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Theorem prmgapprmo 15813
Description: Alternate proof of prmgap 15810: in contrast to prmgap 15810, 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 ≤ (𝑞𝑝) ∧ ∀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.
StepHypRef Expression
1 id 22 . . 3 (𝑛 ∈ ℕ → 𝑛 ∈ ℕ)
2 eqid 2651 . . . . . 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 12812 . . . . . . 7 (𝑗 ∈ ℕ → (1...𝑗) ∈ Fin)
4 eqidd 2652 . . . . . . . . 9 ((𝑗 ∈ ℕ ∧ 𝑘 ∈ (1...𝑗)) → (𝑚 ∈ ℕ ↦ if(𝑚 ∈ ℙ, 𝑚, 1)) = (𝑚 ∈ ℕ ↦ if(𝑚 ∈ ℙ, 𝑚, 1)))
5 eleq1 2718 . . . . . . . . . . 11 (𝑚 = 𝑘 → (𝑚 ∈ ℙ ↔ 𝑘 ∈ ℙ))
6 id 22 . . . . . . . . . . 11 (𝑚 = 𝑘𝑚 = 𝑘)
75, 6ifbieq1d 4142 . . . . . . . . . 10 (𝑚 = 𝑘 → if(𝑚 ∈ ℙ, 𝑚, 1) = if(𝑘 ∈ ℙ, 𝑘, 1))
87adantl 481 . . . . . . . . 9 (((𝑗 ∈ ℕ ∧ 𝑘 ∈ (1...𝑗)) ∧ 𝑚 = 𝑘) → if(𝑚 ∈ ℙ, 𝑚, 1) = if(𝑘 ∈ ℙ, 𝑘, 1))
9 elfznn 12408 . . . . . . . . . 10 (𝑘 ∈ (1...𝑗) → 𝑘 ∈ ℕ)
109adantl 481 . . . . . . . . 9 ((𝑗 ∈ ℕ ∧ 𝑘 ∈ (1...𝑗)) → 𝑘 ∈ ℕ)
11 1nn 11069 . . . . . . . . . . . 12 1 ∈ ℕ
1211a1i 11 . . . . . . . . . . 11 (𝑘 ∈ (1...𝑗) → 1 ∈ ℕ)
139, 12ifcld 4164 . . . . . . . . . 10 (𝑘 ∈ (1...𝑗) → if(𝑘 ∈ ℙ, 𝑘, 1) ∈ ℕ)
1413adantl 481 . . . . . . . . 9 ((𝑗 ∈ ℕ ∧ 𝑘 ∈ (1...𝑗)) → if(𝑘 ∈ ℙ, 𝑘, 1) ∈ ℕ)
154, 8, 10, 14fvmptd 6327 . . . . . . . 8 ((𝑗 ∈ ℕ ∧ 𝑘 ∈ (1...𝑗)) → ((𝑚 ∈ ℕ ↦ if(𝑚 ∈ ℙ, 𝑚, 1))‘𝑘) = if(𝑘 ∈ ℙ, 𝑘, 1))
1615, 14eqeltrd 2730 . . . . . . 7 ((𝑗 ∈ ℕ ∧ 𝑘 ∈ (1...𝑗)) → ((𝑚 ∈ ℕ ↦ if(𝑚 ∈ ℙ, 𝑚, 1))‘𝑘) ∈ ℕ)
173, 16fprodnncl 14729 . . . . . 6 (𝑗 ∈ ℕ → ∏𝑘 ∈ (1...𝑗)((𝑚 ∈ ℕ ↦ if(𝑚 ∈ ℙ, 𝑚, 1))‘𝑘) ∈ ℕ)
182, 17fmpti 6423 . . . . 5 (𝑗 ∈ ℕ ↦ ∏𝑘 ∈ (1...𝑗)((𝑚 ∈ ℕ ↦ if(𝑚 ∈ ℙ, 𝑚, 1))‘𝑘)):ℕ⟶ℕ
19 nnex 11064 . . . . . 6 ℕ ∈ V
2019, 19elmap 7928 . . . . 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)
2118, 20mpbir 221 . . . 4 (𝑗 ∈ ℕ ↦ ∏𝑘 ∈ (1...𝑗)((𝑚 ∈ ℕ ↦ if(𝑚 ∈ ℙ, 𝑚, 1))‘𝑘)) ∈ (ℕ ↑𝑚 ℕ)
