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Theorem 2sqreunnlem1 26797

Description: Lemma 1 for 2sqreunn 26805. (Contributed by AV, 11-Jun-2023.)

**Assertion**
Ref	Expression
2sqreunnlem1	⊢ ((𝑃 ∈ ℙ ∧ (𝑃 mod 4) = 1) → ∃!𝑎 ∈ ℕ ∃!𝑏 ∈ ℕ (𝑎 ≤ 𝑏 ∧ ((𝑎↑2) + (𝑏↑2)) = 𝑃))

Distinct variable group: 𝑃,𝑎,𝑏

Proof of Theorem 2sqreunnlem1 Dummy variables 𝑐 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		2sqnn 26787	. . 3 ⊢ ((𝑃 ∈ ℙ ∧ (𝑃 mod 4) = 1) → ∃𝑥 ∈ ℕ ∃𝑦 ∈ ℕ 𝑃 = ((𝑥↑2) + (𝑦↑2)))
2		simpll 765	. . . . . . . . 9 ⊢ (((𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ) ∧ 𝑃 = ((𝑥↑2) + (𝑦↑2))) → 𝑥 ∈ ℕ)
3	2	adantl 482	. . . . . . . 8 ⊢ ((𝑥 ≤ 𝑦 ∧ ((𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ) ∧ 𝑃 = ((𝑥↑2) + (𝑦↑2)))) → 𝑥 ∈ ℕ)
4		breq1 5108	. . . . . . . . . . 11 ⊢ (𝑎 = 𝑥 → (𝑎 ≤ 𝑏 ↔ 𝑥 ≤ 𝑏))
5		oveq1 7364	. . . . . . . . . . . . 13 ⊢ (𝑎 = 𝑥 → (𝑎↑2) = (𝑥↑2))
6	5	oveq1d 7372	. . . . . . . . . . . 12 ⊢ (𝑎 = 𝑥 → ((𝑎↑2) + (𝑏↑2)) = ((𝑥↑2) + (𝑏↑2)))
7	6	eqeq1d 2738	. . . . . . . . . . 11 ⊢ (𝑎 = 𝑥 → (((𝑎↑2) + (𝑏↑2)) = 𝑃 ↔ ((𝑥↑2) + (𝑏↑2)) = 𝑃))
8	4, 7	anbi12d 631	. . . . . . . . . 10 ⊢ (𝑎 = 𝑥 → ((𝑎 ≤ 𝑏 ∧ ((𝑎↑2) + (𝑏↑2)) = 𝑃) ↔ (𝑥 ≤ 𝑏 ∧ ((𝑥↑2) + (𝑏↑2)) = 𝑃)))
9	8	reubidv 3371	. . . . . . . . 9 ⊢ (𝑎 = 𝑥 → (∃!𝑏 ∈ ℕ (𝑎 ≤ 𝑏 ∧ ((𝑎↑2) + (𝑏↑2)) = 𝑃) ↔ ∃!𝑏 ∈ ℕ (𝑥 ≤ 𝑏 ∧ ((𝑥↑2) + (𝑏↑2)) = 𝑃)))
10	9	adantl 482	. . . . . . . 8 ⊢ (((𝑥 ≤ 𝑦 ∧ ((𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ) ∧ 𝑃 = ((𝑥↑2) + (𝑦↑2)))) ∧ 𝑎 = 𝑥) → (∃!𝑏 ∈ ℕ (𝑎 ≤ 𝑏 ∧ ((𝑎↑2) + (𝑏↑2)) = 𝑃) ↔ ∃!𝑏 ∈ ℕ (𝑥 ≤ 𝑏 ∧ ((𝑥↑2) + (𝑏↑2)) = 𝑃)))
11		simpr 485	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ) → 𝑦 ∈ ℕ)
12	11	adantr 481	. . . . . . . . . . . . . 14 ⊢ (((𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ) ∧ 𝑥 ≤ 𝑦) → 𝑦 ∈ ℕ)
13		breq2 5109	. . . . . . . . . . . . . . . . 17 ⊢ (𝑏 = 𝑦 → (𝑥 ≤ 𝑏 ↔ 𝑥 ≤ 𝑦))
14		oveq1 7364	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑏 = 𝑦 → (𝑏↑2) = (𝑦↑2))
15	14	oveq2d 7373	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑏 = 𝑦 → ((𝑥↑2) + (𝑏↑2)) = ((𝑥↑2) + (𝑦↑2)))
16	15	eqeq1d 2738	. . . . . . . . . . . . . . . . 17 ⊢ (𝑏 = 𝑦 → (((𝑥↑2) + (𝑏↑2)) = ((𝑥↑2) + (𝑦↑2)) ↔ ((𝑥↑2) + (𝑦↑2)) = ((𝑥↑2) + (𝑦↑2))))
17	13, 16	anbi12d 631	. . . . . . . . . . . . . . . 16 ⊢ (𝑏 = 𝑦 → ((𝑥 ≤ 𝑏 ∧ ((𝑥↑2) + (𝑏↑2)) = ((𝑥↑2) + (𝑦↑2))) ↔ (𝑥 ≤ 𝑦 ∧ ((𝑥↑2) + (𝑦↑2)) = ((𝑥↑2) + (𝑦↑2)))))
18		equequ1 2028	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑏 = 𝑦 → (𝑏 = 𝑐 ↔ 𝑦 = 𝑐))
19	18	imbi2d 340	. . . . . . . . . . . . . . . . 17 ⊢ (𝑏 = 𝑦 → (((𝑥 ≤ 𝑐 ∧ ((𝑥↑2) + (𝑐↑2)) = ((𝑥↑2) + (𝑦↑2))) → 𝑏 = 𝑐) ↔ ((𝑥 ≤ 𝑐 ∧ ((𝑥↑2) + (𝑐↑2)) = ((𝑥↑2) + (𝑦↑2))) → 𝑦 = 𝑐)))
20	19	ralbidv 3174	. . . . . . . . . . . . . . . 16 ⊢ (𝑏 = 𝑦 → (∀𝑐 ∈ ℕ ((𝑥 ≤ 𝑐 ∧ ((𝑥↑2) + (𝑐↑2)) = ((𝑥↑2) + (𝑦↑2))) → 𝑏 = 𝑐) ↔ ∀𝑐 ∈ ℕ ((𝑥 ≤ 𝑐 ∧ ((𝑥↑2) + (𝑐↑2)) = ((𝑥↑2) + (𝑦↑2))) → 𝑦 = 𝑐)))
21	17, 20	anbi12d 631	. . . . . . . . . . . . . . 15 ⊢ (𝑏 = 𝑦 → (((𝑥 ≤ 𝑏 ∧ ((𝑥↑2) + (𝑏↑2)) = ((𝑥↑2) + (𝑦↑2))) ∧ ∀𝑐 ∈ ℕ ((𝑥 ≤ 𝑐 ∧ ((𝑥↑2) + (𝑐↑2)) = ((𝑥↑2) + (𝑦↑2))) → 𝑏 = 𝑐)) ↔ ((𝑥 ≤ 𝑦 ∧ ((𝑥↑2) + (𝑦↑2)) = ((𝑥↑2) + (𝑦↑2))) ∧ ∀𝑐 ∈ ℕ ((𝑥 ≤ 𝑐 ∧ ((𝑥↑2) + (𝑐↑2)) = ((𝑥↑2) + (𝑦↑2))) → 𝑦 = 𝑐))))
22	21	adantl 482	. . . . . . . . . . . . . 14 ⊢ ((((𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ) ∧ 𝑥 ≤ 𝑦) ∧ 𝑏 = 𝑦) → (((𝑥 ≤ 𝑏 ∧ ((𝑥↑2) + (𝑏↑2)) = ((𝑥↑2) + (𝑦↑2))) ∧ ∀𝑐 ∈ ℕ ((𝑥 ≤ 𝑐 ∧ ((𝑥↑2) + (𝑐↑2)) = ((𝑥↑2) + (𝑦↑2))) → 𝑏 = 𝑐)) ↔ ((𝑥 ≤ 𝑦 ∧ ((𝑥↑2) + (𝑦↑2)) = ((𝑥↑2) + (𝑦↑2))) ∧ ∀𝑐 ∈ ℕ ((𝑥 ≤ 𝑐 ∧ ((𝑥↑2) + (𝑐↑2)) = ((𝑥↑2) + (𝑦↑2))) → 𝑦 = 𝑐))))
23		simpr 485	. . . . . . . . . . . . . . 15 ⊢ (((𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ) ∧ 𝑥 ≤ 𝑦) → 𝑥 ≤ 𝑦)
