isprm3 - Intuitionistic Logic Explorer

	Intuitionistic Logic Explorer		< Previous Next > Nearby theorems
Mirrors > Home > ILE Home > Th. List > isprm3		GIF version

Theorem isprm3 12843

Description: The predicate "is a prime number". A prime number is an integer greater than or equal to 2 with no divisors strictly between 1 and itself. (Contributed by Paul Chapman, 26-Oct-2012.)

**Assertion**
Ref	Expression
isprm3	⊢ (𝑃 ∈ ℙ ↔ (𝑃 ∈ (ℤ_≥‘2) ∧ ∀𝑧 ∈ (2...(𝑃 − 1)) ¬ 𝑧 ∥ 𝑃))

Distinct variable group: 𝑧,𝑃

**Proof of Theorem isprm3**
Step	Hyp	Ref	Expression
1		isprm2 12842	. 2 ⊢ (𝑃 ∈ ℙ ↔ (𝑃 ∈ (ℤ_≥‘2) ∧ ∀𝑧 ∈ ℕ (𝑧 ∥ 𝑃 → (𝑧 = 1 ∨ 𝑧 = 𝑃))))
2		dvdszrcl 12506	. . . . . . . . . . 11 ⊢ (𝑧 ∥ 𝑃 → (𝑧 ∈ ℤ ∧ 𝑃 ∈ ℤ))
3	2	simpld 112	. . . . . . . . . 10 ⊢ (𝑧 ∥ 𝑃 → 𝑧 ∈ ℤ)
4		1zzd 9624	. . . . . . . . . 10 ⊢ ((𝑃 ∈ (ℤ_≥‘2) ∧ 𝑧 ∥ 𝑃) → 1 ∈ ℤ)
5		zdceq 9673	. . . . . . . . . 10 ⊢ ((𝑧 ∈ ℤ ∧ 1 ∈ ℤ) → DECID 𝑧 = 1)
6	3, 4, 5	syl2an2 598	. . . . . . . . 9 ⊢ ((𝑃 ∈ (ℤ_≥‘2) ∧ 𝑧 ∥ 𝑃) → DECID 𝑧 = 1)
7	2	simprd 114	. . . . . . . . . . 11 ⊢ (𝑧 ∥ 𝑃 → 𝑃 ∈ ℤ)
8	7	adantl 277	. . . . . . . . . 10 ⊢ ((𝑃 ∈ (ℤ_≥‘2) ∧ 𝑧 ∥ 𝑃) → 𝑃 ∈ ℤ)
9		zdceq 9673	. . . . . . . . . 10 ⊢ ((𝑧 ∈ ℤ ∧ 𝑃 ∈ ℤ) → DECID 𝑧 = 𝑃)
10	3, 8, 9	syl2an2 598	. . . . . . . . 9 ⊢ ((𝑃 ∈ (ℤ_≥‘2) ∧ 𝑧 ∥ 𝑃) → DECID 𝑧 = 𝑃)
11		dcor 944	. . . . . . . . 9 ⊢ (DECID 𝑧 = 1 → (DECID 𝑧 = 𝑃 → DECID (𝑧 = 1 ∨ 𝑧 = 𝑃)))
12	6, 10, 11	sylc 62	. . . . . . . 8 ⊢ ((𝑃 ∈ (ℤ_≥‘2) ∧ 𝑧 ∥ 𝑃) → DECID (𝑧 = 1 ∨ 𝑧 = 𝑃))
13		imandc 897	. . . . . . . 8 ⊢ (DECID (𝑧 = 1 ∨ 𝑧 = 𝑃) → ((𝑧 ∈ ℕ → (𝑧 = 1 ∨ 𝑧 = 𝑃)) ↔ ¬ (𝑧 ∈ ℕ ∧ ¬ (𝑧 = 1 ∨ 𝑧 = 𝑃))))
14	12, 13	syl 14	. . . . . . 7 ⊢ ((𝑃 ∈ (ℤ_≥‘2) ∧ 𝑧 ∥ 𝑃) → ((𝑧 ∈ ℕ → (𝑧 = 1 ∨ 𝑧 = 𝑃)) ↔ ¬ (𝑧 ∈ ℕ ∧ ¬ (𝑧 = 1 ∨ 𝑧 = 𝑃))))
15		eluz2nn 9919	. . . . . . . . . . . . . . . 16 ⊢ (𝑃 ∈ (ℤ_≥‘2) → 𝑃 ∈ ℕ)
16		nnz 9616	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑧 ∈ ℕ → 𝑧 ∈ ℤ)
