Theorem dvdsppwf1o 15235

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 4037	. . 3 ⊢ (𝑥 = (𝑃↑𝑛) → (𝑥 ∥ (𝑃↑𝐴) ↔ (𝑃↑𝑛) ∥ (𝑃↑𝐴)))
3		prmnn 12288	. . . . 5 ⊢ (𝑃 ∈ ℙ → 𝑃 ∈ ℕ)
4	3	adantr 276	. . . 4 ⊢ ((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℕ0) → 𝑃 ∈ ℕ)
5		elfznn0 10191	. . . 4 ⊢ (𝑛 ∈ (0...𝐴) → 𝑛 ∈ ℕ0)
6		nnexpcl 10646	. . . 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 12289	. . . . 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 10099	. . . . 5 ⊢ (𝑛 ∈ (0...𝐴) → 𝐴 ∈ (ℤ≥‘𝑛))
12	11	adantl 277	. . . 4 ⊢ (((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℕ0) ∧ 𝑛 ∈ (0...𝐴)) → 𝐴 ∈ (ℤ≥‘𝑛))
13		dvdsexp 12028	. . . 4 ⊢ ((𝑃 ∈ ℤ ∧ 𝑛 ∈ ℕ0 ∧ 𝐴 ∈ (ℤ≥‘𝑛)) → (𝑃↑𝑛) ∥ (𝑃↑𝐴))
14	9, 10, 12, 13	syl3anc 1249	. . 3 ⊢ (((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℕ0) ∧ 𝑛 ∈ (0...𝐴)) → (𝑃↑𝑛) ∥ (𝑃↑𝐴))
15	2, 7, 14	elrabd 2922	. 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 2917	. . . 4 ⊢ (𝑚 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ (𝑃↑𝐴)} → 𝑚 ∈ ℕ)
18		pccl 12478	. . . 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 9449	. . . . 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 528	. . . . . 6 ⊢ (((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℕ0) ∧ 𝑚 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ (𝑃↑𝐴)}) → 𝐴 ∈ ℕ0)
25		zexpcl 10648	. . . . . 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 4037	. . . . . . . 8 ⊢ (𝑥 = 𝑚 → (𝑥 ∥ (𝑃↑𝐴) ↔ 𝑚 ∥ (𝑃↑𝐴)))
28	27	elrab 2920	. . . . . . 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 12506	. . . . 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 1251	. . . 4 ⊢ (((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℕ0) ∧ 𝑚 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ (𝑃↑𝐴)}) → (𝑃 pCnt 𝑚) ≤ (𝑃 pCnt (𝑃↑𝐴)))
33		pcidlem 12502	. . . . 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 4060	. . 3 ⊢ (((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℕ0) ∧ 𝑚 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ (𝑃↑𝐴)}) → (𝑃 pCnt 𝑚) ≤ 𝐴)
36		fznn0 10190	. . . 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 946	. 2 ⊢ (((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℕ0) ∧ 𝑚 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ (𝑃↑𝐴)}) → (𝑃 pCnt 𝑚) ∈ (0...𝐴))
39		oveq2 5931	. . . . . . . . 9 ⊢ (𝑛 = 𝐴 → (𝑃↑𝑛) = (𝑃↑𝐴))
40	39	breq2d 4046	. . . . . . . 8 ⊢ (𝑛 = 𝐴 → (𝑚 ∥ (𝑃↑𝑛) ↔ 𝑚 ∥ (𝑃↑𝐴)))
41	40	rspcev 2868	. . . . . . 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 12512	. . . . . . 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 5931	. . . . 5 ⊢ (𝑛 = (𝑃 pCnt 𝑚) → (𝑃↑𝑛) = (𝑃↑(𝑃 pCnt 𝑚)))
48	47	eqeq2d 2208	. . . 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 10102	. . . . . . 7 ⊢ (𝑛 ∈ (0...𝐴) → 𝑛 ∈ ℤ)
51		pcid 12503	. . . . . . 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 2202	. . . . 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 5931	. . . . 5 ⊢ (𝑚 = (𝑃↑𝑛) → (𝑃 pCnt 𝑚) = (𝑃 pCnt (𝑃↑𝑛)))
56	55	eqeq2d 2208	. . . 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 6129	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 1364   ∈ wcel 2167  ∃wrex 2476  {crab 2479   class class class wbr 4034   ↦ cmpt 4095  –1-1-onto→wf1o 5258  ‘cfv 5259  (class class class)co 5923  0cc0 7881   ≤ cle 8064  ℕcn 8992  ℕ₀cn0 9251  ℤcz 9328  ℤ_≥cuz 9603  ...cfz 10085  ↑cexp 10632   ∥ cdvds 11954  ℙcprime 12285   pCnt cpc 12463

