isprm5 - Metamath Proof Explorer

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Theorem isprm5 15257

Description: One need only check prime divisors of 𝑃 up to √𝑃 in order to ensure primality. (Contributed by Mario Carneiro, 18-Feb-2014.)

**Assertion**
Ref	Expression
isprm5	⊢ (𝑃 ∈ ℙ ↔ (𝑃 ∈ (ℤ_≥‘2) ∧ ∀𝑧 ∈ ℙ ((𝑧↑2) ≤ 𝑃 → ¬ 𝑧 ∥ 𝑃)))

Distinct variable group: 𝑧,𝑃

Proof of Theorem isprm5 Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		isprm4 15235	. 2 ⊢ (𝑃 ∈ ℙ ↔ (𝑃 ∈ (ℤ_≥‘2) ∧ ∀𝑧 ∈ (ℤ_≥‘2)(𝑧 ∥ 𝑃 → 𝑧 = 𝑃)))
2		prmuz2 15246	. . . . . . . 8 ⊢ (𝑧 ∈ ℙ → 𝑧 ∈ (ℤ_≥‘2))
3	2	a1i 11	. . . . . . 7 ⊢ (𝑃 ∈ (ℤ_≥‘2) → (𝑧 ∈ ℙ → 𝑧 ∈ (ℤ_≥‘2)))
4		eluz2b2 11637	. . . . . . . . . . . . 13 ⊢ (𝑃 ∈ (ℤ_≥‘2) ↔ (𝑃 ∈ ℕ ∧ 1 < 𝑃))
5	4	simprbi 479	. . . . . . . . . . . 12 ⊢ (𝑃 ∈ (ℤ_≥‘2) → 1 < 𝑃)
6		eluzelre 11574	. . . . . . . . . . . . 13 ⊢ (𝑃 ∈ (ℤ_≥‘2) → 𝑃 ∈ ℝ)
7		eluz2nn 11602	. . . . . . . . . . . . . 14 ⊢ (𝑃 ∈ (ℤ_≥‘2) → 𝑃 ∈ ℕ)
8	7	nngt0d 10941	. . . . . . . . . . . . 13 ⊢ (𝑃 ∈ (ℤ_≥‘2) → 0 < 𝑃)
9		ltmulgt11 10762	. . . . . . . . . . . . 13 ⊢ ((𝑃 ∈ ℝ ∧ 𝑃 ∈ ℝ ∧ 0 < 𝑃) → (1 < 𝑃 ↔ 𝑃 < (𝑃 · 𝑃)))
10	6, 6, 8, 9	syl3anc 1318	. . . . . . . . . . . 12 ⊢ (𝑃 ∈ (ℤ_≥‘2) → (1 < 𝑃 ↔ 𝑃 < (𝑃 · 𝑃)))
11	5, 10	mpbid 221	. . . . . . . . . . 11 ⊢ (𝑃 ∈ (ℤ_≥‘2) → 𝑃 < (𝑃 · 𝑃))
12	6, 6	remulcld 9949	. . . . . . . . . . . 12 ⊢ (𝑃 ∈ (ℤ_≥‘2) → (𝑃 · 𝑃) ∈ ℝ)
13	6, 12	ltnled 10063	. . . . . . . . . . 11 ⊢ (𝑃 ∈ (ℤ_≥‘2) → (𝑃 < (𝑃 · 𝑃) ↔ ¬ (𝑃 · 𝑃) ≤ 𝑃))
14	11, 13	mpbid 221	. . . . . . . . . 10 ⊢ (𝑃 ∈ (ℤ_≥‘2) → ¬ (𝑃 · 𝑃) ≤ 𝑃)
15		oveq12 6558	. . . . . . . . . . . . 13 ⊢ ((𝑧 = 𝑃 ∧ 𝑧 = 𝑃) → (𝑧 · 𝑧) = (𝑃 · 𝑃))
16	15	anidms 675	. . . . . . . . . . . 12 ⊢ (𝑧 = 𝑃 → (𝑧 · 𝑧) = (𝑃 · 𝑃))
17	16	breq1d 4593	. . . . . . . . . . 11 ⊢ (𝑧 = 𝑃 → ((𝑧 · 𝑧) ≤ 𝑃 ↔ (𝑃 · 𝑃) ≤ 𝑃))
18	17	notbid 307	. . . . . . . . . 10 ⊢ (𝑧 = 𝑃 → (¬ (𝑧 · 𝑧) ≤ 𝑃 ↔ ¬ (𝑃 · 𝑃) ≤ 𝑃))
19	14, 18	syl5ibrcom 236	. . . . . . . . 9 ⊢ (𝑃 ∈ (ℤ_≥‘2) → (𝑧 = 𝑃 → ¬ (𝑧 · 𝑧) ≤ 𝑃))
20	19	imim2d 55	. . . . . . . 8 ⊢ (𝑃 ∈ (ℤ_≥‘2) → ((𝑧 ∥ 𝑃 → 𝑧 = 𝑃) → (𝑧 ∥ 𝑃 → ¬ (𝑧 · 𝑧) ≤ 𝑃)))
21		con2 129	. . . . . . . 8 ⊢ ((𝑧 ∥ 𝑃 → ¬ (𝑧 · 𝑧) ≤ 𝑃) → ((𝑧 · 𝑧) ≤ 𝑃 → ¬ 𝑧 ∥ 𝑃))
22	20, 21	syl6 34	. . . . . . 7 ⊢ (𝑃 ∈ (ℤ_≥‘2) → ((𝑧 ∥ 𝑃 → 𝑧 = 𝑃) → ((𝑧 · 𝑧) ≤ 𝑃 → ¬ 𝑧 ∥ 𝑃)))
23	3, 22	imim12d 79	. . . . . 6 ⊢ (𝑃 ∈ (ℤ_≥‘2) → ((𝑧 ∈ (ℤ_≥‘2) → (𝑧 ∥ 𝑃 → 𝑧 = 𝑃)) → (𝑧 ∈ ℙ → ((𝑧 · 𝑧) ≤ 𝑃 → ¬ 𝑧 ∥ 𝑃))))
24	23	ralimdv2 2944	. . . . 5 ⊢ (𝑃 ∈ (ℤ_≥‘2) → (∀𝑧 ∈ (ℤ_≥‘2)(𝑧 ∥ 𝑃 → 𝑧 = 𝑃) → ∀𝑧 ∈ ℙ ((𝑧 · 𝑧) ≤ 𝑃 → ¬ 𝑧 ∥ 𝑃)))
