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Mirrors > Home > MPE Home > Th. List > infpnlem2 | Structured version Visualization version GIF version |
Description: Lemma for infpn 16428. 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 12062 | . . . . . . 7 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℕ0) | |
3 | 2 | faccld 13815 | . . . . . 6 ⊢ (𝑁 ∈ ℕ → (!‘𝑁) ∈ ℕ) |
4 | 3 | peano2nnd 11812 | . . . . 5 ⊢ (𝑁 ∈ ℕ → ((!‘𝑁) + 1) ∈ ℕ) |
5 | 1, 4 | eqeltrid 2835 | . . . 4 ⊢ (𝑁 ∈ ℕ → 𝐾 ∈ ℕ) |
6 | 3 | nnge1d 11843 | . . . . . 6 ⊢ (𝑁 ∈ ℕ → 1 ≤ (!‘𝑁)) |
7 | 1nn 11806 | . . . . . . 7 ⊢ 1 ∈ ℕ | |
8 | nnleltp1 12197 | . . . . . . 7 ⊢ ((1 ∈ ℕ ∧ (!‘𝑁) ∈ ℕ) → (1 ≤ (!‘𝑁) ↔ 1 < ((!‘𝑁) + 1))) | |
9 | 7, 3, 8 | sylancr 590 | . . . . . 6 ⊢ (𝑁 ∈ ℕ → (1 ≤ (!‘𝑁) ↔ 1 < ((!‘𝑁) + 1))) |
10 | 6, 9 | mpbid 235 | . . . . 5 ⊢ (𝑁 ∈ ℕ → 1 < ((!‘𝑁) + 1)) |
11 | 10, 1 | breqtrrdi 5081 | . . . 4 ⊢ (𝑁 ∈ ℕ → 1 < 𝐾) |
12 | nncn 11803 | . . . . . . 7 ⊢ (𝐾 ∈ ℕ → 𝐾 ∈ ℂ) | |
13 | nnne0 11829 | . . . . . . 7 ⊢ (𝐾 ∈ ℕ → 𝐾 ≠ 0) | |
14 | 12, 13 | jca 515 | . . . . . 6 ⊢ (𝐾 ∈ ℕ → (𝐾 ∈ ℂ ∧ 𝐾 ≠ 0)) |
15 | divid 11484 | . . . . . 6 ⊢ ((𝐾 ∈ ℂ ∧ 𝐾 ≠ 0) → (𝐾 / 𝐾) = 1) | |
16 | 5, 14, 15 | 3syl 18 | . . . . 5 ⊢ (𝑁 ∈ ℕ → (𝐾 / 𝐾) = 1) |
17 | 16, 7 | eqeltrdi 2839 | . . . 4 ⊢ (𝑁 ∈ ℕ → (𝐾 / 𝐾) ∈ ℕ) |
18 | breq2 5043 | . . . . . 6 ⊢ (𝑗 = 𝐾 → (1 < 𝑗 ↔ 1 < 𝐾)) | |
19 | oveq2 7199 | . . . . . . 7 ⊢ (𝑗 = 𝐾 → (𝐾 / 𝑗) = (𝐾 / 𝐾)) | |
20 | 19 | eleq1d 2815 | . . . . . 6 ⊢ (𝑗 = 𝐾 → ((𝐾 / 𝑗) ∈ ℕ ↔ (𝐾 / 𝐾) ∈ ℕ)) |
21 | 18, 20 | anbi12d 634 | . . . . 5 ⊢ (𝑗 = 𝐾 → ((1 < 𝑗 ∧ (𝐾 / 𝑗) ∈ ℕ) ↔ (1 < 𝐾 ∧ (𝐾 / 𝐾) ∈ ℕ))) |
22 | 21 | rspcev 3527 | . . . 4 ⊢ ((𝐾 ∈ ℕ ∧ (1 < 𝐾 ∧ (𝐾 / 𝐾) ∈ ℕ)) → ∃𝑗 ∈ ℕ (1 < 𝑗 ∧ (𝐾 / 𝑗) ∈ ℕ)) |
23 | 5, 11, 17, 22 | syl12anc 837 | . . 3 ⊢ (𝑁 ∈ ℕ → ∃𝑗 ∈ ℕ (1 < 𝑗 ∧ (𝐾 / 𝑗) ∈ ℕ)) |
24 | breq2 5043 | . . . . 5 ⊢ (𝑗 = 𝑘 → (1 < 𝑗 ↔ 1 < 𝑘)) | |
25 | oveq2 7199 | . . . . . 6 ⊢ (𝑗 = 𝑘 → (𝐾 / 𝑗) = (𝐾 / 𝑘)) | |
26 | 25 | eleq1d 2815 | . . . . 5 ⊢ (𝑗 = 𝑘 → ((𝐾 / 𝑗) ∈ ℕ ↔ (𝐾 / 𝑘) ∈ ℕ)) |
27 | 24, 26 | anbi12d 634 | . . . 4 ⊢ (𝑗 = 𝑘 → ((1 < 𝑗 ∧ (𝐾 / 𝑗) ∈ ℕ) ↔ (1 < 𝑘 ∧ (𝐾 / 𝑘) ∈ ℕ))) |
