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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 9606 | . . . . . . 7 ⊢ ¬ 1 ∈ (ℤ≥‘2) | |
2 | eleq1 2240 | . . . . . . 7 ⊢ (𝑁 = 1 → (𝑁 ∈ (ℤ≥‘2) ↔ 1 ∈ (ℤ≥‘2))) | |
3 | 1, 2 | mtbiri 675 | . . . . . 6 ⊢ (𝑁 = 1 → ¬ 𝑁 ∈ (ℤ≥‘2)) |
4 | 3 | orim1i 760 | . . . . 5 ⊢ ((𝑁 = 1 ∨ 𝑁 ∈ (ℤ≥‘2)) → (¬ 𝑁 ∈ (ℤ≥‘2) ∨ 𝑁 ∈ (ℤ≥‘2))) |
5 | 4 | orcomd 729 | . . . 4 ⊢ ((𝑁 = 1 ∨ 𝑁 ∈ (ℤ≥‘2)) → (𝑁 ∈ (ℤ≥‘2) ∨ ¬ 𝑁 ∈ (ℤ≥‘2))) |
6 | elnn1uz2 9607 | . . . 4 ⊢ (𝑁 ∈ ℕ ↔ (𝑁 = 1 ∨ 𝑁 ∈ (ℤ≥‘2))) | |
7 | df-dc 835 | . . . 4 ⊢ (DECID 𝑁 ∈ (ℤ≥‘2) ↔ (𝑁 ∈ (ℤ≥‘2) ∨ ¬ 𝑁 ∈ (ℤ≥‘2))) | |
8 | 5, 6, 7 | 3imtr4i 201 | . . 3 ⊢ (𝑁 ∈ ℕ → DECID 𝑁 ∈ (ℤ≥‘2)) |
9 | 2z 9281 | . . . . . 6 ⊢ 2 ∈ ℤ | |
10 | 9 | a1i 9 | . . . . 5 ⊢ (𝑁 ∈ ℕ → 2 ∈ ℤ) |
11 | nnz 9272 | . . . . . 6 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℤ) | |
12 | peano2zm 9291 | . . . . . 6 ⊢ (𝑁 ∈ ℤ → (𝑁 − 1) ∈ ℤ) | |
13 | 11, 12 | syl 14 | . . . . 5 ⊢ (𝑁 ∈ ℕ → (𝑁 − 1) ∈ ℤ) |
14 | 10, 13 | fzfigd 10431 | . . . 4 ⊢ (𝑁 ∈ ℕ → (2...(𝑁 − 1)) ∈ Fin) |
15 | elfzelz 10025 | . . . . . . . . 9 ⊢ (𝑥 ∈ (2...(𝑁 − 1)) → 𝑥 ∈ ℤ) | |
16 | 15 | adantl 277 | . . . . . . . 8 ⊢ ((𝑁 ∈ ℕ ∧ 𝑥 ∈ (2...(𝑁 − 1))) → 𝑥 ∈ ℤ) |
17 | 1red 7972 | . . . . . . . . 9 ⊢ ((𝑁 ∈ ℕ ∧ 𝑥 ∈ (2...(𝑁 − 1))) → 1 ∈ ℝ) | |
18 | 2re 8989 | . . . . . . . . . 10 ⊢ 2 ∈ ℝ | |
19 | 18 | a1i 9 | . . . . . . . . 9 ⊢ ((𝑁 ∈ ℕ ∧ 𝑥 ∈ (2...(𝑁 − 1))) → 2 ∈ ℝ) |
20 | 16 | zred 9375 | . . . . . . . . 9 ⊢ ((𝑁 ∈ ℕ ∧ 𝑥 ∈ (2...(𝑁 − 1))) → 𝑥 ∈ ℝ) |
21 | 1le2 9127 | . . . . . . . . . 10 ⊢ 1 ≤ 2 | |
22 | 21 | a1i 9 | . . . . . . . . 9 ⊢ ((𝑁 ∈ ℕ ∧ 𝑥 ∈ (2...(𝑁 − 1))) → 1 ≤ 2) |
23 | elfzle1 10027 | . . . . . . . . . 10 ⊢ (𝑥 ∈ (2...(𝑁 − 1)) → 2 ≤ 𝑥) | |
24 | 23 | adantl 277 | . . . . . . . . 9 ⊢ ((𝑁 ∈ ℕ ∧ 𝑥 ∈ (2...(𝑁 − 1))) → 2 ≤ 𝑥) |
25 | 17, 19, 20, 22, 24 | letrd 8081 | . . . . . . . 8 ⊢ ((𝑁 ∈ ℕ ∧ 𝑥 ∈ (2...(𝑁 − 1))) → 1 ≤ 𝑥) |