24		eqidd 2737	. . . . . . . . . . . . . . 15 ⊢ (((𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ) ∧ 𝑥 ≤ 𝑦) → ((𝑥↑2) + (𝑦↑2)) = ((𝑥↑2) + (𝑦↑2)))
25		nnre 12160	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑐 ∈ ℕ → 𝑐 ∈ ℝ)
26	25	resqcld 14030	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑐 ∈ ℕ → (𝑐↑2) ∈ ℝ)
27	26	adantl 482	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ) ∧ 𝑥 ≤ 𝑦) ∧ 𝑐 ∈ ℕ) → (𝑐↑2) ∈ ℝ)
28		nnre 12160	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑦 ∈ ℕ → 𝑦 ∈ ℝ)
29	28	resqcld 14030	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑦 ∈ ℕ → (𝑦↑2) ∈ ℝ)
30	29	adantl 482	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ) → (𝑦↑2) ∈ ℝ)
31	30	ad2antrr 724	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ) ∧ 𝑥 ≤ 𝑦) ∧ 𝑐 ∈ ℕ) → (𝑦↑2) ∈ ℝ)
32		nnre 12160	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑥 ∈ ℕ → 𝑥 ∈ ℝ)
33	32	resqcld 14030	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑥 ∈ ℕ → (𝑥↑2) ∈ ℝ)
34	33	adantr 481	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ) → (𝑥↑2) ∈ ℝ)
35	34	ad2antrr 724	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ) ∧ 𝑥 ≤ 𝑦) ∧ 𝑐 ∈ ℕ) → (𝑥↑2) ∈ ℝ)
36		readdcan 11329	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑐↑2) ∈ ℝ ∧ (𝑦↑2) ∈ ℝ ∧ (𝑥↑2) ∈ ℝ) → (((𝑥↑2) + (𝑐↑2)) = ((𝑥↑2) + (𝑦↑2)) ↔ (𝑐↑2) = (𝑦↑2)))
37	27, 31, 35, 36	syl3anc 1371	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ) ∧ 𝑥 ≤ 𝑦) ∧ 𝑐 ∈ ℕ) → (((𝑥↑2) + (𝑐↑2)) = ((𝑥↑2) + (𝑦↑2)) ↔ (𝑐↑2) = (𝑦↑2)))
38	28	ad4antlr 731	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((((𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ) ∧ 𝑥 ≤ 𝑦) ∧ 𝑐 ∈ ℕ) ∧ (𝑐↑2) = (𝑦↑2)) → 𝑦 ∈ ℝ)
39	25	ad2antlr 725	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((((𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ) ∧ 𝑥 ≤ 𝑦) ∧ 𝑐 ∈ ℕ) ∧ (𝑐↑2) = (𝑦↑2)) → 𝑐 ∈ ℝ)
40		nnnn0 12420	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑦 ∈ ℕ → 𝑦 ∈ ℕ₀)
41	40	nn0ge0d 12476	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑦 ∈ ℕ → 0 ≤ 𝑦)
42	41	ad4antlr 731	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((((𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ) ∧ 𝑥 ≤ 𝑦) ∧ 𝑐 ∈ ℕ) ∧ (𝑐↑2) = (𝑦↑2)) → 0 ≤ 𝑦)
43		nnnn0 12420	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑐 ∈ ℕ → 𝑐 ∈ ℕ₀)
44	43	nn0ge0d 12476	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑐 ∈ ℕ → 0 ≤ 𝑐)
45	44	ad2antlr 725	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((((𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ) ∧ 𝑥 ≤ 𝑦) ∧ 𝑐 ∈ ℕ) ∧ (𝑐↑2) = (𝑦↑2)) → 0 ≤ 𝑐)
46		simpr 485	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((((𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ) ∧ 𝑥 ≤ 𝑦) ∧ 𝑐 ∈ ℕ) ∧ (𝑐↑2) = (𝑦↑2)) → (𝑐↑2) = (𝑦↑2))
47	46	eqcomd 2742	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((((𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ) ∧ 𝑥 ≤ 𝑦) ∧ 𝑐 ∈ ℕ) ∧ (𝑐↑2) = (𝑦↑2)) → (𝑦↑2) = (𝑐↑2))
48	38, 39, 42, 45, 47	sq11d 14161	. . . . . . . . . . . . . . . . . . 19 ⊢ (((((𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ) ∧ 𝑥 ≤ 𝑦) ∧ 𝑐 ∈ ℕ) ∧ (𝑐↑2) = (𝑦↑2)) → 𝑦 = 𝑐)
49	48	ex 413	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ) ∧ 𝑥 ≤ 𝑦) ∧ 𝑐 ∈ ℕ) → ((𝑐↑2) = (𝑦↑2) → 𝑦 = 𝑐))
50	37, 49	sylbid 239	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ) ∧ 𝑥 ≤ 𝑦) ∧ 𝑐 ∈ ℕ) → (((𝑥↑2) + (𝑐↑2)) = ((𝑥↑2) + (𝑦↑2)) → 𝑦 = 𝑐))
51	50	adantld 491	. . . . . . . . . . . . . . . 16 ⊢ ((((𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ) ∧ 𝑥 ≤ 𝑦) ∧ 𝑐 ∈ ℕ) → ((𝑥 ≤ 𝑐 ∧ ((𝑥↑2) + (𝑐↑2)) = ((𝑥↑2) + (𝑦↑2))) → 𝑦 = 𝑐))