25		annim 440	. . . . . . . . 9 ⊢ ((𝑧 ∥ 𝑃 ∧ ¬ 𝑧 = 𝑃) ↔ ¬ (𝑧 ∥ 𝑃 → 𝑧 = 𝑃))
26		oveq12 6558	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑥 = 𝑧 ∧ 𝑥 = 𝑧) → (𝑥 · 𝑥) = (𝑧 · 𝑧))
27	26	anidms 675	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 = 𝑧 → (𝑥 · 𝑥) = (𝑧 · 𝑧))
28	27	breq1d 4593	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = 𝑧 → ((𝑥 · 𝑥) ≤ 𝑃 ↔ (𝑧 · 𝑧) ≤ 𝑃))
29		breq1 4586	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = 𝑧 → (𝑥 ∥ 𝑃 ↔ 𝑧 ∥ 𝑃))
30	28, 29	anbi12d 743	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = 𝑧 → (((𝑥 · 𝑥) ≤ 𝑃 ∧ 𝑥 ∥ 𝑃) ↔ ((𝑧 · 𝑧) ≤ 𝑃 ∧ 𝑧 ∥ 𝑃)))
31	30	rspcev 3282	. . . . . . . . . . . . . 14 ⊢ ((𝑧 ∈ (ℤ_≥‘2) ∧ ((𝑧 · 𝑧) ≤ 𝑃 ∧ 𝑧 ∥ 𝑃)) → ∃𝑥 ∈ (ℤ_≥‘2)((𝑥 · 𝑥) ≤ 𝑃 ∧ 𝑥 ∥ 𝑃))
32	31	ancom2s 840	. . . . . . . . . . . . 13 ⊢ ((𝑧 ∈ (ℤ_≥‘2) ∧ (𝑧 ∥ 𝑃 ∧ (𝑧 · 𝑧) ≤ 𝑃)) → ∃𝑥 ∈ (ℤ_≥‘2)((𝑥 · 𝑥) ≤ 𝑃 ∧ 𝑥 ∥ 𝑃))
33	32	expr 641	. . . . . . . . . . . 12 ⊢ ((𝑧 ∈ (ℤ_≥‘2) ∧ 𝑧 ∥ 𝑃) → ((𝑧 · 𝑧) ≤ 𝑃 → ∃𝑥 ∈ (ℤ_≥‘2)((𝑥 · 𝑥) ≤ 𝑃 ∧ 𝑥 ∥ 𝑃)))
34	33	ad2ant2lr 780	. . . . . . . . . . 11 ⊢ (((𝑃 ∈ (ℤ_≥‘2) ∧ 𝑧 ∈ (ℤ_≥‘2)) ∧ (𝑧 ∥ 𝑃 ∧ ¬ 𝑧 = 𝑃)) → ((𝑧 · 𝑧) ≤ 𝑃 → ∃𝑥 ∈ (ℤ_≥‘2)((𝑥 · 𝑥) ≤ 𝑃 ∧ 𝑥 ∥ 𝑃)))
35		simprl 790	. . . . . . . . . . . . . 14 ⊢ (((𝑃 ∈ (ℤ_≥‘2) ∧ 𝑧 ∈ (ℤ_≥‘2)) ∧ (𝑧 ∥ 𝑃 ∧ ¬ 𝑧 = 𝑃)) → 𝑧 ∥ 𝑃)
36		eluzelz 11573	. . . . . . . . . . . . . . . 16 ⊢ (𝑧 ∈ (ℤ_≥‘2) → 𝑧 ∈ ℤ)
37	36	ad2antlr 759	. . . . . . . . . . . . . . 15 ⊢ (((𝑃 ∈ (ℤ_≥‘2) ∧ 𝑧 ∈ (ℤ_≥‘2)) ∧ (𝑧 ∥ 𝑃 ∧ ¬ 𝑧 = 𝑃)) → 𝑧 ∈ ℤ)
38		eluz2nn 11602	. . . . . . . . . . . . . . . . 17 ⊢ (𝑧 ∈ (ℤ_≥‘2) → 𝑧 ∈ ℕ)
39	38	ad2antlr 759	. . . . . . . . . . . . . . . 16 ⊢ (((𝑃 ∈ (ℤ_≥‘2) ∧ 𝑧 ∈ (ℤ_≥‘2)) ∧ (𝑧 ∥ 𝑃 ∧ ¬ 𝑧 = 𝑃)) → 𝑧 ∈ ℕ)
40	39	nnne0d 10942	. . . . . . . . . . . . . . 15 ⊢ (((𝑃 ∈ (ℤ_≥‘2) ∧ 𝑧 ∈ (ℤ_≥‘2)) ∧ (𝑧 ∥ 𝑃 ∧ ¬ 𝑧 = 𝑃)) → 𝑧 ≠ 0)
41		eluzelz 11573	. . . . . . . . . . . . . . . 16 ⊢ (𝑃 ∈ (ℤ_≥‘2) → 𝑃 ∈ ℤ)
42	41	ad2antrr 758	. . . . . . . . . . . . . . 15 ⊢ (((𝑃 ∈ (ℤ_≥‘2) ∧ 𝑧 ∈ (ℤ_≥‘2)) ∧ (𝑧 ∥ 𝑃 ∧ ¬ 𝑧 = 𝑃)) → 𝑃 ∈ ℤ)
43		dvdsval2 14824	. . . . . . . . . . . . . . 15 ⊢ ((𝑧 ∈ ℤ ∧ 𝑧 ≠ 0 ∧ 𝑃 ∈ ℤ) → (𝑧 ∥ 𝑃 ↔ (𝑃 / 𝑧) ∈ ℤ))
44	37, 40, 42, 43	syl3anc 1318	. . . . . . . . . . . . . 14 ⊢ (((𝑃 ∈ (ℤ_≥‘2) ∧ 𝑧 ∈ (ℤ_≥‘2)) ∧ (𝑧 ∥ 𝑃 ∧ ¬ 𝑧 = 𝑃)) → (𝑧 ∥ 𝑃 ↔ (𝑃 / 𝑧) ∈ ℤ))
45	35, 44	mpbid 221	. . . . . . . . . . . . 13 ⊢ (((𝑃 ∈ (ℤ_≥‘2) ∧ 𝑧 ∈ (ℤ_≥‘2)) ∧ (𝑧 ∥ 𝑃 ∧ ¬ 𝑧 = 𝑃)) → (𝑃 / 𝑧) ∈ ℤ)
46		eluzelre 11574	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑧 ∈ (ℤ_≥‘2) → 𝑧 ∈ ℝ)
47	46	ad2antlr 759	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑃 ∈ (ℤ_≥‘2) ∧ 𝑧 ∈ (ℤ_≥‘2)) ∧ (𝑧 ∥ 𝑃 ∧ ¬ 𝑧 = 𝑃)) → 𝑧 ∈ ℝ)
