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Mirrors > Home > ILE Home > Th. List > prmdc | GIF version |
Description: Primality is decidable. (Contributed by Jim Kingdon, 30-Sep-2024.) |
Ref | Expression |
---|---|
prmdc | ⊢ (𝑁 ∈ ℕ → DECID 𝑁 ∈ ℙ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 1nuz2 9535 | . . . . . . 7 ⊢ ¬ 1 ∈ (ℤ≥‘2) | |
2 | eleq1 2227 | . . . . . . 7 ⊢ (𝑁 = 1 → (𝑁 ∈ (ℤ≥‘2) ↔ 1 ∈ (ℤ≥‘2))) | |
3 | 1, 2 | mtbiri 665 | . . . . . 6 ⊢ (𝑁 = 1 → ¬ 𝑁 ∈ (ℤ≥‘2)) |
4 | 3 | orim1i 750 | . . . . 5 ⊢ ((𝑁 = 1 ∨ 𝑁 ∈ (ℤ≥‘2)) → (¬ 𝑁 ∈ (ℤ≥‘2) ∨ 𝑁 ∈ (ℤ≥‘2))) |
5 | 4 | orcomd 719 | . . . 4 ⊢ ((𝑁 = 1 ∨ 𝑁 ∈ (ℤ≥‘2)) → (𝑁 ∈ (ℤ≥‘2) ∨ ¬ 𝑁 ∈ (ℤ≥‘2))) |
6 | elnn1uz2 9536 | . . . 4 ⊢ (𝑁 ∈ ℕ ↔ (𝑁 = 1 ∨ 𝑁 ∈ (ℤ≥‘2))) | |
7 | df-dc 825 | . . . 4 ⊢ (DECID 𝑁 ∈ (ℤ≥‘2) ↔ (𝑁 ∈ (ℤ≥‘2) ∨ ¬ 𝑁 ∈ (ℤ≥‘2))) | |
8 | 5, 6, 7 | 3imtr4i 200 | . . 3 ⊢ (𝑁 ∈ ℕ → DECID 𝑁 ∈ (ℤ≥‘2)) |
9 | 2z 9210 | . . . . . 6 ⊢ 2 ∈ ℤ | |
10 | 9 | a1i 9 | . . . . 5 ⊢ (𝑁 ∈ ℕ → 2 ∈ ℤ) |
11 | nnz 9201 | . . . . . 6 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℤ) | |
12 | peano2zm 9220 | . . . . . 6 ⊢ (𝑁 ∈ ℤ → (𝑁 − 1) ∈ ℤ) | |
13 | 11, 12 | syl 14 | . . . . 5 ⊢ (𝑁 ∈ ℕ → (𝑁 − 1) ∈ ℤ) |
14 | 10, 13 | fzfigd 10356 | . . . 4 ⊢ (𝑁 ∈ ℕ → (2...(𝑁 − 1)) ∈ Fin) |
15 | elfzelz 9951 | . . . . . . . . 9 ⊢ (𝑥 ∈ (2...(𝑁 − 1)) → 𝑥 ∈ ℤ) | |
16 | 15 | adantl 275 | . . . . . . . 8 ⊢ ((𝑁 ∈ ℕ ∧ 𝑥 ∈ (2...(𝑁 − 1))) → 𝑥 ∈ ℤ) |
17 | 1red 7905 | . . . . . . . . 9 ⊢ ((𝑁 ∈ ℕ ∧ 𝑥 ∈ (2...(𝑁 − 1))) → 1 ∈ ℝ) | |
18 | 2re 8918 | . . . . . . . . . 10 ⊢ 2 ∈ ℝ | |
19 | 18 | a1i 9 | . . . . . . . . 9 ⊢ ((𝑁 ∈ ℕ ∧ 𝑥 ∈ (2...(𝑁 − 1))) → 2 ∈ ℝ) |
20 | 16 | zred 9304 | . . . . . . . . 9 ⊢ ((𝑁 ∈ ℕ ∧ 𝑥 ∈ (2...(𝑁 − 1))) → 𝑥 ∈ ℝ) |
21 | 1le2 9056 | . . . . . . . . . 10 ⊢ 1 ≤ 2 | |
22 | 21 | a1i 9 | . . . . . . . . 9 ⊢ ((𝑁 ∈ ℕ ∧ 𝑥 ∈ (2...(𝑁 − 1))) → 1 ≤ 2) |
23 | elfzle1 9952 | . . . . . . . . . 10 ⊢ (𝑥 ∈ (2...(𝑁 − 1)) → 2 ≤ 𝑥) | |
24 | 23 | adantl 275 | . . . . . . . . 9 ⊢ ((𝑁 ∈ ℕ ∧ 𝑥 ∈ (2...(𝑁 − 1))) → 2 ≤ 𝑥) |