2221a1i 11 . . 3 (𝑛 ∈ ℕ → (𝑗 ∈ ℕ ↦ ∏𝑘 ∈ (1...𝑗)((𝑚 ∈ ℕ ↦ if(𝑚 ∈ ℙ, 𝑚, 1))‘𝑘)) ∈ (ℕ ↑𝑚 ℕ))
23 prmgapprmolem 15812 . . . . 5 ((𝑛 ∈ ℕ ∧ 𝑖 ∈ (2...𝑛)) → 1 < (((#p𝑛) + 𝑖) gcd 𝑖))
24 eqidd 2652 . . . . . . . . . . . 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)))
257adantl 481 . . . . . . . . . . . 12 (((((𝑛 ∈ ℕ ∧ 𝑖 ∈ (2...𝑛)) ∧ 𝑗 ∈ ℕ) ∧ 𝑘 ∈ (1...𝑗)) ∧ 𝑚 = 𝑘) → if(𝑚 ∈ ℙ, 𝑚, 1) = if(𝑘 ∈ ℙ, 𝑘, 1))
269adantl 481 . . . . . . . . . . . 12 ((((𝑛 ∈ ℕ ∧ 𝑖 ∈ (2...𝑛)) ∧ 𝑗 ∈ ℕ) ∧ 𝑘 ∈ (1...𝑗)) → 𝑘 ∈ ℕ)
27 elfzelz 12380 . . . . . . . . . . . . . 14 (𝑘 ∈ (1...𝑗) → 𝑘 ∈ ℤ)
28 1zzd 11446 . . . . . . . . . . . . . 14 (𝑘 ∈ (1...𝑗) → 1 ∈ ℤ)
2927, 28ifcld 4164 . . . . . . . . . . . . 13 (𝑘 ∈ (1...𝑗) → if(𝑘 ∈ ℙ, 𝑘, 1) ∈ ℤ)
3029adantl 481 . . . . . . . . . . . 12 ((((𝑛 ∈ ℕ ∧ 𝑖 ∈ (2...𝑛)) ∧ 𝑗 ∈ ℕ) ∧ 𝑘 ∈ (1...𝑗)) → if(𝑘 ∈ ℙ, 𝑘, 1) ∈ ℤ)
3124, 25, 26, 30fvmptd 6327 . . . . . . . . . . 11 ((((𝑛 ∈ ℕ ∧ 𝑖 ∈ (2...𝑛)) ∧ 𝑗 ∈ ℕ) ∧ 𝑘 ∈ (1...𝑗)) → ((𝑚 ∈ ℕ ↦ if(𝑚 ∈ ℙ, 𝑚, 1))‘𝑘) = if(𝑘 ∈ ℙ, 𝑘, 1))
3231prodeq2dv 14697 . . . . . . . . . 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))
3332mpteq2dva 4777 . . . . . . . . 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 6698 . . . . . . . . . . 11 (𝑗 = 𝑛 → (1...𝑗) = (1...𝑛))
3534prodeq1d 14695 . . . . . . . . . 10 (𝑗 = 𝑛 → ∏𝑘 ∈ (1...𝑗)if(𝑘 ∈ ℙ, 𝑘, 1) = ∏𝑘 ∈ (1...𝑛)if(𝑘 ∈ ℙ, 𝑘, 1))
3635adantl 481 . . . . . . . . 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 472 . . . . . . . . 9 ((𝑛 ∈ ℕ ∧ 𝑖 ∈ (2...𝑛)) → 𝑛 ∈ ℕ)
38 fzfid 12812 . . . . . . . . . 10 ((𝑛 ∈ ℕ ∧ 𝑖 ∈ (2...𝑛)) → (1...𝑛) ∈ Fin)
39 elfznn 12408 . . . . . . . . . . . 12 (𝑘 ∈ (1...𝑛) → 𝑘 ∈ ℕ)
4011a1i 11 . . . . . . . . . . . 12 (𝑘 ∈ (1...𝑛) → 1 ∈ ℕ)
4139, 40ifcld 4164 . . . . . . . . . . 11 (𝑘 ∈ (1...𝑛) → if(𝑘 ∈ ℙ, 𝑘, 1) ∈ ℕ)
4241adantl 481 . . . . . . . . . 10 (((𝑛 ∈ ℕ ∧ 𝑖 ∈ (2...𝑛)) ∧ 𝑘 ∈ (1...𝑛)) → if(𝑘 ∈ ℙ, 𝑘, 1) ∈ ℕ)
