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Mirrors > Home > MPE Home > Th. List > prmgaplcm | Structured version Visualization version GIF version |
Description: Alternate proof of prmgap 17093: in contrast to prmgap 17093, where the gap starts at n! , the factorial of n, the gap starts at the least common multiple of all positive integers less than or equal to n. (Contributed by AV, 13-Aug-2020.) (Revised by AV, 27-Aug-2020.) (Proof modification is discouraged.) (New usage is discouraged.) |
Ref | Expression |
---|---|
prmgaplcm | ⊢ ∀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 | fzssz 13563 | . . . . . . . 8 ⊢ (1...𝑥) ⊆ ℤ | |
3 | 2 | a1i 11 | . . . . . . 7 ⊢ (𝑥 ∈ ℕ → (1...𝑥) ⊆ ℤ) |
4 | fzfi 14010 | . . . . . . . 8 ⊢ (1...𝑥) ∈ Fin | |
5 | 4 | a1i 11 | . . . . . . 7 ⊢ (𝑥 ∈ ℕ → (1...𝑥) ∈ Fin) |
6 | 0nelfz1 13580 | . . . . . . . 8 ⊢ 0 ∉ (1...𝑥) | |
7 | 6 | a1i 11 | . . . . . . 7 ⊢ (𝑥 ∈ ℕ → 0 ∉ (1...𝑥)) |
8 | lcmfn0cl 16660 | . . . . . . 7 ⊢ (((1...𝑥) ⊆ ℤ ∧ (1...𝑥) ∈ Fin ∧ 0 ∉ (1...𝑥)) → (lcm‘(1...𝑥)) ∈ ℕ) | |
9 | 3, 5, 7, 8 | syl3anc 1370 | . . . . . 6 ⊢ (𝑥 ∈ ℕ → (lcm‘(1...𝑥)) ∈ ℕ) |
10 | 9 | adantl 481 | . . . . 5 ⊢ ((𝑛 ∈ ℕ ∧ 𝑥 ∈ ℕ) → (lcm‘(1...𝑥)) ∈ ℕ) |
11 | eqid 2735 | . . . . 5 ⊢ (𝑥 ∈ ℕ ↦ (lcm‘(1...𝑥))) = (𝑥 ∈ ℕ ↦ (lcm‘(1...𝑥))) | |
12 | 10, 11 | fmptd 7134 | . . . 4 ⊢ (𝑛 ∈ ℕ → (𝑥 ∈ ℕ ↦ (lcm‘(1...𝑥))):ℕ⟶ℕ) |
13 | nnex 12270 | . . . . . 6 ⊢ ℕ ∈ V | |
14 | 13, 13 | pm3.2i 470 | . . . . 5 ⊢ (ℕ ∈ V ∧ ℕ ∈ V) |
15 | elmapg 8878 | . . . . 5 ⊢ ((ℕ ∈ V ∧ ℕ ∈ V) → ((𝑥 ∈ ℕ ↦ (lcm‘(1...𝑥))) ∈ (ℕ ↑m ℕ) ↔ (𝑥 ∈ ℕ ↦ (lcm‘(1...𝑥))):ℕ⟶ℕ)) | |
16 | 14, 15 | mp1i 13 | . . . 4 ⊢ (𝑛 ∈ ℕ → ((𝑥 ∈ ℕ ↦ (lcm‘(1...𝑥))) ∈ (ℕ ↑m ℕ) ↔ (𝑥 ∈ ℕ ↦ (lcm‘(1...𝑥))):ℕ⟶ℕ)) |
17 | 12, 16 | mpbird 257 | . . 3 ⊢ (𝑛 ∈ ℕ → (𝑥 ∈ ℕ ↦ (lcm‘(1...𝑥))) ∈ (ℕ ↑m ℕ)) |
18 | prmgaplcmlem2 17086 | . . . . 5 ⊢ ((𝑛 ∈ ℕ ∧ 𝑖 ∈ (2...𝑛)) → 1 < (((lcm‘(1...𝑛)) + 𝑖) gcd 𝑖)) | |
19 | eqidd 2736 | . . . . . . . 8 ⊢ ((𝑛 ∈ ℕ ∧ 𝑖 ∈ (2...𝑛)) → (𝑥 ∈ ℕ ↦ (lcm‘(1...𝑥))) = (𝑥 ∈ ℕ ↦ (lcm‘(1...𝑥)))) | |
20 | oveq2 7439 | . . . . . . . . . 10 ⊢ (𝑥 = 𝑛 → (1...𝑥) = (1...𝑛)) | |
21 | 20 | fveq2d 6911 | . . . . . . . . 9 ⊢ (𝑥 = 𝑛 → (lcm‘(1...𝑥)) = (lcm‘(1...𝑛))) |
22 | 21 | adantl 481 | . . . . . . . 8 ⊢ (((𝑛 ∈ ℕ ∧ 𝑖 ∈ (2...𝑛)) ∧ 𝑥 = 𝑛) → (lcm‘(1...𝑥)) = (lcm‘(1...𝑛))) |
23 | simpl 482 | . . . . . . . 8 ⊢ ((𝑛 ∈ ℕ ∧ 𝑖 ∈ (2...𝑛)) → 𝑛 ∈ ℕ) | |
24 | fzssz 13563 | . . . . . . . . . 10 ⊢ (1...𝑛) ⊆ ℤ | |
25 | fzfi 14010 | . . . . . . . . . 10 ⊢ (1...𝑛) ∈ Fin | |
26 | 24, 25 | pm3.2i 470 | . . . . . . . . 9 ⊢ ((1...𝑛) ⊆ ℤ ∧ (1...𝑛) ∈ Fin) |
27 | lcmfcl 16662 | . . . . . . . . 9 ⊢ (((1...𝑛) ⊆ ℤ ∧ (1...𝑛) ∈ Fin) → (lcm‘(1...𝑛)) ∈ ℕ0) | |
28 | 26, 27 | mp1i 13 | . . . . . . . 8 ⊢ ((𝑛 ∈ ℕ ∧ 𝑖 ∈ (2...𝑛)) → (lcm‘(1...𝑛)) ∈ ℕ0) |
