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Theorem prmgapprmo 16374
Description: Alternate proof of prmgap 16371: in contrast to prmgap 16371, 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 2820 . . . . . 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 13323 . . . . . . 7 (𝑗 ∈ ℕ → (1...𝑗) ∈ Fin)
4 eqidd 2821 . . . . . . . . 9 ((𝑗 ∈ ℕ ∧ 𝑘 ∈ (1...𝑗)) → (𝑚 ∈ ℕ ↦ if(𝑚 ∈ ℙ, 𝑚, 1)) = (𝑚 ∈ ℕ ↦ if(𝑚 ∈ ℙ, 𝑚, 1)))
5 eleq1 2898 . . . . . . . . . . 11 (𝑚 = 𝑘 → (𝑚 ∈ ℙ ↔ 𝑘 ∈ ℙ))
6 id 22 . . . . . . . . . . 11 (𝑚 = 𝑘𝑚 = 𝑘)
75, 6ifbieq1d 4464 . . . . . . . . . 10 (𝑚 = 𝑘 → if(𝑚 ∈ ℙ, 𝑚, 1) = if(𝑘 ∈ ℙ, 𝑘, 1))
87adantl 484 . . . . . . . . 9 (((𝑗 ∈ ℕ ∧ 𝑘 ∈ (1...𝑗)) ∧ 𝑚 = 𝑘) → if(𝑚 ∈ ℙ, 𝑚, 1) = if(𝑘 ∈ ℙ, 𝑘, 1))
9 elfznn 12918 . . . . . . . . . 10 (𝑘 ∈ (1...𝑗) → 𝑘 ∈ ℕ)
109adantl 484 . . . . . . . . 9 ((𝑗 ∈ ℕ ∧ 𝑘 ∈ (1...𝑗)) → 𝑘 ∈ ℕ)
11 1nn 11625 . . . . . . . . . . . 12 1 ∈ ℕ
1211a1i 11 . . . . . . . . . . 11 (𝑘 ∈ (1...𝑗) → 1 ∈ ℕ)
139, 12ifcld 4486 . . . . . . . . . 10 (𝑘 ∈ (1...𝑗) → if(𝑘 ∈ ℙ, 𝑘, 1) ∈ ℕ)
1413adantl 484 . . . . . . . . 9 ((𝑗 ∈ ℕ ∧ 𝑘 ∈ (1...𝑗)) → if(𝑘 ∈ ℙ, 𝑘, 1) ∈ ℕ)
154, 8, 10, 14fvmptd 6749 . . . . . . . 8 ((𝑗 ∈ ℕ ∧ 𝑘 ∈ (1...𝑗)) → ((𝑚 ∈ ℕ ↦ if(𝑚 ∈ ℙ, 𝑚, 1))‘𝑘) = if(𝑘 ∈ ℙ, 𝑘, 1))
1615, 14eqeltrd 2911 . . . . . . 7 ((𝑗 ∈ ℕ ∧ 𝑘 ∈ (1...𝑗)) → ((𝑚 ∈ ℕ ↦ if(𝑚 ∈ ℙ, 𝑚, 1))‘𝑘) ∈ ℕ)
173, 16fprodnncl 15287 . . . . . 6 (𝑗 ∈ ℕ → ∏𝑘 ∈ (1...𝑗)((𝑚 ∈ ℕ ↦ if(𝑚 ∈ ℙ, 𝑚, 1))‘𝑘) ∈ ℕ)
182, 17fmpti 6850 . . . . 5 (𝑗 ∈ ℕ ↦ ∏𝑘 ∈ (1...𝑗)((𝑚 ∈ ℕ ↦ if(𝑚 ∈ ℙ, 𝑚, 1))‘𝑘)):ℕ⟶ℕ
19 nnex 11620 . . . . . 6 ℕ ∈ V
2019, 19elmap 8411 . . . . 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 233 . . . 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 16373 . . . . 5 ((𝑛 ∈ ℕ ∧ 𝑖 ∈ (2...𝑛)) → 1 < (((#p𝑛) + 𝑖) gcd 𝑖))
24 eqidd 2821 . . . . . . . . . . . 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 484 . . . . . . . . . . . 12 (((((𝑛 ∈ ℕ ∧ 𝑖 ∈ (2...𝑛)) ∧ 𝑗 ∈ ℕ) ∧ 𝑘 ∈ (1...𝑗)) ∧ 𝑚 = 𝑘) → if(𝑚 ∈ ℙ, 𝑚, 1) = if(𝑘 ∈ ℙ, 𝑘, 1))
269adantl 484 . . . . . . . . . . . 12 ((((𝑛 ∈ ℕ ∧ 𝑖 ∈ (2...𝑛)) ∧ 𝑗 ∈ ℕ) ∧ 𝑘 ∈ (1...𝑗)) → 𝑘 ∈ ℕ)
27 elfzelz 12890 . . . . . . . . . . . . . 14 (𝑘 ∈ (1...𝑗) → 𝑘 ∈ ℤ)
28 1zzd 11990 . . . . . . . . . . . . . 14 (𝑘 ∈ (1...𝑗) → 1 ∈ ℤ)
2927, 28ifcld 4486 . . . . . . . . . . . . 13 (𝑘 ∈ (1...𝑗) → if(𝑘 ∈ ℙ, 𝑘, 1) ∈ ℤ)
3029adantl 484 . . . . . . . . . . . 12 ((((𝑛 ∈ ℕ ∧ 𝑖 ∈ (2...𝑛)) ∧ 𝑗 ∈ ℕ) ∧ 𝑘 ∈ (1...𝑗)) → if(𝑘 ∈ ℙ, 𝑘, 1) ∈ ℤ)