4338, 42fprodnncl 14729 . . . . . . . . 9 ((𝑛 ∈ ℕ ∧ 𝑖 ∈ (2...𝑛)) → ∏𝑘 ∈ (1...𝑛)if(𝑘 ∈ ℙ, 𝑘, 1) ∈ ℕ)
4433, 36, 37, 43fvmptd 6327 . . . . . . . 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 11337 . . . . . . . . . . 11 (𝑛 ∈ ℕ → 𝑛 ∈ ℕ0)
46 prmoval 15784 . . . . . . . . . . 11 (𝑛 ∈ ℕ0 → (#p𝑛) = ∏𝑘 ∈ (1...𝑛)if(𝑘 ∈ ℙ, 𝑘, 1))
4745, 46syl 17 . . . . . . . . . 10 (𝑛 ∈ ℕ → (#p𝑛) = ∏𝑘 ∈ (1...𝑛)if(𝑘 ∈ ℙ, 𝑘, 1))
4847eqcomd 2657 . . . . . . . . 9 (𝑛 ∈ ℕ → ∏𝑘 ∈ (1...𝑛)if(𝑘 ∈ ℙ, 𝑘, 1) = (#p𝑛))
4948adantr 480 . . . . . . . 8 ((𝑛 ∈ ℕ ∧ 𝑖 ∈ (2...𝑛)) → ∏𝑘 ∈ (1...𝑛)if(𝑘 ∈ ℙ, 𝑘, 1) = (#p𝑛))
5044, 49eqtrd 2685 . . . . . . 7 ((𝑛 ∈ ℕ ∧ 𝑖 ∈ (2...𝑛)) → ((𝑗 ∈ ℕ ↦ ∏𝑘 ∈ (1...𝑗)((𝑚 ∈ ℕ ↦ if(𝑚 ∈ ℙ, 𝑚, 1))‘𝑘))‘𝑛) = (#p𝑛))
5150oveq1d 6705 . . . . . 6 ((𝑛 ∈ ℕ ∧ 𝑖 ∈ (2...𝑛)) → (((𝑗 ∈ ℕ ↦ ∏𝑘 ∈ (1...𝑗)((𝑚 ∈ ℕ ↦ if(𝑚 ∈ ℙ, 𝑚, 1))‘𝑘))‘𝑛) + 𝑖) = ((#p𝑛) + 𝑖))
5251oveq1d 6705 . . . . 5 ((𝑛 ∈ ℕ ∧ 𝑖 ∈ (2...𝑛)) → ((((𝑗 ∈ ℕ ↦ ∏𝑘 ∈ (1...𝑗)((𝑚 ∈ ℕ ↦ if(𝑚 ∈ ℙ, 𝑚, 1))‘𝑘))‘𝑛) + 𝑖) gcd 𝑖) = (((#p𝑛) + 𝑖) gcd 𝑖))
5323, 52breqtrrd 4713 . . . 4 ((𝑛 ∈ ℕ ∧ 𝑖 ∈ (2...𝑛)) → 1 < ((((𝑗 ∈ ℕ ↦ ∏𝑘 ∈ (1...𝑗)((𝑚 ∈ ℕ ↦ if(𝑚 ∈ ℙ, 𝑚, 1))‘𝑘))‘𝑛) + 𝑖) gcd 𝑖))
5453ralrimiva 2995 . . 3 (𝑛 ∈ ℕ → ∀𝑖 ∈ (2...𝑛)1 < ((((𝑗 ∈ ℕ ↦ ∏𝑘 ∈ (1...𝑗)((𝑚 ∈ ℕ ↦ if(𝑚 ∈ ℙ, 𝑚, 1))‘𝑘))‘𝑛) + 𝑖) gcd 𝑖))
551, 22, 54prmgaplem8 15809 . 2 (𝑛 ∈ ℕ → ∃𝑝 ∈ ℙ ∃𝑞 ∈ ℙ (𝑛 ≤ (𝑞𝑝) ∧ ∀𝑧 ∈ ((𝑝 + 1)..^𝑞)𝑧 ∉ ℙ))
5655rgen 2951 1 𝑛 ∈ ℕ ∃𝑝 ∈ ℙ ∃𝑞 ∈ ℙ (𝑛 ≤ (𝑞𝑝) ∧ ∀𝑧 ∈ ((𝑝 + 1)..^𝑞)𝑧 ∉ ℙ)
Colors of variables: wff setvar class
Syntax hints:  wa 383   = wceq 1523  wcel 2030  wnel 2926  wral 2941  wrex 2942  ifcif 4119   class class class wbr 4685  cmpt 4762  wf 5922  cfv 5926  (class class class)co 6690  𝑚 cmap 7899  1c1 9975   + caddc 9977   < clt 10112  cle 10113  cmin 10304  cn 11058  2c2 11108  0cn0 11330  cz 11415  ...cfz 12364  ..^cfzo 12504  cprod 14679   gcd cgcd 15263  cprime 15432  #pcprmo 15782
