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Theorem 2sqreunnlem1 27387

Description: Lemma 1 for 2sqreunn 27395. (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 27377	. . 3 ⊢ ((𝑃 ∈ ℙ ∧ (𝑃 mod 4) = 1) → ∃𝑥 ∈ ℕ ∃𝑦 ∈ ℕ 𝑃 = ((𝑥↑2) + (𝑦↑2)))
2		simpll 766	. . . . . . . . 9 ⊢ (((𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ) ∧ 𝑃 = ((𝑥↑2) + (𝑦↑2))) → 𝑥 ∈ ℕ)
3	2	adantl 481	. . . . . . . 8 ⊢ ((𝑥 ≤ 𝑦 ∧ ((𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ) ∧ 𝑃 = ((𝑥↑2) + (𝑦↑2)))) → 𝑥 ∈ ℕ)
4		breq1 5092	. . . . . . . . . . 11 ⊢ (𝑎 = 𝑥 → (𝑎 ≤ 𝑏 ↔ 𝑥 ≤ 𝑏))
5		oveq1 7353	. . . . . . . . . . . . 13 ⊢ (𝑎 = 𝑥 → (𝑎↑2) = (𝑥↑2))
6	5	oveq1d 7361	. . . . . . . . . . . 12 ⊢ (𝑎 = 𝑥 → ((𝑎↑2) + (𝑏↑2)) = ((𝑥↑2) + (𝑏↑2)))
7	6	eqeq1d 2733	. . . . . . . . . . 11 ⊢ (𝑎 = 𝑥 → (((𝑎↑2) + (𝑏↑2)) = 𝑃 ↔ ((𝑥↑2) + (𝑏↑2)) = 𝑃))
8	4, 7	anbi12d 632	. . . . . . . . . 10 ⊢ (𝑎 = 𝑥 → ((𝑎 ≤ 𝑏 ∧ ((𝑎↑2) + (𝑏↑2)) = 𝑃) ↔ (𝑥 ≤ 𝑏 ∧ ((𝑥↑2) + (𝑏↑2)) = 𝑃)))
9	8	reubidv 3362	. . . . . . . . 9 ⊢ (𝑎 = 𝑥 → (∃!𝑏 ∈ ℕ (𝑎 ≤ 𝑏 ∧ ((𝑎↑2) + (𝑏↑2)) = 𝑃) ↔ ∃!𝑏 ∈ ℕ (𝑥 ≤ 𝑏 ∧ ((𝑥↑2) + (𝑏↑2)) = 𝑃)))
10	9	adantl 481	. . . . . . . 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 484	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ) → 𝑦 ∈ ℕ)
12	11	adantr 480	. . . . . . . . . . . . . 14 ⊢ (((𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ) ∧ 𝑥 ≤ 𝑦) → 𝑦 ∈ ℕ)
13		breq2 5093	. . . . . . . . . . . . . . . . 17 ⊢ (𝑏 = 𝑦 → (𝑥 ≤ 𝑏 ↔ 𝑥 ≤ 𝑦))
14		oveq1 7353	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑏 = 𝑦 → (𝑏↑2) = (𝑦↑2))
15	14	oveq2d 7362	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑏 = 𝑦 → ((𝑥↑2) + (𝑏↑2)) = ((𝑥↑2) + (𝑦↑2)))
16	15	eqeq1d 2733	. . . . . . . . . . . . . . . . 17 ⊢ (𝑏 = 𝑦 → (((𝑥↑2) + (𝑏↑2)) = ((𝑥↑2) + (𝑦↑2)) ↔ ((𝑥↑2) + (𝑦↑2)) = ((𝑥↑2) + (𝑦↑2))))
17	13, 16	anbi12d 632	. . . . . . . . . . . . . . . 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 2026	. . . . . . . . . . . . . . . . . 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 3155	. . . . . . . . . . . . . . . 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 632	. . . . . . . . . . . . . . 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 481	. . . . . . . . . . . . . 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 484	. . . . . . . . . . . . . . 15 ⊢ (((𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ) ∧ 𝑥 ≤ 𝑦) → 𝑥 ≤ 𝑦)
24		eqidd 2732	. . . . . . . . . . . . . . 15 ⊢ (((𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ) ∧ 𝑥 ≤ 𝑦) → ((𝑥↑2) + (𝑦↑2)) = ((𝑥↑2) + (𝑦↑2)))
25		nnre 12132	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑐 ∈ ℕ → 𝑐 ∈ ℝ)
