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Mirrors > Home > MPE Home > Th. List > infpnlem2 | Structured version Visualization version GIF version |
Description: Lemma for infpn 16094. 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 11708 | . . . . . . 7 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℕ0) | |
3 | 2 | faccld 13452 | . . . . . 6 ⊢ (𝑁 ∈ ℕ → (!‘𝑁) ∈ ℕ) |
4 | 3 | peano2nnd 11450 | . . . . 5 ⊢ (𝑁 ∈ ℕ → ((!‘𝑁) + 1) ∈ ℕ) |
5 | 1, 4 | syl5eqel 2864 | . . . 4 ⊢ (𝑁 ∈ ℕ → 𝐾 ∈ ℕ) |
6 | 3 | nnge1d 11481 | . . . . . 6 ⊢ (𝑁 ∈ ℕ → 1 ≤ (!‘𝑁)) |
7 | 1nn 11444 | . . . . . . 7 ⊢ 1 ∈ ℕ | |
8 | nnleltp1 11843 | . . . . . . 7 ⊢ ((1 ∈ ℕ ∧ (!‘𝑁) ∈ ℕ) → (1 ≤ (!‘𝑁) ↔ 1 < ((!‘𝑁) + 1))) | |
9 | 7, 3, 8 | sylancr 578 | . . . . . 6 ⊢ (𝑁 ∈ ℕ → (1 ≤ (!‘𝑁) ↔ 1 < ((!‘𝑁) + 1))) |
10 | 6, 9 | mpbid 224 | . . . . 5 ⊢ (𝑁 ∈ ℕ → 1 < ((!‘𝑁) + 1)) |
11 | 10, 1 | syl6breqr 4965 | . . . 4 ⊢ (𝑁 ∈ ℕ → 1 < 𝐾) |
12 | nncn 11440 | . . . . . . 7 ⊢ (𝐾 ∈ ℕ → 𝐾 ∈ ℂ) | |
13 | nnne0 11467 | . . . . . . 7 ⊢ (𝐾 ∈ ℕ → 𝐾 ≠ 0) | |
14 | 12, 13 | jca 504 | . . . . . 6 ⊢ (𝐾 ∈ ℕ → (𝐾 ∈ ℂ ∧ 𝐾 ≠ 0)) |
15 | divid 11120 | . . . . . 6 ⊢ ((𝐾 ∈ ℂ ∧ 𝐾 ≠ 0) → (𝐾 / 𝐾) = 1) | |
16 | 5, 14, 15 | 3syl 18 | . . . . 5 ⊢ (𝑁 ∈ ℕ → (𝐾 / 𝐾) = 1) |
17 | 16, 7 | syl6eqel 2868 | . . . 4 ⊢ (𝑁 ∈ ℕ → (𝐾 / 𝐾) ∈ ℕ) |
18 | breq2 4927 | . . . . . 6 ⊢ (𝑗 = 𝐾 → (1 < 𝑗 ↔ 1 < 𝐾)) | |
19 | oveq2 6978 | . . . . . . 7 ⊢ (𝑗 = 𝐾 → (𝐾 / 𝑗) = (𝐾 / 𝐾)) | |
20 | 19 | eleq1d 2844 | . . . . . 6 ⊢ (𝑗 = 𝐾 → ((𝐾 / 𝑗) ∈ ℕ ↔ (𝐾 / 𝐾) ∈ ℕ)) |
21 | 18, 20 | anbi12d 621 | . . . . 5 ⊢ (𝑗 = 𝐾 → ((1 < 𝑗 ∧ (𝐾 / 𝑗) ∈ ℕ) ↔ (1 < 𝐾 ∧ (𝐾 / 𝐾) ∈ ℕ))) |
22 | 21 | rspcev 3529 | . . . 4 ⊢ ((𝐾 ∈ ℕ ∧ (1 < 𝐾 ∧ (𝐾 / 𝐾) ∈ ℕ)) → ∃𝑗 ∈ ℕ (1 < 𝑗 ∧ (𝐾 / 𝑗) ∈ ℕ)) |
23 | 5, 11, 17, 22 | syl12anc 824 | . . 3 ⊢ (𝑁 ∈ ℕ → ∃𝑗 ∈ ℕ (1 < 𝑗 ∧ (𝐾 / 𝑗) ∈ ℕ)) |
24 | breq2 4927 | . . . . 5 ⊢ (𝑗 = 𝑘 → (1 < 𝑗 ↔ 1 < 𝑘)) | |
25 | oveq2 6978 | . . . . . 6 ⊢ (𝑗 = 𝑘 → (𝐾 / 𝑗) = (𝐾 / 𝑘)) | |
26 | 25 | eleq1d 2844 | . . . . 5 ⊢ (𝑗 = 𝑘 → ((𝐾 / 𝑗) ∈ ℕ ↔ (𝐾 / 𝑘) ∈ ℕ)) |
27 | 24, 26 | anbi12d 621 | . . . 4 ⊢ (𝑗 = 𝑘 → ((1 < 𝑗 ∧ (𝐾 / 𝑗) ∈ ℕ) ↔ (1 < 𝑘 ∧ (𝐾 / 𝑘) ∈ ℕ))) |
28 | 27 | nnwos 12122 | . . 3 ⊢ (∃𝑗 ∈ ℕ (1 < 𝑗 ∧ (𝐾 / 𝑗) ∈ ℕ) → ∃𝑗 ∈ ℕ ((1 < 𝑗 ∧ (𝐾 / 𝑗) ∈ ℕ) ∧ ∀𝑘 ∈ ℕ ((1 < 𝑘 ∧ (𝐾 / 𝑘) ∈ ℕ) → 𝑗 ≤ 𝑘))) |
29 | 23, 28 | syl 17 | . 2 ⊢ (𝑁 ∈ ℕ → ∃𝑗 ∈ ℕ ((1 < 𝑗 ∧ (𝐾 / 𝑗) ∈ ℕ) ∧ ∀𝑘 ∈ ℕ ((1 < 𝑘 ∧ (𝐾 / 𝑘) ∈ ℕ) → 𝑗 ≤ 𝑘))) |