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991  ax-inf2 8576  ax-cnex 10030  ax-resscn 10031  ax-1cn 10032  ax-icn 10033  ax-addcl 10034  ax-addrcl 10035  ax-mulcl 10036  ax-mulrcl 10037  ax-mulcom 10038  ax-addass 10039  ax-mulass 10040  ax-distr 10041  ax-i2m1 10042  ax-1ne0 10043  ax-1rid 10044  ax-rnegex 10045  ax-rrecex 10046  ax-cnre 10047  ax-pre-lttri 10048  ax-pre-lttrn 10049  ax-pre-ltadd 10050  ax-pre-mulgt0 10051  ax-pre-sup 10052
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-fal 1529  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-nel 2927  df-ral 2946  df-rex 2947  df-reu 2948  df-rmo 2949  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-int 4508  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-se 5103  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-pred 5718  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-isom 5935  df-riota 6651  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-om 7108  df-1st 7210  df-2nd 7211  df-wrecs 7452  df-recs 7513  df-rdg 7551  df-1o 7605  df-2o 7606  df-oadd 7609  df-er 7787  df-map 7901  df-en 7998  df-dom 7999  df-sdom 8000  df-fin 8001  df-sup 8389  df-inf 8390  df-oi 8456  df-card 8803  df-pnf 10114  df-mnf 10115  df-xr 10116  df-ltxr 10117  df-le 10118  df-sub 10306  df-neg 10307  df-div 10723  df-nn 11059  df-2 11117  df-3 11118  df-n0 11331  df-z 11416  df-uz 11726  df-rp 11871  df-fz 12365  df-fzo 12505  df-seq 12842  df-exp 12901  df-fac 13101  df-hash 13158  df-cj 13883  df-re 13884  df-im 13885  df-sqrt 14019  df-abs 14020  df-clim 14263  df-prod 14680  df-dvds 15028  df-gcd 15264  df-prm 15433  df-prmo 15783
This theorem is referenced by: (None)
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