isprm3 - Intuitionistic Logic Explorer

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Theorem isprm3 12823

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 12822	. 2 ⊢ (𝑃 ∈ ℙ ↔ (𝑃 ∈ (ℤ_≥‘2) ∧ ∀𝑧 ∈ ℕ (𝑧 ∥ 𝑃 → (𝑧 = 1 ∨ 𝑧 = 𝑃))))
2		dvdszrcl 12486	. . . . . . . . . . 11 ⊢ (𝑧 ∥ 𝑃 → (𝑧 ∈ ℤ ∧ 𝑃 ∈ ℤ))
3	2	simpld 112	. . . . . . . . . 10 ⊢ (𝑧 ∥ 𝑃 → 𝑧 ∈ ℤ)
4		1zzd 9609	. . . . . . . . . 10 ⊢ ((𝑃 ∈ (ℤ_≥‘2) ∧ 𝑧 ∥ 𝑃) → 1 ∈ ℤ)
5		zdceq 9658	. . . . . . . . . 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 9658	. . . . . . . . . 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 9904	. . . . . . . . . . . . . . . 16 ⊢ (𝑃 ∈ (ℤ_≥‘2) → 𝑃 ∈ ℕ)
16		nnz 9601	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑧 ∈ ℕ → 𝑧 ∈ ℤ)
17		dvdsle 12538	. . . . . . . . . . . . . . . . . 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 9265	. . . . . . . . . . . . . . . . . 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 9601	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑃 ∈ ℕ → 𝑃 ∈ ℤ)
24		zre 9586	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑧 ∈ ℤ → 𝑧 ∈ ℝ)
25		1re 8278	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ 1 ∈ ℝ
26		leltap 8904	. . . . . . . . . . . . . . . . . . . . . . . 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 9608	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ 1 ∈ ℤ
30		zapne 9657	. . . . . . . . . . . . . . . . . . . . . . . 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 9586	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑃 ∈ ℤ → 𝑃 ∈ ℝ)
37		leltap 8904	. . . . . . . . . . . . . . . . . . . . . . 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 9657	. . . . . . . . . . . . . . . . . . . . . . 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 9869	. . . . . . . . . . . . . . 15 ⊢ (𝑃 ∈ (ℤ_≥‘2) → 𝑃 ∈ ℤ)
59		zltp1le 9637	. . . . . . . . . . . . . . . . . . . 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 9301	. . . . . . . . . . . . . . . . . . . 20 ⊢ 2 = (1 + 1)
62	61	breq1i 4118	. . . . . . . . . . . . . . . . . . 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 9640	. . . . . . . . . . . . . . . . 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 9620	. . . . . . . . . . . . . . . . 17 ⊢ (𝑃 ∈ ℤ → (𝑃 − 1) ∈ ℤ)
68		2z 9610	. . . . . . . . . . . . . . . . . 18 ⊢ 2 ∈ ℤ
69		elfz 10354	. . . . . . . . . . . . . . . . . 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 10405	. . . . . . . . . . . . 13 ⊢ (2...(𝑃 − 1)) ⊆ (ℤ_≥‘2)
80		2eluzge1 9914	. . . . . . . . . . . . . 14 ⊢ 2 ∈ (ℤ_≥‘1)
81		uzss 9881	. . . . . . . . . . . . . 14 ⊢ (2 ∈ (ℤ_≥‘1) → (ℤ_≥‘2) ⊆ (ℤ_≥‘1))
82	80, 81	ax-mp 5	. . . . . . . . . . . . 13 ⊢ (ℤ_≥‘2) ⊆ (ℤ_≥‘1)
83	79, 82	sstri 3249	. . . . . . . . . . . 12 ⊢ (2...(𝑃 − 1)) ⊆ (ℤ_≥‘1)
84		nnuz 9896	. . . . . . . . . . . 12 ⊢ ℕ = (ℤ_≥‘1)
85	83, 84	sseqtrri 3275	. . . . . . . . . . 11 ⊢ (2...(𝑃 − 1)) ⊆ ℕ
86	85	sseli 3236	. . . . . . . . . 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 3213 class class class wbr 4111 ‘cfv 5354 (class class class)co 6052 ℝcr 8131 1c1 8133 + caddc 8135 < clt 8313 ≤ cle 8314 − cmin 8449 # cap 8860 ℕcn 9242 2c2 9293 ℤcz 9582 ℤ_≥cuz 9859 ...cfz 10348 ∥ cdvds 12481 ℙcprime 12812

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 4227 ax-sep 4230 ax-nul 4238 ax-pow 4289 ax-pr 4324 ax-un 4556 ax-setind 4661 ax-iinf 4712 ax-cnex 8223 ax-resscn 8224 ax-1cn 8225 ax-1re 8226 ax-icn 8227 ax-addcl 8228 ax-addrcl 8229 ax-mulcl 8230 ax-mulrcl 8231 ax-addcom 8232 ax-mulcom 8233 ax-addass 8234 ax-mulass 8235 ax-distr 8236 ax-i2m1 8237 ax-0lt1 8238 ax-1rid 8239 ax-0id 8240 ax-rnegex 8241 ax-precex 8242 ax-cnre 8243 ax-pre-ltirr 8244 ax-pre-ltwlin 8245 ax-pre-lttrn 8246 ax-pre-apti 8247 ax-pre-ltadd 8248 ax-pre-mulgt0 8249 ax-pre-mulext 8250 ax-arch 8251 ax-caucvg 8252

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 3045 df-csb 3141 df-dif 3215 df-un 3217 df-in 3219 df-ss 3226 df-nul 3511 df-if 3623 df-pw 3673 df-sn 3697 df-pr 3698 df-op 3700 df-uni 3917 df-int 3952 df-iun 3995 df-br 4112 df-opab 4174 df-mpt 4175 df-tr 4211 df-id 4416 df-po 4419 df-iso 4420 df-iord 4489 df-on 4491 df-ilim 4492 df-suc 4494 df-iom 4715 df-xp 4757 df-rel 4758 df-cnv 4759 df-co 4760 df-dm 4761 df-rn 4762 df-res 4763 df-ima 4764 df-iota 5314 df-fun 5356 df-fn 5357 df-f 5358 df-f1 5359 df-fo 5360 df-f1o 5361 df-fv 5362 df-riota 6005 df-ov 6055 df-oprab 6056 df-mpo 6057 df-1st 6336 df-2nd 6337 df-recs 6538 df-frec 6624 df-1o 6649 df-2o 6650 df-er 6769 df-en 6978 df-pnf 8315 df-mnf 8316 df-xr 8317 df-ltxr 8318 df-le 8319 df-sub 8451 df-neg 8452 df-reap 8854 df-ap 8861 df-div 8952 df-inn 9243 df-2 9301 df-3 9302 df-4 9303 df-n0 9502 df-z 9583 df-uz 9860 df-q 9958 df-rp 9993 df-fz 10349 df-seqfrec 10817 df-exp 10908 df-cj 11535 df-re 11536 df-im 11537 df-rsqrt 11691 df-abs 11692 df-dvds 12482 df-prm 12813

This theorem is referenced by: prmind2 12825 2prm 12832 3prm 12833 prmdc 12835 isprm5 12847 mersenne 15914

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