48	47	recnd 9947	. . . . . . . . . . . . . . . 16 ⊢ (((𝑃 ∈ (ℤ_≥‘2) ∧ 𝑧 ∈ (ℤ_≥‘2)) ∧ (𝑧 ∥ 𝑃 ∧ ¬ 𝑧 = 𝑃)) → 𝑧 ∈ ℂ)
49	48	mulid2d 9937	. . . . . . . . . . . . . . 15 ⊢ (((𝑃 ∈ (ℤ_≥‘2) ∧ 𝑧 ∈ (ℤ_≥‘2)) ∧ (𝑧 ∥ 𝑃 ∧ ¬ 𝑧 = 𝑃)) → (1 · 𝑧) = 𝑧)
50	7	ad2antrr 758	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑃 ∈ (ℤ_≥‘2) ∧ 𝑧 ∈ (ℤ_≥‘2)) ∧ (𝑧 ∥ 𝑃 ∧ ¬ 𝑧 = 𝑃)) → 𝑃 ∈ ℕ)
51		dvdsle 14870	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑧 ∈ ℤ ∧ 𝑃 ∈ ℕ) → (𝑧 ∥ 𝑃 → 𝑧 ≤ 𝑃))
52	51	imp 444	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑧 ∈ ℤ ∧ 𝑃 ∈ ℕ) ∧ 𝑧 ∥ 𝑃) → 𝑧 ≤ 𝑃)
53	37, 50, 35, 52	syl21anc 1317	. . . . . . . . . . . . . . . 16 ⊢ (((𝑃 ∈ (ℤ_≥‘2) ∧ 𝑧 ∈ (ℤ_≥‘2)) ∧ (𝑧 ∥ 𝑃 ∧ ¬ 𝑧 = 𝑃)) → 𝑧 ≤ 𝑃)
54		simprr 792	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑃 ∈ (ℤ_≥‘2) ∧ 𝑧 ∈ (ℤ_≥‘2)) ∧ (𝑧 ∥ 𝑃 ∧ ¬ 𝑧 = 𝑃)) → ¬ 𝑧 = 𝑃)
55	54	neqned 2789	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑃 ∈ (ℤ_≥‘2) ∧ 𝑧 ∈ (ℤ_≥‘2)) ∧ (𝑧 ∥ 𝑃 ∧ ¬ 𝑧 = 𝑃)) → 𝑧 ≠ 𝑃)
56	55	necomd 2837	. . . . . . . . . . . . . . . 16 ⊢ (((𝑃 ∈ (ℤ_≥‘2) ∧ 𝑧 ∈ (ℤ_≥‘2)) ∧ (𝑧 ∥ 𝑃 ∧ ¬ 𝑧 = 𝑃)) → 𝑃 ≠ 𝑧)
57	6	ad2antrr 758	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑃 ∈ (ℤ_≥‘2) ∧ 𝑧 ∈ (ℤ_≥‘2)) ∧ (𝑧 ∥ 𝑃 ∧ ¬ 𝑧 = 𝑃)) → 𝑃 ∈ ℝ)
58	47, 57	ltlend 10061	. . . . . . . . . . . . . . . 16 ⊢ (((𝑃 ∈ (ℤ_≥‘2) ∧ 𝑧 ∈ (ℤ_≥‘2)) ∧ (𝑧 ∥ 𝑃 ∧ ¬ 𝑧 = 𝑃)) → (𝑧 < 𝑃 ↔ (𝑧 ≤ 𝑃 ∧ 𝑃 ≠ 𝑧)))
59	53, 56, 58	mpbir2and 959	. . . . . . . . . . . . . . 15 ⊢ (((𝑃 ∈ (ℤ_≥‘2) ∧ 𝑧 ∈ (ℤ_≥‘2)) ∧ (𝑧 ∥ 𝑃 ∧ ¬ 𝑧 = 𝑃)) → 𝑧 < 𝑃)
60	49, 59	eqbrtrd 4605	. . . . . . . . . . . . . 14 ⊢ (((𝑃 ∈ (ℤ_≥‘2) ∧ 𝑧 ∈ (ℤ_≥‘2)) ∧ (𝑧 ∥ 𝑃 ∧ ¬ 𝑧 = 𝑃)) → (1 · 𝑧) < 𝑃)
61		1red 9934	. . . . . . . . . . . . . . 15 ⊢ (((𝑃 ∈ (ℤ_≥‘2) ∧ 𝑧 ∈ (ℤ_≥‘2)) ∧ (𝑧 ∥ 𝑃 ∧ ¬ 𝑧 = 𝑃)) → 1 ∈ ℝ)
62	42	zred 11358	. . . . . . . . . . . . . . 15 ⊢ (((𝑃 ∈ (ℤ_≥‘2) ∧ 𝑧 ∈ (ℤ_≥‘2)) ∧ (𝑧 ∥ 𝑃 ∧ ¬ 𝑧 = 𝑃)) → 𝑃 ∈ ℝ)
63		nnre 10904	. . . . . . . . . . . . . . . . 17 ⊢ (𝑧 ∈ ℕ → 𝑧 ∈ ℝ)
64		nngt0 10926	. . . . . . . . . . . . . . . . 17 ⊢ (𝑧 ∈ ℕ → 0 < 𝑧)
65	63, 64	jca 553	. . . . . . . . . . . . . . . 16 ⊢ (𝑧 ∈ ℕ → (𝑧 ∈ ℝ ∧ 0 < 𝑧))
66	39, 65	syl 17	. . . . . . . . . . . . . . 15 ⊢ (((𝑃 ∈ (ℤ_≥‘2) ∧ 𝑧 ∈ (ℤ_≥‘2)) ∧ (𝑧 ∥ 𝑃 ∧ ¬ 𝑧 = 𝑃)) → (𝑧 ∈ ℝ ∧ 0 < 𝑧))
67		ltmuldiv 10775	. . . . . . . . . . . . . . 15 ⊢ ((1 ∈ ℝ ∧ 𝑃 ∈ ℝ ∧ (𝑧 ∈ ℝ ∧ 0 < 𝑧)) → ((1 · 𝑧) < 𝑃 ↔ 1 < (𝑃 / 𝑧)))
68	61, 62, 66, 67	syl3anc 1318	. . . . . . . . . . . . . 14 ⊢ (((𝑃 ∈ (ℤ_≥‘2) ∧ 𝑧 ∈ (ℤ_≥‘2)) ∧ (𝑧 ∥ 𝑃 ∧ ¬ 𝑧 = 𝑃)) → ((1 · 𝑧) < 𝑃 ↔ 1 < (𝑃 / 𝑧)))
