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Mirrors > Home > MPE Home > Th. List > prmgap | Structured version Visualization version GIF version |
Description: The prime gap theorem: for each positive integer there are (at least) two successive primes with a difference ("gap") at least as big as the given integer. (Contributed by AV, 13-Aug-2020.) |
Ref | Expression |
---|---|
prmgap | ⊢ ∀𝑛 ∈ ℕ ∃𝑝 ∈ ℙ ∃𝑞 ∈ ℙ (𝑛 ≤ (𝑞 − 𝑝) ∧ ∀𝑧 ∈ ((𝑝 + 1)..^𝑞)𝑧 ∉ ℙ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | id 22 | . . 3 ⊢ (𝑛 ∈ ℕ → 𝑛 ∈ ℕ) | |
2 | facmapnn 13646 | . . . 4 ⊢ (𝑥 ∈ ℕ ↦ (!‘𝑥)) ∈ (ℕ ↑m ℕ) | |
3 | 2 | a1i 11 | . . 3 ⊢ (𝑛 ∈ ℕ → (𝑥 ∈ ℕ ↦ (!‘𝑥)) ∈ (ℕ ↑m ℕ)) |
4 | prmgaplem2 16386 | . . . . 5 ⊢ ((𝑛 ∈ ℕ ∧ 𝑖 ∈ (2...𝑛)) → 1 < (((!‘𝑛) + 𝑖) gcd 𝑖)) | |
5 | eqidd 2822 | . . . . . . . 8 ⊢ ((𝑛 ∈ ℕ ∧ 𝑖 ∈ (2...𝑛)) → (𝑥 ∈ ℕ ↦ (!‘𝑥)) = (𝑥 ∈ ℕ ↦ (!‘𝑥))) | |
6 | fveq2 6670 | . . . . . . . . 9 ⊢ (𝑥 = 𝑛 → (!‘𝑥) = (!‘𝑛)) | |
7 | 6 | adantl 484 | . . . . . . . 8 ⊢ (((𝑛 ∈ ℕ ∧ 𝑖 ∈ (2...𝑛)) ∧ 𝑥 = 𝑛) → (!‘𝑥) = (!‘𝑛)) |
8 | simpl 485 | . . . . . . . 8 ⊢ ((𝑛 ∈ ℕ ∧ 𝑖 ∈ (2...𝑛)) → 𝑛 ∈ ℕ) | |
9 | fvexd 6685 | . . . . . . . 8 ⊢ ((𝑛 ∈ ℕ ∧ 𝑖 ∈ (2...𝑛)) → (!‘𝑛) ∈ V) | |
10 | 5, 7, 8, 9 | fvmptd 6775 | . . . . . . 7 ⊢ ((𝑛 ∈ ℕ ∧ 𝑖 ∈ (2...𝑛)) → ((𝑥 ∈ ℕ ↦ (!‘𝑥))‘𝑛) = (!‘𝑛)) |
11 | 10 | oveq1d 7171 | . . . . . 6 ⊢ ((𝑛 ∈ ℕ ∧ 𝑖 ∈ (2...𝑛)) → (((𝑥 ∈ ℕ ↦ (!‘𝑥))‘𝑛) + 𝑖) = ((!‘𝑛) + 𝑖)) |
12 | 11 | oveq1d 7171 | . . . . 5 ⊢ ((𝑛 ∈ ℕ ∧ 𝑖 ∈ (2...𝑛)) → ((((𝑥 ∈ ℕ ↦ (!‘𝑥))‘𝑛) + 𝑖) gcd 𝑖) = (((!‘𝑛) + 𝑖) gcd 𝑖)) |
13 | 4, 12 | breqtrrd 5094 | . . . 4 ⊢ ((𝑛 ∈ ℕ ∧ 𝑖 ∈ (2...𝑛)) → 1 < ((((𝑥 ∈ ℕ ↦ (!‘𝑥))‘𝑛) + 𝑖) gcd 𝑖)) |
14 | 13 | ralrimiva 3182 | . . 3 ⊢ (𝑛 ∈ ℕ → ∀𝑖 ∈ (2...𝑛)1 < ((((𝑥 ∈ ℕ ↦ (!‘𝑥))‘𝑛) + 𝑖) gcd 𝑖)) |
15 | 1, 3, 14 | prmgaplem8 16394 | . 2 ⊢ (𝑛 ∈ ℕ → ∃𝑝 ∈ ℙ ∃𝑞 ∈ ℙ (𝑛 ≤ (𝑞 − 𝑝) ∧ ∀𝑧 ∈ ((𝑝 + 1)..^𝑞)𝑧 ∉ ℙ)) |
16 | 15 | rgen 3148 | 1 ⊢ ∀𝑛 ∈ ℕ ∃𝑝 ∈ ℙ ∃𝑞 ∈ ℙ (𝑛 ≤ (𝑞 − 𝑝) ∧ ∀𝑧 ∈ ((𝑝 + 1)..^𝑞)𝑧 ∉ ℙ) |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 398 = wceq 1537 ∈ wcel 2114 ∉ wnel 3123 ∀wral 3138 ∃wrex 3139 Vcvv 3494 class class class wbr 5066 ↦ cmpt 5146 ‘cfv 6355 (class class class)co 7156 ↑m cmap 8406 1c1 10538 + caddc 10540 < clt 10675 ≤ cle 10676 − cmin 10870 ℕcn 11638 2c2 11693 ...cfz 12893 ..^cfzo 13034 !cfa 13634 gcd cgcd 15843 ℙcprime 16015 |
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 2793 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 ax-cnex 10593 ax-resscn 10594 ax-1cn 10595 ax-icn 10596 ax-addcl 10597 ax-addrcl 10598 ax-mulcl 10599 ax-mulrcl 10600 ax-mulcom 10601 ax-addass 10602 ax-mulass 10603 ax-distr 10604 ax-i2m1 10605 ax-1ne0 10606 ax-1rid 10607 ax-rnegex 10608 ax-rrecex 10609 ax-cnre 10610 ax-pre-lttri 10611 ax-pre-lttrn 10612 ax-pre-ltadd 10613 ax-pre-mulgt0 10614 ax-pre-sup 10615 |
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 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4839 df-iun 4921 df-br 5067 df-opab 5129 df-mpt 5147 df-tr 5173 df-id 5460 df-eprel 5465 df-po 5474 df-so 5475 df-fr 5514 df-we 5516 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-pred 6148 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-riota 7114 df-ov 7159 df-oprab 7160 df-mpo 7161 df-om 7581 df-1st 7689 df-2nd 7690 df-wrecs 7947 df-recs 8008 df-rdg 8046 df-1o 8102 df-2o 8103 df-er 8289 df-map 8408 df-en 8510 df-dom 8511 df-sdom 8512 df-fin 8513 df-sup 8906 df-inf 8907 df-pnf 10677 df-mnf 10678 df-xr 10679 df-ltxr 10680 df-le 10681 df-sub 10872 df-neg 10873 df-div 11298 df-nn 11639 df-2 11701 df-3 11702 df-n0 11899 df-z 11983 df-uz 12245 df-rp 12391 df-fz 12894 df-fzo 13035 df-seq 13371 df-exp 13431 df-fac 13635 df-cj 14458 df-re 14459 df-im 14460 df-sqrt 14594 df-abs 14595 df-dvds 15608 df-gcd 15844 df-prm 16016 |
This theorem is referenced by: (None) |
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