3124, 25, 26, 30fvmptd 6749 . . . . . . . . . . 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 15255 . . . . . . . . . 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 5135 . . . . . . . . 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 7139 . . . . . . . . . . 11 (𝑗 = 𝑛 → (1...𝑗) = (1...𝑛))
3534prodeq1d 15253 . . . . . . . . . 10 (𝑗 = 𝑛 → ∏𝑘 ∈ (1...𝑗)if(𝑘 ∈ ℙ, 𝑘, 1) = ∏𝑘 ∈ (1...𝑛)if(𝑘 ∈ ℙ, 𝑘, 1))
3635adantl 484 . . . . . . . . 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 485 . . . . . . . . 9 ((𝑛 ∈ ℕ ∧ 𝑖 ∈ (2...𝑛)) → 𝑛 ∈ ℕ)
38 fzfid 13323 . . . . . . . . . 10 ((𝑛 ∈ ℕ ∧ 𝑖 ∈ (2...𝑛)) → (1...𝑛) ∈ Fin)
39 elfznn 12918 . . . . . . . . . . . 12 (𝑘 ∈ (1...𝑛) → 𝑘 ∈ ℕ)
4011a1i 11 . . . . . . . . . . . 12 (𝑘 ∈ (1...𝑛) → 1 ∈ ℕ)
4139, 40ifcld 4486 . . . . . . . . . . 11 (𝑘 ∈ (1...𝑛) → if(𝑘 ∈ ℙ, 𝑘, 1) ∈ ℕ)
4241adantl 484 . . . . . . . . . 10 (((𝑛 ∈ ℕ ∧ 𝑖 ∈ (2...𝑛)) ∧ 𝑘 ∈ (1...𝑛)) → if(𝑘 ∈ ℙ, 𝑘, 1) ∈ ℕ)
4338, 42fprodnncl 15287 . . . . . . . . 9 ((𝑛 ∈ ℕ ∧ 𝑖 ∈ (2...𝑛)) → ∏𝑘 ∈ (1...𝑛)if(𝑘 ∈ ℙ, 𝑘, 1) ∈ ℕ)
4433, 36, 37, 43fvmptd 6749 . . . . . . . 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 11881 . . . . . . . . . . 11 (𝑛 ∈ ℕ → 𝑛 ∈ ℕ0)
46 prmoval 16345 . . . . . . . . . . 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 2826 . . . . . . . . 9 (𝑛 ∈ ℕ → ∏𝑘 ∈ (1...𝑛)if(𝑘 ∈ ℙ, 𝑘, 1) = (#p𝑛))
4948adantr 483 . . . . . . . 8 ((𝑛 ∈ ℕ ∧ 𝑖 ∈ (2...𝑛)) → ∏𝑘 ∈ (1...𝑛)if(𝑘 ∈ ℙ, 𝑘, 1) = (#p𝑛))
5044, 49eqtrd 2855 . . . . . . 7 ((𝑛 ∈ ℕ ∧ 𝑖 ∈ (2...𝑛)) → ((𝑗 ∈ ℕ ↦ ∏𝑘 ∈ (1...𝑗)((𝑚 ∈ ℕ ↦ if(𝑚 ∈ ℙ, 𝑚, 1))‘𝑘))‘𝑛) = (#p𝑛))
5150oveq1d 7146 . . . . . 6 ((𝑛 ∈ ℕ ∧ 𝑖 ∈ (2...𝑛)) → (((𝑗 ∈ ℕ ↦ ∏𝑘 ∈ (1...𝑗)((𝑚 ∈ ℕ ↦ if(𝑚 ∈ ℙ, 𝑚, 1))‘𝑘))‘𝑛) + 𝑖) = ((#p𝑛) + 𝑖))
5251oveq1d 7146 . . . . 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 5068 . . . 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 3169 . . 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 16370 . 2 (𝑛 ∈ ℕ → ∃𝑝 ∈ ℙ ∃𝑞 ∈ ℙ (𝑛 ≤ (𝑞𝑝) ∧ ∀𝑧 ∈ ((𝑝 + 1)..^𝑞)𝑧 ∉ ℙ))
5655rgen 3135 1 𝑛 ∈ ℕ ∃𝑝 ∈ ℙ ∃𝑞 ∈ ℙ (𝑛 ≤ (𝑞𝑝) ∧ ∀𝑧 ∈ ((𝑝 + 1)..^𝑞)𝑧 ∉ ℙ)
Colors of variables: wff setvar class
Syntax hints:  wa 398   = wceq 1537  wcel 2114  wnel 3110  wral 3125  wrex 3126  ifcif 4441   class class class wbr 5040  cmpt 5120  wf 6325  cfv 6329  (class class class)co 7131  m cmap 8382  1c1 10514   + caddc 10516   < clt 10651  cle 10652  cmin 10846  cn 11614  2c2 11669  0cn0 11874  cz 11958  ...cfz 12874  ..^cfzo 13015  cprod 15237   gcd cgcd 15819  cprime 15991  #pcprmo 16343