69	60, 68	mpbid 221	. . . . . . . . . . . . 13 ⊢ (((𝑃 ∈ (ℤ_≥‘2) ∧ 𝑧 ∈ (ℤ_≥‘2)) ∧ (𝑧 ∥ 𝑃 ∧ ¬ 𝑧 = 𝑃)) → 1 < (𝑃 / 𝑧))
70		eluz2b1 11635	. . . . . . . . . . . . 13 ⊢ ((𝑃 / 𝑧) ∈ (ℤ_≥‘2) ↔ ((𝑃 / 𝑧) ∈ ℤ ∧ 1 < (𝑃 / 𝑧)))
71	45, 69, 70	sylanbrc 695	. . . . . . . . . . . 12 ⊢ (((𝑃 ∈ (ℤ_≥‘2) ∧ 𝑧 ∈ (ℤ_≥‘2)) ∧ (𝑧 ∥ 𝑃 ∧ ¬ 𝑧 = 𝑃)) → (𝑃 / 𝑧) ∈ (ℤ_≥‘2))
72	47, 47	remulcld 9949	. . . . . . . . . . . . . . . 16 ⊢ (((𝑃 ∈ (ℤ_≥‘2) ∧ 𝑧 ∈ (ℤ_≥‘2)) ∧ (𝑧 ∥ 𝑃 ∧ ¬ 𝑧 = 𝑃)) → (𝑧 · 𝑧) ∈ ℝ)
73	39, 39	nnmulcld 10945	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑃 ∈ (ℤ_≥‘2) ∧ 𝑧 ∈ (ℤ_≥‘2)) ∧ (𝑧 ∥ 𝑃 ∧ ¬ 𝑧 = 𝑃)) → (𝑧 · 𝑧) ∈ ℕ)
74		nnrp 11718	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑃 ∈ ℕ → 𝑃 ∈ ℝ⁺)
75		nnrp 11718	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑧 · 𝑧) ∈ ℕ → (𝑧 · 𝑧) ∈ ℝ⁺)
76		rpdivcl 11732	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑃 ∈ ℝ⁺ ∧ (𝑧 · 𝑧) ∈ ℝ⁺) → (𝑃 / (𝑧 · 𝑧)) ∈ ℝ⁺)
77	74, 75, 76	syl2an 493	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑃 ∈ ℕ ∧ (𝑧 · 𝑧) ∈ ℕ) → (𝑃 / (𝑧 · 𝑧)) ∈ ℝ⁺)
78	50, 73, 77	syl2anc 691	. . . . . . . . . . . . . . . 16 ⊢ (((𝑃 ∈ (ℤ_≥‘2) ∧ 𝑧 ∈ (ℤ_≥‘2)) ∧ (𝑧 ∥ 𝑃 ∧ ¬ 𝑧 = 𝑃)) → (𝑃 / (𝑧 · 𝑧)) ∈ ℝ⁺)
79	57, 72, 78	lemul1d 11791	. . . . . . . . . . . . . . 15 ⊢ (((𝑃 ∈ (ℤ_≥‘2) ∧ 𝑧 ∈ (ℤ_≥‘2)) ∧ (𝑧 ∥ 𝑃 ∧ ¬ 𝑧 = 𝑃)) → (𝑃 ≤ (𝑧 · 𝑧) ↔ (𝑃 · (𝑃 / (𝑧 · 𝑧))) ≤ ((𝑧 · 𝑧) · (𝑃 / (𝑧 · 𝑧)))))
80	57	recnd 9947	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑃 ∈ (ℤ_≥‘2) ∧ 𝑧 ∈ (ℤ_≥‘2)) ∧ (𝑧 ∥ 𝑃 ∧ ¬ 𝑧 = 𝑃)) → 𝑃 ∈ ℂ)
81	80, 48, 80, 48, 40, 40	divmuldivd 10721	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑃 ∈ (ℤ_≥‘2) ∧ 𝑧 ∈ (ℤ_≥‘2)) ∧ (𝑧 ∥ 𝑃 ∧ ¬ 𝑧 = 𝑃)) → ((𝑃 / 𝑧) · (𝑃 / 𝑧)) = ((𝑃 · 𝑃) / (𝑧 · 𝑧)))
82	73	nncnd 10913	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑃 ∈ (ℤ_≥‘2) ∧ 𝑧 ∈ (ℤ_≥‘2)) ∧ (𝑧 ∥ 𝑃 ∧ ¬ 𝑧 = 𝑃)) → (𝑧 · 𝑧) ∈ ℂ)
83	73	nnne0d 10942	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑃 ∈ (ℤ_≥‘2) ∧ 𝑧 ∈ (ℤ_≥‘2)) ∧ (𝑧 ∥ 𝑃 ∧ ¬ 𝑧 = 𝑃)) → (𝑧 · 𝑧) ≠ 0)
84	80, 80, 82, 83	divassd 10715	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑃 ∈ (ℤ_≥‘2) ∧ 𝑧 ∈ (ℤ_≥‘2)) ∧ (𝑧 ∥ 𝑃 ∧ ¬ 𝑧 = 𝑃)) → ((𝑃 · 𝑃) / (𝑧 · 𝑧)) = (𝑃 · (𝑃 / (𝑧 · 𝑧))))
85	81, 84	eqtrd 2644	. . . . . . . . . . . . . . . 16 ⊢ (((𝑃 ∈ (ℤ_≥‘2) ∧ 𝑧 ∈ (ℤ_≥‘2)) ∧ (𝑧 ∥ 𝑃 ∧ ¬ 𝑧 = 𝑃)) → ((𝑃 / 𝑧) · (𝑃 / 𝑧)) = (𝑃 · (𝑃 / (𝑧 · 𝑧))))
86	80, 82, 83	divcan2d 10682	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑃 ∈ (ℤ_≥‘2) ∧ 𝑧 ∈ (ℤ_≥‘2)) ∧ (𝑧 ∥ 𝑃 ∧ ¬ 𝑧 = 𝑃)) → ((𝑧 · 𝑧) · (𝑃 / (𝑧 · 𝑧))) = 𝑃)
