prmdc - Intuitionistic Logic Explorer

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Theorem prmdc 12144

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 9620	. . . . . . 7 ⊢ ¬ 1 ∈ (ℤ_≥‘2)
2		eleq1 2250	. . . . . . 7 ⊢ (𝑁 = 1 → (𝑁 ∈ (ℤ_≥‘2) ↔ 1 ∈ (ℤ_≥‘2)))
3	1, 2	mtbiri 676	. . . . . 6 ⊢ (𝑁 = 1 → ¬ 𝑁 ∈ (ℤ_≥‘2))
4	3	orim1i 761	. . . . 5 ⊢ ((𝑁 = 1 ∨ 𝑁 ∈ (ℤ_≥‘2)) → (¬ 𝑁 ∈ (ℤ_≥‘2) ∨ 𝑁 ∈ (ℤ_≥‘2)))
5	4	orcomd 730	. . . 4 ⊢ ((𝑁 = 1 ∨ 𝑁 ∈ (ℤ_≥‘2)) → (𝑁 ∈ (ℤ_≥‘2) ∨ ¬ 𝑁 ∈ (ℤ_≥‘2)))
6		elnn1uz2 9621	. . . 4 ⊢ (𝑁 ∈ ℕ ↔ (𝑁 = 1 ∨ 𝑁 ∈ (ℤ_≥‘2)))
7		df-dc 836	. . . 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 9295	. . . . . 6 ⊢ 2 ∈ ℤ
10	9	a1i 9	. . . . 5 ⊢ (𝑁 ∈ ℕ → 2 ∈ ℤ)
11		nnz 9286	. . . . . 6 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℤ)
12		peano2zm 9305	. . . . . 6 ⊢ (𝑁 ∈ ℤ → (𝑁 − 1) ∈ ℤ)
13	11, 12	syl 14	. . . . 5 ⊢ (𝑁 ∈ ℕ → (𝑁 − 1) ∈ ℤ)
14	10, 13	fzfigd 10445	. . . 4 ⊢ (𝑁 ∈ ℕ → (2...(𝑁 − 1)) ∈ Fin)
15		elfzelz 10039	. . . . . . . . 9 ⊢ (𝑥 ∈ (2...(𝑁 − 1)) → 𝑥 ∈ ℤ)
16	15	adantl 277	. . . . . . . 8 ⊢ ((𝑁 ∈ ℕ ∧ 𝑥 ∈ (2...(𝑁 − 1))) → 𝑥 ∈ ℤ)
17		1red 7986	. . . . . . . . 9 ⊢ ((𝑁 ∈ ℕ ∧ 𝑥 ∈ (2...(𝑁 − 1))) → 1 ∈ ℝ)
18		2re 9003	. . . . . . . . . 10 ⊢ 2 ∈ ℝ
19	18	a1i 9	. . . . . . . . 9 ⊢ ((𝑁 ∈ ℕ ∧ 𝑥 ∈ (2...(𝑁 − 1))) → 2 ∈ ℝ)
20	16	zred 9389	. . . . . . . . 9 ⊢ ((𝑁 ∈ ℕ ∧ 𝑥 ∈ (2...(𝑁 − 1))) → 𝑥 ∈ ℝ)
21		1le2 9141	. . . . . . . . . 10 ⊢ 1 ≤ 2
22	21	a1i 9	. . . . . . . . 9 ⊢ ((𝑁 ∈ ℕ ∧ 𝑥 ∈ (2...(𝑁 − 1))) → 1 ≤ 2)
23		elfzle1 10041	. . . . . . . . . 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 8095	. . . . . . . 8 ⊢ ((𝑁 ∈ ℕ ∧ 𝑥 ∈ (2...(𝑁 − 1))) → 1 ≤ 𝑥)
26		elnnz1 9290	. . . . . . . 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 11819	. . . . . . 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 843	. . . . . 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 2560	. . . 4 ⊢ (𝑁 ∈ ℕ → ∀𝑥 ∈ (2...(𝑁 − 1))DECID ¬ 𝑥 ∥ 𝑁)
34		dcfi 6994	. . . 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 935	. . 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 12132	. . 3 ⊢ (𝑁 ∈ ℙ ↔ (𝑁 ∈ (ℤ_≥‘2) ∧ ∀𝑥 ∈ (2...(𝑁 − 1)) ¬ 𝑥 ∥ 𝑁))
39	38	dcbii 841	. 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 709 DECID wdc 835 = wceq 1363 ∈ wcel 2158 ∀wral 2465 class class class wbr 4015 ‘cfv 5228 (class class class)co 5888 Fincfn 6754 ℝcr 7824 1c1 7826 ≤ cle 8007 − cmin 8142 ℕcn 8933 2c2 8984 ℤcz 9267 ℤ_≥cuz 9542 ...cfz 10022 ∥ cdvds 11808 ℙcprime 12121

