isprm3 - Intuitionistic Logic Explorer

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Theorem isprm3 12648

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 12647	. 2 ⊢ (𝑃 ∈ ℙ ↔ (𝑃 ∈ (ℤ_≥‘2) ∧ ∀𝑧 ∈ ℕ (𝑧 ∥ 𝑃 → (𝑧 = 1 ∨ 𝑧 = 𝑃))))
2		dvdszrcl 12311	. . . . . . . . . . 11 ⊢ (𝑧 ∥ 𝑃 → (𝑧 ∈ ℤ ∧ 𝑃 ∈ ℤ))
3	2	simpld 112	. . . . . . . . . 10 ⊢ (𝑧 ∥ 𝑃 → 𝑧 ∈ ℤ)
4		1zzd 9481	. . . . . . . . . 10 ⊢ ((𝑃 ∈ (ℤ_≥‘2) ∧ 𝑧 ∥ 𝑃) → 1 ∈ ℤ)
5		zdceq 9530	. . . . . . . . . 10 ⊢ ((𝑧 ∈ ℤ ∧ 1 ∈ ℤ) → DECID 𝑧 = 1)
6	3, 4, 5	syl2an2 596	. . . . . . . . 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 9530	. . . . . . . . . 10 ⊢ ((𝑧 ∈ ℤ ∧ 𝑃 ∈ ℤ) → DECID 𝑧 = 𝑃)
10	3, 8, 9	syl2an2 596	. . . . . . . . 9 ⊢ ((𝑃 ∈ (ℤ_≥‘2) ∧ 𝑧 ∥ 𝑃) → DECID 𝑧 = 𝑃)
11		dcor 941	. . . . . . . . 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 894	. . . . . . . 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 9769	. . . . . . . . . . . . . . . 16 ⊢ (𝑃 ∈ (ℤ_≥‘2) → 𝑃 ∈ ℕ)
16		nnz 9473	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑧 ∈ ℕ → 𝑧 ∈ ℤ)
17		dvdsle 12363	. . . . . . . . . . . . . . . . . 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 9141	. . . . . . . . . . . . . . . . . 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 9473	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑃 ∈ ℕ → 𝑃 ∈ ℤ)
24		zre 9458	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑧 ∈ ℤ → 𝑧 ∈ ℝ)
25		1re 8153	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ 1 ∈ ℝ
26		leltap 8780	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((1 ∈ ℝ ∧ 𝑧 ∈ ℝ ∧ 1 ≤ 𝑧) → (1 < 𝑧 ↔ 𝑧 # 1))
27	25, 26	mp3an1 1358	. . . . . . . . . . . . . . . . . . . . . . 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 9480	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ 1 ∈ ℤ
30		zapne 9529	. . . . . . . . . . . . . . . . . . . . . . . 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 1040	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑧 ∈ ℤ ∧ 𝑃 ∈ ℤ ∧ 1 ≤ 𝑧) → (1 < 𝑧 ↔ 𝑧 ≠ 1))
35	34	3expia 1229	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑧 ∈ ℤ ∧ 𝑃 ∈ ℤ) → (1 ≤ 𝑧 → (1 < 𝑧 ↔ 𝑧 ≠ 1)))
36		zre 9458	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑃 ∈ ℤ → 𝑃 ∈ ℝ)
37		leltap 8780	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑧 ∈ ℝ ∧ 𝑃 ∈ ℝ ∧ 𝑧 ≤ 𝑃) → (𝑧 < 𝑃 ↔ 𝑃 # 𝑧))
38	24, 37	syl3an1 1304	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑧 ∈ ℤ ∧ 𝑃 ∈ ℝ ∧ 𝑧 ≤ 𝑃) → (𝑧 < 𝑃 ↔ 𝑃 # 𝑧))
39	36, 38	syl3an2 1305	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑧 ∈ ℤ ∧ 𝑃 ∈ ℤ ∧ 𝑧 ≤ 𝑃) → (𝑧 < 𝑃 ↔ 𝑃 # 𝑧))