26	25	resqcld 14032	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑐 ∈ ℕ → (𝑐↑2) ∈ ℝ)
27	26	adantl 481	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ) ∧ 𝑥 ≤ 𝑦) ∧ 𝑐 ∈ ℕ) → (𝑐↑2) ∈ ℝ)
28		nnre 12132	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑦 ∈ ℕ → 𝑦 ∈ ℝ)
29	28	resqcld 14032	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑦 ∈ ℕ → (𝑦↑2) ∈ ℝ)
30	29	adantl 481	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ) → (𝑦↑2) ∈ ℝ)
31	30	ad2antrr 726	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ) ∧ 𝑥 ≤ 𝑦) ∧ 𝑐 ∈ ℕ) → (𝑦↑2) ∈ ℝ)
32		nnre 12132	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑥 ∈ ℕ → 𝑥 ∈ ℝ)
33	32	resqcld 14032	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑥 ∈ ℕ → (𝑥↑2) ∈ ℝ)
34	33	adantr 480	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ) → (𝑥↑2) ∈ ℝ)
35	34	ad2antrr 726	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ) ∧ 𝑥 ≤ 𝑦) ∧ 𝑐 ∈ ℕ) → (𝑥↑2) ∈ ℝ)
36		readdcan 11287	. . . . . . . . . . . . . . . . . . 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 1373	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ) ∧ 𝑥 ≤ 𝑦) ∧ 𝑐 ∈ ℕ) → (((𝑥↑2) + (𝑐↑2)) = ((𝑥↑2) + (𝑦↑2)) ↔ (𝑐↑2) = (𝑦↑2)))
38	28	ad4antlr 733	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((((𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ) ∧ 𝑥 ≤ 𝑦) ∧ 𝑐 ∈ ℕ) ∧ (𝑐↑2) = (𝑦↑2)) → 𝑦 ∈ ℝ)
39	25	ad2antlr 727	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((((𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ) ∧ 𝑥 ≤ 𝑦) ∧ 𝑐 ∈ ℕ) ∧ (𝑐↑2) = (𝑦↑2)) → 𝑐 ∈ ℝ)
40		nnnn0 12388	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑦 ∈ ℕ → 𝑦 ∈ ℕ₀)
41	40	nn0ge0d 12445	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑦 ∈ ℕ → 0 ≤ 𝑦)
42	41	ad4antlr 733	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((((𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ) ∧ 𝑥 ≤ 𝑦) ∧ 𝑐 ∈ ℕ) ∧ (𝑐↑2) = (𝑦↑2)) → 0 ≤ 𝑦)
43		nnnn0 12388	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑐 ∈ ℕ → 𝑐 ∈ ℕ₀)
44	43	nn0ge0d 12445	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑐 ∈ ℕ → 0 ≤ 𝑐)
45	44	ad2antlr 727	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((((𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ) ∧ 𝑥 ≤ 𝑦) ∧ 𝑐 ∈ ℕ) ∧ (𝑐↑2) = (𝑦↑2)) → 0 ≤ 𝑐)
46		simpr 484	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((((𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ) ∧ 𝑥 ≤ 𝑦) ∧ 𝑐 ∈ ℕ) ∧ (𝑐↑2) = (𝑦↑2)) → (𝑐↑2) = (𝑦↑2))
47	46	eqcomd 2737	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((((𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ) ∧ 𝑥 ≤ 𝑦) ∧ 𝑐 ∈ ℕ) ∧ (𝑐↑2) = (𝑦↑2)) → (𝑦↑2) = (𝑐↑2))