29 | 19, 22, 23, 28 | fvmptd 7023 | . . . . . . 7 ⊢ ((𝑛 ∈ ℕ ∧ 𝑖 ∈ (2...𝑛)) → ((𝑥 ∈ ℕ ↦ (lcm‘(1...𝑥)))‘𝑛) = (lcm‘(1...𝑛))) |
30 | 29 | oveq1d 7446 | . . . . . 6 ⊢ ((𝑛 ∈ ℕ ∧ 𝑖 ∈ (2...𝑛)) → (((𝑥 ∈ ℕ ↦ (lcm‘(1...𝑥)))‘𝑛) + 𝑖) = ((lcm‘(1...𝑛)) + 𝑖)) |
31 | 30 | oveq1d 7446 | . . . . 5 ⊢ ((𝑛 ∈ ℕ ∧ 𝑖 ∈ (2...𝑛)) → ((((𝑥 ∈ ℕ ↦ (lcm‘(1...𝑥)))‘𝑛) + 𝑖) gcd 𝑖) = (((lcm‘(1...𝑛)) + 𝑖) gcd 𝑖)) |
32 | 18, 31 | breqtrrd 5176 | . . . 4 ⊢ ((𝑛 ∈ ℕ ∧ 𝑖 ∈ (2...𝑛)) → 1 < ((((𝑥 ∈ ℕ ↦ (lcm‘(1...𝑥)))‘𝑛) + 𝑖) gcd 𝑖)) |
33 | 32 | ralrimiva 3144 | . . 3 ⊢ (𝑛 ∈ ℕ → ∀𝑖 ∈ (2...𝑛)1 < ((((𝑥 ∈ ℕ ↦ (lcm‘(1...𝑥)))‘𝑛) + 𝑖) gcd 𝑖)) |
34 | 1, 17, 33 | prmgaplem8 17092 | . 2 ⊢ (𝑛 ∈ ℕ → ∃𝑝 ∈ ℙ ∃𝑞 ∈ ℙ (𝑛 ≤ (𝑞 − 𝑝) ∧ ∀𝑧 ∈ ((𝑝 + 1)..^𝑞)𝑧 ∉ ℙ)) |
35 | 34 | rgen 3061 | 1 ⊢ ∀𝑛 ∈ ℕ ∃𝑝 ∈ ℙ ∃𝑞 ∈ ℙ (𝑛 ≤ (𝑞 − 𝑝) ∧ ∀𝑧 ∈ ((𝑝 + 1)..^𝑞)𝑧 ∉ ℙ) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 206 ∧ wa 395 = wceq 1537 ∈ wcel 2106 ∉ wnel 3044 ∀wral 3059 ∃wrex 3068 Vcvv 3478 ⊆ wss 3963 class class class wbr 5148 ↦ cmpt 5231 ⟶wf 6559 ‘cfv 6563 (class class class)co 7431 ↑m cmap 8865 Fincfn 8984 0cc0 11153 1c1 11154 + caddc 11156 < clt 11293 ≤ cle 11294 − cmin 11490 ℕcn 12264 2c2 12319 ℕ0cn0 12524 ℤcz 12611 ...cfz 13544 ..^cfzo 13691 gcd cgcd 16528 lcmclcmf 16623 ℙcprime 16705 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-rep 5285 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-inf2 9679 ax-cnex 11209 ax-resscn 11210 ax-1cn 11211 ax-icn 11212 ax-addcl 11213 ax-addrcl 11214 ax-mulcl 11215 ax-mulrcl 11216 ax-mulcom 11217 ax-addass 11218 ax-mulass 11219 ax-distr 11220 ax-i2m1 11221 ax-1ne0 11222 ax-1rid 11223 ax-rnegex 11224 ax-rrecex 11225 ax-cnre 11226 ax-pre-lttri 11227 ax-pre-lttrn 11228 ax-pre-ltadd 11229 ax-pre-mulgt0 11230 ax-pre-sup 11231 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-rmo 3378 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-int 4952 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-se 5642 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-pred 6323 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-isom 6572 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8013 df-2nd 8014 df-frecs 8305 df-wrecs 8336 df-recs 8410 df-rdg 8449 df-1o 8505 df-2o 8506 df-er 8744 df-map 8867 df-en 8985 df-dom 8986 df-sdom 8987 df-fin 8988 df-sup 9480 df-inf 9481 df-oi 9548 df-card 9977 df-pnf 11295 df-mnf 11296 df-xr 11297 df-ltxr 11298 df-le 11299 df-sub 11492 df-neg 11493 df-div 11919 df-nn 12265 df-2 12327 df-3 12328 df-n0 12525 df-z 12612 df-uz 12877 df-rp 13033 df-fz 13545 df-fzo 13692 df-seq 14040 df-exp 14100 df-fac 14310 df-hash 14367 df-cj 15135 df-re 15136 df-im 15137 df-sqrt 15271 df-abs 15272 df-clim 15521 df-prod 15937 df-dvds 16288 df-gcd 16529 df-lcmf 16625 df-prm 16706 |
This theorem is referenced by: (None) |
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