40		zapne 9529	. . . . . . . . . . . . . . . . . . . . . . 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 1041	. . . . . . . . . . . . . . . . . . . . 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 1229	. . . . . . . . . . . . . . . . . . 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		pm4.38 607	. . . . . . . . . . . . . . . . . 18 ⊢ (((1 < 𝑧 ↔ 𝑧 ≠ 1) ∧ (𝑧 < 𝑃 ↔ 𝑃 ≠ 𝑧)) → ((1 < 𝑧 ∧ 𝑧 < 𝑃) ↔ (𝑧 ≠ 1 ∧ 𝑃 ≠ 𝑧)))
48		df-ne 2401	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑧 ≠ 1 ↔ ¬ 𝑧 = 1)
49		nesym 2445	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑃 ≠ 𝑧 ↔ ¬ 𝑧 = 𝑃)
50	48, 49	anbi12i 460	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑧 ≠ 1 ∧ 𝑃 ≠ 𝑧) ↔ (¬ 𝑧 = 1 ∧ ¬ 𝑧 = 𝑃))
51		ioran 757	. . . . . . . . . . . . . . . . . . 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 9739	. . . . . . . . . . . . . . 15 ⊢ (𝑃 ∈ (ℤ_≥‘2) → 𝑃 ∈ ℤ)
59		zltp1le 9509	. . . . . . . . . . . . . . . . . . . 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 9177	. . . . . . . . . . . . . . . . . . . 20 ⊢ 2 = (1 + 1)
62	61	breq1i 4090	. . . . . . . . . . . . . . . . . . 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 9512	. . . . . . . . . . . . . . . . 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 9492	. . . . . . . . . . . . . . . . 17 ⊢ (𝑃 ∈ ℤ → (𝑃 − 1) ∈ ℤ)
68		2z 9482	. . . . . . . . . . . . . . . . . 18 ⊢ 2 ∈ ℤ
69		elfz 10218	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑧 ∈ ℤ ∧ 2 ∈ ℤ ∧ (𝑃 − 1) ∈ ℤ) → (𝑧 ∈ (2...(𝑃 − 1)) ↔ (2 ≤ 𝑧 ∧ 𝑧 ≤ (𝑃 − 1))))
70	68, 69	mp3an2 1359	. . . . . . . . . . . . . . . . 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 10269	. . . . . . . . . . . . 13 ⊢ (2...(𝑃 − 1)) ⊆ (ℤ_≥‘2)
80		2eluzge1 9779	. . . . . . . . . . . . . 14 ⊢ 2 ∈ (ℤ_≥‘1)
81		uzss 9751	. . . . . . . . . . . . . 14 ⊢ (2 ∈ (ℤ_≥‘1) → (ℤ_≥‘2) ⊆ (ℤ_≥‘1))
82	80, 81	ax-mp 5	. . . . . . . . . . . . 13 ⊢ (ℤ_≥‘2) ⊆ (ℤ_≥‘1)
83	79, 82	sstri 3233	. . . . . . . . . . . 12 ⊢ (2...(𝑃 − 1)) ⊆ (ℤ_≥‘1)
84		nnuz 9766	. . . . . . . . . . . 12 ⊢ ℕ = (ℤ_≥‘1)
85	83, 84	sseqtrri 3259	. . . . . . . . . . 11 ⊢ (2...(𝑃 − 1)) ⊆ ℕ
86	85	sseli 3220	. . . . . . . . . 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 671	. . . . . . 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 673	. . . . 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 2532	. . 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 713 DECID wdc 839 ∧ w3a 1002 = wceq 1395 ∈ wcel 2200 ≠ wne 2400 ∀wral 2508 ⊆ wss 3197 class class class wbr 4083 ‘cfv 5318 (class class class)co 6007 ℝcr 8006 1c1 8008 + caddc 8010 < clt 8189 ≤ cle 8190 − cmin 8325 # cap 8736 ℕcn 9118 2c2 9169 ℤcz 9454 ℤ_≥cuz 9730 ...cfz 10212 ∥ cdvds 12306 ℙcprime 12637