25 | 17, 19, 20, 22, 24 | letrd 8013 | . . . . . . . 8 ⊢ ((𝑁 ∈ ℕ ∧ 𝑥 ∈ (2...(𝑁 − 1))) → 1 ≤ 𝑥) |
26 | elnnz1 9205 | . . . . . . . 8 ⊢ (𝑥 ∈ ℕ ↔ (𝑥 ∈ ℤ ∧ 1 ≤ 𝑥)) | |
27 | 16, 25, 26 | sylanbrc 414 | . . . . . . 7 ⊢ ((𝑁 ∈ ℕ ∧ 𝑥 ∈ (2...(𝑁 − 1))) → 𝑥 ∈ ℕ) |
28 | 11 | adantr 274 | . . . . . . 7 ⊢ ((𝑁 ∈ ℕ ∧ 𝑥 ∈ (2...(𝑁 − 1))) → 𝑁 ∈ ℤ) |
29 | dvdsdc 11724 | . . . . . . 7 ⊢ ((𝑥 ∈ ℕ ∧ 𝑁 ∈ ℤ) → DECID 𝑥 ∥ 𝑁) | |
30 | 27, 28, 29 | syl2anc 409 | . . . . . 6 ⊢ ((𝑁 ∈ ℕ ∧ 𝑥 ∈ (2...(𝑁 − 1))) → DECID 𝑥 ∥ 𝑁) |
31 | dcn 832 | . . . . . 6 ⊢ (DECID 𝑥 ∥ 𝑁 → DECID ¬ 𝑥 ∥ 𝑁) | |
32 | 30, 31 | syl 14 | . . . . 5 ⊢ ((𝑁 ∈ ℕ ∧ 𝑥 ∈ (2...(𝑁 − 1))) → DECID ¬ 𝑥 ∥ 𝑁) |
33 | 32 | ralrimiva 2537 | . . . 4 ⊢ (𝑁 ∈ ℕ → ∀𝑥 ∈ (2...(𝑁 − 1))DECID ¬ 𝑥 ∥ 𝑁) |
34 | dcfi 6937 | . . . 4 ⊢ (((2...(𝑁 − 1)) ∈ Fin ∧ ∀𝑥 ∈ (2...(𝑁 − 1))DECID ¬ 𝑥 ∥ 𝑁) → DECID ∀𝑥 ∈ (2...(𝑁 − 1)) ¬ 𝑥 ∥ 𝑁) | |
35 | 14, 33, 34 | syl2anc 409 | . . 3 ⊢ (𝑁 ∈ ℕ → DECID ∀𝑥 ∈ (2...(𝑁 − 1)) ¬ 𝑥 ∥ 𝑁) |
36 | dcan 923 | . . 3 ⊢ (DECID 𝑁 ∈ (ℤ≥‘2) → (DECID ∀𝑥 ∈ (2...(𝑁 − 1)) ¬ 𝑥 ∥ 𝑁 → DECID (𝑁 ∈ (ℤ≥‘2) ∧ ∀𝑥 ∈ (2...(𝑁 − 1)) ¬ 𝑥 ∥ 𝑁))) | |
37 | 8, 35, 36 | sylc 62 | . 2 ⊢ (𝑁 ∈ ℕ → DECID (𝑁 ∈ (ℤ≥‘2) ∧ ∀𝑥 ∈ (2...(𝑁 − 1)) ¬ 𝑥 ∥ 𝑁)) |
38 | isprm3 12029 | . . 3 ⊢ (𝑁 ∈ ℙ ↔ (𝑁 ∈ (ℤ≥‘2) ∧ ∀𝑥 ∈ (2...(𝑁 − 1)) ¬ 𝑥 ∥ 𝑁)) | |
39 | 38 | dcbii 830 | . 2 ⊢ (DECID 𝑁 ∈ ℙ ↔ DECID (𝑁 ∈ (ℤ≥‘2) ∧ ∀𝑥 ∈ (2...(𝑁 − 1)) ¬ 𝑥 ∥ 𝑁)) |
40 | 37, 39 | sylibr 133 | 1 ⊢ (𝑁 ∈ ℕ → DECID 𝑁 ∈ ℙ) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 103 ∨ wo 698 DECID wdc 824 = wceq 1342 ∈ wcel 2135 ∀wral 2442 class class class wbr 3976 ‘cfv 5182 (class class class)co 5836 Fincfn 6697 ℝcr 7743 1c1 7745 ≤ cle 7925 − cmin 8060 ℕcn 8848 2c2 8899 ℤcz 9182 ℤ≥cuz 9457 ...cfz 9935 ∥ cdvds 11713 ℙcprime 12018 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 604 ax-in2 605 ax-io 699 ax-5 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-13 2137 ax-14 2138 ax-ext 2146 ax-coll 4091 ax-sep 4094 ax-nul 4102 ax-pow 4147 