52	51	ralrimiva 3143	. . . . . . . . . . . . . . 15 ⊢ (((𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ) ∧ 𝑥 ≤ 𝑦) → ∀𝑐 ∈ ℕ ((𝑥 ≤ 𝑐 ∧ ((𝑥↑2) + (𝑐↑2)) = ((𝑥↑2) + (𝑦↑2))) → 𝑦 = 𝑐))
53	23, 24, 52	jca31 515	. . . . . . . . . . . . . 14 ⊢ (((𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ) ∧ 𝑥 ≤ 𝑦) → ((𝑥 ≤ 𝑦 ∧ ((𝑥↑2) + (𝑦↑2)) = ((𝑥↑2) + (𝑦↑2))) ∧ ∀𝑐 ∈ ℕ ((𝑥 ≤ 𝑐 ∧ ((𝑥↑2) + (𝑐↑2)) = ((𝑥↑2) + (𝑦↑2))) → 𝑦 = 𝑐)))
54	12, 22, 53	rspcedvd 3583	. . . . . . . . . . . . 13 ⊢ (((𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ) ∧ 𝑥 ≤ 𝑦) → ∃𝑏 ∈ ℕ ((𝑥 ≤ 𝑏 ∧ ((𝑥↑2) + (𝑏↑2)) = ((𝑥↑2) + (𝑦↑2))) ∧ ∀𝑐 ∈ ℕ ((𝑥 ≤ 𝑐 ∧ ((𝑥↑2) + (𝑐↑2)) = ((𝑥↑2) + (𝑦↑2))) → 𝑏 = 𝑐)))
55		breq2 5109	. . . . . . . . . . . . . . 15 ⊢ (𝑏 = 𝑐 → (𝑥 ≤ 𝑏 ↔ 𝑥 ≤ 𝑐))
56		oveq1 7364	. . . . . . . . . . . . . . . . 17 ⊢ (𝑏 = 𝑐 → (𝑏↑2) = (𝑐↑2))
57	56	oveq2d 7373	. . . . . . . . . . . . . . . 16 ⊢ (𝑏 = 𝑐 → ((𝑥↑2) + (𝑏↑2)) = ((𝑥↑2) + (𝑐↑2)))
58	57	eqeq1d 2738	. . . . . . . . . . . . . . 15 ⊢ (𝑏 = 𝑐 → (((𝑥↑2) + (𝑏↑2)) = ((𝑥↑2) + (𝑦↑2)) ↔ ((𝑥↑2) + (𝑐↑2)) = ((𝑥↑2) + (𝑦↑2))))
59	55, 58	anbi12d 631	. . . . . . . . . . . . . 14 ⊢ (𝑏 = 𝑐 → ((𝑥 ≤ 𝑏 ∧ ((𝑥↑2) + (𝑏↑2)) = ((𝑥↑2) + (𝑦↑2))) ↔ (𝑥 ≤ 𝑐 ∧ ((𝑥↑2) + (𝑐↑2)) = ((𝑥↑2) + (𝑦↑2)))))
60	59	reu8 3691	. . . . . . . . . . . . 13 ⊢ (∃!𝑏 ∈ ℕ (𝑥 ≤ 𝑏 ∧ ((𝑥↑2) + (𝑏↑2)) = ((𝑥↑2) + (𝑦↑2))) ↔ ∃𝑏 ∈ ℕ ((𝑥 ≤ 𝑏 ∧ ((𝑥↑2) + (𝑏↑2)) = ((𝑥↑2) + (𝑦↑2))) ∧ ∀𝑐 ∈ ℕ ((𝑥 ≤ 𝑐 ∧ ((𝑥↑2) + (𝑐↑2)) = ((𝑥↑2) + (𝑦↑2))) → 𝑏 = 𝑐)))
61	54, 60	sylibr 233	. . . . . . . . . . . 12 ⊢ (((𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ) ∧ 𝑥 ≤ 𝑦) → ∃!𝑏 ∈ ℕ (𝑥 ≤ 𝑏 ∧ ((𝑥↑2) + (𝑏↑2)) = ((𝑥↑2) + (𝑦↑2))))
62	61	ex 413	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ) → (𝑥 ≤ 𝑦 → ∃!𝑏 ∈ ℕ (𝑥 ≤ 𝑏 ∧ ((𝑥↑2) + (𝑏↑2)) = ((𝑥↑2) + (𝑦↑2)))))
63	62	adantr 481	. . . . . . . . . 10 ⊢ (((𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ) ∧ 𝑃 = ((𝑥↑2) + (𝑦↑2))) → (𝑥 ≤ 𝑦 → ∃!𝑏 ∈ ℕ (𝑥 ≤ 𝑏 ∧ ((𝑥↑2) + (𝑏↑2)) = ((𝑥↑2) + (𝑦↑2)))))
64	63	impcom 408	. . . . . . . . 9 ⊢ ((𝑥 ≤ 𝑦 ∧ ((𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ) ∧ 𝑃 = ((𝑥↑2) + (𝑦↑2)))) → ∃!𝑏 ∈ ℕ (𝑥 ≤ 𝑏 ∧ ((𝑥↑2) + (𝑏↑2)) = ((𝑥↑2) + (𝑦↑2))))
65		eqeq2 2748	. . . . . . . . . . . . 13 ⊢ (𝑃 = ((𝑥↑2) + (𝑦↑2)) → (((𝑥↑2) + (𝑏↑2)) = 𝑃 ↔ ((𝑥↑2) + (𝑏↑2)) = ((𝑥↑2) + (𝑦↑2))))
66	65	anbi2d 629	. . . . . . . . . . . 12 ⊢ (𝑃 = ((𝑥↑2) + (𝑦↑2)) → ((𝑥 ≤ 𝑏 ∧ ((𝑥↑2) + (𝑏↑2)) = 𝑃) ↔ (𝑥 ≤ 𝑏 ∧ ((𝑥↑2) + (𝑏↑2)) = ((𝑥↑2) + (𝑦↑2)))))
67	66	reubidv 3371	. . . . . . . . . . 11 ⊢ (𝑃 = ((𝑥↑2) + (𝑦↑2)) → (∃!𝑏 ∈ ℕ (𝑥 ≤ 𝑏 ∧ ((𝑥↑2) + (𝑏↑2)) = 𝑃) ↔ ∃!𝑏 ∈ ℕ (𝑥 ≤ 𝑏 ∧ ((𝑥↑2) + (𝑏↑2)) = ((𝑥↑2) + (𝑦↑2)))))
68	67	adantl 482	. . . . . . . . . 10 ⊢ (((𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ) ∧ 𝑃 = ((𝑥↑2) + (𝑦↑2))) → (∃!𝑏 ∈ ℕ (𝑥 ≤ 𝑏 ∧ ((𝑥↑2) + (𝑏↑2)) = 𝑃) ↔ ∃!𝑏 ∈ ℕ (𝑥 ≤ 𝑏 ∧ ((𝑥↑2) + (𝑏↑2)) = ((𝑥↑2) + (𝑦↑2)))))
69	68	adantl 482	. . . . . . . . 9 ⊢ ((𝑥 ≤ 𝑦 ∧ ((𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ) ∧ 𝑃 = ((𝑥↑2) + (𝑦↑2)))) → (∃!𝑏 ∈ ℕ (𝑥 ≤ 𝑏 ∧ ((𝑥↑2) + (𝑏↑2)) = 𝑃) ↔ ∃!𝑏 ∈ ℕ (𝑥 ≤ 𝑏 ∧ ((𝑥↑2) + (𝑏↑2)) = ((𝑥↑2) + (𝑦↑2)))))
70	64, 69	mpbird 256	. . . . . . . 8 ⊢ ((𝑥 ≤ 𝑦 ∧ ((𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ) ∧ 𝑃 = ((𝑥↑2) + (𝑦↑2)))) → ∃!𝑏 ∈ ℕ (𝑥 ≤ 𝑏 ∧ ((𝑥↑2) + (𝑏↑2)) = 𝑃))
71	3, 10, 70	rspcedvd 3583	. . . . . . 7 ⊢ ((𝑥 ≤ 𝑦 ∧ ((𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ) ∧ 𝑃 = ((𝑥↑2) + (𝑦↑2)))) → ∃𝑎 ∈ ℕ ∃!𝑏 ∈ ℕ (𝑎 ≤ 𝑏 ∧ ((𝑎↑2) + (𝑏↑2)) = 𝑃))
72	11	adantr 481	. . . . . . . . 9 ⊢ (((𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ) ∧ 𝑃 = ((𝑥↑2) + (𝑦↑2))) → 𝑦 ∈ ℕ)