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 617 ax-in2 618 ax-io 714 ax-5 1493 ax-7 1494 ax-gen 1495 ax-ie1 1539 ax-ie2 1540 ax-8 1550 ax-10 1551 ax-11 1552 ax-i12 1553 ax-bndl 1555 ax-4 1556 ax-17 1572 ax-i9 1576 ax-ial 1580 ax-i5r 1581 ax-13 2202 ax-14 2203 ax-ext 2211 ax-coll 4199 ax-sep 4202 ax-nul 4210 ax-pow 4258 ax-pr 4293 ax-un 4524 ax-setind 4629 ax-iinf 4680 ax-cnex 8098 ax-resscn 8099 ax-1cn 8100 ax-1re 8101 ax-icn 8102 ax-addcl 8103 ax-addrcl 8104 ax-mulcl 8105 ax-mulrcl 8106 ax-addcom 8107 ax-mulcom 8108 ax-addass 8109 ax-mulass 8110 ax-distr 8111 ax-i2m1 8112 ax-0lt1 8113 ax-1rid 8114 ax-0id 8115 ax-rnegex 8116 ax-precex 8117 ax-cnre 8118 ax-pre-ltirr 8119 ax-pre-ltwlin 8120 ax-pre-lttrn 8121 ax-pre-apti 8122 ax-pre-ltadd 8123 ax-pre-mulgt0 8124 ax-pre-mulext 8125 ax-arch 8126 ax-caucvg 8127

This theorem depends on definitions: df-bi 117 df-stab 836 df-dc 840 df-3or 1003 df-3an 1004 df-tru 1398 df-fal 1401 df-nf 1507 df-sb 1809 df-eu 2080 df-mo 2081 df-clab 2216 df-cleq 2222 df-clel 2225 df-nfc 2361 df-ne 2401 df-nel 2496 df-ral 2513 df-rex 2514 df-reu 2515 df-rmo 2516 df-rab 2517 df-v 2801 df-sbc 3029 df-csb 3125 df-dif 3199 df-un 3201 df-in 3203 df-ss 3210 df-nul 3492 df-if 3603 df-pw 3651 df-sn 3672 df-pr 3673 df-op 3675 df-uni 3889 df-int 3924 df-iun 3967 df-br 4084 df-opab 4146 df-mpt 4147 df-tr 4183 df-id 4384 df-po 4387 df-iso 4388 df-iord 4457 df-on 4459 df-ilim 4460 df-suc 4462 df-iom 4683 df-xp 4725 df-rel 4726 df-cnv 4727 df-co 4728 df-dm 4729 df-rn 4730 df-res 4731 df-ima 4732 df-iota 5278 df-fun 5320 df-fn 5321 df-f 5322 df-f1 5323 df-fo 5324 df-f1o 5325 df-fv 5326 df-riota 5960 df-ov 6010 df-oprab 6011 df-mpo 6012 df-1st 6292 df-2nd 6293 df-recs 6457 df-frec 6543 df-1o 6568 df-2o 6569 df-er 6688 df-en 6896 df-pnf 8191 df-mnf 8192 df-xr 8193 df-ltxr 8194 df-le 8195 df-sub 8327 df-neg 8328 df-reap 8730 df-ap 8737 df-div 8828 df-inn 9119 df-2 9177 df-3 9178 df-4 9179 df-n0 9378 df-z 9455 df-uz 9731 df-q 9823 df-rp 9858 df-fz 10213 df-seqfrec 10678 df-exp 10769 df-cj 11361 df-re 11362 df-im 11363 df-rsqrt 11517 df-abs 11518 df-dvds 12307 df-prm 12638

This theorem is referenced by: prmind2 12650 2prm 12657 3prm 12658 prmdc 12660 isprm5 12672 mersenne 15679

W3C validator