48	38, 39, 42, 45, 47	sq11d 14165	. . . . . . . . . . . . . . . . . . 19 ⊢ (((((𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ) ∧ 𝑥 ≤ 𝑦) ∧ 𝑐 ∈ ℕ) ∧ (𝑐↑2) = (𝑦↑2)) → 𝑦 = 𝑐)
49	48	ex 412	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ) ∧ 𝑥 ≤ 𝑦) ∧ 𝑐 ∈ ℕ) → ((𝑐↑2) = (𝑦↑2) → 𝑦 = 𝑐))
50	37, 49	sylbid 240	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ) ∧ 𝑥 ≤ 𝑦) ∧ 𝑐 ∈ ℕ) → (((𝑥↑2) + (𝑐↑2)) = ((𝑥↑2) + (𝑦↑2)) → 𝑦 = 𝑐))
51	50	adantld 490	. . . . . . . . . . . . . . . 16 ⊢ ((((𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ) ∧ 𝑥 ≤ 𝑦) ∧ 𝑐 ∈ ℕ) → ((𝑥 ≤ 𝑐 ∧ ((𝑥↑2) + (𝑐↑2)) = ((𝑥↑2) + (𝑦↑2))) → 𝑦 = 𝑐))
52	51	ralrimiva 3124	. . . . . . . . . . . . . . 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 514	. . . . . . . . . . . . . 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 3574	. . . . . . . . . . . . 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 5093	. . . . . . . . . . . . . . 15 ⊢ (𝑏 = 𝑐 → (𝑥 ≤ 𝑏 ↔ 𝑥 ≤ 𝑐))
56		oveq1 7353	. . . . . . . . . . . . . . . . 17 ⊢ (𝑏 = 𝑐 → (𝑏↑2) = (𝑐↑2))
57	56	oveq2d 7362	. . . . . . . . . . . . . . . 16 ⊢ (𝑏 = 𝑐 → ((𝑥↑2) + (𝑏↑2)) = ((𝑥↑2) + (𝑐↑2)))
58	57	eqeq1d 2733	. . . . . . . . . . . . . . 15 ⊢ (𝑏 = 𝑐 → (((𝑥↑2) + (𝑏↑2)) = ((𝑥↑2) + (𝑦↑2)) ↔ ((𝑥↑2) + (𝑐↑2)) = ((𝑥↑2) + (𝑦↑2))))
59	55, 58	anbi12d 632	. . . . . . . . . . . . . 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 3687	. . . . . . . . . . . . 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 234	. . . . . . . . . . . 12 ⊢ (((𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ) ∧ 𝑥 ≤ 𝑦) → ∃!𝑏 ∈ ℕ (𝑥 ≤ 𝑏 ∧ ((𝑥↑2) + (𝑏↑2)) = ((𝑥↑2) + (𝑦↑2))))
62	61	ex 412	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ) → (𝑥 ≤ 𝑦 → ∃!𝑏 ∈ ℕ (𝑥 ≤ 𝑏 ∧ ((𝑥↑2) + (𝑏↑2)) = ((𝑥↑2) + (𝑦↑2)))))
63	62	adantr 480	. . . . . . . . . 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 407	. . . . . . . . 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 2743	. . . . . . . . . . . . 13 ⊢ (𝑃 = ((𝑥↑2) + (𝑦↑2)) → (((𝑥↑2) + (𝑏↑2)) = 𝑃 ↔ ((𝑥↑2) + (𝑏↑2)) = ((𝑥↑2) + (𝑦↑2))))
66	65	anbi2d 630	. . . . . . . . . . . 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 3362	. . . . . . . . . . 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 481	. . . . . . . . . 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 481	. . . . . . . . 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 257	. . . . . . . 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 3574	. . . . . . 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 480	. . . . . . . . 9 ⊢ (((𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ) ∧ 𝑃 = ((𝑥↑2) + (𝑦↑2))) → 𝑦 ∈ ℕ)
