Theorem prmgapprmo 16763

Description: Alternate proof of prmgap 16760: in contrast to prmgap 16760, where the gap starts at n! , the factorial of n, the gap starts at n#, the primorial of n. (Contributed by AV, 15-Aug-2020.) (Revised by AV, 29-Aug-2020.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
prmgapprmo	⊢ ∀𝑛 ∈ ℕ ∃𝑝 ∈ ℙ ∃𝑞 ∈ ℙ (𝑛 ≤ (𝑞 − 𝑝) ∧ ∀𝑧 ∈ ((𝑝 + 1)..^𝑞)𝑧 ∉ ℙ)

Distinct variable group:   𝑛,𝑝,𝑞,𝑧

Proof of Theorem prmgapprmo

Dummy variables 𝑖 𝑗 𝑘 𝑚 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		id 22	. . 3 ⊢ (𝑛 ∈ ℕ → 𝑛 ∈ ℕ)
2		eqid 2738	. . . . . 6 ⊢ (𝑗 ∈ ℕ ↦ ∏𝑘 ∈ (1...𝑗)((𝑚 ∈ ℕ ↦ if(𝑚 ∈ ℙ, 𝑚, 1))‘𝑘)) = (𝑗 ∈ ℕ ↦ ∏𝑘 ∈ (1...𝑗)((𝑚 ∈ ℕ ↦ if(𝑚 ∈ ℙ, 𝑚, 1))‘𝑘))
3		fzfid 13693	. . . . . . 7 ⊢ (𝑗 ∈ ℕ → (1...𝑗) ∈ Fin)
4		eqidd 2739	. . . . . . . . 9 ⊢ ((𝑗 ∈ ℕ ∧ 𝑘 ∈ (1...𝑗)) → (𝑚 ∈ ℕ ↦ if(𝑚 ∈ ℙ, 𝑚, 1)) = (𝑚 ∈ ℕ ↦ if(𝑚 ∈ ℙ, 𝑚, 1)))
5		eleq1 2826	. . . . . . . . . . 11 ⊢ (𝑚 = 𝑘 → (𝑚 ∈ ℙ ↔ 𝑘 ∈ ℙ))
6		id 22	. . . . . . . . . . 11 ⊢ (𝑚 = 𝑘 → 𝑚 = 𝑘)
7	5, 6	ifbieq1d 4483	. . . . . . . . . 10 ⊢ (𝑚 = 𝑘 → if(𝑚 ∈ ℙ, 𝑚, 1) = if(𝑘 ∈ ℙ, 𝑘, 1))
8	7	adantl 482	. . . . . . . . 9 ⊢ (((𝑗 ∈ ℕ ∧ 𝑘 ∈ (1...𝑗)) ∧ 𝑚 = 𝑘) → if(𝑚 ∈ ℙ, 𝑚, 1) = if(𝑘 ∈ ℙ, 𝑘, 1))
9		elfznn 13285	. . . . . . . . . 10 ⊢ (𝑘 ∈ (1...𝑗) → 𝑘 ∈ ℕ)
10	9	adantl 482	. . . . . . . . 9 ⊢ ((𝑗 ∈ ℕ ∧ 𝑘 ∈ (1...𝑗)) → 𝑘 ∈ ℕ)
11		1nn 11984	. . . . . . . . . . . 12 ⊢ 1 ∈ ℕ
12	11	a1i 11	. . . . . . . . . . 11 ⊢ (𝑘 ∈ (1...𝑗) → 1 ∈ ℕ)
13	9, 12	ifcld 4505	. . . . . . . . . 10 ⊢ (𝑘 ∈ (1...𝑗) → if(𝑘 ∈ ℙ, 𝑘, 1) ∈ ℕ)
14	13	adantl 482	. . . . . . . . 9 ⊢ ((𝑗 ∈ ℕ ∧ 𝑘 ∈ (1...𝑗)) → if(𝑘 ∈ ℙ, 𝑘, 1) ∈ ℕ)
15	4, 8, 10, 14	fvmptd 6882	. . . . . . . 8 ⊢ ((𝑗 ∈ ℕ ∧ 𝑘 ∈ (1...𝑗)) → ((𝑚 ∈ ℕ ↦ if(𝑚 ∈ ℙ, 𝑚, 1))‘𝑘) = if(𝑘 ∈ ℙ, 𝑘, 1))