73	72	adantl 482	. . . . . . . 8 ⊢ ((¬ 𝑥 ≤ 𝑦 ∧ ((𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ) ∧ 𝑃 = ((𝑥↑2) + (𝑦↑2)))) → 𝑦 ∈ ℕ)
74		breq1 5108	. . . . . . . . . . 11 ⊢ (𝑎 = 𝑦 → (𝑎 ≤ 𝑏 ↔ 𝑦 ≤ 𝑏))
75		oveq1 7364	. . . . . . . . . . . . 13 ⊢ (𝑎 = 𝑦 → (𝑎↑2) = (𝑦↑2))
76	75	oveq1d 7372	. . . . . . . . . . . 12 ⊢ (𝑎 = 𝑦 → ((𝑎↑2) + (𝑏↑2)) = ((𝑦↑2) + (𝑏↑2)))
77	76	eqeq1d 2738	. . . . . . . . . . 11 ⊢ (𝑎 = 𝑦 → (((𝑎↑2) + (𝑏↑2)) = 𝑃 ↔ ((𝑦↑2) + (𝑏↑2)) = 𝑃))
78	74, 77	anbi12d 631	. . . . . . . . . 10 ⊢ (𝑎 = 𝑦 → ((𝑎 ≤ 𝑏 ∧ ((𝑎↑2) + (𝑏↑2)) = 𝑃) ↔ (𝑦 ≤ 𝑏 ∧ ((𝑦↑2) + (𝑏↑2)) = 𝑃)))
79	78	reubidv 3371	. . . . . . . . 9 ⊢ (𝑎 = 𝑦 → (∃!𝑏 ∈ ℕ (𝑎 ≤ 𝑏 ∧ ((𝑎↑2) + (𝑏↑2)) = 𝑃) ↔ ∃!𝑏 ∈ ℕ (𝑦 ≤ 𝑏 ∧ ((𝑦↑2) + (𝑏↑2)) = 𝑃)))
80	79	adantl 482	. . . . . . . 8 ⊢ (((¬ 𝑥 ≤ 𝑦 ∧ ((𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ) ∧ 𝑃 = ((𝑥↑2) + (𝑦↑2)))) ∧ 𝑎 = 𝑦) → (∃!𝑏 ∈ ℕ (𝑎 ≤ 𝑏 ∧ ((𝑎↑2) + (𝑏↑2)) = 𝑃) ↔ ∃!𝑏 ∈ ℕ (𝑦 ≤ 𝑏 ∧ ((𝑦↑2) + (𝑏↑2)) = 𝑃)))
81		simpll 765	. . . . . . . . . . . . . 14 ⊢ (((𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ) ∧ ¬ 𝑥 ≤ 𝑦) → 𝑥 ∈ ℕ)
82		breq2 5109	. . . . . . . . . . . . . . . . 17 ⊢ (𝑏 = 𝑥 → (𝑦 ≤ 𝑏 ↔ 𝑦 ≤ 𝑥))
83		oveq1 7364	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑏 = 𝑥 → (𝑏↑2) = (𝑥↑2))
84	83	oveq2d 7373	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑏 = 𝑥 → ((𝑦↑2) + (𝑏↑2)) = ((𝑦↑2) + (𝑥↑2)))
85	84	eqeq1d 2738	. . . . . . . . . . . . . . . . 17 ⊢ (𝑏 = 𝑥 → (((𝑦↑2) + (𝑏↑2)) = ((𝑥↑2) + (𝑦↑2)) ↔ ((𝑦↑2) + (𝑥↑2)) = ((𝑥↑2) + (𝑦↑2))))
86	82, 85	anbi12d 631	. . . . . . . . . . . . . . . 16 ⊢ (𝑏 = 𝑥 → ((𝑦 ≤ 𝑏 ∧ ((𝑦↑2) + (𝑏↑2)) = ((𝑥↑2) + (𝑦↑2))) ↔ (𝑦 ≤ 𝑥 ∧ ((𝑦↑2) + (𝑥↑2)) = ((𝑥↑2) + (𝑦↑2)))))
87		equequ1 2028	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑏 = 𝑥 → (𝑏 = 𝑐 ↔ 𝑥 = 𝑐))
88	87	imbi2d 340	. . . . . . . . . . . . . . . . 17 ⊢ (𝑏 = 𝑥 → (((𝑦 ≤ 𝑐 ∧ ((𝑦↑2) + (𝑐↑2)) = ((𝑥↑2) + (𝑦↑2))) → 𝑏 = 𝑐) ↔ ((𝑦 ≤ 𝑐 ∧ ((𝑦↑2) + (𝑐↑2)) = ((𝑥↑2) + (𝑦↑2))) → 𝑥 = 𝑐)))
89	88	ralbidv 3174	. . . . . . . . . . . . . . . 16 ⊢ (𝑏 = 𝑥 → (∀𝑐 ∈ ℕ ((𝑦 ≤ 𝑐 ∧ ((𝑦↑2) + (𝑐↑2)) = ((𝑥↑2) + (𝑦↑2))) → 𝑏 = 𝑐) ↔ ∀𝑐 ∈ ℕ ((𝑦 ≤ 𝑐 ∧ ((𝑦↑2) + (𝑐↑2)) = ((𝑥↑2) + (𝑦↑2))) → 𝑥 = 𝑐)))
90	86, 89	anbi12d 631	. . . . . . . . . . . . . . 15 ⊢ (𝑏 = 𝑥 → (((𝑦 ≤ 𝑏 ∧ ((𝑦↑2) + (𝑏↑2)) = ((𝑥↑2) + (𝑦↑2))) ∧ ∀𝑐 ∈ ℕ ((𝑦 ≤ 𝑐 ∧ ((𝑦↑2) + (𝑐↑2)) = ((𝑥↑2) + (𝑦↑2))) → 𝑏 = 𝑐)) ↔ ((𝑦 ≤ 𝑥 ∧ ((𝑦↑2) + (𝑥↑2)) = ((𝑥↑2) + (𝑦↑2))) ∧ ∀𝑐 ∈ ℕ ((𝑦 ≤ 𝑐 ∧ ((𝑦↑2) + (𝑐↑2)) = ((𝑥↑2) + (𝑦↑2))) → 𝑥 = 𝑐))))
91	90	adantl 482	. . . . . . . . . . . . . 14 ⊢ ((((𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ) ∧ ¬ 𝑥 ≤ 𝑦) ∧ 𝑏 = 𝑥) → (((𝑦 ≤ 𝑏 ∧ ((𝑦↑2) + (𝑏↑2)) = ((𝑥↑2) + (𝑦↑2))) ∧ ∀𝑐 ∈ ℕ ((𝑦 ≤ 𝑐 ∧ ((𝑦↑2) + (𝑐↑2)) = ((𝑥↑2) + (𝑦↑2))) → 𝑏 = 𝑐)) ↔ ((𝑦 ≤ 𝑥 ∧ ((𝑦↑2) + (𝑥↑2)) = ((𝑥↑2) + (𝑦↑2))) ∧ ∀𝑐 ∈ ℕ ((𝑦 ≤ 𝑐 ∧ ((𝑦↑2) + (𝑐↑2)) = ((𝑥↑2) + (𝑦↑2))) → 𝑥 = 𝑐))))
92		ltnle 11234	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑦 ∈ ℝ ∧ 𝑥 ∈ ℝ) → (𝑦 < 𝑥 ↔ ¬ 𝑥 ≤ 𝑦))
93	28, 32, 92	syl2anr 597	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ) → (𝑦 < 𝑥 ↔ ¬ 𝑥 ≤ 𝑦))
94	28	ad2antlr 725	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ) ∧ 𝑦 < 𝑥) → 𝑦 ∈ ℝ)
95	32	ad2antrr 724	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ) ∧ 𝑦 < 𝑥) → 𝑥 ∈ ℝ)
96		simpr 485	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ) ∧ 𝑦 < 𝑥) → 𝑦 < 𝑥)
97	94, 95, 96	ltled 11303	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ) ∧ 𝑦 < 𝑥) → 𝑦 ≤ 𝑥)
98	97	ex 413	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ) → (𝑦 < 𝑥 → 𝑦 ≤ 𝑥))