17		dvdsle 12558	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑧 ∈ ℤ ∧ 𝑃 ∈ ℕ) → (𝑧 ∥ 𝑃 → 𝑧 ≤ 𝑃))
18	16, 17	sylan 283	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑧 ∈ ℕ ∧ 𝑃 ∈ ℕ) → (𝑧 ∥ 𝑃 → 𝑧 ≤ 𝑃))
19		nnge1 9280	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑧 ∈ ℕ → 1 ≤ 𝑧)
20	19	adantr 276	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑧 ∈ ℕ ∧ 𝑃 ∈ ℕ) → 1 ≤ 𝑧)
21	18, 20	jctild 316	. . . . . . . . . . . . . . . 16 ⊢ ((𝑧 ∈ ℕ ∧ 𝑃 ∈ ℕ) → (𝑧 ∥ 𝑃 → (1 ≤ 𝑧 ∧ 𝑧 ≤ 𝑃)))
22	15, 21	sylan2 286	. . . . . . . . . . . . . . 15 ⊢ ((𝑧 ∈ ℕ ∧ 𝑃 ∈ (ℤ_≥‘2)) → (𝑧 ∥ 𝑃 → (1 ≤ 𝑧 ∧ 𝑧 ≤ 𝑃)))
23		nnz 9616	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑃 ∈ ℕ → 𝑃 ∈ ℤ)
24		zre 9601	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑧 ∈ ℤ → 𝑧 ∈ ℝ)
25		1re 8289	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ 1 ∈ ℝ
26		leltap 8917	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((1 ∈ ℝ ∧ 𝑧 ∈ ℝ ∧ 1 ≤ 𝑧) → (1 < 𝑧 ↔ 𝑧 # 1))
27	25, 26	mp3an1 1361	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑧 ∈ ℝ ∧ 1 ≤ 𝑧) → (1 < 𝑧 ↔ 𝑧 # 1))
28	24, 27	sylan 283	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑧 ∈ ℤ ∧ 1 ≤ 𝑧) → (1 < 𝑧 ↔ 𝑧 # 1))
29		1z 9623	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ 1 ∈ ℤ
30		zapne 9672	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝑧 ∈ ℤ ∧ 1 ∈ ℤ) → (𝑧 # 1 ↔ 𝑧 ≠ 1))
31	29, 30	mpan2 425	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑧 ∈ ℤ → (𝑧 # 1 ↔ 𝑧 ≠ 1))
32	31	adantr 276	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑧 ∈ ℤ ∧ 1 ≤ 𝑧) → (𝑧 # 1 ↔ 𝑧 ≠ 1))
33	28, 32	bitrd 188	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑧 ∈ ℤ ∧ 1 ≤ 𝑧) → (1 < 𝑧 ↔ 𝑧 ≠ 1))
34	33	3adant2 1043	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑧 ∈ ℤ ∧ 𝑃 ∈ ℤ ∧ 1 ≤ 𝑧) → (1 < 𝑧 ↔ 𝑧 ≠ 1))
35	34	3expia 1232	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑧 ∈ ℤ ∧ 𝑃 ∈ ℤ) → (1 ≤ 𝑧 → (1 < 𝑧 ↔ 𝑧 ≠ 1)))