16	15, 14	eqeltrd 2839	. . . . . . 7 ⊢ ((𝑗 ∈ ℕ ∧ 𝑘 ∈ (1...𝑗)) → ((𝑚 ∈ ℕ ↦ if(𝑚 ∈ ℙ, 𝑚, 1))‘𝑘) ∈ ℕ)
17	3, 16	fprodnncl 15665	. . . . . 6 ⊢ (𝑗 ∈ ℕ → ∏𝑘 ∈ (1...𝑗)((𝑚 ∈ ℕ ↦ if(𝑚 ∈ ℙ, 𝑚, 1))‘𝑘) ∈ ℕ)
18	2, 17	fmpti 6986	. . . . 5 ⊢ (𝑗 ∈ ℕ ↦ ∏𝑘 ∈ (1...𝑗)((𝑚 ∈ ℕ ↦ if(𝑚 ∈ ℙ, 𝑚, 1))‘𝑘)):ℕ⟶ℕ
19		nnex 11979	. . . . . 6 ⊢ ℕ ∈ V
20	19, 19	elmap 8659	. . . . 5 ⊢ ((𝑗 ∈ ℕ ↦ ∏𝑘 ∈ (1...𝑗)((𝑚 ∈ ℕ ↦ if(𝑚 ∈ ℙ, 𝑚, 1))‘𝑘)) ∈ (ℕ ↑m ℕ) ↔ (𝑗 ∈ ℕ ↦ ∏𝑘 ∈ (1...𝑗)((𝑚 ∈ ℕ ↦ if(𝑚 ∈ ℙ, 𝑚, 1))‘𝑘)):ℕ⟶ℕ)
21	18, 20	mpbir 230	. . . 4 ⊢ (𝑗 ∈ ℕ ↦ ∏𝑘 ∈ (1...𝑗)((𝑚 ∈ ℕ ↦ if(𝑚 ∈ ℙ, 𝑚, 1))‘𝑘)) ∈ (ℕ ↑m ℕ)
22	21	a1i 11	. . 3 ⊢ (𝑛 ∈ ℕ → (𝑗 ∈ ℕ ↦ ∏𝑘 ∈ (1...𝑗)((𝑚 ∈ ℕ ↦ if(𝑚 ∈ ℙ, 𝑚, 1))‘𝑘)) ∈ (ℕ ↑m ℕ))
23		prmgapprmolem 16762	. . . . 5 ⊢ ((𝑛 ∈ ℕ ∧ 𝑖 ∈ (2...𝑛)) → 1 < (((#p‘𝑛) + 𝑖) gcd 𝑖))
24		eqidd 2739	. . . . . . . . . . . 12 ⊢ ((((𝑛 ∈ ℕ ∧ 𝑖 ∈ (2...𝑛)) ∧ 𝑗 ∈ ℕ) ∧ 𝑘 ∈ (1...𝑗)) → (𝑚 ∈ ℕ ↦ if(𝑚 ∈ ℙ, 𝑚, 1)) = (𝑚 ∈ ℕ ↦ if(𝑚 ∈ ℙ, 𝑚, 1)))
25	7	adantl 482	. . . . . . . . . . . 12 ⊢ (((((𝑛 ∈ ℕ ∧ 𝑖 ∈ (2...𝑛)) ∧ 𝑗 ∈ ℕ) ∧ 𝑘 ∈ (1...𝑗)) ∧ 𝑚 = 𝑘) → if(𝑚 ∈ ℙ, 𝑚, 1) = if(𝑘 ∈ ℙ, 𝑘, 1))
26	9	adantl 482	. . . . . . . . . . . 12 ⊢ ((((𝑛 ∈ ℕ ∧ 𝑖 ∈ (2...𝑛)) ∧ 𝑗 ∈ ℕ) ∧ 𝑘 ∈ (1...𝑗)) → 𝑘 ∈ ℕ)
27		elfzelz 13256	. . . . . . . . . . . . . 14 ⊢ (𝑘 ∈ (1...𝑗) → 𝑘 ∈ ℤ)
28		1zzd 12351	. . . . . . . . . . . . . 14 ⊢ (𝑘 ∈ (1...𝑗) → 1 ∈ ℤ)
29	27, 28	ifcld 4505	. . . . . . . . . . . . 13 ⊢ (𝑘 ∈ (1...𝑗) → if(𝑘 ∈ ℙ, 𝑘, 1) ∈ ℤ)
30	29	adantl 482	. . . . . . . . . . . 12 ⊢ ((((𝑛 ∈ ℕ ∧ 𝑖 ∈ (2...𝑛)) ∧ 𝑗 ∈ ℕ) ∧ 𝑘 ∈ (1...𝑗)) → if(𝑘 ∈ ℙ, 𝑘, 1) ∈ ℤ)
31	24, 25, 26, 30	fvmptd 6882	. . . . . . . . . . 11 ⊢ ((((𝑛 ∈ ℕ ∧ 𝑖 ∈ (2...𝑛)) ∧ 𝑗 ∈ ℕ) ∧ 𝑘 ∈ (1...𝑗)) → ((𝑚 ∈ ℕ ↦ if(𝑚 ∈ ℙ, 𝑚, 1))‘𝑘) = if(𝑘 ∈ ℙ, 𝑘, 1))