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 615  ax-in2 616  ax-io 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-coll 4149  ax-sep 4152  ax-nul 4160  ax-pow 4208  ax-pr 4243  ax-un 4469  ax-setind 4574  ax-iinf 4625  ax-cnex 7972  ax-resscn 7973  ax-1cn 7974  ax-1re 7975  ax-icn 7976  ax-addcl 7977  ax-addrcl 7978  ax-mulcl 7979  ax-mulrcl 7980  ax-addcom 7981  ax-mulcom 7982  ax-addass 7983  ax-mulass 7984  ax-distr 7985  ax-i2m1 7986  ax-0lt1 7987  ax-1rid 7988  ax-0id 7989  ax-rnegex 7990  ax-precex 7991  ax-cnre 7992  ax-pre-ltirr 7993  ax-pre-ltwlin 7994  ax-pre-lttrn 7995  ax-pre-apti 7996  ax-pre-ltadd 7997  ax-pre-mulgt0 7998  ax-pre-mulext 7999  ax-arch 8000  ax-caucvg 8001

This theorem depends on definitions:  df-bi 117  df-stab 832  df-dc 836  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-nel 2463  df-ral 2480  df-rex 2481  df-reu 2482  df-rmo 2483  df-rab 2484  df-v 2765  df-sbc 2990  df-csb 3085  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-nul 3452  df-if 3563  df-pw 3608  df-sn 3629  df-pr 3630  df-op 3632  df-uni 3841  df-int 3876  df-iun 3919  df-br 4035  df-opab 4096  df-mpt 4097  df-tr 4133  df-id 4329  df-po 4332  df-iso 4333  df-iord 4402  df-on 4404  df-ilim 4405  df-suc 4407  df-iom 4628  df-xp 4670  df-rel 4671  df-cnv 4672  df-co 4673  df-dm 4674  df-rn 4675  df-res 4676  df-ima 4677  df-iota 5220  df-fun 5261  df-fn 5262  df-f 5263  df-f1 5264  df-fo 5265  df-f1o 5266  df-fv 5267  df-isom 5268  df-riota 5878  df-ov 5926  df-oprab 5927  df-mpo 5928  df-1st 6199  df-2nd 6200  df-recs 6364  df-frec 6450  df-1o 6475  df-2o 6476  df-er 6593  df-en 6801  df-sup 7051  df-inf 7052  df-pnf 8065  df-mnf 8066  df-xr 8067  df-ltxr 8068  df-le 8069  df-sub 8201  df-neg 8202  df-reap 8604  df-ap 8611  df-div 8702  df-inn 8993  df-2 9051  df-3 9052  df-4 9053  df-n0 9252  df-xnn0 9315  df-z 9329  df-uz 9604  df-q 9696  df-rp 9731  df-fz 10086  df-fzo 10220  df-fl 10362  df-mod 10417  df-seqfrec 10542  df-exp 10633  df-cj 11009  df-re 11010  df-im 11011  df-rsqrt 11165  df-abs 11166  df-dvds 11955  df-gcd 12131  df-prm 12286  df-pc 12464

This theorem is referenced by:  sgmppw  15238  0sgmppw  15239