36		zre 9601	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑃 ∈ ℤ → 𝑃 ∈ ℝ)
37		leltap 8917	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑧 ∈ ℝ ∧ 𝑃 ∈ ℝ ∧ 𝑧 ≤ 𝑃) → (𝑧 < 𝑃 ↔ 𝑃 # 𝑧))
38	24, 37	syl3an1 1307	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑧 ∈ ℤ ∧ 𝑃 ∈ ℝ ∧ 𝑧 ≤ 𝑃) → (𝑧 < 𝑃 ↔ 𝑃 # 𝑧))
39	36, 38	syl3an2 1308	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑧 ∈ ℤ ∧ 𝑃 ∈ ℤ ∧ 𝑧 ≤ 𝑃) → (𝑧 < 𝑃 ↔ 𝑃 # 𝑧))
40		zapne 9672	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑃 ∈ ℤ ∧ 𝑧 ∈ ℤ) → (𝑃 # 𝑧 ↔ 𝑃 ≠ 𝑧))
41	40	ancoms 268	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑧 ∈ ℤ ∧ 𝑃 ∈ ℤ) → (𝑃 # 𝑧 ↔ 𝑃 ≠ 𝑧))
42	41	3adant3 1044	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑧 ∈ ℤ ∧ 𝑃 ∈ ℤ ∧ 𝑧 ≤ 𝑃) → (𝑃 # 𝑧 ↔ 𝑃 ≠ 𝑧))
43	39, 42	bitrd 188	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑧 ∈ ℤ ∧ 𝑃 ∈ ℤ ∧ 𝑧 ≤ 𝑃) → (𝑧 < 𝑃 ↔ 𝑃 ≠ 𝑧))
44	43	3expia 1232	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑧 ∈ ℤ ∧ 𝑃 ∈ ℤ) → (𝑧 ≤ 𝑃 → (𝑧 < 𝑃 ↔ 𝑃 ≠ 𝑧)))
45	35, 44	anim12d 335	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑧 ∈ ℤ ∧ 𝑃 ∈ ℤ) → ((1 ≤ 𝑧 ∧ 𝑧 ≤ 𝑃) → ((1 < 𝑧 ↔ 𝑧 ≠ 1) ∧ (𝑧 < 𝑃 ↔ 𝑃 ≠ 𝑧))))
46	23, 45	sylan2 286	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑧 ∈ ℤ ∧ 𝑃 ∈ ℕ) → ((1 ≤ 𝑧 ∧ 𝑧 ≤ 𝑃) → ((1 < 𝑧 ↔ 𝑧 ≠ 1) ∧ (𝑧 < 𝑃 ↔ 𝑃 ≠ 𝑧))))
47		pm4.38 609	. . . . . . . . . . . . . . . . . 18 ⊢ (((1 < 𝑧 ↔ 𝑧 ≠ 1) ∧ (𝑧 < 𝑃 ↔ 𝑃 ≠ 𝑧)) → ((1 < 𝑧 ∧ 𝑧 < 𝑃) ↔ (𝑧 ≠ 1 ∧ 𝑃 ≠ 𝑧)))
48		df-ne 2415	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑧 ≠ 1 ↔ ¬ 𝑧 = 1)
49		nesym 2459	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑃 ≠ 𝑧 ↔ ¬ 𝑧 = 𝑃)
50	48, 49	anbi12i 460	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑧 ≠ 1 ∧ 𝑃 ≠ 𝑧) ↔ (¬ 𝑧 = 1 ∧ ¬ 𝑧 = 𝑃))
51		ioran 760	. . . . . . . . . . . . . . . . . . 19 ⊢ (¬ (𝑧 = 1 ∨ 𝑧 = 𝑃) ↔ (¬ 𝑧 = 1 ∧ ¬ 𝑧 = 𝑃))