99	93, 98	sylbird 259	. . . . . . . . . . . . . . . 16 ⊢ ((𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ) → (¬ 𝑥 ≤ 𝑦 → 𝑦 ≤ 𝑥))
100	99	imp 407	. . . . . . . . . . . . . . 15 ⊢ (((𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ) ∧ ¬ 𝑥 ≤ 𝑦) → 𝑦 ≤ 𝑥)
101	29	recnd 11183	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑦 ∈ ℕ → (𝑦↑2) ∈ ℂ)
102	101	adantl 482	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ) → (𝑦↑2) ∈ ℂ)
103	33	recnd 11183	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 ∈ ℕ → (𝑥↑2) ∈ ℂ)
104	103	adantr 481	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ) → (𝑥↑2) ∈ ℂ)
105	102, 104	addcomd 11357	. . . . . . . . . . . . . . . 16 ⊢ ((𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ) → ((𝑦↑2) + (𝑥↑2)) = ((𝑥↑2) + (𝑦↑2)))
106	105	adantr 481	. . . . . . . . . . . . . . 15 ⊢ (((𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ) ∧ ¬ 𝑥 ≤ 𝑦) → ((𝑦↑2) + (𝑥↑2)) = ((𝑥↑2) + (𝑦↑2)))
107	34	recnd 11183	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ) → (𝑥↑2) ∈ ℂ)
108	107	adantr 481	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ) ∧ 𝑐 ∈ ℕ) → (𝑥↑2) ∈ ℂ)
109	30	recnd 11183	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ) → (𝑦↑2) ∈ ℂ)
110	109	adantr 481	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ) ∧ 𝑐 ∈ ℕ) → (𝑦↑2) ∈ ℂ)
111	108, 110	addcomd 11357	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ) ∧ 𝑐 ∈ ℕ) → ((𝑥↑2) + (𝑦↑2)) = ((𝑦↑2) + (𝑥↑2)))
112	111	eqeq2d 2747	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ) ∧ 𝑐 ∈ ℕ) → (((𝑦↑2) + (𝑐↑2)) = ((𝑥↑2) + (𝑦↑2)) ↔ ((𝑦↑2) + (𝑐↑2)) = ((𝑦↑2) + (𝑥↑2))))
113	26	adantl 482	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ) ∧ 𝑐 ∈ ℕ) → (𝑐↑2) ∈ ℝ)
114	33	ad2antrr 724	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ) ∧ 𝑐 ∈ ℕ) → (𝑥↑2) ∈ ℝ)
115	29	ad2antlr 725	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ) ∧ 𝑐 ∈ ℕ) → (𝑦↑2) ∈ ℝ)
116		readdcan 11329	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝑐↑2) ∈ ℝ ∧ (𝑥↑2) ∈ ℝ ∧ (𝑦↑2) ∈ ℝ) → (((𝑦↑2) + (𝑐↑2)) = ((𝑦↑2) + (𝑥↑2)) ↔ (𝑐↑2) = (𝑥↑2)))
117	113, 114, 115, 116	syl3anc 1371	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ) ∧ 𝑐 ∈ ℕ) → (((𝑦↑2) + (𝑐↑2)) = ((𝑦↑2) + (𝑥↑2)) ↔ (𝑐↑2) = (𝑥↑2)))
118	112, 117	bitrd 278	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ) ∧ 𝑐 ∈ ℕ) → (((𝑦↑2) + (𝑐↑2)) = ((𝑥↑2) + (𝑦↑2)) ↔ (𝑐↑2) = (𝑥↑2)))
119	25	ad2antlr 725	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ) ∧ 𝑐 ∈ ℕ) ∧ (𝑐↑2) = (𝑥↑2)) → 𝑐 ∈ ℝ)
120	32	adantr 481	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ) → 𝑥 ∈ ℝ)
121	120	ad2antrr 724	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ) ∧ 𝑐 ∈ ℕ) ∧ (𝑐↑2) = (𝑥↑2)) → 𝑥 ∈ ℝ)
122	44	ad2antlr 725	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ) ∧ 𝑐 ∈ ℕ) ∧ (𝑐↑2) = (𝑥↑2)) → 0 ≤ 𝑐)
123		nnnn0 12420	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑥 ∈ ℕ → 𝑥 ∈ ℕ₀)
124	123	nn0ge0d 12476	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑥 ∈ ℕ → 0 ≤ 𝑥)
125	124	adantr 481	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ) → 0 ≤ 𝑥)
126	125	ad2antrr 724	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ) ∧ 𝑐 ∈ ℕ) ∧ (𝑐↑2) = (𝑥↑2)) → 0 ≤ 𝑥)
127		simpr 485	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ) ∧ 𝑐 ∈ ℕ) ∧ (𝑐↑2) = (𝑥↑2)) → (𝑐↑2) = (𝑥↑2))
128	119, 121, 122, 126, 127	sq11d 14161	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ) ∧ 𝑐 ∈ ℕ) ∧ (𝑐↑2) = (𝑥↑2)) → 𝑐 = 𝑥)
129	128	eqcomd 2742	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ) ∧ 𝑐 ∈ ℕ) ∧ (𝑐↑2) = (𝑥↑2)) → 𝑥 = 𝑐)
