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Mirrors > Home > ILE Home > Th. List > prmunb | GIF version |
Description: The primes are unbounded. (Contributed by Paul Chapman, 28-Nov-2012.) |
Ref | Expression |
---|---|
prmunb | ⊢ (𝑁 ∈ ℕ → ∃𝑝 ∈ ℙ 𝑁 < 𝑝) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nnnn0 9183 | . 2 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℕ0) | |
2 | faccl 10715 | . . . 4 ⊢ (𝑁 ∈ ℕ0 → (!‘𝑁) ∈ ℕ) | |
3 | elnnuz 9564 | . . . . 5 ⊢ ((!‘𝑁) ∈ ℕ ↔ (!‘𝑁) ∈ (ℤ≥‘1)) | |
4 | eluzp1p1 9553 | . . . . . 6 ⊢ ((!‘𝑁) ∈ (ℤ≥‘1) → ((!‘𝑁) + 1) ∈ (ℤ≥‘(1 + 1))) | |
5 | df-2 8978 | . . . . . . 7 ⊢ 2 = (1 + 1) | |
6 | 5 | fveq2i 5519 | . . . . . 6 ⊢ (ℤ≥‘2) = (ℤ≥‘(1 + 1)) |
7 | 4, 6 | eleqtrrdi 2271 | . . . . 5 ⊢ ((!‘𝑁) ∈ (ℤ≥‘1) → ((!‘𝑁) + 1) ∈ (ℤ≥‘2)) |
8 | 3, 7 | sylbi 121 | . . . 4 ⊢ ((!‘𝑁) ∈ ℕ → ((!‘𝑁) + 1) ∈ (ℤ≥‘2)) |
9 | exprmfct 12138 | . . . 4 ⊢ (((!‘𝑁) + 1) ∈ (ℤ≥‘2) → ∃𝑝 ∈ ℙ 𝑝 ∥ ((!‘𝑁) + 1)) | |
10 | 2, 8, 9 | 3syl 17 | . . 3 ⊢ (𝑁 ∈ ℕ0 → ∃𝑝 ∈ ℙ 𝑝 ∥ ((!‘𝑁) + 1)) |
11 | prmz 12111 | . . . . . . . . 9 ⊢ (𝑝 ∈ ℙ → 𝑝 ∈ ℤ) | |
12 | nn0z 9273 | . . . . . . . . 9 ⊢ (𝑁 ∈ ℕ0 → 𝑁 ∈ ℤ) | |
13 | eluz 9541 | . . . . . . . . 9 ⊢ ((𝑝 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑁 ∈ (ℤ≥‘𝑝) ↔ 𝑝 ≤ 𝑁)) | |
14 | 11, 12, 13 | syl2an 289 | . . . . . . . 8 ⊢ ((𝑝 ∈ ℙ ∧ 𝑁 ∈ ℕ0) → (𝑁 ∈ (ℤ≥‘𝑝) ↔ 𝑝 ≤ 𝑁)) |
15 | prmuz2 12131 | . . . . . . . . . . . . . . . . 17 ⊢ (𝑝 ∈ ℙ → 𝑝 ∈ (ℤ≥‘2)) | |
16 | eluz2b2 9603 | . . . . . . . . . . . . . . . . 17 ⊢ (𝑝 ∈ (ℤ≥‘2) ↔ (𝑝 ∈ ℕ ∧ 1 < 𝑝)) | |
17 | 15, 16 | sylib 122 | . . . . . . . . . . . . . . . 16 ⊢ (𝑝 ∈ ℙ → (𝑝 ∈ ℕ ∧ 1 < 𝑝)) |
18 | 17 | adantr 276 | . . . . . . . . . . . . . . 15 ⊢ ((𝑝 ∈ ℙ ∧ 𝑁 ∈ (ℤ≥‘𝑝)) → (𝑝 ∈ ℕ ∧ 1 < 𝑝)) |
19 | 18 | simpld 112 | . . . . . . . . . . . . . 14 ⊢ ((𝑝 ∈ ℙ ∧ 𝑁 ∈ (ℤ≥‘𝑝)) → 𝑝 ∈ ℕ) |
20 | 19 | nnnn0d 9229 | . . . . . . . . . . . . 13 ⊢ ((𝑝 ∈ ℙ ∧ 𝑁 ∈ (ℤ≥‘𝑝)) → 𝑝 ∈ ℕ0) |
21 | eluznn0 9599 | . . . . . . . . . . . . 13 ⊢ ((𝑝 ∈ ℕ0 ∧ 𝑁 ∈ (ℤ≥‘𝑝)) → 𝑁 ∈ ℕ0) | |
22 | 20, 21 | sylancom 420 | . . . . . . . . . . . 12 ⊢ ((𝑝 ∈ ℙ ∧ 𝑁 ∈ (ℤ≥‘𝑝)) → 𝑁 ∈ ℕ0) |
