Theorem prmgapprmo 16997

Description: Alternate proof of prmgap 16994: in contrast to prmgap 16994, 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 2732	. . . . . 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 13940	. . . . . . 7 ⊢ (𝑗 ∈ ℕ → (1...𝑗) ∈ Fin)
4		eqidd 2733	. . . . . . . . 9 ⊢ ((𝑗 ∈ ℕ ∧ 𝑘 ∈ (1...𝑗)) → (𝑚 ∈ ℕ ↦ if(𝑚 ∈ ℙ, 𝑚, 1)) = (𝑚 ∈ ℕ ↦ if(𝑚 ∈ ℙ, 𝑚, 1)))
5		eleq1 2821	. . . . . . . . . . 11 ⊢ (𝑚 = 𝑘 → (𝑚 ∈ ℙ ↔ 𝑘 ∈ ℙ))
6		id 22	. . . . . . . . . . 11 ⊢ (𝑚 = 𝑘 → 𝑚 = 𝑘)
7	5, 6	ifbieq1d 4552	. . . . . . . . . 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 13532	. . . . . . . . . 10 ⊢ (𝑘 ∈ (1...𝑗) → 𝑘 ∈ ℕ)
10	9	adantl 482	. . . . . . . . 9 ⊢ ((𝑗 ∈ ℕ ∧ 𝑘 ∈ (1...𝑗)) → 𝑘 ∈ ℕ)
11		1nn 12225	. . . . . . . . . . . 12 ⊢ 1 ∈ ℕ
12	11	a1i 11	. . . . . . . . . . 11 ⊢ (𝑘 ∈ (1...𝑗) → 1 ∈ ℕ)
13	9, 12	ifcld 4574	. . . . . . . . . 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 7005	. . . . . . . 8 ⊢ ((𝑗 ∈ ℕ ∧ 𝑘 ∈ (1...𝑗)) → ((𝑚 ∈ ℕ ↦ if(𝑚 ∈ ℙ, 𝑚, 1))‘𝑘) = if(𝑘 ∈ ℙ, 𝑘, 1))
16	15, 14	eqeltrd 2833	. . . . . . 7 ⊢ ((𝑗 ∈ ℕ ∧ 𝑘 ∈ (1...𝑗)) → ((𝑚 ∈ ℕ ↦ if(𝑚 ∈ ℙ, 𝑚, 1))‘𝑘) ∈ ℕ)
17	3, 16	fprodnncl 15901	. . . . . 6 ⊢ (𝑗 ∈ ℕ → ∏𝑘 ∈ (1...𝑗)((𝑚 ∈ ℕ ↦ if(𝑚 ∈ ℙ, 𝑚, 1))‘𝑘) ∈ ℕ)
18	2, 17	fmpti 7113	. . . . 5 ⊢ (𝑗 ∈ ℕ ↦ ∏𝑘 ∈ (1...𝑗)((𝑚 ∈ ℕ ↦ if(𝑚 ∈ ℙ, 𝑚, 1))‘𝑘)):ℕ⟶ℕ
19		nnex 12220	. . . . . 6 ⊢ ℕ ∈ V
20	19, 19	elmap 8867	. . . . 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 16996	. . . . 5 ⊢ ((𝑛 ∈ ℕ ∧ 𝑖 ∈ (2...𝑛)) → 1 < (((#p‘𝑛) + 𝑖) gcd 𝑖))
24		eqidd 2733	. . . . . . . . . . . 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 13503	. . . . . . . . . . . . . 14 ⊢ (𝑘 ∈ (1...𝑗) → 𝑘 ∈ ℤ)
28		1zzd 12595	. . . . . . . . . . . . . 14 ⊢ (𝑘 ∈ (1...𝑗) → 1 ∈ ℤ)
29	27, 28	ifcld 4574	. . . . . . . . . . . . 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 7005	. . . . . . . . . . 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 15869	. . . . . . . . . 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 5248	. . . . . . . . 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 7419	. . . . . . . . . . 11 ⊢ (𝑗 = 𝑛 → (1...𝑗) = (1...𝑛))
35	34	prodeq1d 15867	. . . . . . . . . 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 13940	. . . . . . . . . 10 ⊢ ((𝑛 ∈ ℕ ∧ 𝑖 ∈ (2...𝑛)) → (1...𝑛) ∈ Fin)
39		elfznn 13532	. . . . . . . . . . . 12 ⊢ (𝑘 ∈ (1...𝑛) → 𝑘 ∈ ℕ)
40	11	a1i 11	. . . . . . . . . . . 12 ⊢ (𝑘 ∈ (1...𝑛) → 1 ∈ ℕ)
41	39, 40	ifcld 4574	. . . . . . . . . . 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 15901	. . . . . . . . 9 ⊢ ((𝑛 ∈ ℕ ∧ 𝑖 ∈ (2...𝑛)) → ∏𝑘 ∈ (1...𝑛)if(𝑘 ∈ ℙ, 𝑘, 1) ∈ ℕ)
44	33, 36, 37, 43	fvmptd 7005	. . . . . . . 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 12481	. . . . . . . . . . 11 ⊢ (𝑛 ∈ ℕ → 𝑛 ∈ ℕ0)
46		prmoval 16968	. . . . . . . . . . 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 2738	. . . . . . . . 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 2772	. . . . . . 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 7426	. . . . . 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 7426	. . . . 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 5176	. . . 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 3146	. . 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 16993	. 2 ⊢ (𝑛 ∈ ℕ → ∃𝑝 ∈ ℙ ∃𝑞 ∈ ℙ (𝑛 ≤ (𝑞 − 𝑝) ∧ ∀𝑧 ∈ ((𝑝 + 1)..^𝑞)𝑧 ∉ ℙ))
56	55	rgen 3063	1 ⊢ ∀𝑛 ∈ ℕ ∃𝑝 ∈ ℙ ∃𝑞 ∈ ℙ (𝑛 ≤ (𝑞 − 𝑝) ∧ ∀𝑧 ∈ ((𝑝 + 1)..^𝑞)𝑧 ∉ ℙ)