28 | 27 | nnwos 12476 | . . 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 16426 | . . 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 3183 | . 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 209 ∧ wa 399 ∨ wo 847 = wceq 1543 ∈ wcel 2112 ≠ wne 2932 ∀wral 3051 ∃wrex 3052 class class class wbr 5039 ‘cfv 6358 (class class class)co 7191 ℂcc 10692 0cc0 10694 1c1 10695 + caddc 10697 < clt 10832 ≤ cle 10833 / cdiv 11454 ℕcn 11795 !cfa 13804 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2018 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2160 ax-12 2177 ax-ext 2708 ax-sep 5177 ax-nul 5184 ax-pow 5243 ax-pr 5307 ax-un 7501 ax-cnex 10750 ax-resscn 10751 ax-1cn 10752 ax-icn 10753 ax-addcl 10754 ax-addrcl 10755 ax-mulcl 10756 ax-mulrcl 10757 ax-mulcom 10758 ax-addass 10759 ax-mulass 10760 ax-distr 10761 ax-i2m1 10762 ax-1ne0 10763 ax-1rid 10764 ax-rnegex 10765 ax-rrecex 10766 ax-cnre 10767 ax-pre-lttri 10768 ax-pre-lttrn 10769 ax-pre-ltadd 10770 ax-pre-mulgt0 10771 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2073 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2728 df-clel 2809 df-nfc 2879 df-ne 2933 df-nel 3037 df-ral 3056 df-rex 3057 df-reu 3058 df-rmo 3059 df-rab 3060 df-v 3400 df-sbc 3684 df-csb 3799 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-pss 3872 df-nul 4224 df-if 4426 df-pw 4501 df-sn 4528 df-pr 4530 df-tp 4532 df-op 4534 df-uni 4806 df-iun 4892 df-br 5040 df-opab 5102 df-mpt 5121 df-tr 5147 df-id 5440 df-eprel 5445 df-po 5453 df-so 5454 df-fr 5494 df-we 5496 df-xp 5542 df-rel 5543 df-cnv 5544 df-co 5545 df-dm 5546 df-rn 5547 df-res 5548 df-ima 5549 df-pred 6140 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6316 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-riota 7148 df-ov 7194 df-oprab 7195 df-mpo 7196 df-om 7623 df-2nd 7740 df-wrecs 8025 df-recs 8086 df-rdg 8124 df-er 8369 df-en 8605 df-dom 8606 df-sdom 8607 df-pnf 10834 df-mnf 10835 df-xr 10836 df-ltxr 10837 df-le 10838 df-sub 11029 df-neg 11030 df-div 11455 df-nn 11796 df-n0 12056 df-z 12142 df-uz 12404 df-seq 13540 df-fac 13805 |
This theorem is referenced by: infpn 16428 |
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