130	129	ex 413	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ) ∧ 𝑐 ∈ ℕ) → ((𝑐↑2) = (𝑥↑2) → 𝑥 = 𝑐))
131	118, 130	sylbid 239	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ) ∧ 𝑐 ∈ ℕ) → (((𝑦↑2) + (𝑐↑2)) = ((𝑥↑2) + (𝑦↑2)) → 𝑥 = 𝑐))
132	131	adantld 491	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ) ∧ 𝑐 ∈ ℕ) → ((𝑦 ≤ 𝑐 ∧ ((𝑦↑2) + (𝑐↑2)) = ((𝑥↑2) + (𝑦↑2))) → 𝑥 = 𝑐))
133	132	ralrimiva 3143	. . . . . . . . . . . . . . . 16 ⊢ ((𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ) → ∀𝑐 ∈ ℕ ((𝑦 ≤ 𝑐 ∧ ((𝑦↑2) + (𝑐↑2)) = ((𝑥↑2) + (𝑦↑2))) → 𝑥 = 𝑐))
134	133	adantr 481	. . . . . . . . . . . . . . 15 ⊢ (((𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ) ∧ ¬ 𝑥 ≤ 𝑦) → ∀𝑐 ∈ ℕ ((𝑦 ≤ 𝑐 ∧ ((𝑦↑2) + (𝑐↑2)) = ((𝑥↑2) + (𝑦↑2))) → 𝑥 = 𝑐))
135	100, 106, 134	jca31 515	. . . . . . . . . . . . . 14 ⊢ (((𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ) ∧ ¬ 𝑥 ≤ 𝑦) → ((𝑦 ≤ 𝑥 ∧ ((𝑦↑2) + (𝑥↑2)) = ((𝑥↑2) + (𝑦↑2))) ∧ ∀𝑐 ∈ ℕ ((𝑦 ≤ 𝑐 ∧ ((𝑦↑2) + (𝑐↑2)) = ((𝑥↑2) + (𝑦↑2))) → 𝑥 = 𝑐)))
136	81, 91, 135	rspcedvd 3583	. . . . . . . . . . . . 13 ⊢ (((𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ) ∧ ¬ 𝑥 ≤ 𝑦) → ∃𝑏 ∈ ℕ ((𝑦 ≤ 𝑏 ∧ ((𝑦↑2) + (𝑏↑2)) = ((𝑥↑2) + (𝑦↑2))) ∧ ∀𝑐 ∈ ℕ ((𝑦 ≤ 𝑐 ∧ ((𝑦↑2) + (𝑐↑2)) = ((𝑥↑2) + (𝑦↑2))) → 𝑏 = 𝑐)))
137		breq2 5109	. . . . . . . . . . . . . . 15 ⊢ (𝑏 = 𝑐 → (𝑦 ≤ 𝑏 ↔ 𝑦 ≤ 𝑐))
138	56	oveq2d 7373	. . . . . . . . . . . . . . . 16 ⊢ (𝑏 = 𝑐 → ((𝑦↑2) + (𝑏↑2)) = ((𝑦↑2) + (𝑐↑2)))
139	138	eqeq1d 2738	. . . . . . . . . . . . . . 15 ⊢ (𝑏 = 𝑐 → (((𝑦↑2) + (𝑏↑2)) = ((𝑥↑2) + (𝑦↑2)) ↔ ((𝑦↑2) + (𝑐↑2)) = ((𝑥↑2) + (𝑦↑2))))
140	137, 139	anbi12d 631	. . . . . . . . . . . . . 14 ⊢ (𝑏 = 𝑐 → ((𝑦 ≤ 𝑏 ∧ ((𝑦↑2) + (𝑏↑2)) = ((𝑥↑2) + (𝑦↑2))) ↔ (𝑦 ≤ 𝑐 ∧ ((𝑦↑2) + (𝑐↑2)) = ((𝑥↑2) + (𝑦↑2)))))
141	140	reu8 3691	. . . . . . . . . . . . 13 ⊢ (∃!𝑏 ∈ ℕ (𝑦 ≤ 𝑏 ∧ ((𝑦↑2) + (𝑏↑2)) = ((𝑥↑2) + (𝑦↑2))) ↔ ∃𝑏 ∈ ℕ ((𝑦 ≤ 𝑏 ∧ ((𝑦↑2) + (𝑏↑2)) = ((𝑥↑2) + (𝑦↑2))) ∧ ∀𝑐 ∈ ℕ ((𝑦 ≤ 𝑐 ∧ ((𝑦↑2) + (𝑐↑2)) = ((𝑥↑2) + (𝑦↑2))) → 𝑏 = 𝑐)))
142	136, 141	sylibr 233	. . . . . . . . . . . 12 ⊢ (((𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ) ∧ ¬ 𝑥 ≤ 𝑦) → ∃!𝑏 ∈ ℕ (𝑦 ≤ 𝑏 ∧ ((𝑦↑2) + (𝑏↑2)) = ((𝑥↑2) + (𝑦↑2))))
143	142	ex 413	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ) → (¬ 𝑥 ≤ 𝑦 → ∃!𝑏 ∈ ℕ (𝑦 ≤ 𝑏 ∧ ((𝑦↑2) + (𝑏↑2)) = ((𝑥↑2) + (𝑦↑2)))))
144	143	adantr 481	. . . . . . . . . 10 ⊢ (((𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ) ∧ 𝑃 = ((𝑥↑2) + (𝑦↑2))) → (¬ 𝑥 ≤ 𝑦 → ∃!𝑏 ∈ ℕ (𝑦 ≤ 𝑏 ∧ ((𝑦↑2) + (𝑏↑2)) = ((𝑥↑2) + (𝑦↑2)))))
145	144	impcom 408	. . . . . . . . 9 ⊢ ((¬ 𝑥 ≤ 𝑦 ∧ ((𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ) ∧ 𝑃 = ((𝑥↑2) + (𝑦↑2)))) → ∃!𝑏 ∈ ℕ (𝑦 ≤ 𝑏 ∧ ((𝑦↑2) + (𝑏↑2)) = ((𝑥↑2) + (𝑦↑2))))
146		eqeq2 2748	. . . . . . . . . . . . 13 ⊢ (𝑃 = ((𝑥↑2) + (𝑦↑2)) → (((𝑦↑2) + (𝑏↑2)) = 𝑃 ↔ ((𝑦↑2) + (𝑏↑2)) = ((𝑥↑2) + (𝑦↑2))))
147	146	anbi2d 629	. . . . . . . . . . . 12 ⊢ (𝑃 = ((𝑥↑2) + (𝑦↑2)) → ((𝑦 ≤ 𝑏 ∧ ((𝑦↑2) + (𝑏↑2)) = 𝑃) ↔ (𝑦 ≤ 𝑏 ∧ ((𝑦↑2) + (𝑏↑2)) = ((𝑥↑2) + (𝑦↑2)))))
148	147	reubidv 3371	. . . . . . . . . . 11 ⊢ (𝑃 = ((𝑥↑2) + (𝑦↑2)) → (∃!𝑏 ∈ ℕ (𝑦 ≤ 𝑏 ∧ ((𝑦↑2) + (𝑏↑2)) = 𝑃) ↔ ∃!𝑏 ∈ ℕ (𝑦 ≤ 𝑏 ∧ ((𝑦↑2) + (𝑏↑2)) = ((𝑥↑2) + (𝑦↑2)))))
149	148	adantl 482	. . . . . . . . . 10 ⊢ (((𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ) ∧ 𝑃 = ((𝑥↑2) + (𝑦↑2))) → (∃!𝑏 ∈ ℕ (𝑦 ≤ 𝑏 ∧ ((𝑦↑2) + (𝑏↑2)) = 𝑃) ↔ ∃!𝑏 ∈ ℕ (𝑦 ≤ 𝑏 ∧ ((𝑦↑2) + (𝑏↑2)) = ((𝑥↑2) + (𝑦↑2)))))
150	149	adantl 482	. . . . . . . . 9 ⊢ ((¬ 𝑥 ≤ 𝑦 ∧ ((𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ) ∧ 𝑃 = ((𝑥↑2) + (𝑦↑2)))) → (∃!𝑏 ∈ ℕ (𝑦 ≤ 𝑏 ∧ ((𝑦↑2) + (𝑏↑2)) = 𝑃) ↔ ∃!𝑏 ∈ ℕ (𝑦 ≤ 𝑏 ∧ ((𝑦↑2) + (𝑏↑2)) = ((𝑥↑2) + (𝑦↑2)))))