73	72	adantl 481	. . . . . . . 8 ⊢ ((¬ 𝑥 ≤ 𝑦 ∧ ((𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ) ∧ 𝑃 = ((𝑥↑2) + (𝑦↑2)))) → 𝑦 ∈ ℕ)
74		breq1 5092	. . . . . . . . . . 11 ⊢ (𝑎 = 𝑦 → (𝑎 ≤ 𝑏 ↔ 𝑦 ≤ 𝑏))
75		oveq1 7353	. . . . . . . . . . . . 13 ⊢ (𝑎 = 𝑦 → (𝑎↑2) = (𝑦↑2))
76	75	oveq1d 7361	. . . . . . . . . . . 12 ⊢ (𝑎 = 𝑦 → ((𝑎↑2) + (𝑏↑2)) = ((𝑦↑2) + (𝑏↑2)))
77	76	eqeq1d 2733	. . . . . . . . . . 11 ⊢ (𝑎 = 𝑦 → (((𝑎↑2) + (𝑏↑2)) = 𝑃 ↔ ((𝑦↑2) + (𝑏↑2)) = 𝑃))
78	74, 77	anbi12d 632	. . . . . . . . . 10 ⊢ (𝑎 = 𝑦 → ((𝑎 ≤ 𝑏 ∧ ((𝑎↑2) + (𝑏↑2)) = 𝑃) ↔ (𝑦 ≤ 𝑏 ∧ ((𝑦↑2) + (𝑏↑2)) = 𝑃)))
79	78	reubidv 3362	. . . . . . . . 9 ⊢ (𝑎 = 𝑦 → (∃!𝑏 ∈ ℕ (𝑎 ≤ 𝑏 ∧ ((𝑎↑2) + (𝑏↑2)) = 𝑃) ↔ ∃!𝑏 ∈ ℕ (𝑦 ≤ 𝑏 ∧ ((𝑦↑2) + (𝑏↑2)) = 𝑃)))
80	79	adantl 481	. . . . . . . 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 766	. . . . . . . . . . . . . 14 ⊢ (((𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ) ∧ ¬ 𝑥 ≤ 𝑦) → 𝑥 ∈ ℕ)
82		breq2 5093	. . . . . . . . . . . . . . . . 17 ⊢ (𝑏 = 𝑥 → (𝑦 ≤ 𝑏 ↔ 𝑦 ≤ 𝑥))
83		oveq1 7353	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑏 = 𝑥 → (𝑏↑2) = (𝑥↑2))
84	83	oveq2d 7362	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑏 = 𝑥 → ((𝑦↑2) + (𝑏↑2)) = ((𝑦↑2) + (𝑥↑2)))
85	84	eqeq1d 2733	. . . . . . . . . . . . . . . . 17 ⊢ (𝑏 = 𝑥 → (((𝑦↑2) + (𝑏↑2)) = ((𝑥↑2) + (𝑦↑2)) ↔ ((𝑦↑2) + (𝑥↑2)) = ((𝑥↑2) + (𝑦↑2))))
86	82, 85	anbi12d 632	. . . . . . . . . . . . . . . 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 2026	. . . . . . . . . . . . . . . . . 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 3155	. . . . . . . . . . . . . . . 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 632	. . . . . . . . . . . . . . 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 481	. . . . . . . . . . . . . 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 11192	. . . . . . . . . . . . . . . . . 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 727	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ) ∧ 𝑦 < 𝑥) → 𝑦 ∈ ℝ)
95	32	ad2antrr 726	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ) ∧ 𝑦 < 𝑥) → 𝑥 ∈ ℝ)
96		simpr 484	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ) ∧ 𝑦 < 𝑥) → 𝑦 < 𝑥)
97	94, 95, 96	ltled 11261	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ) ∧ 𝑦 < 𝑥) → 𝑦 ≤ 𝑥)
98	97	ex 412	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ) → (𝑦 < 𝑥 → 𝑦 ≤ 𝑥))
99	93, 98	sylbird 260	. . . . . . . . . . . . . . . 16 ⊢ ((𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ) → (¬ 𝑥 ≤ 𝑦 → 𝑦 ≤ 𝑥))
100	99	imp 406	. . . . . . . . . . . . . . 15 ⊢ (((𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ) ∧ ¬ 𝑥 ≤ 𝑦) → 𝑦 ≤ 𝑥)
