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Mirrors > Home > MPE Home > Th. List > infpnlem2 | Structured version Visualization version GIF version |
Description: Lemma for infpn 16248. 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 11905 | . . . . . . 7 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℕ0) | |
3 | 2 | faccld 13645 | . . . . . 6 ⊢ (𝑁 ∈ ℕ → (!‘𝑁) ∈ ℕ) |
4 | 3 | peano2nnd 11655 | . . . . 5 ⊢ (𝑁 ∈ ℕ → ((!‘𝑁) + 1) ∈ ℕ) |
5 | 1, 4 | eqeltrid 2917 | . . . 4 ⊢ (𝑁 ∈ ℕ → 𝐾 ∈ ℕ) |
6 | 3 | nnge1d 11686 | . . . . . 6 ⊢ (𝑁 ∈ ℕ → 1 ≤ (!‘𝑁)) |
7 | 1nn 11649 | . . . . . . 7 ⊢ 1 ∈ ℕ | |
8 | nnleltp1 12038 | . . . . . . 7 ⊢ ((1 ∈ ℕ ∧ (!‘𝑁) ∈ ℕ) → (1 ≤ (!‘𝑁) ↔ 1 < ((!‘𝑁) + 1))) | |
9 | 7, 3, 8 | sylancr 589 | . . . . . 6 ⊢ (𝑁 ∈ ℕ → (1 ≤ (!‘𝑁) ↔ 1 < ((!‘𝑁) + 1))) |
10 | 6, 9 | mpbid 234 | . . . . 5 ⊢ (𝑁 ∈ ℕ → 1 < ((!‘𝑁) + 1)) |
11 | 10, 1 | breqtrrdi 5108 | . . . 4 ⊢ (𝑁 ∈ ℕ → 1 < 𝐾) |
12 | nncn 11646 | . . . . . . 7 ⊢ (𝐾 ∈ ℕ → 𝐾 ∈ ℂ) | |
13 | nnne0 11672 | . . . . . . 7 ⊢ (𝐾 ∈ ℕ → 𝐾 ≠ 0) | |
14 | 12, 13 | jca 514 | . . . . . 6 ⊢ (𝐾 ∈ ℕ → (𝐾 ∈ ℂ ∧ 𝐾 ≠ 0)) |
15 | divid 11327 | . . . . . 6 ⊢ ((𝐾 ∈ ℂ ∧ 𝐾 ≠ 0) → (𝐾 / 𝐾) = 1) | |
16 | 5, 14, 15 | 3syl 18 | . . . . 5 ⊢ (𝑁 ∈ ℕ → (𝐾 / 𝐾) = 1) |
17 | 16, 7 | eqeltrdi 2921 | . . . 4 ⊢ (𝑁 ∈ ℕ → (𝐾 / 𝐾) ∈ ℕ) |
18 | breq2 5070 | . . . . . 6 ⊢ (𝑗 = 𝐾 → (1 < 𝑗 ↔ 1 < 𝐾)) | |
19 | oveq2 7164 | . . . . . . 7 ⊢ (𝑗 = 𝐾 → (𝐾 / 𝑗) = (𝐾 / 𝐾)) | |
20 | 19 | eleq1d 2897 | . . . . . 6 ⊢ (𝑗 = 𝐾 → ((𝐾 / 𝑗) ∈ ℕ ↔ (𝐾 / 𝐾) ∈ ℕ)) |
21 | 18, 20 | anbi12d 632 | . . . . 5 ⊢ (𝑗 = 𝐾 → ((1 < 𝑗 ∧ (𝐾 / 𝑗) ∈ ℕ) ↔ (1 < 𝐾 ∧ (𝐾 / 𝐾) ∈ ℕ))) |
22 | 21 | rspcev 3623 | . . . 4 ⊢ ((𝐾 ∈ ℕ ∧ (1 < 𝐾 ∧ (𝐾 / 𝐾) ∈ ℕ)) → ∃𝑗 ∈ ℕ (1 < 𝑗 ∧ (𝐾 / 𝑗) ∈ ℕ)) |
23 | 5, 11, 17, 22 | syl12anc 834 | . . 3 ⊢ (𝑁 ∈ ℕ → ∃𝑗 ∈ ℕ (1 < 𝑗 ∧ (𝐾 / 𝑗) ∈ ℕ)) |
24 | breq2 5070 | . . . . 5 ⊢ (𝑗 = 𝑘 → (1 < 𝑗 ↔ 1 < 𝑘)) | |
25 | oveq2 7164 | . . . . . 6 ⊢ (𝑗 = 𝑘 → (𝐾 / 𝑗) = (𝐾 / 𝑘)) | |
26 | 25 | eleq1d 2897 | . . . . 5 ⊢ (𝑗 = 𝑘 → ((𝐾 / 𝑗) ∈ ℕ ↔ (𝐾 / 𝑘) ∈ ℕ)) |
27 | 24, 26 | anbi12d 632 | . . . 4 ⊢ (𝑗 = 𝑘 → ((1 < 𝑗 ∧ (𝐾 / 𝑗) ∈ ℕ) ↔ (1 < 𝑘 ∧ (𝐾 / 𝑘) ∈ ℕ))) |
28 | 27 | nnwos 12316 | . . 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 16246 | . . 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 3274 | . 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 208 ∧ wa 398 ∨ wo 843 = wceq 1537 ∈ wcel 2114 ≠ wne 3016 ∀wral 3138 ∃wrex 3139 class class class wbr 5066 ‘cfv 6355 (class class class)co 7156 ℂcc 10535 0cc0 10537 1c1 10538 + caddc 10540 < clt 10675 ≤ cle 10676 / cdiv 11297 ℕcn 11638 !cfa 13634 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 ax-cnex 10593 ax-resscn 10594 ax-1cn 10595 ax-icn 10596 ax-addcl 10597 ax-addrcl 10598 ax-mulcl 10599 ax-mulrcl 10600 ax-mulcom 10601 ax-addass 10602 ax-mulass 10603 ax-distr 10604 ax-i2m1 10605 ax-1ne0 10606 ax-1rid 10607 ax-rnegex 10608 ax-rrecex 10609 ax-cnre 10610 ax-pre-lttri 10611 ax-pre-lttrn 10612 ax-pre-ltadd 10613 ax-pre-mulgt0 10614 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4839 df-iun 4921 df-br 5067 df-opab 5129 df-mpt 5147 df-tr 5173 df-id 5460 df-eprel 5465 df-po 5474 df-so 5475 df-fr 5514 df-we 5516 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-pred 6148 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-riota 7114 df-ov 7159 df-oprab 7160 df-mpo 7161 df-om 7581 df-2nd 7690 df-wrecs 7947 df-recs 8008 df-rdg 8046 df-er 8289 df-en 8510 df-dom 8511 df-sdom 8512 df-pnf 10677 df-mnf 10678 df-xr 10679 df-ltxr 10680 df-le 10681 df-sub 10872 df-neg 10873 df-div 11298 df-nn 11639 df-n0 11899 df-z 11983 df-uz 12245 df-seq 13371 df-fac 13635 |
This theorem is referenced by: infpn 16248 |
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