52	50, 51	bitr4i 187	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑧 ≠ 1 ∧ 𝑃 ≠ 𝑧) ↔ ¬ (𝑧 = 1 ∨ 𝑧 = 𝑃))
53	47, 52	bitrdi 196	. . . . . . . . . . . . . . . . 17 ⊢ (((1 < 𝑧 ↔ 𝑧 ≠ 1) ∧ (𝑧 < 𝑃 ↔ 𝑃 ≠ 𝑧)) → ((1 < 𝑧 ∧ 𝑧 < 𝑃) ↔ ¬ (𝑧 = 1 ∨ 𝑧 = 𝑃)))
54	46, 53	syl6 33	. . . . . . . . . . . . . . . 16 ⊢ ((𝑧 ∈ ℤ ∧ 𝑃 ∈ ℕ) → ((1 ≤ 𝑧 ∧ 𝑧 ≤ 𝑃) → ((1 < 𝑧 ∧ 𝑧 < 𝑃) ↔ ¬ (𝑧 = 1 ∨ 𝑧 = 𝑃))))
55	16, 15, 54	syl2an 289	. . . . . . . . . . . . . . 15 ⊢ ((𝑧 ∈ ℕ ∧ 𝑃 ∈ (ℤ_≥‘2)) → ((1 ≤ 𝑧 ∧ 𝑧 ≤ 𝑃) → ((1 < 𝑧 ∧ 𝑧 < 𝑃) ↔ ¬ (𝑧 = 1 ∨ 𝑧 = 𝑃))))
56	22, 55	syld 45	. . . . . . . . . . . . . 14 ⊢ ((𝑧 ∈ ℕ ∧ 𝑃 ∈ (ℤ_≥‘2)) → (𝑧 ∥ 𝑃 → ((1 < 𝑧 ∧ 𝑧 < 𝑃) ↔ ¬ (𝑧 = 1 ∨ 𝑧 = 𝑃))))
57	56	imp 124	. . . . . . . . . . . . 13 ⊢ (((𝑧 ∈ ℕ ∧ 𝑃 ∈ (ℤ_≥‘2)) ∧ 𝑧 ∥ 𝑃) → ((1 < 𝑧 ∧ 𝑧 < 𝑃) ↔ ¬ (𝑧 = 1 ∨ 𝑧 = 𝑃)))
58		eluzelz 9884	. . . . . . . . . . . . . . 15 ⊢ (𝑃 ∈ (ℤ_≥‘2) → 𝑃 ∈ ℤ)
59		zltp1le 9652	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((1 ∈ ℤ ∧ 𝑧 ∈ ℤ) → (1 < 𝑧 ↔ (1 + 1) ≤ 𝑧))
60	29, 59	mpan 424	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑧 ∈ ℤ → (1 < 𝑧 ↔ (1 + 1) ≤ 𝑧))
61		df-2 9316	. . . . . . . . . . . . . . . . . . . 20 ⊢ 2 = (1 + 1)
62	61	breq1i 4121	. . . . . . . . . . . . . . . . . . 19 ⊢ (2 ≤ 𝑧 ↔ (1 + 1) ≤ 𝑧)
63	60, 62	bitr4di 198	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑧 ∈ ℤ → (1 < 𝑧 ↔ 2 ≤ 𝑧))
64	63	adantr 276	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑧 ∈ ℤ ∧ 𝑃 ∈ ℤ) → (1 < 𝑧 ↔ 2 ≤ 𝑧))
65		zltlem1 9655	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑧 ∈ ℤ ∧ 𝑃 ∈ ℤ) → (𝑧 < 𝑃 ↔ 𝑧 ≤ (𝑃 − 1)))
66	64, 65	anbi12d 473	. . . . . . . . . . . . . . . 16 ⊢ ((𝑧 ∈ ℤ ∧ 𝑃 ∈ ℤ) → ((1 < 𝑧 ∧ 𝑧 < 𝑃) ↔ (2 ≤ 𝑧 ∧ 𝑧 ≤ (𝑃 − 1))))
67		peano2zm 9635	. . . . . . . . . . . . . . . . 17 ⊢ (𝑃 ∈ ℤ → (𝑃 − 1) ∈ ℤ)
68		2z 9625	. . . . . . . . . . . . . . . . . 18 ⊢ 2 ∈ ℤ
69		elfz 10370	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑧 ∈ ℤ ∧ 2 ∈ ℤ ∧ (𝑃 − 1) ∈ ℤ) → (𝑧 ∈ (2...(𝑃 − 1)) ↔ (2 ≤ 𝑧 ∧ 𝑧 ≤ (𝑃 − 1))))