101	29	recnd 11140	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑦 ∈ ℕ → (𝑦↑2) ∈ ℂ)
102	101	adantl 481	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ) → (𝑦↑2) ∈ ℂ)
103	33	recnd 11140	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 ∈ ℕ → (𝑥↑2) ∈ ℂ)
104	103	adantr 480	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ) → (𝑥↑2) ∈ ℂ)
105	102, 104	addcomd 11315	. . . . . . . . . . . . . . . 16 ⊢ ((𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ) → ((𝑦↑2) + (𝑥↑2)) = ((𝑥↑2) + (𝑦↑2)))
106	105	adantr 480	. . . . . . . . . . . . . . 15 ⊢ (((𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ) ∧ ¬ 𝑥 ≤ 𝑦) → ((𝑦↑2) + (𝑥↑2)) = ((𝑥↑2) + (𝑦↑2)))
107	34	recnd 11140	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ) → (𝑥↑2) ∈ ℂ)
108	107	adantr 480	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ) ∧ 𝑐 ∈ ℕ) → (𝑥↑2) ∈ ℂ)
109	30	recnd 11140	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ) → (𝑦↑2) ∈ ℂ)
110	109	adantr 480	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ) ∧ 𝑐 ∈ ℕ) → (𝑦↑2) ∈ ℂ)
111	108, 110	addcomd 11315	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ) ∧ 𝑐 ∈ ℕ) → ((𝑥↑2) + (𝑦↑2)) = ((𝑦↑2) + (𝑥↑2)))
112	111	eqeq2d 2742	. . . . . . . . . . . . . . . . . . . 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 481	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ) ∧ 𝑐 ∈ ℕ) → (𝑐↑2) ∈ ℝ)
114	33	ad2antrr 726	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ) ∧ 𝑐 ∈ ℕ) → (𝑥↑2) ∈ ℝ)
115	29	ad2antlr 727	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ) ∧ 𝑐 ∈ ℕ) → (𝑦↑2) ∈ ℝ)
116		readdcan 11287	. . . . . . . . . . . . . . . . . . . . 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 1373	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ) ∧ 𝑐 ∈ ℕ) → (((𝑦↑2) + (𝑐↑2)) = ((𝑦↑2) + (𝑥↑2)) ↔ (𝑐↑2) = (𝑥↑2)))
118	112, 117	bitrd 279	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ) ∧ 𝑐 ∈ ℕ) → (((𝑦↑2) + (𝑐↑2)) = ((𝑥↑2) + (𝑦↑2)) ↔ (𝑐↑2) = (𝑥↑2)))
119	25	ad2antlr 727	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ) ∧ 𝑐 ∈ ℕ) ∧ (𝑐↑2) = (𝑥↑2)) → 𝑐 ∈ ℝ)
120	32	adantr 480	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ) → 𝑥 ∈ ℝ)
121	120	ad2antrr 726	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ) ∧ 𝑐 ∈ ℕ) ∧ (𝑐↑2) = (𝑥↑2)) → 𝑥 ∈ ℝ)
122	44	ad2antlr 727	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ) ∧ 𝑐 ∈ ℕ) ∧ (𝑐↑2) = (𝑥↑2)) → 0 ≤ 𝑐)
123		nnnn0 12388	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑥 ∈ ℕ → 𝑥 ∈ ℕ₀)
124	123	nn0ge0d 12445	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑥 ∈ ℕ → 0 ≤ 𝑥)
125	124	adantr 480	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ) → 0 ≤ 𝑥)