87	86	eqcomd 2616	. . . . . . . . . . . . . . . 16 ⊢ (((𝑃 ∈ (ℤ_≥‘2) ∧ 𝑧 ∈ (ℤ_≥‘2)) ∧ (𝑧 ∥ 𝑃 ∧ ¬ 𝑧 = 𝑃)) → 𝑃 = ((𝑧 · 𝑧) · (𝑃 / (𝑧 · 𝑧))))
88	85, 87	breq12d 4596	. . . . . . . . . . . . . . 15 ⊢ (((𝑃 ∈ (ℤ_≥‘2) ∧ 𝑧 ∈ (ℤ_≥‘2)) ∧ (𝑧 ∥ 𝑃 ∧ ¬ 𝑧 = 𝑃)) → (((𝑃 / 𝑧) · (𝑃 / 𝑧)) ≤ 𝑃 ↔ (𝑃 · (𝑃 / (𝑧 · 𝑧))) ≤ ((𝑧 · 𝑧) · (𝑃 / (𝑧 · 𝑧)))))
89	79, 88	bitr4d 270	. . . . . . . . . . . . . 14 ⊢ (((𝑃 ∈ (ℤ_≥‘2) ∧ 𝑧 ∈ (ℤ_≥‘2)) ∧ (𝑧 ∥ 𝑃 ∧ ¬ 𝑧 = 𝑃)) → (𝑃 ≤ (𝑧 · 𝑧) ↔ ((𝑃 / 𝑧) · (𝑃 / 𝑧)) ≤ 𝑃))
90	89	biimpd 218	. . . . . . . . . . . . 13 ⊢ (((𝑃 ∈ (ℤ_≥‘2) ∧ 𝑧 ∈ (ℤ_≥‘2)) ∧ (𝑧 ∥ 𝑃 ∧ ¬ 𝑧 = 𝑃)) → (𝑃 ≤ (𝑧 · 𝑧) → ((𝑃 / 𝑧) · (𝑃 / 𝑧)) ≤ 𝑃))
91	80, 48, 40	divcan2d 10682	. . . . . . . . . . . . . 14 ⊢ (((𝑃 ∈ (ℤ_≥‘2) ∧ 𝑧 ∈ (ℤ_≥‘2)) ∧ (𝑧 ∥ 𝑃 ∧ ¬ 𝑧 = 𝑃)) → (𝑧 · (𝑃 / 𝑧)) = 𝑃)
92		dvds0lem 14830	. . . . . . . . . . . . . 14 ⊢ (((𝑧 ∈ ℤ ∧ (𝑃 / 𝑧) ∈ ℤ ∧ 𝑃 ∈ ℤ) ∧ (𝑧 · (𝑃 / 𝑧)) = 𝑃) → (𝑃 / 𝑧) ∥ 𝑃)
93	37, 45, 42, 91, 92	syl31anc 1321	. . . . . . . . . . . . 13 ⊢ (((𝑃 ∈ (ℤ_≥‘2) ∧ 𝑧 ∈ (ℤ_≥‘2)) ∧ (𝑧 ∥ 𝑃 ∧ ¬ 𝑧 = 𝑃)) → (𝑃 / 𝑧) ∥ 𝑃)
94	90, 93	jctird 565	. . . . . . . . . . . 12 ⊢ (((𝑃 ∈ (ℤ_≥‘2) ∧ 𝑧 ∈ (ℤ_≥‘2)) ∧ (𝑧 ∥ 𝑃 ∧ ¬ 𝑧 = 𝑃)) → (𝑃 ≤ (𝑧 · 𝑧) → (((𝑃 / 𝑧) · (𝑃 / 𝑧)) ≤ 𝑃 ∧ (𝑃 / 𝑧) ∥ 𝑃)))
95		oveq12 6558	. . . . . . . . . . . . . . . 16 ⊢ ((𝑥 = (𝑃 / 𝑧) ∧ 𝑥 = (𝑃 / 𝑧)) → (𝑥 · 𝑥) = ((𝑃 / 𝑧) · (𝑃 / 𝑧)))
96	95	anidms 675	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = (𝑃 / 𝑧) → (𝑥 · 𝑥) = ((𝑃 / 𝑧) · (𝑃 / 𝑧)))
97	96	breq1d 4593	. . . . . . . . . . . . . 14 ⊢ (𝑥 = (𝑃 / 𝑧) → ((𝑥 · 𝑥) ≤ 𝑃 ↔ ((𝑃 / 𝑧) · (𝑃 / 𝑧)) ≤ 𝑃))
98		breq1 4586	. . . . . . . . . . . . . 14 ⊢ (𝑥 = (𝑃 / 𝑧) → (𝑥 ∥ 𝑃 ↔ (𝑃 / 𝑧) ∥ 𝑃))
99	97, 98	anbi12d 743	. . . . . . . . . . . . 13 ⊢ (𝑥 = (𝑃 / 𝑧) → (((𝑥 · 𝑥) ≤ 𝑃 ∧ 𝑥 ∥ 𝑃) ↔ (((𝑃 / 𝑧) · (𝑃 / 𝑧)) ≤ 𝑃 ∧ (𝑃 / 𝑧) ∥ 𝑃)))
100	99	rspcev 3282	. . . . . . . . . . . 12 ⊢ (((𝑃 / 𝑧) ∈ (ℤ_≥‘2) ∧ (((𝑃 / 𝑧) · (𝑃 / 𝑧)) ≤ 𝑃 ∧ (𝑃 / 𝑧) ∥ 𝑃)) → ∃𝑥 ∈ (ℤ_≥‘2)((𝑥 · 𝑥) ≤ 𝑃 ∧ 𝑥 ∥ 𝑃))
101	71, 94, 100	syl6an 566	. . . . . . . . . . 11 ⊢ (((𝑃 ∈ (ℤ_≥‘2) ∧ 𝑧 ∈ (ℤ_≥‘2)) ∧ (𝑧 ∥ 𝑃 ∧ ¬ 𝑧 = 𝑃)) → (𝑃 ≤ (𝑧 · 𝑧) → ∃𝑥 ∈ (ℤ_≥‘2)((𝑥 · 𝑥) ≤ 𝑃 ∧ 𝑥 ∥ 𝑃)))
102	72, 57	letrid 10068	. . . . . . . . . . 11 ⊢ (((𝑃 ∈ (ℤ_≥‘2) ∧ 𝑧 ∈ (ℤ_≥‘2)) ∧ (𝑧 ∥ 𝑃 ∧ ¬ 𝑧 = 𝑃)) → ((𝑧 · 𝑧) ≤ 𝑃 ∨ 𝑃 ≤ (𝑧 · 𝑧)))
103	34, 101, 102	mpjaod 395	. . . . . . . . . 10 ⊢ (((𝑃 ∈ (ℤ_≥‘2) ∧ 𝑧 ∈ (ℤ_≥‘2)) ∧ (𝑧 ∥ 𝑃 ∧ ¬ 𝑧 = 𝑃)) → ∃𝑥 ∈ (ℤ_≥‘2)((𝑥 · 𝑥) ≤ 𝑃 ∧ 𝑥 ∥ 𝑃))
104	103	ex 449	. . . . . . . . 9 ⊢ ((𝑃 ∈ (ℤ_≥‘2) ∧ 𝑧 ∈ (ℤ_≥‘2)) → ((𝑧 ∥ 𝑃 ∧ ¬ 𝑧 = 𝑃) → ∃𝑥 ∈ (ℤ_≥‘2)((𝑥 · 𝑥) ≤ 𝑃 ∧ 𝑥 ∥ 𝑃)))