70	68, 69	mp3an2 1362	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑧 ∈ ℤ ∧ (𝑃 − 1) ∈ ℤ) → (𝑧 ∈ (2...(𝑃 − 1)) ↔ (2 ≤ 𝑧 ∧ 𝑧 ≤ (𝑃 − 1))))
71	67, 70	sylan2 286	. . . . . . . . . . . . . . . 16 ⊢ ((𝑧 ∈ ℤ ∧ 𝑃 ∈ ℤ) → (𝑧 ∈ (2...(𝑃 − 1)) ↔ (2 ≤ 𝑧 ∧ 𝑧 ≤ (𝑃 − 1))))
72	66, 71	bitr4d 191	. . . . . . . . . . . . . . 15 ⊢ ((𝑧 ∈ ℤ ∧ 𝑃 ∈ ℤ) → ((1 < 𝑧 ∧ 𝑧 < 𝑃) ↔ 𝑧 ∈ (2...(𝑃 − 1))))
73	16, 58, 72	syl2an 289	. . . . . . . . . . . . . 14 ⊢ ((𝑧 ∈ ℕ ∧ 𝑃 ∈ (ℤ_≥‘2)) → ((1 < 𝑧 ∧ 𝑧 < 𝑃) ↔ 𝑧 ∈ (2...(𝑃 − 1))))
74	73	adantr 276	. . . . . . . . . . . . 13 ⊢ (((𝑧 ∈ ℕ ∧ 𝑃 ∈ (ℤ_≥‘2)) ∧ 𝑧 ∥ 𝑃) → ((1 < 𝑧 ∧ 𝑧 < 𝑃) ↔ 𝑧 ∈ (2...(𝑃 − 1))))
75	57, 74	bitr3d 190	. . . . . . . . . . . 12 ⊢ (((𝑧 ∈ ℕ ∧ 𝑃 ∈ (ℤ_≥‘2)) ∧ 𝑧 ∥ 𝑃) → (¬ (𝑧 = 1 ∨ 𝑧 = 𝑃) ↔ 𝑧 ∈ (2...(𝑃 − 1))))
76	75	anasss 399	. . . . . . . . . . 11 ⊢ ((𝑧 ∈ ℕ ∧ (𝑃 ∈ (ℤ_≥‘2) ∧ 𝑧 ∥ 𝑃)) → (¬ (𝑧 = 1 ∨ 𝑧 = 𝑃) ↔ 𝑧 ∈ (2...(𝑃 − 1))))
77	76	expcom 116	. . . . . . . . . 10 ⊢ ((𝑃 ∈ (ℤ_≥‘2) ∧ 𝑧 ∥ 𝑃) → (𝑧 ∈ ℕ → (¬ (𝑧 = 1 ∨ 𝑧 = 𝑃) ↔ 𝑧 ∈ (2...(𝑃 − 1)))))
78	77	pm5.32d 450	. . . . . . . . 9 ⊢ ((𝑃 ∈ (ℤ_≥‘2) ∧ 𝑧 ∥ 𝑃) → ((𝑧 ∈ ℕ ∧ ¬ (𝑧 = 1 ∨ 𝑧 = 𝑃)) ↔ (𝑧 ∈ ℕ ∧ 𝑧 ∈ (2...(𝑃 − 1)))))
79		fzssuz 10423	. . . . . . . . . . . . 13 ⊢ (2...(𝑃 − 1)) ⊆ (ℤ_≥‘2)
80		2eluzge1 9929	. . . . . . . . . . . . . 14 ⊢ 2 ∈ (ℤ_≥‘1)
81		uzss 9896	. . . . . . . . . . . . . 14 ⊢ (2 ∈ (ℤ_≥‘1) → (ℤ_≥‘2) ⊆ (ℤ_≥‘1))
82	80, 81	ax-mp 5	. . . . . . . . . . . . 13 ⊢ (ℤ_≥‘2) ⊆ (ℤ_≥‘1)
83	79, 82	sstri 3251	. . . . . . . . . . . 12 ⊢ (2...(𝑃 − 1)) ⊆ (ℤ_≥‘1)
84		nnuz 9911	. . . . . . . . . . . 12 ⊢ ℕ = (ℤ_≥‘1)
85	83, 84	sseqtrri 3277	. . . . . . . . . . 11 ⊢ (2...(𝑃 − 1)) ⊆ ℕ
86	85	sseli 3238	. . . . . . . . . 10 ⊢ (𝑧 ∈ (2...(𝑃 − 1)) → 𝑧 ∈ ℕ)
87	86	pm4.71ri 392	. . . . . . . . 9 ⊢ (𝑧 ∈ (2...(𝑃 − 1)) ↔ (𝑧 ∈ ℕ ∧ 𝑧 ∈ (2...(𝑃 − 1))))