126	125	ad2antrr 726	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ) ∧ 𝑐 ∈ ℕ) ∧ (𝑐↑2) = (𝑥↑2)) → 0 ≤ 𝑥)
127		simpr 484	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ) ∧ 𝑐 ∈ ℕ) ∧ (𝑐↑2) = (𝑥↑2)) → (𝑐↑2) = (𝑥↑2))
128	119, 121, 122, 126, 127	sq11d 14165	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ) ∧ 𝑐 ∈ ℕ) ∧ (𝑐↑2) = (𝑥↑2)) → 𝑐 = 𝑥)
129	128	eqcomd 2737	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ) ∧ 𝑐 ∈ ℕ) ∧ (𝑐↑2) = (𝑥↑2)) → 𝑥 = 𝑐)
130	129	ex 412	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ) ∧ 𝑐 ∈ ℕ) → ((𝑐↑2) = (𝑥↑2) → 𝑥 = 𝑐))
131	118, 130	sylbid 240	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ) ∧ 𝑐 ∈ ℕ) → (((𝑦↑2) + (𝑐↑2)) = ((𝑥↑2) + (𝑦↑2)) → 𝑥 = 𝑐))
132	131	adantld 490	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ) ∧ 𝑐 ∈ ℕ) → ((𝑦 ≤ 𝑐 ∧ ((𝑦↑2) + (𝑐↑2)) = ((𝑥↑2) + (𝑦↑2))) → 𝑥 = 𝑐))
133	132	ralrimiva 3124	. . . . . . . . . . . . . . . 16 ⊢ ((𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ) → ∀𝑐 ∈ ℕ ((𝑦 ≤ 𝑐 ∧ ((𝑦↑2) + (𝑐↑2)) = ((𝑥↑2) + (𝑦↑2))) → 𝑥 = 𝑐))
134	133	adantr 480	. . . . . . . . . . . . . . 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 514	. . . . . . . . . . . . . 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 3574	. . . . . . . . . . . . 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 5093	. . . . . . . . . . . . . . 15 ⊢ (𝑏 = 𝑐 → (𝑦 ≤ 𝑏 ↔ 𝑦 ≤ 𝑐))
138	56	oveq2d 7362	. . . . . . . . . . . . . . . 16 ⊢ (𝑏 = 𝑐 → ((𝑦↑2) + (𝑏↑2)) = ((𝑦↑2) + (𝑐↑2)))
139	138	eqeq1d 2733	. . . . . . . . . . . . . . 15 ⊢ (𝑏 = 𝑐 → (((𝑦↑2) + (𝑏↑2)) = ((𝑥↑2) + (𝑦↑2)) ↔ ((𝑦↑2) + (𝑐↑2)) = ((𝑥↑2) + (𝑦↑2))))
140	137, 139	anbi12d 632	. . . . . . . . . . . . . 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 3687	. . . . . . . . . . . . 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 234	. . . . . . . . . . . 12 ⊢ (((𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ) ∧ ¬ 𝑥 ≤ 𝑦) → ∃!𝑏 ∈ ℕ (𝑦 ≤ 𝑏 ∧ ((𝑦↑2) + (𝑏↑2)) = ((𝑥↑2) + (𝑦↑2))))
143	142	ex 412	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ) → (¬ 𝑥 ≤ 𝑦 → ∃!𝑏 ∈ ℕ (𝑦 ≤ 𝑏 ∧ ((𝑦↑2) + (𝑏↑2)) = ((𝑥↑2) + (𝑦↑2)))))
144	143	adantr 480	. . . . . . . . . 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 407	. . . . . . . . 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 2743	. . . . . . . . . . . . 13 ⊢ (𝑃 = ((𝑥↑2) + (𝑦↑2)) → (((𝑦↑2) + (𝑏↑2)) = 𝑃 ↔ ((𝑦↑2) + (𝑏↑2)) = ((𝑥↑2) + (𝑦↑2))))
