prmdc - Intuitionistic Logic Explorer

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Theorem prmdc 12084

Description: Primality is decidable. (Contributed by Jim Kingdon, 30-Sep-2024.)

**Assertion**
Ref	Expression
prmdc	⊢ (𝑁 ∈ ℕ → DECID 𝑁 ∈ ℙ)

Proof of Theorem prmdc Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		1nuz2 9565	. . . . . . 7 ⊢ ¬ 1 ∈ (ℤ_≥‘2)
2		eleq1 2233	. . . . . . 7 ⊢ (𝑁 = 1 → (𝑁 ∈ (ℤ_≥‘2) ↔ 1 ∈ (ℤ_≥‘2)))
3	1, 2	mtbiri 670	. . . . . 6 ⊢ (𝑁 = 1 → ¬ 𝑁 ∈ (ℤ_≥‘2))
4	3	orim1i 755	. . . . 5 ⊢ ((𝑁 = 1 ∨ 𝑁 ∈ (ℤ_≥‘2)) → (¬ 𝑁 ∈ (ℤ_≥‘2) ∨ 𝑁 ∈ (ℤ_≥‘2)))
5	4	orcomd 724	. . . 4 ⊢ ((𝑁 = 1 ∨ 𝑁 ∈ (ℤ_≥‘2)) → (𝑁 ∈ (ℤ_≥‘2) ∨ ¬ 𝑁 ∈ (ℤ_≥‘2)))
6		elnn1uz2 9566	. . . 4 ⊢ (𝑁 ∈ ℕ ↔ (𝑁 = 1 ∨ 𝑁 ∈ (ℤ_≥‘2)))
7		df-dc 830	. . . 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 9240	. . . . . 6 ⊢ 2 ∈ ℤ
10	9	a1i 9	. . . . 5 ⊢ (𝑁 ∈ ℕ → 2 ∈ ℤ)
11		nnz 9231	. . . . . 6 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℤ)
12		peano2zm 9250	. . . . . 6 ⊢ (𝑁 ∈ ℤ → (𝑁 − 1) ∈ ℤ)
13	11, 12	syl 14	. . . . 5 ⊢ (𝑁 ∈ ℕ → (𝑁 − 1) ∈ ℤ)
14	10, 13	fzfigd 10387	. . . 4 ⊢ (𝑁 ∈ ℕ → (2...(𝑁 − 1)) ∈ Fin)
15		elfzelz 9981	. . . . . . . . 9 ⊢ (𝑥 ∈ (2...(𝑁 − 1)) → 𝑥 ∈ ℤ)
16	15	adantl 275	. . . . . . . 8 ⊢ ((𝑁 ∈ ℕ ∧ 𝑥 ∈ (2...(𝑁 − 1))) → 𝑥 ∈ ℤ)
17		1red 7935	. . . . . . . . 9 ⊢ ((𝑁 ∈ ℕ ∧ 𝑥 ∈ (2...(𝑁 − 1))) → 1 ∈ ℝ)
18		2re 8948	. . . . . . . . . 10 ⊢ 2 ∈ ℝ
19	18	a1i 9	. . . . . . . . 9 ⊢ ((𝑁 ∈ ℕ ∧ 𝑥 ∈ (2...(𝑁 − 1))) → 2 ∈ ℝ)
20	16	zred 9334	. . . . . . . . 9 ⊢ ((𝑁 ∈ ℕ ∧ 𝑥 ∈ (2...(𝑁 − 1))) → 𝑥 ∈ ℝ)
21		1le2 9086	. . . . . . . . . 10 ⊢ 1 ≤ 2
22	21	a1i 9	. . . . . . . . 9 ⊢ ((𝑁 ∈ ℕ ∧ 𝑥 ∈ (2...(𝑁 − 1))) → 1 ≤ 2)
23		elfzle1 9983	. . . . . . . . . 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 8043	. . . . . . . 8 ⊢ ((𝑁 ∈ ℕ ∧ 𝑥 ∈ (2...(𝑁 − 1))) → 1 ≤ 𝑥)
26		elnnz1 9235	. . . . . . . 8 ⊢ (𝑥 ∈ ℕ ↔ (𝑥 ∈ ℤ ∧ 1 ≤ 𝑥))
27	16, 25, 26	sylanbrc 415	. . . . . . 7 ⊢ ((𝑁 ∈ ℕ ∧ 𝑥 ∈ (2...(𝑁 − 1))) → 𝑥 ∈ ℕ)
28	11	adantr 274	. . . . . . 7 ⊢ ((𝑁 ∈ ℕ ∧ 𝑥 ∈ (2...(𝑁 − 1))) → 𝑁 ∈ ℤ)
29		dvdsdc 11760	. . . . . . 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 837	. . . . . 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 2543	. . . 4 ⊢ (𝑁 ∈ ℕ → ∀𝑥 ∈ (2...(𝑁 − 1))DECID ¬ 𝑥 ∥ 𝑁)
34		dcfi 6958	. . . 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		dcan2 929	. . 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 12072	. . 3 ⊢ (𝑁 ∈ ℙ ↔ (𝑁 ∈ (ℤ_≥‘2) ∧ ∀𝑥 ∈ (2...(𝑁 − 1)) ¬ 𝑥 ∥ 𝑁))
39	38	dcbii 835	. 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 703 DECID wdc 829 = wceq 1348 ∈ wcel 2141 ∀wral 2448 class class class wbr 3989 ‘cfv 5198 (class class class)co 5853 Fincfn 6718 ℝcr 7773 1c1 7775 ≤ cle 7955 − cmin 8090 ℕcn 8878 2c2 8929 ℤcz 9212 ℤ_≥cuz 9487 ...cfz 9965 ∥ cdvds 11749 ℙcprime 12061

