑𝐴)}) → (𝑃 pCnt 𝑚) ∈ (0...𝐴))
39		oveq2 6026	. . . . . . . . 9 ⊢ (𝑛 = 𝐴 → (𝑃↑𝑛) = (𝑃↑𝐴))
40	39	breq2d 4100	. . . . . . . 8 ⊢ (𝑛 = 𝐴 → (𝑚 ∥ (𝑃↑𝑛) ↔ 𝑚 ∥ (𝑃↑𝐴)))
41	40	rspcev 2910	. . . . . . 7 ⊢ ((𝐴 ∈ ℕ0 ∧ 𝑚 ∥ (𝑃↑𝐴)) → ∃𝑛 ∈ ℕ0 𝑚 ∥ (𝑃↑𝑛))
42	24, 30, 41	syl2anc 411	. . . . . 6 ⊢ (((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℕ0) ∧ 𝑚 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ (𝑃↑𝐴)}) → ∃𝑛 ∈ ℕ0 𝑚 ∥ (𝑃↑𝑛))
43		pcprmpw2 12911	. . . . . . 7 ⊢ ((𝑃 ∈ ℙ ∧ 𝑚 ∈ ℕ) → (∃𝑛 ∈ ℕ0 𝑚 ∥ (𝑃↑𝑛) ↔ 𝑚 = (𝑃↑(𝑃 pCnt 𝑚))))
44	16, 17, 43	syl2an 289	. . . . . 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 147	. . . . 5 ⊢ (((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℕ0) ∧ 𝑚 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ (𝑃↑𝐴)}) → 𝑚 = (𝑃↑(𝑃 pCnt 𝑚)))
46	45	adantrl 478	. . . 4 ⊢ (((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℕ0) ∧ (𝑛 ∈ (0...𝐴) ∧ 𝑚 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ (𝑃↑𝐴)})) → 𝑚 = (𝑃↑(𝑃 pCnt 𝑚)))
47		oveq2 6026	. . . . 5 ⊢ (𝑛 = (𝑃 pCnt 𝑚) → (𝑃↑𝑛) = (𝑃↑(𝑃 pCnt 𝑚)))
48	47	eqeq2d 2243	. . . 4 ⊢ (𝑛 = (𝑃 pCnt 𝑚) → (𝑚 = (𝑃↑𝑛) ↔ 𝑚 = (𝑃↑(𝑃 pCnt 𝑚))))
49	46, 48	syl5ibrcom 157	. . 3 ⊢ (((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℕ0) ∧ (𝑛 ∈ (0...𝐴) ∧ 𝑚 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ (𝑃↑𝐴)})) → (𝑛 = (𝑃 pCnt 𝑚) → 𝑚 = (𝑃↑𝑛)))
50		elfzelz 10260	. . . . . . 7 ⊢ (𝑛 ∈ (0...𝐴) → 𝑛 ∈ ℤ)
51		pcid 12902	. . . . . . 7 ⊢ ((𝑃 ∈ ℙ ∧ 𝑛 ∈ ℤ) → (𝑃 pCnt (𝑃↑𝑛)) = 𝑛)
52	16, 50, 51	syl2an 289	. . . . . 6 ⊢ (((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℕ0) ∧ 𝑛 ∈ (0...𝐴)) → (𝑃 pCnt (𝑃↑𝑛)) = 𝑛)
53	52	eqcomd 2237	. . . . 5 ⊢ (((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℕ0) ∧ 𝑛 ∈ (0...𝐴)) → 𝑛 = (𝑃 pCnt (𝑃↑𝑛)))
54	53	adantrr 479	. . . 4 ⊢ (((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℕ0) ∧ (𝑛 ∈ (0...𝐴) ∧ 𝑚 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ (𝑃↑𝐴)})) → 𝑛 = (𝑃 pCnt (𝑃↑𝑛)))
55		oveq2 6026	. . . . 5 ⊢ (𝑚 = (𝑃↑𝑛) → (𝑃 pCnt 𝑚) = (𝑃 pCnt (𝑃↑𝑛)))
56	55	eqeq2d 2243	. . . 4 ⊢ (𝑚 = (𝑃↑𝑛) → (𝑛 = (𝑃 pCnt 𝑚) ↔ 𝑛 = (𝑃 pCnt (𝑃↑𝑛))))
57	54, 56	syl5ibrcom 157	. . 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 129	. 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 6228	1 ⊢ ((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℕ0) → 𝐹:(0...𝐴)–1-1-onto→{𝑥 ∈ ℕ ∣ 𝑥 ∥ (𝑃↑𝐴)})