105	25, 104	syl5bir 232	. . . . . . . 8 ⊢ ((𝑃 ∈ (ℤ_≥‘2) ∧ 𝑧 ∈ (ℤ_≥‘2)) → (¬ (𝑧 ∥ 𝑃 → 𝑧 = 𝑃) → ∃𝑥 ∈ (ℤ_≥‘2)((𝑥 · 𝑥) ≤ 𝑃 ∧ 𝑥 ∥ 𝑃)))
106	105	rexlimdva 3013	. . . . . . 7 ⊢ (𝑃 ∈ (ℤ_≥‘2) → (∃𝑧 ∈ (ℤ_≥‘2) ¬ (𝑧 ∥ 𝑃 → 𝑧 = 𝑃) → ∃𝑥 ∈ (ℤ_≥‘2)((𝑥 · 𝑥) ≤ 𝑃 ∧ 𝑥 ∥ 𝑃)))
107		prmz 15227	. . . . . . . . . . . . . . 15 ⊢ (𝑧 ∈ ℙ → 𝑧 ∈ ℤ)
108	107	ad2antrl 760	. . . . . . . . . . . . . 14 ⊢ ((((𝑃 ∈ (ℤ_≥‘2) ∧ 𝑥 ∈ (ℤ_≥‘2)) ∧ ((𝑥 · 𝑥) ≤ 𝑃 ∧ 𝑥 ∥ 𝑃)) ∧ (𝑧 ∈ ℙ ∧ 𝑧 ∥ 𝑥)) → 𝑧 ∈ ℤ)
109	108	zred 11358	. . . . . . . . . . . . 13 ⊢ ((((𝑃 ∈ (ℤ_≥‘2) ∧ 𝑥 ∈ (ℤ_≥‘2)) ∧ ((𝑥 · 𝑥) ≤ 𝑃 ∧ 𝑥 ∥ 𝑃)) ∧ (𝑧 ∈ ℙ ∧ 𝑧 ∥ 𝑥)) → 𝑧 ∈ ℝ)
110	109, 109	remulcld 9949	. . . . . . . . . . . 12 ⊢ ((((𝑃 ∈ (ℤ_≥‘2) ∧ 𝑥 ∈ (ℤ_≥‘2)) ∧ ((𝑥 · 𝑥) ≤ 𝑃 ∧ 𝑥 ∥ 𝑃)) ∧ (𝑧 ∈ ℙ ∧ 𝑧 ∥ 𝑥)) → (𝑧 · 𝑧) ∈ ℝ)
111		eluzelz 11573	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ∈ (ℤ_≥‘2) → 𝑥 ∈ ℤ)
112	111	ad3antlr 763	. . . . . . . . . . . . . 14 ⊢ ((((𝑃 ∈ (ℤ_≥‘2) ∧ 𝑥 ∈ (ℤ_≥‘2)) ∧ ((𝑥 · 𝑥) ≤ 𝑃 ∧ 𝑥 ∥ 𝑃)) ∧ (𝑧 ∈ ℙ ∧ 𝑧 ∥ 𝑥)) → 𝑥 ∈ ℤ)
113	112	zred 11358	. . . . . . . . . . . . 13 ⊢ ((((𝑃 ∈ (ℤ_≥‘2) ∧ 𝑥 ∈ (ℤ_≥‘2)) ∧ ((𝑥 · 𝑥) ≤ 𝑃 ∧ 𝑥 ∥ 𝑃)) ∧ (𝑧 ∈ ℙ ∧ 𝑧 ∥ 𝑥)) → 𝑥 ∈ ℝ)
114	113, 113	remulcld 9949	. . . . . . . . . . . 12 ⊢ ((((𝑃 ∈ (ℤ_≥‘2) ∧ 𝑥 ∈ (ℤ_≥‘2)) ∧ ((𝑥 · 𝑥) ≤ 𝑃 ∧ 𝑥 ∥ 𝑃)) ∧ (𝑧 ∈ ℙ ∧ 𝑧 ∥ 𝑥)) → (𝑥 · 𝑥) ∈ ℝ)
115	41	ad3antrrr 762	. . . . . . . . . . . . 13 ⊢ ((((𝑃 ∈ (ℤ_≥‘2) ∧ 𝑥 ∈ (ℤ_≥‘2)) ∧ ((𝑥 · 𝑥) ≤ 𝑃 ∧ 𝑥 ∥ 𝑃)) ∧ (𝑧 ∈ ℙ ∧ 𝑧 ∥ 𝑥)) → 𝑃 ∈ ℤ)
116	115	zred 11358	. . . . . . . . . . . 12 ⊢ ((((𝑃 ∈ (ℤ_≥‘2) ∧ 𝑥 ∈ (ℤ_≥‘2)) ∧ ((𝑥 · 𝑥) ≤ 𝑃 ∧ 𝑥 ∥ 𝑃)) ∧ (𝑧 ∈ ℙ ∧ 𝑧 ∥ 𝑥)) → 𝑃 ∈ ℝ)
117		eluz2nn 11602	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ∈ (ℤ_≥‘2) → 𝑥 ∈ ℕ)
118	117	ad3antlr 763	. . . . . . . . . . . . . 14 ⊢ ((((𝑃 ∈ (ℤ_≥‘2) ∧ 𝑥 ∈ (ℤ_≥‘2)) ∧ ((𝑥 · 𝑥) ≤ 𝑃 ∧ 𝑥 ∥ 𝑃)) ∧ (𝑧 ∈ ℙ ∧ 𝑧 ∥ 𝑥)) → 𝑥 ∈ ℕ)
119		simprr 792	. . . . . . . . . . . . . 14 ⊢ ((((𝑃 ∈ (ℤ_≥‘2) ∧ 𝑥 ∈ (ℤ_≥‘2)) ∧ ((𝑥 · 𝑥) ≤ 𝑃 ∧ 𝑥 ∥ 𝑃)) ∧ (𝑧 ∈ ℙ ∧ 𝑧 ∥ 𝑥)) → 𝑧 ∥ 𝑥)
120		dvdsle 14870	. . . . . . . . . . . . . . 15 ⊢ ((𝑧 ∈ ℤ ∧ 𝑥 ∈ ℕ) → (𝑧 ∥ 𝑥 → 𝑧 ≤ 𝑥))
121	120	imp 444	. . . . . . . . . . . . . 14 ⊢ (((𝑧 ∈ ℤ ∧ 𝑥 ∈ ℕ) ∧ 𝑧 ∥ 𝑥) → 𝑧 ≤ 𝑥)
122	108, 118, 119, 121	syl21anc 1317	. . . . . . . . . . . . 13 ⊢ ((((𝑃 ∈ (ℤ_≥‘2) ∧ 𝑥 ∈ (ℤ_≥‘2)) ∧ ((𝑥 · 𝑥) ≤ 𝑃 ∧ 𝑥 ∥ 𝑃)) ∧ (𝑧 ∈ ℙ ∧ 𝑧 ∥ 𝑥)) → 𝑧 ≤ 𝑥)
123		eluzge2nn0 11603	. . . . . . . . . . . . . . . . 17 ⊢ (𝑧 ∈ (ℤ_≥‘2) → 𝑧 ∈ ℕ₀)
124	123	nn0ge0d 11231	. . . . . . . . . . . . . . . 16 ⊢ (𝑧 ∈ (ℤ_≥‘2) → 0 ≤ 𝑧)
125	2, 124	syl 17	. . . . . . . . . . . . . . 15 ⊢ (𝑧 ∈ ℙ → 0 ≤ 𝑧)