32	31	prodeq2dv 15633	. . . . . . . . . 10 ⊢ (((𝑛 ∈ ℕ ∧ 𝑖 ∈ (2...𝑛)) ∧ 𝑗 ∈ ℕ) → ∏𝑘 ∈ (1...𝑗)((𝑚 ∈ ℕ ↦ if(𝑚 ∈ ℙ, 𝑚, 1))‘𝑘) = ∏𝑘 ∈ (1...𝑗)if(𝑘 ∈ ℙ, 𝑘, 1))
33	32	mpteq2dva 5174	. . . . . . . . 9 ⊢ ((𝑛 ∈ ℕ ∧ 𝑖 ∈ (2...𝑛)) → (𝑗 ∈ ℕ ↦ ∏𝑘 ∈ (1...𝑗)((𝑚 ∈ ℕ ↦ if(𝑚 ∈ ℙ, 𝑚, 1))‘𝑘)) = (𝑗 ∈ ℕ ↦ ∏𝑘 ∈ (1...𝑗)if(𝑘 ∈ ℙ, 𝑘, 1)))
34		oveq2 7283	. . . . . . . . . . 11 ⊢ (𝑗 = 𝑛 → (1...𝑗) = (1...𝑛))
35	34	prodeq1d 15631	. . . . . . . . . 10 ⊢ (𝑗 = 𝑛 → ∏𝑘 ∈ (1...𝑗)if(𝑘 ∈ ℙ, 𝑘, 1) = ∏𝑘 ∈ (1...𝑛)if(𝑘 ∈ ℙ, 𝑘, 1))
36	35	adantl 482	. . . . . . . . 9 ⊢ (((𝑛 ∈ ℕ ∧ 𝑖 ∈ (2...𝑛)) ∧ 𝑗 = 𝑛) → ∏𝑘 ∈ (1...𝑗)if(𝑘 ∈ ℙ, 𝑘, 1) = ∏𝑘 ∈ (1...𝑛)if(𝑘 ∈ ℙ, 𝑘, 1))
37		simpl 483	. . . . . . . . 9 ⊢ ((𝑛 ∈ ℕ ∧ 𝑖 ∈ (2...𝑛)) → 𝑛 ∈ ℕ)
38		fzfid 13693	. . . . . . . . . 10 ⊢ ((𝑛 ∈ ℕ ∧ 𝑖 ∈ (2...𝑛)) → (1...𝑛) ∈ Fin)
39		elfznn 13285	. . . . . . . . . . . 12 ⊢ (𝑘 ∈ (1...𝑛) → 𝑘 ∈ ℕ)
40	11	a1i 11	. . . . . . . . . . . 12 ⊢ (𝑘 ∈ (1...𝑛) → 1 ∈ ℕ)
41	39, 40	ifcld 4505	. . . . . . . . . . 11 ⊢ (𝑘 ∈ (1...𝑛) → if(𝑘 ∈ ℙ, 𝑘, 1) ∈ ℕ)
42	41	adantl 482	. . . . . . . . . 10 ⊢ (((𝑛 ∈ ℕ ∧ 𝑖 ∈ (2...𝑛)) ∧ 𝑘 ∈ (1...𝑛)) → if(𝑘 ∈ ℙ, 𝑘, 1) ∈ ℕ)
43	38, 42	fprodnncl 15665	. . . . . . . . 9 ⊢ ((𝑛 ∈ ℕ ∧ 𝑖 ∈ (2...𝑛)) → ∏𝑘 ∈ (1...𝑛)if(𝑘 ∈ ℙ, 𝑘, 1) ∈ ℕ)
44	33, 36, 37, 43	fvmptd 6882	. . . . . . . 8 ⊢ ((𝑛 ∈ ℕ ∧ 𝑖 ∈ (2...𝑛)) → ((𝑗 ∈ ℕ ↦ ∏𝑘 ∈ (1...𝑗)((𝑚 ∈ ℕ ↦ if(𝑚 ∈ ℙ, 𝑚, 1))‘𝑘))‘𝑛) = ∏𝑘 ∈ (1...𝑛)if(𝑘 ∈ ℙ, 𝑘, 1))
45		nnnn0 12240	. . . . . . . . . . 11 ⊢ (𝑛 ∈ ℕ → 𝑛 ∈ ℕ0)
46		prmoval 16734	. . . . . . . . . . 11 ⊢ (𝑛 ∈ ℕ0 → (#p‘𝑛) = ∏𝑘 ∈ (1...𝑛)if(𝑘 ∈ ℙ, 𝑘, 1))
47	45, 46	syl 17	. . . . . . . . . 10 ⊢ (𝑛 ∈ ℕ → (#p‘𝑛) = ∏𝑘 ∈ (1...𝑛)if(𝑘 ∈ ℙ, 𝑘, 1))
48	47	eqcomd 2744	. . . . . . . . 9 ⊢ (𝑛 ∈ ℕ → ∏𝑘 ∈ (1...𝑛)if(𝑘 ∈ ℙ, 𝑘, 1) = (#p‘𝑛))