ax-pr 4181 ax-un 4405 ax-setind 4508 ax-iinf 4559 ax-cnex 7835 ax-resscn 7836 ax-1cn 7837 ax-1re 7838 ax-icn 7839 ax-addcl 7840 ax-addrcl 7841 ax-mulcl 7842 ax-mulrcl 7843 ax-addcom 7844 ax-mulcom 7845 ax-addass 7846 ax-mulass 7847 ax-distr 7848 ax-i2m1 7849 ax-0lt1 7850 ax-1rid 7851 ax-0id 7852 ax-rnegex 7853 ax-precex 7854 ax-cnre 7855 ax-pre-ltirr 7856 ax-pre-ltwlin 7857 ax-pre-lttrn 7858 ax-pre-apti 7859 ax-pre-ltadd 7860 ax-pre-mulgt0 7861 ax-pre-mulext 7862 ax-arch 7863 ax-caucvg 7864 |
This theorem depends on definitions: df-bi 116 df-stab 821 df-dc 825 df-3or 968 df-3an 969 df-tru 1345 df-fal 1348 df-nf 1448 df-sb 1750 df-eu 2016 df-mo 2017 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-ne 2335 df-nel 2430 df-ral 2447 df-rex 2448 df-reu 2449 df-rmo 2450 df-rab 2451 df-v 2723 df-sbc 2947 df-csb 3041 df-dif 3113 df-un 3115 df-in 3117 df-ss 3124 df-nul 3405 df-if 3516 df-pw 3555 df-sn 3576 df-pr 3577 df-op 3579 df-uni 3784 df-int 3819 df-iun 3862 df-br 3977 df-opab 4038 df-mpt 4039 df-tr 4075 df-id 4265 df-po 4268 df-iso 4269 df-iord 4338 df-on 4340 df-ilim 4341 df-suc 4343 df-iom 4562 df-xp 4604 df-rel 4605 df-cnv 4606 df-co 4607 df-dm 4608 df-rn 4609 df-res 4610 df-ima 4611 df-iota 5147 df-fun 5184 df-fn 5185 df-f 5186 df-f1 5187 df-fo 5188 df-f1o 5189 df-fv 5190 df-riota 5792 df-ov 5839 df-oprab 5840 df-mpo 5841 df-1st 6100 df-2nd 6101 df-recs 6264 df-frec 6350 df-1o 6375 df-2o 6376 df-er 6492 df-en 6698 df-fin 6700 df-pnf 7926 df-mnf 7927 df-xr 7928 df-ltxr 7929 df-le 7930 df-sub 8062 df-neg 8063 df-reap 8464 df-ap 8471 df-div 8560 df-inn 8849 df-2 8907 df-3 8908 df-4 8909 df-n0 9106 df-z 9183 df-uz 9458 df-q 9549 df-rp 9581 df-fz 9936 df-fl 10195 df-mod 10248 df-seqfrec 10371 df-exp 10445 df-cj 10770 df-re 10771 df-im 10772 df-rsqrt 10926 df-abs 10927 df-dvds 11714 df-prm 12019 |
This theorem is referenced by: pcmptcl 12249 pcmpt 12250 |
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