126	125	ad2antrl 760	. . . . . . . . . . . . . 14 ⊢ ((((𝑃 ∈ (ℤ_≥‘2) ∧ 𝑥 ∈ (ℤ_≥‘2)) ∧ ((𝑥 · 𝑥) ≤ 𝑃 ∧ 𝑥 ∥ 𝑃)) ∧ (𝑧 ∈ ℙ ∧ 𝑧 ∥ 𝑥)) → 0 ≤ 𝑧)
127		nnnn0 11176	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 ∈ ℕ → 𝑥 ∈ ℕ₀)
128	127	nn0ge0d 11231	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ∈ ℕ → 0 ≤ 𝑥)
129	118, 128	syl 17	. . . . . . . . . . . . . 14 ⊢ ((((𝑃 ∈ (ℤ_≥‘2) ∧ 𝑥 ∈ (ℤ_≥‘2)) ∧ ((𝑥 · 𝑥) ≤ 𝑃 ∧ 𝑥 ∥ 𝑃)) ∧ (𝑧 ∈ ℙ ∧ 𝑧 ∥ 𝑥)) → 0 ≤ 𝑥)
130		le2msq 10802	. . . . . . . . . . . . . 14 ⊢ (((𝑧 ∈ ℝ ∧ 0 ≤ 𝑧) ∧ (𝑥 ∈ ℝ ∧ 0 ≤ 𝑥)) → (𝑧 ≤ 𝑥 ↔ (𝑧 · 𝑧) ≤ (𝑥 · 𝑥)))
131	109, 126, 113, 129, 130	syl22anc 1319	. . . . . . . . . . . . 13 ⊢ ((((𝑃 ∈ (ℤ_≥‘2) ∧ 𝑥 ∈ (ℤ_≥‘2)) ∧ ((𝑥 · 𝑥) ≤ 𝑃 ∧ 𝑥 ∥ 𝑃)) ∧ (𝑧 ∈ ℙ ∧ 𝑧 ∥ 𝑥)) → (𝑧 ≤ 𝑥 ↔ (𝑧 · 𝑧) ≤ (𝑥 · 𝑥)))
132	122, 131	mpbid 221	. . . . . . . . . . . 12 ⊢ ((((𝑃 ∈ (ℤ_≥‘2) ∧ 𝑥 ∈ (ℤ_≥‘2)) ∧ ((𝑥 · 𝑥) ≤ 𝑃 ∧ 𝑥 ∥ 𝑃)) ∧ (𝑧 ∈ ℙ ∧ 𝑧 ∥ 𝑥)) → (𝑧 · 𝑧) ≤ (𝑥 · 𝑥))
133		simplrl 796	. . . . . . . . . . . 12 ⊢ ((((𝑃 ∈ (ℤ_≥‘2) ∧ 𝑥 ∈ (ℤ_≥‘2)) ∧ ((𝑥 · 𝑥) ≤ 𝑃 ∧ 𝑥 ∥ 𝑃)) ∧ (𝑧 ∈ ℙ ∧ 𝑧 ∥ 𝑥)) → (𝑥 · 𝑥) ≤ 𝑃)
134	110, 114, 116, 132, 133	letrd 10073	. . . . . . . . . . 11 ⊢ ((((𝑃 ∈ (ℤ_≥‘2) ∧ 𝑥 ∈ (ℤ_≥‘2)) ∧ ((𝑥 · 𝑥) ≤ 𝑃 ∧ 𝑥 ∥ 𝑃)) ∧ (𝑧 ∈ ℙ ∧ 𝑧 ∥ 𝑥)) → (𝑧 · 𝑧) ≤ 𝑃)
135		simplrr 797	. . . . . . . . . . . 12 ⊢ ((((𝑃 ∈ (ℤ_≥‘2) ∧ 𝑥 ∈ (ℤ_≥‘2)) ∧ ((𝑥 · 𝑥) ≤ 𝑃 ∧ 𝑥 ∥ 𝑃)) ∧ (𝑧 ∈ ℙ ∧ 𝑧 ∥ 𝑥)) → 𝑥 ∥ 𝑃)
136		dvdstr 14856	. . . . . . . . . . . . 13 ⊢ ((𝑧 ∈ ℤ ∧ 𝑥 ∈ ℤ ∧ 𝑃 ∈ ℤ) → ((𝑧 ∥ 𝑥 ∧ 𝑥 ∥ 𝑃) → 𝑧 ∥ 𝑃))
137	108, 112, 115, 136	syl3anc 1318	. . . . . . . . . . . 12 ⊢ ((((𝑃 ∈ (ℤ_≥‘2) ∧ 𝑥 ∈ (ℤ_≥‘2)) ∧ ((𝑥 · 𝑥) ≤ 𝑃 ∧ 𝑥 ∥ 𝑃)) ∧ (𝑧 ∈ ℙ ∧ 𝑧 ∥ 𝑥)) → ((𝑧 ∥ 𝑥 ∧ 𝑥 ∥ 𝑃) → 𝑧 ∥ 𝑃))
138	119, 135, 137	mp2and 711	. . . . . . . . . . 11 ⊢ ((((𝑃 ∈ (ℤ_≥‘2) ∧ 𝑥 ∈ (ℤ_≥‘2)) ∧ ((𝑥 · 𝑥) ≤ 𝑃 ∧ 𝑥 ∥ 𝑃)) ∧ (𝑧 ∈ ℙ ∧ 𝑧 ∥ 𝑥)) → 𝑧 ∥ 𝑃)
139	134, 138	jc 158	. . . . . . . . . 10 ⊢ ((((𝑃 ∈ (ℤ_≥‘2) ∧ 𝑥 ∈ (ℤ_≥‘2)) ∧ ((𝑥 · 𝑥) ≤ 𝑃 ∧ 𝑥 ∥ 𝑃)) ∧ (𝑧 ∈ ℙ ∧ 𝑧 ∥ 𝑥)) → ¬ ((𝑧 · 𝑧) ≤ 𝑃 → ¬ 𝑧 ∥ 𝑃))
140		exprmfct 15254	. . . . . . . . . . 11 ⊢ (𝑥 ∈ (ℤ_≥‘2) → ∃𝑧 ∈ ℙ 𝑧 ∥ 𝑥)
141	140	ad2antlr 759	. . . . . . . . . 10 ⊢ (((𝑃 ∈ (ℤ_≥‘2) ∧ 𝑥 ∈ (ℤ_≥‘2)) ∧ ((𝑥 · 𝑥) ≤ 𝑃 ∧ 𝑥 ∥ 𝑃)) → ∃𝑧 ∈ ℙ 𝑧 ∥ 𝑥)
142	139, 141	reximddv 3001	. . . . . . . . 9 ⊢ (((𝑃 ∈ (ℤ_≥‘2) ∧ 𝑥 ∈ (ℤ_≥‘2)) ∧ ((𝑥 · 𝑥) ≤ 𝑃 ∧ 𝑥 ∥ 𝑃)) → ∃𝑧 ∈ ℙ ¬ ((𝑧 · 𝑧) ≤ 𝑃 → ¬ 𝑧 ∥ 𝑃))
143	142	ex 449	. . . . . . . 8 ⊢ ((𝑃 ∈ (ℤ_≥‘2) ∧ 𝑥 ∈ (ℤ_≥‘2)) → (((𝑥 · 𝑥) ≤ 𝑃 ∧ 𝑥 ∥ 𝑃) → ∃𝑧 ∈ ℙ ¬ ((𝑧 · 𝑧) ≤ 𝑃 → ¬ 𝑧 ∥ 𝑃)))
144	143	rexlimdva 3013	. . . . . . 7 ⊢ (𝑃 ∈ (ℤ_≥‘2) → (∃𝑥 ∈ (ℤ_≥‘2)((𝑥 · 𝑥) ≤ 𝑃 ∧ 𝑥 ∥ 𝑃) → ∃𝑧 ∈ ℙ ¬ ((𝑧 · 𝑧) ≤ 𝑃 → ¬ 𝑧 ∥ 𝑃)))