151	145, 150	mpbird 256	. . . . . . . 8 ⊢ ((¬ 𝑥 ≤ 𝑦 ∧ ((𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ) ∧ 𝑃 = ((𝑥↑2) + (𝑦↑2)))) → ∃!𝑏 ∈ ℕ (𝑦 ≤ 𝑏 ∧ ((𝑦↑2) + (𝑏↑2)) = 𝑃))
152	73, 80, 151	rspcedvd 3583	. . . . . . 7 ⊢ ((¬ 𝑥 ≤ 𝑦 ∧ ((𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ) ∧ 𝑃 = ((𝑥↑2) + (𝑦↑2)))) → ∃𝑎 ∈ ℕ ∃!𝑏 ∈ ℕ (𝑎 ≤ 𝑏 ∧ ((𝑎↑2) + (𝑏↑2)) = 𝑃))
153	71, 152	pm2.61ian 810	. . . . . 6 ⊢ (((𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ) ∧ 𝑃 = ((𝑥↑2) + (𝑦↑2))) → ∃𝑎 ∈ ℕ ∃!𝑏 ∈ ℕ (𝑎 ≤ 𝑏 ∧ ((𝑎↑2) + (𝑏↑2)) = 𝑃))
154	153	ex 413	. . . . 5 ⊢ ((𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ) → (𝑃 = ((𝑥↑2) + (𝑦↑2)) → ∃𝑎 ∈ ℕ ∃!𝑏 ∈ ℕ (𝑎 ≤ 𝑏 ∧ ((𝑎↑2) + (𝑏↑2)) = 𝑃)))
155	154	adantl 482	. . . 4 ⊢ (((𝑃 ∈ ℙ ∧ (𝑃 mod 4) = 1) ∧ (𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ)) → (𝑃 = ((𝑥↑2) + (𝑦↑2)) → ∃𝑎 ∈ ℕ ∃!𝑏 ∈ ℕ (𝑎 ≤ 𝑏 ∧ ((𝑎↑2) + (𝑏↑2)) = 𝑃)))
156	155	rexlimdvva 3205	. . 3 ⊢ ((𝑃 ∈ ℙ ∧ (𝑃 mod 4) = 1) → (∃𝑥 ∈ ℕ ∃𝑦 ∈ ℕ 𝑃 = ((𝑥↑2) + (𝑦↑2)) → ∃𝑎 ∈ ℕ ∃!𝑏 ∈ ℕ (𝑎 ≤ 𝑏 ∧ ((𝑎↑2) + (𝑏↑2)) = 𝑃)))
157	1, 156	mpd 15	. 2 ⊢ ((𝑃 ∈ ℙ ∧ (𝑃 mod 4) = 1) → ∃𝑎 ∈ ℕ ∃!𝑏 ∈ ℕ (𝑎 ≤ 𝑏 ∧ ((𝑎↑2) + (𝑏↑2)) = 𝑃))
158		reurex 3357	. . . . 5 ⊢ (∃!𝑏 ∈ ℕ (𝑎 ≤ 𝑏 ∧ ((𝑎↑2) + (𝑏↑2)) = 𝑃) → ∃𝑏 ∈ ℕ (𝑎 ≤ 𝑏 ∧ ((𝑎↑2) + (𝑏↑2)) = 𝑃))
159	158	a1i 11	. . . 4 ⊢ (((𝑃 ∈ ℙ ∧ (𝑃 mod 4) = 1) ∧ 𝑎 ∈ ℕ) → (∃!𝑏 ∈ ℕ (𝑎 ≤ 𝑏 ∧ ((𝑎↑2) + (𝑏↑2)) = 𝑃) → ∃𝑏 ∈ ℕ (𝑎 ≤ 𝑏 ∧ ((𝑎↑2) + (𝑏↑2)) = 𝑃)))
160	159	ralrimiva 3143	. . 3 ⊢ ((𝑃 ∈ ℙ ∧ (𝑃 mod 4) = 1) → ∀𝑎 ∈ ℕ (∃!𝑏 ∈ ℕ (𝑎 ≤ 𝑏 ∧ ((𝑎↑2) + (𝑏↑2)) = 𝑃) → ∃𝑏 ∈ ℕ (𝑎 ≤ 𝑏 ∧ ((𝑎↑2) + (𝑏↑2)) = 𝑃)))
161		2sqmo 26785	. . . . 5 ⊢ (𝑃 ∈ ℙ → ∃*𝑎 ∈ ℕ₀ ∃𝑏 ∈ ℕ₀ (𝑎 ≤ 𝑏 ∧ ((𝑎↑2) + (𝑏↑2)) = 𝑃))
162		nnssnn0 12416	. . . . . 6 ⊢ ℕ ⊆ ℕ₀
163		nfcv 2907	. . . . . . 7 ⊢ Ⅎ𝑎ℕ
164		nfcv 2907	. . . . . . 7 ⊢ Ⅎ𝑎ℕ₀
165	163, 164	ssrmof 4009	. . . . . 6 ⊢ (ℕ ⊆ ℕ₀ → (∃𝑎 ∈ ℕ₀ ∃𝑏 ∈ ℕ₀ (𝑎 ≤ 𝑏 ∧ ((𝑎↑2) + (𝑏↑2)) = 𝑃) → ∃𝑎 ∈ ℕ ∃𝑏 ∈ ℕ₀ (𝑎 ≤ 𝑏 ∧ ((𝑎↑2) + (𝑏↑2)) = 𝑃)))
166	162, 165	ax-mp 5	. . . . 5 ⊢ (∃𝑎 ∈ ℕ₀ ∃𝑏 ∈ ℕ₀ (𝑎 ≤ 𝑏 ∧ ((𝑎↑2) + (𝑏↑2)) = 𝑃) → ∃𝑎 ∈ ℕ ∃𝑏 ∈ ℕ₀ (𝑎 ≤ 𝑏 ∧ ((𝑎↑2) + (𝑏↑2)) = 𝑃))
167		ssrexv 4011	. . . . . . 7 ⊢ (ℕ ⊆ ℕ₀ → (∃𝑏 ∈ ℕ (𝑎 ≤ 𝑏 ∧ ((𝑎↑2) + (𝑏↑2)) = 𝑃) → ∃𝑏 ∈ ℕ₀ (𝑎 ≤ 𝑏 ∧ ((𝑎↑2) + (𝑏↑2)) = 𝑃)))
168	162, 167	ax-mp 5	. . . . . 6 ⊢ (∃𝑏 ∈ ℕ (𝑎 ≤ 𝑏 ∧ ((𝑎↑2) + (𝑏↑2)) = 𝑃) → ∃𝑏 ∈ ℕ₀ (𝑎 ≤ 𝑏 ∧ ((𝑎↑2) + (𝑏↑2)) = 𝑃))
169	168	rmoimi 3700	. . . . 5 ⊢ (∃𝑎 ∈ ℕ ∃𝑏 ∈ ℕ₀ (𝑎 ≤ 𝑏 ∧ ((𝑎↑2) + (𝑏↑2)) = 𝑃) → ∃𝑎 ∈ ℕ ∃𝑏 ∈ ℕ (𝑎 ≤ 𝑏 ∧ ((𝑎↑2) + (𝑏↑2)) = 𝑃))
170	161, 166, 169	3syl 18	. . . 4 ⊢ (𝑃 ∈ ℙ → ∃*𝑎 ∈ ℕ ∃𝑏 ∈ ℕ (𝑎 ≤ 𝑏 ∧ ((𝑎↑2) + (𝑏↑2)) = 𝑃))
171	170	adantr 481	. . 3 ⊢ ((𝑃 ∈ ℙ ∧ (𝑃 mod 4) = 1) → ∃*𝑎 ∈ ℕ ∃𝑏 ∈ ℕ (𝑎 ≤ 𝑏 ∧ ((𝑎↑2) + (𝑏↑2)) = 𝑃))
172		rmoim 3698	. . 3 ⊢ (∀𝑎 ∈ ℕ (∃!𝑏 ∈ ℕ (𝑎 ≤ 𝑏 ∧ ((𝑎↑2) + (𝑏↑2)) = 𝑃) → ∃𝑏 ∈ ℕ (𝑎 ≤ 𝑏 ∧ ((𝑎↑2) + (𝑏↑2)) = 𝑃)) → (∃𝑎 ∈ ℕ ∃𝑏 ∈ ℕ (𝑎 ≤ 𝑏 ∧ ((𝑎↑2) + (𝑏↑2)) = 𝑃) → ∃𝑎 ∈ ℕ ∃!𝑏 ∈ ℕ (𝑎 ≤ 𝑏 ∧ ((𝑎↑2) + (𝑏↑2)) = 𝑃)))
173	160, 171, 172	sylc 65	. 2 ⊢ ((𝑃 ∈ ℙ ∧ (𝑃 mod 4) = 1) → ∃*𝑎 ∈ ℕ ∃!𝑏 ∈ ℕ (𝑎 ≤ 𝑏 ∧ ((𝑎↑2) + (𝑏↑2)) = 𝑃))
174		reu5 3355	. 2 ⊢ (∃!𝑎 ∈ ℕ ∃!𝑏 ∈ ℕ (𝑎 ≤ 𝑏 ∧ ((𝑎↑2) + (𝑏↑2)) = 𝑃) ↔ (∃𝑎 ∈ ℕ ∃!𝑏 ∈ ℕ (𝑎 ≤ 𝑏 ∧ ((𝑎↑2) + (𝑏↑2)) = 𝑃) ∧ ∃*𝑎 ∈ ℕ ∃!𝑏 ∈ ℕ (𝑎 ≤ 𝑏 ∧ ((𝑎↑2) + (𝑏↑2)) = 𝑃)))
175	157, 173, 174	sylanbrc 583	1 ⊢ ((𝑃 ∈ ℙ ∧ (𝑃 mod 4) = 1) → ∃!𝑎 ∈ ℕ ∃!𝑏 ∈ ℕ (𝑎 ≤ 𝑏 ∧ ((𝑎↑2) + (𝑏↑2)) = 𝑃))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 396 = wceq 1541 ∈ wcel 2106 ∀wral 3064 ∃wrex 3073 ∃!wreu 3351 ∃*wrmo 3352 ⊆ wss 3910 class class class wbr 5105 (class class class)co 7357 ℂcc 11049 ℝcr 11050 0cc0 11051 1c1 11052 + caddc 11054 < clt 11189 ≤ cle 11190 ℕcn 12153 2c2 12208 4c4 12210 ℕ₀cn0 12413 mod cmo 13774 ↑cexp 13967 ℙcprime 16547