23 | nnz 9272 | . . . . . . . . . . . 12 ⊢ ((!‘𝑁) ∈ ℕ → (!‘𝑁) ∈ ℤ) | |
24 | 22, 2, 23 | 3syl 17 | . . . . . . . . . . 11 ⊢ ((𝑝 ∈ ℙ ∧ 𝑁 ∈ (ℤ≥‘𝑝)) → (!‘𝑁) ∈ ℤ) |
25 | 18 | simprd 114 | . . . . . . . . . . 11 ⊢ ((𝑝 ∈ ℙ ∧ 𝑁 ∈ (ℤ≥‘𝑝)) → 1 < 𝑝) |
26 | dvdsfac 11866 | . . . . . . . . . . . 12 ⊢ ((𝑝 ∈ ℕ ∧ 𝑁 ∈ (ℤ≥‘𝑝)) → 𝑝 ∥ (!‘𝑁)) | |
27 | 19, 26 | sylancom 420 | . . . . . . . . . . 11 ⊢ ((𝑝 ∈ ℙ ∧ 𝑁 ∈ (ℤ≥‘𝑝)) → 𝑝 ∥ (!‘𝑁)) |
28 | ndvdsp1 11937 | . . . . . . . . . . . 12 ⊢ (((!‘𝑁) ∈ ℤ ∧ 𝑝 ∈ ℕ ∧ 1 < 𝑝) → (𝑝 ∥ (!‘𝑁) → ¬ 𝑝 ∥ ((!‘𝑁) + 1))) | |
29 | 28 | imp 124 | . . . . . . . . . . 11 ⊢ ((((!‘𝑁) ∈ ℤ ∧ 𝑝 ∈ ℕ ∧ 1 < 𝑝) ∧ 𝑝 ∥ (!‘𝑁)) → ¬ 𝑝 ∥ ((!‘𝑁) + 1)) |
30 | 24, 19, 25, 27, 29 | syl31anc 1241 | . . . . . . . . . 10 ⊢ ((𝑝 ∈ ℙ ∧ 𝑁 ∈ (ℤ≥‘𝑝)) → ¬ 𝑝 ∥ ((!‘𝑁) + 1)) |
31 | 30 | ex 115 | . . . . . . . . 9 ⊢ (𝑝 ∈ ℙ → (𝑁 ∈ (ℤ≥‘𝑝) → ¬ 𝑝 ∥ ((!‘𝑁) + 1))) |
32 | 31 | adantr 276 | . . . . . . . 8 ⊢ ((𝑝 ∈ ℙ ∧ 𝑁 ∈ ℕ0) → (𝑁 ∈ (ℤ≥‘𝑝) → ¬ 𝑝 ∥ ((!‘𝑁) + 1))) |
33 | 14, 32 | sylbird 170 | . . . . . . 7 ⊢ ((𝑝 ∈ ℙ ∧ 𝑁 ∈ ℕ0) → (𝑝 ≤ 𝑁 → ¬ 𝑝 ∥ ((!‘𝑁) + 1))) |
34 | 33 | con2d 624 | . . . . . 6 ⊢ ((𝑝 ∈ ℙ ∧ 𝑁 ∈ ℕ0) → (𝑝 ∥ ((!‘𝑁) + 1) → ¬ 𝑝 ≤ 𝑁)) |
35 | 34 | ancoms 268 | . . . . 5 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑝 ∈ ℙ) → (𝑝 ∥ ((!‘𝑁) + 1) → ¬ 𝑝 ≤ 𝑁)) |
36 | zltnle 9299 | . . . . . 6 ⊢ ((𝑁 ∈ ℤ ∧ 𝑝 ∈ ℤ) → (𝑁 < 𝑝 ↔ ¬ 𝑝 ≤ 𝑁)) | |
37 | 12, 11, 36 | syl2an 289 | . . . . 5 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑝 ∈ ℙ) → (𝑁 < 𝑝 ↔ ¬ 𝑝 ≤ 𝑁)) |
38 | 35, 37 | sylibrd 169 | . . . 4 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑝 ∈ ℙ) → (𝑝 ∥ ((!‘𝑁) + 1) → 𝑁 < 𝑝)) |
39 | 38 | reximdva 2579 | . . 3 ⊢ (𝑁 ∈ ℕ0 → (∃𝑝 ∈ ℙ 𝑝 ∥ ((!‘𝑁) + 1) → ∃𝑝 ∈ ℙ 𝑁 < 𝑝)) |
40 | 10, 39 | mpd 13 | . 2 ⊢ (𝑁 ∈ ℕ0 → ∃𝑝 ∈ ℙ 𝑁 < 𝑝) |
41 | 1, 40 | syl 14 | 1 ⊢ (𝑁 ∈ ℕ → ∃𝑝 ∈ ℙ 𝑁 < 𝑝) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 104 ↔ wb 105 ∧ w3a 978 ∈ wcel 2148 ∃wrex 2456 class class class wbr 4004 ‘cfv 5217 (class class class)co 5875 1c1 7812 + caddc 7814 < clt 7992 ≤ cle 7993 ℕcn 8919 2c2 8970 ℕ0cn0 9176 ℤcz 9253 ℤ≥cuz 9528 !cfa 10705 ∥ cdvds 11794 ℙcprime 12107 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 