Syntax hints:   ∧ wa 396   = wceq 1541   ∈ wcel 2106   ∉ wnel 3046  ∀wral 3061  ∃wrex 3070  ifcif 4528   class class class wbr 5148   ↦ cmpt 5231  ⟶wf 6539  ‘cfv 6543  (class class class)co 7411   ↑_m cmap 8822  1c1 11113   + caddc 11115   < clt 11250   ≤ cle 11251   − cmin 11446  ℕcn 12214  2c2 12269  ℕ₀cn0 12474  ℤcz 12560  ...cfz 13486  ..^cfzo 13629  ∏cprod 15851   gcd cgcd 16437  ℙcprime 16610  #_pcprmo 16966

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  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 2703  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7727  ax-inf2 9638  ax-cnex 11168  ax-resscn 11169  ax-1cn 11170  ax-icn 11171  ax-addcl 11172  ax-addrcl 11173  ax-mulcl 11174  ax-mulrcl 11175  ax-mulcom 11176  ax-addass 11177  ax-mulass 11178  ax-distr 11179  ax-i2m1 11180  ax-1ne0 11181  ax-1rid 11182  ax-rnegex 11183  ax-rrecex 11184  ax-cnre 11185  ax-pre-lttri 11186  ax-pre-lttrn 11187  ax-pre-ltadd 11188  ax-pre-mulgt0 11189  ax-pre-sup 11190

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3376  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-pss 3967  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-int 4951  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5574  df-eprel 5580  df-po 5588  df-so 5589  df-fr 5631  df-se 5632  df-we 5633  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-pred 6300  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-isom 6552  df-riota 7367  df-ov 7414  df-oprab 7415  df-mpo 7416  df-om 7858  df-1st 7977  df-2nd 7978  df-frecs 8268  df-wrecs 8299  df-recs 8373  df-rdg 8412  df-1o 8468  df-2o 8469  df-er 8705  df-map 8824  df-en 8942  df-dom 8943  df-sdom 8944  df-fin 8945  df-sup 9439  df-inf 9440  df-oi 9507  df-card 9936  df-pnf 11252  df-mnf 11253  df-xr 11254  df-ltxr 11255  df-le 11256  df-sub 11448  df-neg 11449  df-div 11874  df-nn 12215  df-2 12277  df-3 12278  df-n0 12475  df-z 12561  df-uz 12825  df-rp 12977  df-fz 13487  df-fzo 13630  df-seq 13969  df-exp 14030  df-fac 14236  df-hash 14293  df-cj 15048  df-re 15049  df-im 15050  df-sqrt 15184  df-abs 15185  df-clim 15434  df-prod 15852  df-dvds 16200  df-gcd 16438  df-prm 16611  df-prmo 16967

This theorem is referenced by: (None)