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2707 ax-rep 5242 ax-sep 5256 ax-nul 5263 ax-pow 5320 ax-pr 5384 ax-un 7672 ax-cnex 11107 ax-resscn 11108 ax-1cn 11109 ax-icn 11110 ax-addcl 11111 ax-addrcl 11112 ax-mulcl 11113 ax-mulrcl 11114 ax-mulcom 11115 ax-addass 11116 ax-mulass 11117 ax-distr 11118 ax-i2m1 11119 ax-1ne0 11120 ax-1rid 11121 ax-rnegex 11122 ax-rrecex 11123 ax-cnre 11124 ax-pre-lttri 11125 ax-pre-lttrn 11126 ax-pre-ltadd 11127 ax-pre-mulgt0 11128 ax-pre-sup 11129 ax-addf 11130 ax-mulf 11131

This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3065 df-rex 3074 df-rmo 3353 df-reu 3354 df-rab 3408 df-v 3447 df-sbc 3740 df-csb 3856 df-dif 3913 df-un 3915 df-in 3917 df-ss 3927 df-pss 3929 df-nul 4283 df-if 4487 df-pw 4562 df-sn 4587 df-pr 4589 df-tp 4591 df-op 4593 df-uni 4866 df-int 4908 df-iun 4956 df-iin 4957 df-br 5106 df-opab 5168 df-mpt 5189 df-tr 5223 df-id 5531 df-eprel 5537 df-po 5545 df-so 5546 df-fr 5588 df-se 5589 df-we 5590 df-xp 5639 df-rel 5640 df-cnv 5641 df-co 5642 df-dm 5643 df-rn 5644 df-res 5645 df-ima 5646 df-pred 6253 df-ord 6320 df-on 6321 df-lim 6322 df-suc 6323 df-iota 6448 df-fun 6498 df-fn 6499 df-f 6500 df-f1 6501 df-fo 6502 df-f1o 6503 df-fv 6504 df-isom 6505 df-riota 7313 df-ov 7360 df-oprab 7361 df-mpo 7362 df-of 7617 df-ofr 7618 df-om 7803 df-1st 7921 df-2nd 7922 df-supp 8093 df-tpos 8157 df-frecs 8212 df-wrecs 8243 df-recs 8317 df-rdg 8356 df-1o 8412 df-2o 8413 df-oadd 8416 df-er 8648 df-ec 8650 df-qs 8654 df-map 8767 df-pm 8768 df-ixp 8836 df-en 8884 df-dom 8885 df-sdom 8886 df-fin 8887 df-fsupp 9306 df-sup 9378 df-inf 9379 df-oi 9446 df-dju 9837 df-card 9875 df-pnf 11191 df-mnf 11192 df-xr 11193 df-ltxr 11194 df-le 11195 df-sub 11387 df-neg 11388 df-div 11813 df-nn 12154 df-2 12216 df-3 12217 df-4 12218 df-5 12219 df-6 12220 df-7 12221 df-8 12222 df-9 12223 df-n0 12414 df-xnn0 12486 df-z 12500 df-dec 12619 df-uz 12764 df-q 12874 df-rp 12916 df-fz 13425 df-fzo 13568 df-fl 13697 df-mod 13775 df-seq 13907 df-exp 13968 df-hash 14231 df-cj 14984 df-re 14985 df-im 14986 df-sqrt 15120 df-abs 15121 df-dvds 16137 df-gcd 16375 df-prm 16548 df-phi 16638 df-pc 16709 df-gz 16802 df-struct 17019 df-sets 17036 df-slot 17054 df-ndx 17066 df-base 17084 df-ress 17113 df-plusg 17146 df-mulr 17147 df-starv 17148 df-sca 17149 df-vsca 17150 df-ip 17151 df-tset 17152 df-ple 17153 df-ds 17155 df-unif 17156 df-hom 17157 df-cco 17158 df-0g 17323 df-gsum 17324 df-prds 17329 df-pws 17331 df-imas 17390 df-qus 17391 df-mre 17466 df-mrc 17467 df-acs 17469 df-mgm 18497 df-sgrp 18546 df-mnd 18557 df-mhm 18601 df-submnd 18602 df-grp 18751 df-minusg 18752 df-sbg 18753 df-mulg 18873 df-subg 18925 df-nsg 18926 df-eqg 18927 df-ghm 19006 df-cntz 19097 df-cmn 19564 df-abl 19565 df-mgp 19897 df-ur 19914 df-srg 19918 df-ring 19966 df-cring 19967 df-oppr 20049 df-dvdsr 20070 df-unit 20071 df-invr 20101 df-dvr 20112 df-rnghom 20146 df-drng 20187 df-field 20188 df-subrg 20220 df-lmod 20324 df-lss 20393 df-lsp 20433 df-sra 20633 df-rgmod 20634 df-lidl 20635 df-rsp 20636 df-2idl 20702 df-nzr 20728 df-rlreg 20753 df-domn 20754 df-idom 20755 df-cnfld 20797 df-zring 20870 df-zrh 20904 df-zn 20907 df-assa 21259 df-asp 21260 df-ascl 21261 df-psr 21311 df-mvr 21312 df-mpl 21313 df-opsr 21315 df-evls 21482 df-evl 21483 df-psr1 21551 df-vr1 21552 df-ply1 21553 df-coe1 21554 df-evl1 21682 df-mdeg 25417 df-deg1 25418 df-mon1 25495 df-uc1p 25496 df-q1p 25497 df-r1p 25498 df-lgs 26643

This theorem is referenced by: 2sqreunnltlem 26798 2sqreunn 26805

W3C validator