88	78, 87	bitr4di 198	. . . . . . . 8 ⊢ ((𝑃 ∈ (ℤ_≥‘2) ∧ 𝑧 ∥ 𝑃) → ((𝑧 ∈ ℕ ∧ ¬ (𝑧 = 1 ∨ 𝑧 = 𝑃)) ↔ 𝑧 ∈ (2...(𝑃 − 1))))
89	88	notbid 673	. . . . . . 7 ⊢ ((𝑃 ∈ (ℤ_≥‘2) ∧ 𝑧 ∥ 𝑃) → (¬ (𝑧 ∈ ℕ ∧ ¬ (𝑧 = 1 ∨ 𝑧 = 𝑃)) ↔ ¬ 𝑧 ∈ (2...(𝑃 − 1))))
90	14, 89	bitrd 188	. . . . . 6 ⊢ ((𝑃 ∈ (ℤ_≥‘2) ∧ 𝑧 ∥ 𝑃) → ((𝑧 ∈ ℕ → (𝑧 = 1 ∨ 𝑧 = 𝑃)) ↔ ¬ 𝑧 ∈ (2...(𝑃 − 1))))
91	90	pm5.74da 443	. . . . 5 ⊢ (𝑃 ∈ (ℤ_≥‘2) → ((𝑧 ∥ 𝑃 → (𝑧 ∈ ℕ → (𝑧 = 1 ∨ 𝑧 = 𝑃))) ↔ (𝑧 ∥ 𝑃 → ¬ 𝑧 ∈ (2...(𝑃 − 1)))))
92		bi2.04 248	. . . . 5 ⊢ ((𝑧 ∥ 𝑃 → (𝑧 ∈ ℕ → (𝑧 = 1 ∨ 𝑧 = 𝑃))) ↔ (𝑧 ∈ ℕ → (𝑧 ∥ 𝑃 → (𝑧 = 1 ∨ 𝑧 = 𝑃))))
93		con2b 675	. . . . 5 ⊢ ((𝑧 ∥ 𝑃 → ¬ 𝑧 ∈ (2...(𝑃 − 1))) ↔ (𝑧 ∈ (2...(𝑃 − 1)) → ¬ 𝑧 ∥ 𝑃))
94	91, 92, 93	3bitr3g 222	. . . 4 ⊢ (𝑃 ∈ (ℤ_≥‘2) → ((𝑧 ∈ ℕ → (𝑧 ∥ 𝑃 → (𝑧 = 1 ∨ 𝑧 = 𝑃))) ↔ (𝑧 ∈ (2...(𝑃 − 1)) → ¬ 𝑧 ∥ 𝑃)))
95	94	ralbidv2 2546	. . 3 ⊢ (𝑃 ∈ (ℤ_≥‘2) → (∀𝑧 ∈ ℕ (𝑧 ∥ 𝑃 → (𝑧 = 1 ∨ 𝑧 = 𝑃)) ↔ ∀𝑧 ∈ (2...(𝑃 − 1)) ¬ 𝑧 ∥ 𝑃))
96	95	pm5.32i 454	. 2 ⊢ ((𝑃 ∈ (ℤ_≥‘2) ∧ ∀𝑧 ∈ ℕ (𝑧 ∥ 𝑃 → (𝑧 = 1 ∨ 𝑧 = 𝑃))) ↔ (𝑃 ∈ (ℤ_≥‘2) ∧ ∀𝑧 ∈ (2...(𝑃 − 1)) ¬ 𝑧 ∥ 𝑃))
97	1, 96	bitri 184	1 ⊢ (𝑃 ∈ ℙ ↔ (𝑃 ∈ (ℤ_≥‘2) ∧ ∀𝑧 ∈ (2...(𝑃 − 1)) ¬ 𝑧 ∥ 𝑃))

Colors of variables: wff set class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 104 ↔ wb 105 ∨ wo 716 DECID wdc 842 ∧ w3a 1005 = wceq 1398 ∈ wcel 2205 ≠ wne 2414 ∀wral 2522 ⊆ wss 3214 class class class wbr 4114 ‘cfv 5357 (class class class)co 6058 ℝcr 8142 1c1 8144 + caddc 8146 < clt 8324 ≤ cle 8325 − cmin 8461 # cap 8873 ℕcn 9257 2c2 9308 ℤcz 9597 ℤ_≥cuz 9874 ...cfz 10364 ∥ cdvds 12501 ℙcprime 12832

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 619 ax-in2 620 ax-io 717 ax-5 1496 ax-7 1497 ax-gen 1498 ax-ie1 1542 ax-ie2 1543 ax-8 1553 ax-10 1554 ax-11 1555 ax-i12 1556 ax-bndl 1558 ax-4 1559 ax-17 1575 ax-i9 1579 ax-ial 1583 ax-i5r 1584 ax-13 2207 ax-14 2208 ax-ext 2216 ax-coll 4230 ax-sep 4233 ax-nul 4241 ax-pow 4292 ax-pr 4327 ax-un 4559 ax-setind 4664 ax-iinf 4715 ax-cnex 8234 ax-resscn 