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Mirrors > Home > MPE Home > Th. List > prmgaplcm | Structured version Visualization version GIF version |
Description: Alternate proof of prmgap 16931: in contrast to prmgap 16931, 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 13443 | . . . . . . . 8 ⊢ (1...𝑥) ⊆ ℤ | |
3 | 2 | a1i 11 | . . . . . . 7 ⊢ (𝑥 ∈ ℕ → (1...𝑥) ⊆ ℤ) |
4 | fzfi 13877 | . . . . . . . 8 ⊢ (1...𝑥) ∈ Fin | |
5 | 4 | a1i 11 | . . . . . . 7 ⊢ (𝑥 ∈ ℕ → (1...𝑥) ∈ Fin) |
6 | 0nelfz1 13460 | . . . . . . . 8 ⊢ 0 ∉ (1...𝑥) | |
7 | 6 | a1i 11 | . . . . . . 7 ⊢ (𝑥 ∈ ℕ → 0 ∉ (1...𝑥)) |
8 | lcmfn0cl 16502 | . . . . . . 7 ⊢ (((1...𝑥) ⊆ ℤ ∧ (1...𝑥) ∈ Fin ∧ 0 ∉ (1...𝑥)) → (lcm‘(1...𝑥)) ∈ ℕ) | |
9 | 3, 5, 7, 8 | syl3anc 1371 | . . . . . 6 ⊢ (𝑥 ∈ ℕ → (lcm‘(1...𝑥)) ∈ ℕ) |
10 | 9 | adantl 482 | . . . . 5 ⊢ ((𝑛 ∈ ℕ ∧ 𝑥 ∈ ℕ) → (lcm‘(1...𝑥)) ∈ ℕ) |
11 | eqid 2736 | . . . . 5 ⊢ (𝑥 ∈ ℕ ↦ (lcm‘(1...𝑥))) = (𝑥 ∈ ℕ ↦ (lcm‘(1...𝑥))) | |
12 | 10, 11 | fmptd 7062 | . . . 4 ⊢ (𝑛 ∈ ℕ → (𝑥 ∈ ℕ ↦ (lcm‘(1...𝑥))):ℕ⟶ℕ) |
13 | nnex 12159 | . . . . . 6 ⊢ ℕ ∈ V | |
14 | 13, 13 | pm3.2i 471 | . . . . 5 ⊢ (ℕ ∈ V ∧ ℕ ∈ V) |
15 | elmapg 8778 | . . . . 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 256 | . . 3 ⊢ (𝑛 ∈ ℕ → (𝑥 ∈ ℕ ↦ (lcm‘(1...𝑥))) ∈ (ℕ ↑m ℕ)) |
18 | prmgaplcmlem2 16924 | . . . . 5 ⊢ ((𝑛 ∈ ℕ ∧ 𝑖 ∈ (2...𝑛)) → 1 < (((lcm‘(1...𝑛)) + 𝑖) gcd 𝑖)) | |
19 | eqidd 2737 | . . . . . . . 8 ⊢ ((𝑛 ∈ ℕ ∧ 𝑖 ∈ (2...𝑛)) → (𝑥 ∈ ℕ ↦ (lcm‘(1...𝑥))) = (𝑥 ∈ ℕ ↦ (lcm‘(1...𝑥)))) | |
20 | oveq2 7365 | . . . . . . . . . 10 ⊢ (𝑥 = 𝑛 → (1...𝑥) = (1...𝑛)) | |
21 | 20 | fveq2d 6846 | . . . . . . . . 9 ⊢ (𝑥 = 𝑛 → (lcm‘(1...𝑥)) = (lcm‘(1...𝑛))) |
22 | 21 | adantl 482 | . . . . . . . 8 ⊢ (((𝑛 ∈ ℕ ∧ 𝑖 ∈ (2...𝑛)) ∧ 𝑥 = 𝑛) → (lcm‘(1...𝑥)) = (lcm‘(1...𝑛))) |
23 | simpl 483 | . . . . . . . 8 ⊢ ((𝑛 ∈ ℕ ∧ 𝑖 ∈ (2...𝑛)) → 𝑛 ∈ ℕ) | |
24 | fzssz 13443 | . . . . . . . . . 10 ⊢ (1...𝑛) ⊆ ℤ | |
25 | fzfi 13877 | . . . . . . . . . 10 ⊢ (1...𝑛) ∈ Fin | |
26 | 24, 25 | pm3.2i 471 | . . . . . . . . 9 ⊢ ((1...𝑛) ⊆ ℤ ∧ (1...𝑛) ∈ Fin) |
27 | lcmfcl 16504 | . . . . . . . . 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 6955 | . . . . . . 7 ⊢ ((𝑛 ∈ ℕ ∧ 𝑖 ∈ (2...𝑛)) → ((𝑥 ∈ ℕ ↦ (lcm‘(1...𝑥)))‘𝑛) = (lcm‘(1...𝑛))) |
30 | 29 | oveq1d 7372 | . . . . . 6 ⊢ ((𝑛 ∈ ℕ ∧ 𝑖 ∈ (2...𝑛)) → (((𝑥 ∈ ℕ ↦ (lcm‘(1...𝑥)))‘𝑛) + 𝑖) = ((lcm‘(1...𝑛)) + 𝑖)) |
31 | 30 | oveq1d 7372 | . . . . 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 5133 | . . . 4 ⊢ ((𝑛 ∈ ℕ ∧ 𝑖 ∈ (2...𝑛)) → 1 < ((((𝑥 ∈ ℕ ↦ (lcm‘(1...𝑥)))‘𝑛) + 𝑖) gcd 𝑖)) |
33 | 32 | ralrimiva 3143 | . . 3 ⊢ (𝑛 ∈ ℕ → ∀𝑖 ∈ (2...𝑛)1 < ((((𝑥 ∈ ℕ ↦ (lcm‘(1...𝑥)))‘𝑛) + 𝑖) gcd 𝑖)) |
