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Mirrors > Home > MPE Home > Th. List > infpnlem2 | Structured version Visualization version GIF version |
Description: Lemma for infpn 15994. For any positive integer 𝑁, there exists a prime number 𝑗 greater than 𝑁. (Contributed by NM, 5-May-2005.) |
Ref | Expression |
---|---|
infpnlem.1 | ⊢ 𝐾 = ((!‘𝑁) + 1) |
Ref | Expression |
---|---|
infpnlem2 | ⊢ (𝑁 ∈ ℕ → ∃𝑗 ∈ ℕ (𝑁 < 𝑗 ∧ ∀𝑘 ∈ ℕ ((𝑗 / 𝑘) ∈ ℕ → (𝑘 = 1 ∨ 𝑘 = 𝑗)))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | infpnlem.1 | . . . . 5 ⊢ 𝐾 = ((!‘𝑁) + 1) | |
2 | nnnn0 11633 | . . . . . . 7 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℕ0) | |
3 | faccl 13370 | . . . . . . 7 ⊢ (𝑁 ∈ ℕ0 → (!‘𝑁) ∈ ℕ) | |
4 | 2, 3 | syl 17 | . . . . . 6 ⊢ (𝑁 ∈ ℕ → (!‘𝑁) ∈ ℕ) |
5 | 4 | peano2nnd 11376 | . . . . 5 ⊢ (𝑁 ∈ ℕ → ((!‘𝑁) + 1) ∈ ℕ) |
6 | 1, 5 | syl5eqel 2910 | . . . 4 ⊢ (𝑁 ∈ ℕ → 𝐾 ∈ ℕ) |
7 | 4 | nnge1d 11406 | . . . . . 6 ⊢ (𝑁 ∈ ℕ → 1 ≤ (!‘𝑁)) |
8 | 1nn 11370 | . . . . . . 7 ⊢ 1 ∈ ℕ | |
9 | nnleltp1 11767 | . . . . . . 7 ⊢ ((1 ∈ ℕ ∧ (!‘𝑁) ∈ ℕ) → (1 ≤ (!‘𝑁) ↔ 1 < ((!‘𝑁) + 1))) | |
10 | 8, 4, 9 | sylancr 581 | . . . . . 6 ⊢ (𝑁 ∈ ℕ → (1 ≤ (!‘𝑁) ↔ 1 < ((!‘𝑁) + 1))) |
11 | 7, 10 | mpbid 224 | . . . . 5 ⊢ (𝑁 ∈ ℕ → 1 < ((!‘𝑁) + 1)) |
12 | 11, 1 | syl6breqr 4917 | . . . 4 ⊢ (𝑁 ∈ ℕ → 1 < 𝐾) |
13 | nncn 11366 | . . . . . . 7 ⊢ (𝐾 ∈ ℕ → 𝐾 ∈ ℂ) | |
14 | nnne0 11393 | . . . . . . 7 ⊢ (𝐾 ∈ ℕ → 𝐾 ≠ 0) | |
15 | 13, 14 | jca 507 | . . . . . 6 ⊢ (𝐾 ∈ ℕ → (𝐾 ∈ ℂ ∧ 𝐾 ≠ 0)) |
16 | divid 11046 | . . . . . 6 ⊢ ((𝐾 ∈ ℂ ∧ 𝐾 ≠ 0) → (𝐾 / 𝐾) = 1) | |
17 | 6, 15, 16 | 3syl 18 | . . . . 5 ⊢ (𝑁 ∈ ℕ → (𝐾 / 𝐾) = 1) |
18 | 17, 8 | syl6eqel 2914 | . . . 4 ⊢ (𝑁 ∈ ℕ → (𝐾 / 𝐾) ∈ ℕ) |
19 | breq2 4879 | . . . . . 6 ⊢ (𝑗 = 𝐾 → (1 < 𝑗 ↔ 1 < 𝐾)) | |
20 | oveq2 6918 | . . . . . . 7 ⊢ (𝑗 = 𝐾 → (𝐾 / 𝑗) = (𝐾 / 𝐾)) | |