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 609 ax-in2 610 ax-io 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-13 2143 ax-14 2144 ax-ext 2152 ax-coll 4104 ax-sep 4107 ax-nul 4115 ax-pow 4160 ax-pr 4194 ax-un 4418 ax-setind 4521 ax-iinf 4572 ax-cnex 7865 ax-resscn 7866 ax-1cn 7867 ax-1re 7868 ax-icn 7869 ax-addcl 7870 ax-addrcl 7871 ax-mulcl 7872 ax-mulrcl 7873 ax-addcom 7874 ax-mulcom 7875 ax-addass 7876 ax-mulass 7877 ax-distr 7878 ax-i2m1 7879 ax-0lt1 7880 ax-1rid 7881 ax-0id 7882 ax-rnegex 7883 ax-precex 7884 ax-cnre 7885 ax-pre-ltirr 7886 ax-pre-ltwlin 7887 ax-pre-lttrn 7888 ax-pre-apti 7889 ax-pre-ltadd 7890 ax-pre-mulgt0 7891 ax-pre-mulext 7892 ax-arch 7893 ax-caucvg 7894

This theorem depends on definitions: df-bi 116 df-stab 826 df-dc 830 df-3or 974 df-3an 975 df-tru 1351 df-fal 1354 df-nf 1454 df-sb 1756 df-eu 2022 df-mo 2023 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-ne 2341 df-nel 2436 df-ral 2453 df-rex 2454 df-reu 2455 df-rmo 2456 df-rab 2457 df-v 2732 df-sbc 2956 df-csb 3050 df-dif 3123 df-un 3125 df-in 3127 df-ss 3134 df-nul 3415 df-if 3527 df-pw 3568 df-sn 3589 df-pr 3590 df-op 3592 df-uni 3797 df-int 3832 df-iun 3875 df-br 3990 df-opab 4051 df-mpt 4052 df-tr 4088 df-id 4278 df-po 4281 df-iso 4282 df-iord 4351 df-on 4353 df-ilim 4354 df-suc 4356 df-iom 4575 df-xp 4617 df-rel 4618 df-cnv 4619 df-co 4620 df-dm 4621 df-rn 4622 df-res 4623 df-ima 4624 df-iota 5160 df-fun 5200 df-fn 5201 df-f 5202 df-f1 5203 df-fo 5204 df-f1o 5205 df-fv 5206 df-riota 5809 df-ov 5856 df-oprab 5857 df-mpo 5858 df-1st 6119 df-2nd 6120 df-recs 6284 df-frec 6370 df-1o 6395 df-2o 6396 df-er 6513 df-en 6719 df-fin 6721 df-pnf 7956 df-mnf 7957 df-xr 7958 df-ltxr 7959 df-le 7960 df-sub 8092 df-neg 8093 df-reap 8494 df-ap 8501 df-div 8590 df-inn 8879 df-2 8937 df-3 8938 df-4 8939 df-n0 9136 df-z 9213 df-uz 9488 df-q 9579 df-rp 9611 df-fz 9966 df-fl 10226 df-mod 10279 df-seqfrec 10402 df-exp 10476 df-cj 10806 df-re 10807 df-im 10808 df-rsqrt 10962 df-abs 10963 df-dvds 11750 df-prm 12062

This theorem is referenced by: pcmptcl 12294 pcmpt 12295 1arith 12319 prminf 12410 lgsval 13699 lgsfvalg 13700 lgsfcl2 13701 lgsval2lem 13705 lgsval4lem 13706 lgsneg 13719 lgsmod 13721 lgsdir 13730 lgsdilem2 13731 lgsdi 13732 lgsne0 13733

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