49	48	adantr 481	. . . . . . . 8 ⊢ ((𝑛 ∈ ℕ ∧ 𝑖 ∈ (2...𝑛)) → ∏𝑘 ∈ (1...𝑛)if(𝑘 ∈ ℙ, 𝑘, 1) = (#p‘𝑛))
50	44, 49	eqtrd 2778	. . . . . . 7 ⊢ ((𝑛 ∈ ℕ ∧ 𝑖 ∈ (2...𝑛)) → ((𝑗 ∈ ℕ ↦ ∏𝑘 ∈ (1...𝑗)((𝑚 ∈ ℕ ↦ if(𝑚 ∈ ℙ, 𝑚, 1))‘𝑘))‘𝑛) = (#p‘𝑛))
51	50	oveq1d 7290	. . . . . 6 ⊢ ((𝑛 ∈ ℕ ∧ 𝑖 ∈ (2...𝑛)) → (((𝑗 ∈ ℕ ↦ ∏𝑘 ∈ (1...𝑗)((𝑚 ∈ ℕ ↦ if(𝑚 ∈ ℙ, 𝑚, 1))‘𝑘))‘𝑛) + 𝑖) = ((#p‘𝑛) + 𝑖))
52	51	oveq1d 7290	. . . . 5 ⊢ ((𝑛 ∈ ℕ ∧ 𝑖 ∈ (2...𝑛)) → ((((𝑗 ∈ ℕ ↦ ∏𝑘 ∈ (1...𝑗)((𝑚 ∈ ℕ ↦ if(𝑚 ∈ ℙ, 𝑚, 1))‘𝑘))‘𝑛) + 𝑖) gcd 𝑖) = (((#p‘𝑛) + 𝑖) gcd 𝑖))
53	23, 52	breqtrrd 5102	. . . 4 ⊢ ((𝑛 ∈ ℕ ∧ 𝑖 ∈ (2...𝑛)) → 1 < ((((𝑗 ∈ ℕ ↦ ∏𝑘 ∈ (1...𝑗)((𝑚 ∈ ℕ ↦ if(𝑚 ∈ ℙ, 𝑚, 1))‘𝑘))‘𝑛) + 𝑖) gcd 𝑖))
54	53	ralrimiva 3103	. . 3 ⊢ (𝑛 ∈ ℕ → ∀𝑖 ∈ (2...𝑛)1 < ((((𝑗 ∈ ℕ ↦ ∏𝑘 ∈ (1...𝑗)((𝑚 ∈ ℕ ↦ if(𝑚 ∈ ℙ, 𝑚, 1))‘𝑘))‘𝑛) + 𝑖) gcd 𝑖))
55	1, 22, 54	prmgaplem8 16759	. 2 ⊢ (𝑛 ∈ ℕ → ∃𝑝 ∈ ℙ ∃𝑞 ∈ ℙ (𝑛 ≤ (𝑞 − 𝑝) ∧ ∀𝑧 ∈ ((𝑝 + 1)..^𝑞)𝑧 ∉ ℙ))
56	55	rgen 3074	1 ⊢ ∀𝑛 ∈ ℕ ∃𝑝 ∈ ℙ ∃𝑞 ∈ ℙ (𝑛 ≤ (𝑞 − 𝑝) ∧ ∀𝑧 ∈ ((𝑝 + 1)..^𝑞)𝑧 ∉ ℙ)

Syntax hints:   ∧ wa 396   = wceq 1539   ∈ wcel 2106   ∉ wnel 3049  ∀wral 3064  ∃wrex 3065  ifcif 4459   class class class wbr 5074   ↦ cmpt 5157  ⟶wf 6429  ‘cfv 6433  (class class class)co 7275   ↑_m cmap 8615  1c1 10872   + caddc 10874   < clt 11009   ≤ cle 11010   − cmin 11205  ℕcn 11973  2c2 12028  ℕ₀cn0 12233  ℤcz 12319  ...cfz 13239  ..^cfzo 13382  ∏cprod 15615   gcd cgcd 16201  ℙcprime 16376  #_pcprmo 16732

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-rep 5209  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352  ax-un 7588  ax-inf2 9399  ax-cnex 10927  ax-resscn 10928  ax-1cn 10929  ax-icn 10930  ax-addcl 10931  ax-addrcl 10932  ax-mulcl 10933  