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 615 ax-in2 616 ax-io 710 ax-5 1457 ax-7 1458 ax-gen 1459 ax-ie1 1503 ax-ie2 1504 ax-8 1514 ax-10 1515 ax-11 1516 ax-i12 1517 ax-bndl 1519 ax-4 1520 ax-17 1536 ax-i9 1540 ax-ial 1544 ax-i5r 1545 ax-13 2160 ax-14 2161 ax-ext 2169 ax-coll 4130 ax-sep 4133 ax-nul 4141 ax-pow 4186 ax-pr 4221 ax-un 4445 ax-setind 4548 ax-iinf 4599 ax-cnex 7916 ax-resscn 7917 ax-1cn 7918 ax-1re 7919 ax-icn 7920 ax-addcl 7921 ax-addrcl 7922 ax-mulcl 7923 ax-mulrcl 7924 ax-addcom 7925 ax-mulcom 7926 ax-addass 7927 ax-mulass 7928 ax-distr 7929 ax-i2m1 7930 ax-0lt1 7931 ax-1rid 7932 ax-0id 7933 ax-rnegex 7934 ax-precex 7935 ax-cnre 7936 ax-pre-ltirr 7937 ax-pre-ltwlin 7938 ax-pre-lttrn 7939 ax-pre-apti 7940 ax-pre-ltadd 7941 ax-pre-mulgt0 7942 ax-pre-mulext 7943 ax-arch 7944 ax-caucvg 7945

This theorem depends on definitions: df-bi 117 df-stab 832 df-dc 836 df-3or 980 df-3an 981 df-tru 1366 df-fal 1369 df-nf 1471 df-sb 1773 df-eu 2039 df-mo 2040 df-clab 2174 df-cleq 2180 df-clel 2183 df-nfc 2318 df-ne 2358 df-nel 2453 df-ral 2470 df-rex 2471 df-reu 2472 df-rmo 2473 df-rab 2474 df-v 2751 df-sbc 2975 df-csb 3070 df-dif 3143 df-un 3145 df-in 3147 df-ss 3154 df-nul 3435 df-if 3547 df-pw 3589 df-sn 3610 df-pr 3611 df-op 3613 df-uni 3822 df-int 3857 df-iun 3900 df-br 4016 df-opab 4077 df-mpt 4078 df-tr 4114 df-id 4305 df-po 4308 df-iso 4309 df-iord 4378 df-on 4380 df-ilim 4381 df-suc 4383 df-iom 4602 df-xp 4644 df-rel 4645 df-cnv 4646 df-co 4647 df-dm 4648 df-rn 4649 df-res 4650 df-ima 4651 df-iota 5190 df-fun 5230 df-fn 5231 df-f 5232 df-f1 5233 df-fo 5234 df-f1o 5235 df-fv 5236 df-riota 5844 df-ov 5891 df-oprab 5892 df-mpo 5893 df-1st 6155 df-2nd 6156 df-recs 6320 df-frec 6406 df-1o 6431 df-2o 6432 df-er 6549 df-en 6755 df-fin 6757 df-pnf 8008 df-mnf 8009 df-xr 8010 df-ltxr 8011 df-le 8012 df-sub 8144 df-neg 8145 df-reap 8546 df-ap 8553 df-div 8644 df-inn 8934 df-2 8992 df-3 8993 df-4 8994 df-n0 9191 df-z 9268 df-uz 9543 df-q 9634 df-rp 9668 df-fz 10023 df-fl 10284 df-mod 10337 df-seqfrec 10460 df-exp 10534 df-cj 10865 df-re 10866 df-im 10867 df-rsqrt 11021 df-abs 11022 df-dvds 11809 df-prm 12122

This theorem is referenced by: pcmptcl 12354 pcmpt 12355 1arith 12379 prminf 12470 lgsval 14758 lgsfvalg 14759 lgsfcl2 14760 lgsval2lem 14764 lgsval4lem 14765 lgsneg 14778 lgsmod 14780 lgsdir 14789 lgsdilem2 14790 lgsdi 14791 lgsne0 14792

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