21 | 20 | eleq1d 2891 | . . . . . 6 ⊢ (𝑗 = 𝐾 → ((𝐾 / 𝑗) ∈ ℕ ↔ (𝐾 / 𝐾) ∈ ℕ)) |
22 | 19, 21 | anbi12d 624 | . . . . 5 ⊢ (𝑗 = 𝐾 → ((1 < 𝑗 ∧ (𝐾 / 𝑗) ∈ ℕ) ↔ (1 < 𝐾 ∧ (𝐾 / 𝐾) ∈ ℕ))) |
23 | 22 | rspcev 3526 | . . . 4 ⊢ ((𝐾 ∈ ℕ ∧ (1 < 𝐾 ∧ (𝐾 / 𝐾) ∈ ℕ)) → ∃𝑗 ∈ ℕ (1 < 𝑗 ∧ (𝐾 / 𝑗) ∈ ℕ)) |
24 | 6, 12, 18, 23 | syl12anc 870 | . . 3 ⊢ (𝑁 ∈ ℕ → ∃𝑗 ∈ ℕ (1 < 𝑗 ∧ (𝐾 / 𝑗) ∈ ℕ)) |
25 | breq2 4879 | . . . . 5 ⊢ (𝑗 = 𝑘 → (1 < 𝑗 ↔ 1 < 𝑘)) | |
26 | oveq2 6918 | . . . . . 6 ⊢ (𝑗 = 𝑘 → (𝐾 / 𝑗) = (𝐾 / 𝑘)) | |
27 | 26 | eleq1d 2891 | . . . . 5 ⊢ (𝑗 = 𝑘 → ((𝐾 / 𝑗) ∈ ℕ ↔ (𝐾 / 𝑘) ∈ ℕ)) |
28 | 25, 27 | anbi12d 624 | . . . 4 ⊢ (𝑗 = 𝑘 → ((1 < 𝑗 ∧ (𝐾 / 𝑗) ∈ ℕ) ↔ (1 < 𝑘 ∧ (𝐾 / 𝑘) ∈ ℕ))) |
29 | 28 | nnwos 12045 | . . 3 ⊢ (∃𝑗 ∈ ℕ (1 < 𝑗 ∧ (𝐾 / 𝑗) ∈ ℕ) → ∃𝑗 ∈ ℕ ((1 < 𝑗 ∧ (𝐾 / 𝑗) ∈ ℕ) ∧ ∀𝑘 ∈ ℕ ((1 < 𝑘 ∧ (𝐾 / 𝑘) ∈ ℕ) → 𝑗 ≤ 𝑘))) |
30 | 24, 29 | syl 17 | . 2 ⊢ (𝑁 ∈ ℕ → ∃𝑗 ∈ ℕ ((1 < 𝑗 ∧ (𝐾 / 𝑗) ∈ ℕ) ∧ ∀𝑘 ∈ ℕ ((1 < 𝑘 ∧ (𝐾 / 𝑘) ∈ ℕ) → 𝑗 ≤ 𝑘))) |
31 | 1 | infpnlem1 15992 | . . 3 ⊢ ((𝑁 ∈ ℕ ∧ 𝑗 ∈ ℕ) → (((1 < 𝑗 ∧ (𝐾 / 𝑗) ∈ ℕ) ∧ ∀𝑘 ∈ ℕ ((1 < 𝑘 ∧ (𝐾 / 𝑘) ∈ ℕ) → 𝑗 ≤ 𝑘)) → (𝑁 < 𝑗 ∧ ∀𝑘 ∈ ℕ ((𝑗 / 𝑘) ∈ ℕ → (𝑘 = 1 ∨ 𝑘 = 𝑗))))) |
32 | 31 | reximdva 3225 | . 2 ⊢ (𝑁 ∈ ℕ → (∃𝑗 ∈ ℕ ((1 < 𝑗 ∧ (𝐾 / 𝑗) ∈ ℕ) ∧ ∀𝑘 ∈ ℕ ((1 < 𝑘 ∧ (𝐾 / 𝑘) ∈ ℕ) → 𝑗 ≤ 𝑘)) → ∃𝑗 ∈ ℕ (𝑁 < 𝑗 ∧ ∀𝑘 ∈ ℕ ((𝑗 / 𝑘) ∈ ℕ → (𝑘 = 1 ∨ 𝑘 = 𝑗))))) |
33 | 30, 32 | mpd 15 | 1 ⊢ (𝑁 ∈ ℕ → ∃𝑗 ∈ ℕ (𝑁 < 𝑗 ∧ ∀𝑘 ∈ ℕ ((𝑗 / 𝑘) ∈ ℕ → (𝑘 = 1 ∨ 𝑘 = 𝑗)))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 198 ∧ wa 386 ∨ wo 878 = wceq 1656 ∈ wcel 2164 ≠ wne 2999 ∀wral 3117 ∃wrex 3118 class class class wbr 4875 ‘cfv 6127 (class class class)co 6910 ℂcc 10257 0cc0 10259 1c1 10260 + caddc 10262 < clt 10398 ≤ cle 10399 / cdiv 11016 ℕcn 11357 ℕ0cn0 11625 !cfa 13360 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1894 ax-4 1908 ax-5 2009 