ax-mulrcl 10934  ax-mulcom 10935  ax-addass 10936  ax-mulass 10937  ax-distr 10938  ax-i2m1 10939  ax-1ne0 10940  ax-1rid 10941  ax-rnegex 10942  ax-rrecex 10943  ax-cnre 10944  ax-pre-lttri 10945  ax-pre-lttrn 10946  ax-pre-ltadd 10947  ax-pre-mulgt0 10948  ax-pre-sup 10949

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3069  df-rex 3070  df-rmo 3071  df-reu 3072  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-pss 3906  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-int 4880  df-iun 4926  df-br 5075  df-opab 5137  df-mpt 5158  df-tr 5192  df-id 5489  df-eprel 5495  df-po 5503  df-so 5504  df-fr 5544  df-se 5545  df-we 5546  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-pred 6202  df-ord 6269  df-on 6270  df-lim 6271  df-suc 6272  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438  df-fo 6439  df-f1o 6440  df-fv 6441  df-isom 6442  df-riota 7232  df-ov 7278  df-oprab 7279  df-mpo 7280  df-om 7713  df-1st 7831  df-2nd 7832  df-frecs 8097  df-wrecs 8128  df-recs 8202  df-rdg 8241  df-1o 8297  df-2o 8298  df-er 8498  df-map 8617  df-en 8734  df-dom 8735  df-sdom 8736  df-fin 8737  df-sup 9201  df-inf 9202  df-oi 9269  df-card 9697  df-pnf 11011  df-mnf 11012  df-xr 11013  df-ltxr 11014  df-le 11015  df-sub 11207  df-neg 11208  df-div 11633  df-nn 11974  df-2 12036  df-3 12037  df-n0 12234  df-z 12320  df-uz 12583  df-rp 12731  df-fz 13240  df-fzo 13383  df-seq 13722  df-exp 13783  df-fac 13988  df-hash 14045  df-cj 14810  df-re 14811  df-im 14812  df-sqrt 14946  df-abs 14947  df-clim 15197  df-prod 15616  df-dvds 15964  df-gcd 16202  df-prm 16377  df-prmo 16733

This theorem is referenced by: (None)