prmdc - Intuitionistic Logic Explorer

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Theorem prmdc 12129

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 9605	. . . . . . 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 9606	. . . 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 9280	. . . . . 6 ⊢ 2 ∈ ℤ
10	9	a1i 9	. . . . 5 ⊢ (𝑁 ∈ ℕ → 2 ∈ ℤ)
11		nnz 9271	. . . . . 6 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℤ)
12		peano2zm 9290	. . . . . 6 ⊢ (𝑁 ∈ ℤ → (𝑁 − 1) ∈ ℤ)
13	11, 12	syl 14	. . . . 5 ⊢ (𝑁 ∈ ℕ → (𝑁 − 1) ∈ ℤ)
14	10, 13	fzfigd 10430	. . . 4 ⊢ (𝑁 ∈ ℕ → (2...(𝑁 − 1)) ∈ Fin)
15		elfzelz 10024	. . . . . . . . 9 ⊢ (𝑥 ∈ (2...(𝑁 − 1)) → 𝑥 ∈ ℤ)
16	15	adantl 277	. . . . . . . 8 ⊢ ((𝑁 ∈ ℕ ∧ 𝑥 ∈ (2...(𝑁 − 1))) → 𝑥 ∈ ℤ)
17		1red 7971	. . . . . . . . 9 ⊢ ((𝑁 ∈ ℕ ∧ 𝑥 ∈ (2...(𝑁 − 1))) → 1 ∈ ℝ)
18		2re 8988	. . . . . . . . . 10 ⊢ 2 ∈ ℝ
19	18	a1i 9	. . . . . . . . 9 ⊢ ((𝑁 ∈ ℕ ∧ 𝑥 ∈ (2...(𝑁 − 1))) → 2 ∈ ℝ)
20	16	zred 9374	. . . . . . . . 9 ⊢ ((𝑁 ∈ ℕ ∧ 𝑥 ∈ (2...(𝑁 − 1))) → 𝑥 ∈ ℝ)
21		1le2 9126	. . . . . . . . . 10 ⊢ 1 ≤ 2
22	21	a1i 9	. . . . . . . . 9 ⊢ ((𝑁 ∈ ℕ ∧ 𝑥 ∈ (2...(𝑁 − 1))) → 1 ≤ 2)
23		elfzle1 10026	. . . . . . . . . 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 8080	. . . . . . . 8 ⊢ ((𝑁 ∈ ℕ ∧ 𝑥 ∈ (2...(𝑁 − 1))) → 1 ≤ 𝑥)
26		elnnz1 9275	. . . . . . . 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 11804	. . . . . . 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 6979	. . . 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 12117	. . 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 4003 ‘cfv 5216 (class class class)co 5874 Fincfn 6739 ℝcr 7809 1c1 7811 ≤ cle 7992 − cmin 8127 ℕcn 8918 2c2 8969 ℤcz 9252 ℤ_≥cuz 9527 ...cfz 10007 ∥ cdvds 11793 ℙcprime 12106

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 4118 ax-sep 4121 ax-nul 4129 ax-pow 4174 ax-pr 4209 ax-un 4433 ax-setind 4536 ax-iinf 4587 ax-cnex 7901 ax-resscn 7902 ax-1cn 7903 ax-1re 7904 ax-icn 7905 ax-addcl 7906 ax-addrcl 7907 ax-mulcl 7908 ax-mulrcl 7909 ax-addcom 7910 ax-mulcom 7911 ax-addass 7912 ax-mulass 7913 ax-distr 7914 ax-i2m1 7915 ax-0lt1 7916 ax-1rid 7917 ax-0id 7918 ax-rnegex 7919 ax-precex 7920 ax-cnre 7921 ax-pre-ltirr 7922 ax-pre-ltwlin 7923 ax-pre-lttrn 7924 ax-pre-apti 7925 ax-pre-ltadd 7926 ax-pre-mulgt0 7927 ax-pre-mulext 7928 ax-arch 7929 ax-caucvg 7930

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 2739 df-sbc 2963 df-csb 3058 df-dif 3131 df-un 3133 df-in 3135 df-ss 3142 df-nul 3423 df-if 3535 df-pw 3577 df-sn 3598 df-pr 3599 df-op 3601 df-uni 3810 df-int 3845 df-iun 3888 df-br 4004 df-opab 4065 df-mpt 4066 df-tr 4102 df-id 4293 df-po 4296 df-iso 4297 df-iord 4366 df-on 4368 df-ilim 4369 df-suc 4371 df-iom 4590 df-xp 4632 df-rel 4633 df-cnv 4634 df-co 4635 df-dm 4636 df-rn 4637 df-res 4638 df-ima 4639 df-iota 5178 df-fun 5218 df-fn 5219 df-f 5220 df-f1 5221 df-fo 5222 df-f1o 5223 df-fv 5224 df-riota 5830 df-ov 5877 df-oprab 5878 df-mpo 5879 df-1st 6140 df-2nd 6141 df-recs 6305 df-frec 6391 df-1o 6416 df-2o 6417 df-er 6534 df-en 6740 df-fin 6742 df-pnf 7993 df-mnf 7994 df-xr 7995 df-ltxr 7996 df-le 7997 df-sub 8129 df-neg 8130 df-reap 8531 df-ap 8538 df-div 8629 df-inn 8919 df-2 8977 df-3 8978 df-4 8979 df-n0 9176 df-z 9253 df-uz 9528 df-q 9619 df-rp 9653 df-fz 10008 df-fl 10269 df-mod 10322 df-seqfrec 10445 df-exp 10519 df-cj 10850 df-re 10851 df-im 10852 df-rsqrt 11006 df-abs 11007 df-dvds 11794 df-prm 12107

This theorem is referenced by: pcmptcl 12339 pcmpt 12340 1arith 12364 prminf 12455 lgsval 14375 lgsfvalg 14376 lgsfcl2 14377 lgsval2lem 14381 lgsval4lem 14382 lgsneg 14395 lgsmod 14397 lgsdir 14406 lgsdilem2 14407 lgsdi 14408 lgsne0 14409

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