26 | elnnz1 9276 | . . . . . . . 8 ⊢ (𝑥 ∈ ℕ ↔ (𝑥 ∈ ℤ ∧ 1 ≤ 𝑥)) | |
27 | 16, 25, 26 | sylanbrc 417 | . . . . . . 7 ⊢ ((𝑁 ∈ ℕ ∧ 𝑥 ∈ (2...(𝑁 − 1))) → 𝑥 ∈ ℕ) |
28 | 11 | adantr 276 | . . . . . . 7 ⊢ ((𝑁 ∈ ℕ ∧ 𝑥 ∈ (2...(𝑁 − 1))) → 𝑁 ∈ ℤ) |
29 | dvdsdc 11805 | . . . . . . 7 ⊢ ((𝑥 ∈ ℕ ∧ 𝑁 ∈ ℤ) → DECID 𝑥 ∥ 𝑁) | |
30 | 27, 28, 29 | syl2anc 411 | . . . . . 6 ⊢ ((𝑁 ∈ ℕ ∧ 𝑥 ∈ (2...(𝑁 − 1))) → DECID 𝑥 ∥ 𝑁) |
31 | dcn 842 | . . . . . 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 2550 | . . . 4 ⊢ (𝑁 ∈ ℕ → ∀𝑥 ∈ (2...(𝑁 − 1))DECID ¬ 𝑥 ∥ 𝑁) |
34 | dcfi 6980 | . . . 4 ⊢ (((2...(𝑁 − 1)) ∈ Fin ∧ ∀𝑥 ∈ (2...(𝑁 − 1))DECID ¬ 𝑥 ∥ 𝑁) → DECID ∀𝑥 ∈ (2...(𝑁 − 1)) ¬ 𝑥 ∥ 𝑁) | |
35 | 14, 33, 34 | syl2anc 411 | . . 3 ⊢ (𝑁 ∈ ℕ → DECID ∀𝑥 ∈ (2...(𝑁 − 1)) ¬ 𝑥 ∥ 𝑁) |
36 | dcan2 934 | . . 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 12118 | . . 3 ⊢ (𝑁 ∈ ℙ ↔ (𝑁 ∈ (ℤ≥‘2) ∧ ∀𝑥 ∈ (2...(𝑁 − 1)) ¬ 𝑥 ∥ 𝑁)) | |
39 | 38 | dcbii 840 | . 2 ⊢ (DECID 𝑁 ∈ ℙ ↔ DECID (𝑁 ∈ (ℤ≥‘2) ∧ ∀𝑥 ∈ (2...(𝑁 − 1)) ¬ 𝑥 ∥ 𝑁)) |
40 | 37, 39 | sylibr 134 | 1 ⊢ (𝑁 ∈ ℕ → DECID 𝑁 ∈ ℙ) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 104 ∨ wo 708 DECID wdc 834 = wceq 1353 ∈ wcel 2148 ∀wral 2455 class class class wbr 4004 ‘cfv 5217 (class class class)co 5875 Fincfn 6740 ℝcr 7810 1c1 7812 ≤ cle 7993 − cmin 8128 ℕcn 8919 2c2 8970 ℤcz 9253 ℤ≥cuz 9528 ...cfz 10008 ∥ 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-fin 6743 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-fl 10270 df-mod 10323 df-seqfrec 10446 df-exp 10520 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: pcmptcl 12340 pcmpt 12341 1arith 12365 prminf 12456 lgsval 14408 lgsfvalg 14409 lgsfcl2 14410 lgsval2lem 14414 lgsval4lem 14415 lgsneg 14428 lgsmod 14430 lgsdir 14439 lgsdilem2 14440 lgsdi 14441 lgsne0 14442 |
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