8235 ax-1cn 8236 ax-1re 8237 ax-icn 8238 ax-addcl 8239 ax-addrcl 8240 ax-mulcl 8241 ax-mulrcl 8242 ax-addcom 8243 ax-mulcom 8244 ax-addass 8245 ax-mulass 8246 ax-distr 8247 ax-i2m1 8248 ax-0lt1 8249 ax-1rid 8250 ax-0id 8251 ax-rnegex 8252 ax-precex 8253 ax-cnre 8254 ax-pre-ltirr 8255 ax-pre-ltwlin 8256 ax-pre-lttrn 8257 ax-pre-apti 8258 ax-pre-ltadd 8259 ax-pre-mulgt0 8260 ax-pre-mulext 8261 ax-arch 8262 ax-caucvg 8263

This theorem depends on definitions: df-bi 117 df-stab 839 df-dc 843 df-3or 1006 df-3an 1007 df-tru 1401 df-fal 1404 df-nf 1510 df-sb 1812 df-eu 2085 df-mo 2086 df-clab 2221 df-cleq 2227 df-clel 2230 df-nfc 2375 df-ne 2415 df-nel 2510 df-ral 2527 df-rex 2528 df-reu 2529 df-rmo 2530 df-rab 2531 df-v 2817 df-sbc 3046 df-csb 3142 df-dif 3216 df-un 3218 df-in 3220 df-ss 3227 df-nul 3513 df-if 3625 df-pw 3676 df-sn 3700 df-pr 3701 df-op 3703 df-uni 3920 df-int 3955 df-iun 3998 df-br 4115 df-opab 4177 df-mpt 4178 df-tr 4214 df-id 4419 df-po 4422 df-iso 4423 df-iord 4492 df-on 4494 df-ilim 4495 df-suc 4497 df-iom 4718 df-xp 4760 df-rel 4761 df-cnv 4762 df-co 4763 df-dm 4764 df-rn 4765 df-res 4766 df-ima 4767 df-iota 5317 df-fun 5359 df-fn 5360 df-f 5361 df-f1 5362 df-fo 5363 df-f1o 5364 df-fv 5365 df-riota 6011 df-ov 6061 df-oprab 6062 df-mpo 6063 df-1st 6347 df-2nd 6348 df-recs 6549 df-frec 6635 df-1o 6660 df-2o 6661 df-er 6780 df-en 6989 df-pnf 8326 df-mnf 8327 df-xr 8328 df-ltxr 8329 df-le 8330 df-sub 8463 df-neg 8464 df-reap 8867 df-ap 8874 df-div 8967 df-inn 9258 df-2 9316 df-3 9317 df-4 9318 df-n0 9517 df-z 9598 df-uz 9875 df-q 9973 df-rp 10008 df-fz 10365 df-seqfrec 10837 df-exp 10928 df-cj 11555 df-re 11556 df-im 11557 df-rsqrt 11711 df-abs 11712 df-dvds 12502 df-prm 12833

This theorem is referenced by: prmind2 12845 2prm 12852 3prm 12853 prmdc 12855 isprm5 12867 mersenne 15994

W3C validator