Theorem dvdsppwf1o 15671

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 4086	. . 3 ⊢ (𝑥 = (𝑃↑𝑛) → (𝑥 ∥ (𝑃↑𝐴) ↔ (𝑃↑𝑛) ∥ (𝑃↑𝐴)))
3		prmnn 12640	. . . . 5 ⊢ (𝑃 ∈ ℙ → 𝑃 ∈ ℕ)
4	3	adantr 276	. . . 4 ⊢ ((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℕ0) → 𝑃 ∈ ℕ)
5		elfznn0 10318	. . . 4 ⊢ (𝑛 ∈ (0...𝐴) → 𝑛 ∈ ℕ0)
6		nnexpcl 10782	. . . 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 12641	. . . . 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 10226	. . . . 5 ⊢ (𝑛 ∈ (0...𝐴) → 𝐴 ∈ (ℤ≥‘𝑛))
12	11	adantl 277	. . . 4 ⊢ (((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℕ0) ∧ 𝑛 ∈ (0...𝐴)) → 𝐴 ∈ (ℤ≥‘𝑛))
13		dvdsexp 12380	. . . 4 ⊢ ((𝑃 ∈ ℤ ∧ 𝑛 ∈ ℕ0 ∧ 𝐴 ∈ (ℤ≥‘𝑛)) → (𝑃↑𝑛) ∥ (𝑃↑𝐴))
14	9, 10, 12, 13	syl3anc 1271	. . 3 ⊢ (((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℕ0) ∧ 𝑛 ∈ (0...𝐴)) → (𝑃↑𝑛) ∥ (𝑃↑𝐴))
15	2, 7, 14	elrabd 2961	. 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 2956	. . . 4 ⊢ (𝑚 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ (𝑃↑𝐴)} → 𝑚 ∈ ℕ)
18		pccl 12830	. . . 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 9576	. . . . 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 10784	. . . . . 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 4086	. . . . . . . 8 ⊢ (𝑥 = 𝑚 → (𝑥 ∥ (𝑃↑𝐴) ↔ 𝑚 ∥ (𝑃↑𝐴)))
28	27	elrab 2959	. . . . . . 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 12858	. . . . 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 1273	. . . 4 ⊢ (((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℕ0) ∧ 𝑚 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ (𝑃↑𝐴)}) → (𝑃 pCnt 𝑚) ≤ (𝑃 pCnt (𝑃↑𝐴)))
33		pcidlem 12854	. . . . 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 4109	. . 3 ⊢ (((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℕ0) ∧ 𝑚 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ (𝑃↑𝐴)}) → (𝑃 pCnt 𝑚) ≤ 𝐴)
36		fznn0 10317	. . . 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 950	. 2 ⊢ (((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℕ0) ∧ 𝑚 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ (𝑃↑𝐴)}) → (𝑃 pCnt 𝑚) ∈ (0...𝐴))
39		oveq2 6015	. . . . . . . . 9 ⊢ (𝑛 = 𝐴 → (𝑃↑𝑛) = (𝑃↑𝐴))
40	39	breq2d 4095	. . . . . . . 8 ⊢ (𝑛 = 𝐴 → (𝑚 ∥ (𝑃↑𝑛) ↔ 𝑚 ∥ (𝑃↑𝐴)))
41	40	rspcev 2907	. . . . . . 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 12864	. . . . . . 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 6015	. . . . 5 ⊢ (𝑛 = (𝑃 pCnt 𝑚) → (𝑃↑𝑛) = (𝑃↑(𝑃 pCnt 𝑚)))
48	47	eqeq2d 2241	. . . 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 10229	. . . . . . 7 ⊢ (𝑛 ∈ (0...𝐴) → 𝑛 ∈ ℤ)
51		pcid 12855	. . . . . . 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 2235	. . . . 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 6015	. . . . 5 ⊢ (𝑚 = (𝑃↑𝑛) → (𝑃 pCnt 𝑚) = (𝑃 pCnt (𝑃↑𝑛)))
56	55	eqeq2d 2241	. . . 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 6217	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 1395   ∈ wcel 2200  ∃wrex 2509  {crab 2512   class class class wbr 4083   ↦ cmpt 4145  –1-1-onto→wf1o 5317  ‘cfv 5318  (class class class)co 6007  0cc0 8007   ≤ cle 8190  ℕcn 9118  ℕ₀cn0 9377  ℤcz 9454  ℤ_≥cuz 9730  ...cfz 10212  ↑cexp 10768   ∥ cdvds 12306  ℙcprime 12637   pCnt cpc 12815

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-coll 4199  ax-sep 4202  ax-nul 4210  ax-pow 4258  ax-pr 4293  ax-un 4524  ax-setind 4629  ax-iinf 4680  ax-cnex 8098  ax-resscn 8099  ax-1cn 8100  ax-1re 8101  ax-icn 8102  ax-addcl 8103  ax-addrcl 8104  ax-mulcl 8105  ax-mulrcl 8106  ax-addcom 8107  ax-mulcom 8108  ax-addass 8109  ax-mulass 8110  ax-distr 8111  ax-i2m1 8112  ax-0lt1 8113  ax-1rid 8114  ax-0id 8115  ax-rnegex 8116  ax-precex 8117  ax-cnre 8118  ax-pre-ltirr 8119  ax-pre-ltwlin 8120  ax-pre-lttrn 8121  ax-pre-apti 8122  ax-pre-ltadd 8123  ax-pre-mulgt0 8124  ax-pre-mulext 8125  ax-arch 8126  ax-caucvg 8127

This theorem depends on definitions:  df-bi 117  df-stab 836  df-dc 840  df-3or 1003  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-nel 2496  df-ral 2513  df-rex 2514  df-reu 2515  df-rmo 2516  df-rab 2517  df-v 2801  df-sbc 3029  df-csb 3125  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-nul 3492  df-if 3603  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3889  df-int 3924  df-iun 3967  df-br 4084  df-opab 4146  df-mpt 4147  df-tr 4183  df-id 4384  df-po 4387  df-iso 4388  df-iord 4457  df-on 4459  df-ilim 4460  df-suc 4462  df-iom 4683  df-xp 4725  df-rel 4726  df-cnv 4727  df-co 4728  df-dm 4729  df-rn 4730  df-res 4731  df-ima 4732  df-iota 5278  df-fun 5320  df-fn 5321  df-f 5322  df-f1 5323  df-fo 5324  df-f1o 5325  df-fv 5326  df-isom 5327  df-riota 5960  df-ov 6010  df-oprab 6011  df-mpo 6012  df-1st 6292  df-2nd 6293  df-recs 6457  df-frec 6543  df-1o 6568  df-2o 6569  df-er 6688  df-en 6896  df-sup 7159  df-inf 7160  df-pnf 8191  df-mnf 8192  df-xr 8193  df-ltxr 8194  df-le 8195  df-sub 8327  df-neg 8328  df-reap 8730  df-ap 8737  df-div 8828  df-inn 9119  df-2 9177  df-3 9178  df-4 9179  df-n0 9378  df-xnn0 9441  df-z 9455  df-uz 9731  df-q 9823  df-rp 9858  df-fz 10213  df-fzo 10347  df-fl 10498  df-mod 10553  df-seqfrec 10678  df-exp 10769  df-cj 11361  df-re 11362  df-im 11363  df-rsqrt 11517  df-abs 11518  df-dvds 12307  df-gcd 12483  df-prm 12638  df-pc 12816

This theorem is referenced by:  sgmppw  15674  0sgmppw  15675