30 | 1 | infpnlem1 16092 | . . 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))))) |
31 | 30 | reximdva 3213 | . 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))))) |
32 | 29, 31 | 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 387 ∨ wo 833 = wceq 1507 ∈ wcel 2048 ≠ wne 2961 ∀wral 3082 ∃wrex 3083 class class class wbr 4923 ‘cfv 6182 (class class class)co 6970 ℂcc 10325 0cc0 10327 1c1 10328 + caddc 10330 < clt 10466 ≤ cle 10467 / cdiv 11090 ℕcn 11431 !cfa 13441 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1758 ax-4 1772 ax-5 1869 ax-6 1928 ax-7 1964 ax-8 2050 ax-9 2057 ax-10 2077 ax-11 2091 ax-12 2104 ax-13 2299 ax-ext 2745 ax-sep 5054 ax-nul 5061 ax-pow 5113 ax-pr 5180 ax-un 7273 ax-cnex 10383 ax-resscn 10384 ax-1cn 10385 ax-icn 10386 ax-addcl 10387 ax-addrcl 10388 ax-mulcl 10389 ax-mulrcl 10390 ax-mulcom 10391 ax-addass 10392 ax-mulass 10393 ax-distr 10394 ax-i2m1 10395 ax-1ne0 10396 ax-1rid 10397 ax-rnegex 10398 ax-rrecex 10399 ax-cnre 10400 ax-pre-lttri 10401 ax-pre-lttrn 10402 ax-pre-ltadd 10403 ax-pre-mulgt0 10404 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 834 df-3or 1069 df-3an 1070 df-tru 1510 df-ex 1743 df-nf 1747 df-sb 2014 df-mo 2544 df-eu 2580 df-clab 2754 df-cleq 2765 df-clel 2840 df-nfc 2912 df-ne 2962 df-nel 3068 df-ral 3087 df-rex 3088 df-reu 3089 df-rmo 3090 df-rab 3091 df-v 3411 df-sbc 3678 df-csb 3783 df-dif 3828 df-un 3830 df-in 3832 df-ss 3839 df-pss 3841 df-nul 4174 df-if 4345 df-pw 4418 df-sn 4436 df-pr 4438 df-tp 4440 df-op 4442 df-uni 4707 df-iun 4788 df-br 4924 df-opab 4986 df-mpt 5003 df-tr 5025 df-id 5305 df-eprel 5310 df-po 5319 df-so 5320 df-fr 5359 df-we 5361 df-xp 5406 df-rel 5407 df-cnv 5408 df-co 5409 df-dm 5410 df-rn 5411 df-res 5412 df-ima 5413 df-pred 5980 df-ord 6026 df-on 6027 df-lim 6028 df-suc 6029 df-iota 6146 df-fun 6184 df-fn 6185 df-f 6186 df-f1 6187 df-fo 6188 df-f1o 6189 df-fv 6190 df-riota 6931 df-ov 6973 df-oprab 6974 df-mpo 6975 df-om 7391 df-2nd 7495 df-wrecs 7743 df-recs 7805 df-rdg 7843 df-er 8081 df-en 8299 df-dom 8300 df-sdom 8301 df-pnf 10468 df-mnf 10469 df-xr 10470 df-ltxr 10471 df-le 10472 df-sub 10664 df-neg 10665 df-div 11091 df-nn 11432 df-n0 11701 df-z 11787 df-uz 12052 df-seq 13178 df-fac 13442 |
This theorem is referenced by: infpn 16094 |
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