614 ax-in2 615 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-13 2150 ax-14 2151 ax-ext 2159 ax-coll 4119 ax-sep 4122 ax-nul 4130 ax-pow 4175 ax-pr 4210 ax-un 4434 ax-setind 4537 ax-iinf 4588 ax-cnex 7902 ax-resscn 7903 ax-1cn 7904 ax-1re 7905 ax-icn 7906 ax-addcl 7907 ax-addrcl 7908 ax-mulcl 7909 ax-mulrcl 7910 ax-addcom 7911 ax-mulcom 7912 ax-addass 7913 ax-mulass 7914 ax-distr 7915 ax-i2m1 7916 ax-0lt1 7917 ax-1rid 7918 ax-0id 7919 ax-rnegex 7920 ax-precex 7921 ax-cnre 7922 ax-pre-ltirr 7923 ax-pre-ltwlin 7924 ax-pre-lttrn 7925 ax-pre-apti 7926 ax-pre-ltadd 7927 ax-pre-mulgt0 7928 ax-pre-mulext 7929 ax-arch 7930 ax-caucvg 7931 |
This theorem depends on definitions: df-bi 117 df-stab 831 df-dc 835 df-3or 979 df-3an 980 df-tru 1356 df-fal 1359 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ne 2348 df-nel 2443 df-ral 2460 df-rex 2461 df-reu 2462 df-rmo 2463 df-rab 2464 df-v 2740 df-sbc 2964 df-csb 3059 df-dif 3132 df-un 3134 df-in 3136 df-ss 3143 df-nul 3424 df-if 3536 df-pw 3578 df-sn 3599 df-pr 3600 df-op 3602 df-uni 3811 df-int 3846 df-iun 3889 df-br 4005 df-opab 4066 df-mpt 4067 df-tr 4103 df-id 4294 df-po 4297 df-iso 4298 df-iord 4367 df-on 4369 df-ilim 4370 df-suc 4372 df-iom 4591 df-xp 4633 df-rel 4634 df-cnv 4635 df-co 4636 df-dm 4637 df-rn 4638 df-res 4639 df-ima 4640 df-iota 5179 df-fun 5219 df-fn 5220 df-f 5221 df-f1 5222 df-fo 5223 df-f1o 5224 df-fv 5225 df-riota 5831 df-ov 5878 df-oprab 5879 df-mpo 5880 df-1st 6141 df-2nd 6142 df-recs 6306 df-frec 6392 df-1o 6417 df-2o 6418 df-er 6535 df-en 6741 df-pnf 7994 df-mnf 7995 df-xr 7996 df-ltxr 7997 df-le 7998 df-sub 8130 df-neg 8131 df-reap 8532 df-ap 8539 df-div 8630 df-inn 8920 df-2 8978 df-3 8979 df-4 8980 df-n0 9177 df-z 9254 df-uz 9529 df-q 9620 df-rp 9654 df-fz 10009 df-fzo 10143 df-fl 10270 df-mod 10323 df-seqfrec 10446 df-exp 10520 df-fac 10706 df-cj 10851 df-re 10852 df-im 10853 df-rsqrt 11007 df-abs 11008 df-dvds 11795 df-prm 12108 |
This theorem is referenced by: prminf 12456 |
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