ax-6 2075 ax-7 2112 ax-8 2166 ax-9 2173 ax-10 2192 ax-11 2207 ax-12 2220 ax-13 2389 ax-ext 2803 ax-sep 5007 ax-nul 5015 ax-pow 5067 ax-pr 5129 ax-un 7214 ax-cnex 10315 ax-resscn 10316 ax-1cn 10317 ax-icn 10318 ax-addcl 10319 ax-addrcl 10320 ax-mulcl 10321 ax-mulrcl 10322 ax-mulcom 10323 ax-addass 10324 ax-mulass 10325 ax-distr 10326 ax-i2m1 10327 ax-1ne0 10328 ax-1rid 10329 ax-rnegex 10330 ax-rrecex 10331 ax-cnre 10332 ax-pre-lttri 10333 ax-pre-lttrn 10334 ax-pre-ltadd 10335 ax-pre-mulgt0 10336 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 879 df-3or 1112 df-3an 1113 df-tru 1660 df-ex 1879 df-nf 1883 df-sb 2068 df-mo 2605 df-eu 2640 df-clab 2812 df-cleq 2818 df-clel 2821 df-nfc 2958 df-ne 3000 df-nel 3103 df-ral 3122 df-rex 3123 df-reu 3124 df-rmo 3125 df-rab 3126 df-v 3416 df-sbc 3663 df-csb 3758 df-dif 3801 df-un 3803 df-in 3805 df-ss 3812 df-pss 3814 df-nul 4147 df-if 4309 df-pw 4382 df-sn 4400 df-pr 4402 df-tp 4404 df-op 4406 df-uni 4661 df-iun 4744 df-br 4876 df-opab 4938 df-mpt 4955 df-tr 4978 df-id 5252 df-eprel 5257 df-po 5265 df-so 5266 df-fr 5305 df-we 5307 df-xp 5352 df-rel 5353 df-cnv 5354 df-co 5355 df-dm 5356 df-rn 5357 df-res 5358 df-ima 5359 df-pred 5924 df-ord 5970 df-on 5971 df-lim 5972 df-suc 5973 df-iota 6090 df-fun 6129 df-fn 6130 df-f 6131 df-f1 6132 df-fo 6133 df-f1o 6134 df-fv 6135 df-riota 6871 df-ov 6913 df-oprab 6914 df-mpt2 6915 df-om 7332 df-2nd 7434 df-wrecs 7677 df-recs 7739 df-rdg 7777 df-er 8014 df-en 8229 df-dom 8230 df-sdom 8231 df-pnf 10400 df-mnf 10401 df-xr 10402 df-ltxr 10403 df-le 10404 df-sub 10594 df-neg 10595 df-div 11017 df-nn 11358 df-n0 11626 df-z 11712 df-uz 11976 df-seq 13103 df-fac 13361 |
This theorem is referenced by: infpn 15994 |
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