145	106, 144	syld 46	. . . . . 6 ⊢ (𝑃 ∈ (ℤ_≥‘2) → (∃𝑧 ∈ (ℤ_≥‘2) ¬ (𝑧 ∥ 𝑃 → 𝑧 = 𝑃) → ∃𝑧 ∈ ℙ ¬ ((𝑧 · 𝑧) ≤ 𝑃 → ¬ 𝑧 ∥ 𝑃)))
146		rexnal 2978	. . . . . 6 ⊢ (∃𝑧 ∈ (ℤ_≥‘2) ¬ (𝑧 ∥ 𝑃 → 𝑧 = 𝑃) ↔ ¬ ∀𝑧 ∈ (ℤ_≥‘2)(𝑧 ∥ 𝑃 → 𝑧 = 𝑃))
147		rexnal 2978	. . . . . 6 ⊢ (∃𝑧 ∈ ℙ ¬ ((𝑧 · 𝑧) ≤ 𝑃 → ¬ 𝑧 ∥ 𝑃) ↔ ¬ ∀𝑧 ∈ ℙ ((𝑧 · 𝑧) ≤ 𝑃 → ¬ 𝑧 ∥ 𝑃))
148	145, 146, 147	3imtr3g 283	. . . . 5 ⊢ (𝑃 ∈ (ℤ_≥‘2) → (¬ ∀𝑧 ∈ (ℤ_≥‘2)(𝑧 ∥ 𝑃 → 𝑧 = 𝑃) → ¬ ∀𝑧 ∈ ℙ ((𝑧 · 𝑧) ≤ 𝑃 → ¬ 𝑧 ∥ 𝑃)))
149	24, 148	impcon4bid 216	. . . 4 ⊢ (𝑃 ∈ (ℤ_≥‘2) → (∀𝑧 ∈ (ℤ_≥‘2)(𝑧 ∥ 𝑃 → 𝑧 = 𝑃) ↔ ∀𝑧 ∈ ℙ ((𝑧 · 𝑧) ≤ 𝑃 → ¬ 𝑧 ∥ 𝑃)))
150		prmnn 15226	. . . . . . . . 9 ⊢ (𝑧 ∈ ℙ → 𝑧 ∈ ℕ)
151	150	nncnd 10913	. . . . . . . 8 ⊢ (𝑧 ∈ ℙ → 𝑧 ∈ ℂ)
152	151	sqvald 12867	. . . . . . 7 ⊢ (𝑧 ∈ ℙ → (𝑧↑2) = (𝑧 · 𝑧))
153	152	breq1d 4593	. . . . . 6 ⊢ (𝑧 ∈ ℙ → ((𝑧↑2) ≤ 𝑃 ↔ (𝑧 · 𝑧) ≤ 𝑃))
154	153	imbi1d 330	. . . . 5 ⊢ (𝑧 ∈ ℙ → (((𝑧↑2) ≤ 𝑃 → ¬ 𝑧 ∥ 𝑃) ↔ ((𝑧 · 𝑧) ≤ 𝑃 → ¬ 𝑧 ∥ 𝑃)))
155	154	ralbiia 2962	. . . 4 ⊢ (∀𝑧 ∈ ℙ ((𝑧↑2) ≤ 𝑃 → ¬ 𝑧 ∥ 𝑃) ↔ ∀𝑧 ∈ ℙ ((𝑧 · 𝑧) ≤ 𝑃 → ¬ 𝑧 ∥ 𝑃))
156	149, 155	syl6bbr 277	. . 3 ⊢ (𝑃 ∈ (ℤ_≥‘2) → (∀𝑧 ∈ (ℤ_≥‘2)(𝑧 ∥ 𝑃 → 𝑧 = 𝑃) ↔ ∀𝑧 ∈ ℙ ((𝑧↑2) ≤ 𝑃 → ¬ 𝑧 ∥ 𝑃)))
157	156	pm5.32i 667	. 2 ⊢ ((𝑃 ∈ (ℤ_≥‘2) ∧ ∀𝑧 ∈ (ℤ_≥‘2)(𝑧 ∥ 𝑃 → 𝑧 = 𝑃)) ↔ (𝑃 ∈ (ℤ_≥‘2) ∧ ∀𝑧 ∈ ℙ ((𝑧↑2) ≤ 𝑃 → ¬ 𝑧 ∥ 𝑃)))
158	1, 157	bitri 263	1 ⊢ (𝑃 ∈ ℙ ↔ (𝑃 ∈ (ℤ_≥‘2) ∧ ∀𝑧 ∈ ℙ ((𝑧↑2) ≤ 𝑃 → ¬ 𝑧 ∥ 𝑃)))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 195 ∧ wa 383 = wceq 1475 ∈ wcel 1977 ≠ wne 2780 ∀wral 2896 ∃wrex 2897 class class class wbr 4583 ‘cfv 5804 (class class class)co 6549 ℝcr 9814 0cc0 9815 1c1 9816 · cmul 9820 < clt 9953 ≤ cle 9954 / cdiv 10563 ℕcn 10897 2c2 10947 ℤcz 11254 ℤ_≥cuz 11563 ℝ⁺crp 11708 ↑cexp 12722 ∥ cdvds 14821 ℙcprime 15223

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 ax-pre-sup 9893

This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-int 4411 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-1st 7059 df-2nd 7060 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-1o 7447 df-2o 7448 df-oadd 7451 df-er 7629 df-en 7842 df-dom 7843 df-sdom 7844 df-fin 7845 df-sup 8231 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-div 10564 df-nn 10898 df-2 10956 df-3 10957 df-n0 11170 df-z 11255 df-uz 11564 df-rp 11709 df-fz 12198 df-seq 12664 df-exp 12723 df-cj 13687 df-re 13688 df-im 13689 df-sqrt 13823 df-abs 13824 df-dvds 14822 df-prm 15224

This theorem is referenced by: isprm7 15258 pockthg 15448 prmlem1a 15651

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