34 | 1, 17, 33 | prmgaplem8 16930 | . 2 ⊢ (𝑛 ∈ ℕ → ∃𝑝 ∈ ℙ ∃𝑞 ∈ ℙ (𝑛 ≤ (𝑞 − 𝑝) ∧ ∀𝑧 ∈ ((𝑝 + 1)..^𝑞)𝑧 ∉ ℙ)) |
35 | 34 | rgen 3066 | 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 205 ∧ wa 396 = wceq 1541 ∈ wcel 2106 ∉ wnel 3049 ∀wral 3064 ∃wrex 3073 Vcvv 3445 ⊆ wss 3910 class class class wbr 5105 ↦ cmpt 5188 ⟶wf 6492 ‘cfv 6496 (class class class)co 7357 ↑m cmap 8765 Fincfn 8883 0cc0 11051 1c1 11052 + caddc 11054 < clt 11189 ≤ cle 11190 − cmin 11385 ℕcn 12153 2c2 12208 ℕ0cn0 12413 ℤcz 12499 ...cfz 13424 ..^cfzo 13567 gcd cgcd 16374 lcmclcmf 16465 ℙcprime 16547 |
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 2707 ax-rep 5242 ax-sep 5256 ax-nul 5263 ax-pow 5320 ax-pr 5384 ax-un 7672 ax-inf2 9577 ax-cnex 11107 ax-resscn 11108 ax-1cn 11109 ax-icn 11110 ax-addcl 11111 ax-addrcl 11112 ax-mulcl 11113 ax-mulrcl 11114 ax-mulcom 11115 ax-addass 11116 ax-mulass 11117 ax-distr 11118 ax-i2m1 11119 ax-1ne0 11120 ax-1rid 11121 ax-rnegex 11122 ax-rrecex 11123 ax-cnre 11124 ax-pre-lttri 11125 ax-pre-lttrn 11126 ax-pre-ltadd 11127 ax-pre-mulgt0 11128 ax-pre-sup 11129 |
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 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3065 df-rex 3074 df-rmo 3353 df-reu 3354 df-rab 3408 df-v 3447 df-sbc 3740 df-csb 3856 df-dif 3913 df-un 3915 df-in 3917 df-ss 3927 df-pss 3929 df-nul 4283 df-if 4487 df-pw 4562 df-sn 4587 df-pr 4589 df-op 4593 df-uni 4866 df-int 4908 df-iun 4956 df-br 5106 df-opab 5168 df-mpt 5189 df-tr 5223 df-id 5531 df-eprel 5537 df-po 5545 df-so 5546 df-fr 5588 df-se 5589 df-we 5590 df-xp 5639 df-rel 5640 df-cnv 5641 df-co 5642 df-dm 5643 df-rn 5644 df-res 5645 df-ima 5646 df-pred 6253 df-ord 6320 df-on 6321 df-lim 6322 df-suc 6323 df-iota 6448 df-fun 6498 df-fn 6499 df-f 6500 df-f1 6501 df-fo 6502 df-f1o 6503 df-fv 6504 df-isom 6505 df-riota 7313 df-ov 7360 df-oprab 7361 df-mpo 7362 df-om 7803 df-1st 7921 df-2nd 7922 df-frecs 8212 df-wrecs 8243 df-recs 8317 df-rdg 8356 df-1o 8412 df-2o 8413 df-er 8648 df-map 8767 df-en 8884 df-dom 8885 df-sdom 8886 df-fin 8887 df-sup 9378 df-inf 9379 df-oi 9446 df-card 9875 df-pnf 11191 df-mnf 11192 df-xr 11193 df-ltxr 11194 df-le 11195 df-sub 11387 df-neg 11388 df-div 11813 df-nn 12154 df-2 12216 df-3 12217 df-n0 12414 df-z 12500 df-uz 12764 df-rp 12916 df-fz 13425 df-fzo 13568 df-seq 13907 df-exp 13968 df-fac 14174 df-hash 14231 df-cj 14984 df-re 14985 df-im 14986 df-sqrt 15120 df-abs 15121 df-clim 15370 df-prod 15789 df-dvds 16137 df-gcd 16375 df-lcmf 16467 df-prm 16548 |
This theorem is referenced by: (None) |
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