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   = wceq 1397   ∈ wcel 2202  ∃wrex 2511  {crab 2514   class class class wbr 4088   ↦ cmpt 4150  –1-1-onto→wf1o 5325  ‘cfv 5326  (class class class)co 6018  0cc0 8032   ≤ cle 8215  ℕcn 9143  ℕ₀cn0 9402  ℤcz 9479  ℤ_≥cuz 9755  ...cfz 10243  ↑cexp 10801   ∥ cdvds 12353  ℙcprime 12684   pCnt cpc 12862

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 619  ax-in2 620  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-coll 4204  ax-sep 4207  ax-nul 4215  ax-pow 4264  ax-pr 4299  ax-un 4530  ax-setind 4635  ax-iinf 4686  ax-cnex 8123  ax-resscn 8124  ax-1cn 8125  ax-1re 8126  ax-icn 8127  ax-addcl 8128  ax-addrcl 8129  ax-mulcl 8130  ax-mulrcl 8131  ax-addcom 8132  ax-mulcom 8133  ax-addass 8134  ax-mulass 8135  ax-distr 8136  ax-i2m1 8137  ax-0lt1 8138  ax-1rid 8139  ax-0id 8140  ax-rnegex 8141  ax-precex 8142  ax-cnre 8143  ax-pre-ltirr 8144  ax-pre-ltwlin 8145  ax-pre-lttrn 8146  ax-pre-apti 8147  ax-pre-ltadd 8148  ax-pre-mulgt0 8149  ax-pre-mulext 8150  ax-arch 8151  ax-caucvg 8152

This theorem depends on definitions:  df-bi 117  df-stab 838  df-dc 842  df-3or 1005  df-3an 1006  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ne 2403  df-nel 2498  df-ral 2515  df-rex 2516  df-reu 2517  df-rmo 2518  df-rab 2519  df-v 2804  df-sbc 3032  df-csb 3128  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-nul 3495  df-if 3606  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-int 3929  df-iun 3972  df-br 4089  df-opab 4151  df-mpt 4152  df-tr 4188  df-id 4390  df-po 4393  df-iso 4394  df-iord 4463  df-on 4465  df-ilim 4466  df-suc 4468  df-iom 4689  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-res 4737  df-ima 4738  df-iota 5286  df-fun 5328  df-fn 5329  df-f 5330  df-f1 5331  df-fo 5332  df-f1o 5333  df-fv 5334  df-isom 5335  df-riota 5971  df-ov 6021  df-oprab 6022  df-mpo 6023  df-1st 6303  df-2nd 6304  df-recs 6471  df-frec 6557  df-1o 6582  df-2o 6583  df-er 6702  df-en 6910  df-sup 7183  df-inf 7184  df-pnf 8216  df-mnf 8217  df-xr 8218  df-ltxr 8219  df-le 8220  df-sub 8352  df-neg 8353  df-reap 8755  df-ap 8762  df-div 8853  df-inn 9144  df-2 9202  df-3 9203  df-4 9204  df-n0 9403  df-xnn0 9466  df-z 9480  df-uz 9756  df-q 9854  df-rp 9889  df-fz 10244  df-fzo 10378  df-fl 10531  df-mod 10586  df-seqfrec 10711  df-exp 10802  df-cj 11407  df-re 11408  df-im 11409  df-rsqrt 11563  df-abs 11564  df-dvds 12354  df-gcd 12530  df-prm 12685  df-pc 12863

This theorem is referenced by:  sgmppw  15722  0sgmppw  15723