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2792  ax-rep 5164  ax-sep 5177  ax-nul 5184  ax-pow 5240  ax-pr 5304  ax-un 7437  ax-inf2 9080  ax-cnex 10569  ax-resscn 10570  ax-1cn 10571  ax-icn 10572  ax-addcl 10573  ax-addrcl 10574  ax-mulcl 10575  ax-mulrcl 10576  ax-mulcom 10577  ax-addass 10578  ax-mulass 10579  ax-distr 10580  ax-i2m1 10581  ax-1ne0 10582  ax-1rid 10583  ax-rnegex 10584  ax-rrecex 10585  ax-cnre 10586  ax-pre-lttri 10587  ax-pre-lttrn 10588  ax-pre-ltadd 10589  ax-pre-mulgt0 10590  ax-pre-sup 10591
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1540  df-fal 1550  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2653  df-clab 2799  df-cleq 2813  df-clel 2891  df-nfc 2959  df-ne 3007  df-nel 3111  df-ral 3130  df-rex 3131  df-reu 3132  df-rmo 3133  df-rab 3134  df-v 3475  df-sbc 3752  df-csb 3860  df-dif 3915  df-un 3917  df-in 3919  df-ss 3928  df-pss 3930  df-nul 4268  df-if 4442  df-pw 4515  df-sn 4542  df-pr 4544  df-tp 4546  df-op 4548  df-uni 4813  df-int 4851  df-iun 4895  df-br 5041  df-opab 5103  df-mpt 5121  df-tr 5147  df-id 5434  df-eprel 5439  df-po 5448  df-so 5449  df-fr 5488  df-se 5489  df-we 5490  df-xp 5535  df-rel 5536  df-cnv 5537  df-co 5538  df-dm 5539  df-rn 5540  df-res 5541  df-ima 5542  df-pred 6122  df-ord 6168  df-on 6169  df-lim 6170  df-suc 6171  df-iota 6288  df-fun 6331  df-fn 6332  df-f 6333  df-f1 6334  df-fo 6335  df-f1o 6336  df-fv 6337  df-isom 6338  df-riota 7089  df-ov 7134  df-oprab 7135  df-mpo 7136  df-om 7557  df-1st 7665  df-2nd 7666  df-wrecs 7923  df-recs 7984  df-rdg 8022  df-1o 8078  df-2o 8079  df-oadd 8082  df-er 8265  df-map 8384  df-en 8486  df-dom 8487  df-sdom 8488  df-fin 8489  df-sup 8882  df-inf 8883  df-oi 8950  df-card 9344  df-pnf 10653  df-mnf 10654  df-xr 10655  df-ltxr 10656  df-le 10657  df-sub 10848  df-neg 10849  df-div 11274  df-nn 11615  df-2 11677  df-3 11678  df-n0 11875  df-z 11959  df-uz 12221  df-rp 12367  df-fz 12875  df-fzo 13016  df-seq 13352  df-exp 13413  df-fac 13617  df-hash 13674  df-cj 14436  df-re 14437  df-im 14438  df-sqrt 14572  df-abs 14573  df-clim 14823  df-prod 15238  df-dvds 15586  df-gcd 15820  df-prm 15992  df-prmo 16344
This theorem is referenced by: (None)
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