147	146	anbi2d 630	. . . . . . . . . . . 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 3362	. . . . . . . . . . 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 481	. . . . . . . . . 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 481	. . . . . . . . 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 257	. . . . . . . 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 3574	. . . . . . 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 811	. . . . . 6 ⊢ (((𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ) ∧ 𝑃 = ((𝑥↑2) + (𝑦↑2))) → ∃𝑎 ∈ ℕ ∃!𝑏 ∈ ℕ (𝑎 ≤ 𝑏 ∧ ((𝑎↑2) + (𝑏↑2)) = 𝑃))
154	153	ex 412	. . . . 5 ⊢ ((𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ) → (𝑃 = ((𝑥↑2) + (𝑦↑2)) → ∃𝑎 ∈ ℕ ∃!𝑏 ∈ ℕ (𝑎 ≤ 𝑏 ∧ ((𝑎↑2) + (𝑏↑2)) = 𝑃)))
155	154	adantl 481	. . . 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 3189	. . 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 3350	. . . . 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 3124	. . 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 27375	. . . . 5 ⊢ (𝑃 ∈ ℙ → ∃*𝑎 ∈ ℕ₀ ∃𝑏 ∈ ℕ₀ (𝑎 ≤ 𝑏 ∧ ((𝑎↑2) + (𝑏↑2)) = 𝑃))
162		nnssnn0 12384	. . . . . 6 ⊢ ℕ ⊆ ℕ₀
163		nfcv 2894	. . . . . . 7 ⊢ Ⅎ𝑎ℕ
164		nfcv 2894	. . . . . . 7 ⊢ Ⅎ𝑎ℕ₀
165	163, 164	ssrmof 3997	. . . . . 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 3999	. . . . . . 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 3696	. . . . 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 480	. . 3 ⊢ ((𝑃 ∈ ℙ ∧ (𝑃 mod 4) = 1) → ∃*𝑎 ∈ ℕ ∃𝑏 ∈ ℕ (𝑎 ≤ 𝑏 ∧ ((𝑎↑2) + (𝑏↑2)) = 𝑃))
172		rmoim 3694	. . 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 3348	. 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 206 ∧ wa 395 = wceq 1541 ∈ wcel 2111 ∀wral 3047 ∃wrex 3056 ∃!wreu 3344 ∃*wrmo 3345 ⊆ wss 3897 class class class wbr 5089 (class class class)co 7346 ℂcc 11004 ℝcr 11005 0cc0 11006 1c1 11007 + caddc 11009 < clt 11146 ≤ cle 11147 ℕcn 12125 2c2 12180 4c4 12182 ℕ₀cn0 12381 mod cmo 13773 ↑cexp 13968 ℙcprime 16582

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1968 ax-7 2009 ax-8 2113 ax-9 2121 ax-10 2144 ax-11 2160 ax-12 2180 ax-ext 2703 ax-rep 5215 ax-sep 5232 ax-nul 5242 ax-pow 5301 ax-pr 5368 ax-un 7668 ax-cnex 11062 ax-resscn 11063 ax-1cn 11064 ax-icn 11065 ax-addcl 11066 ax-addrcl 11067 ax-mulcl 11068 ax-mulrcl 11069 ax-mulcom 11070 ax-addass 11071 ax-mulass 11072 ax-distr 11073 ax-i2m1 11074 ax-1ne0 11075 ax-1rid 11076 ax-rnegex 11077 ax-rrecex 11078 ax-cnre 11079 ax-pre-lttri 11080 ax-pre-lttrn 11081 ax-pre-ltadd 11082 ax-pre-mulgt0 11083 ax-pre-sup 11084 ax-addf 11085 ax-mulf 11086

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2068 df-mo 2535 df-eu 2564 df-clab 2710 df-cleq 2723 df-clel 2806 df-nfc 2881 df-ne 2929 df-nel 3033 df-ral 3048 df-rex 3057 df-rmo 3346 df-reu 3347 df-rab 3396 df-v 3438 df-sbc 3737 df-csb 3846 df-dif 3900 df-un 3902 df-in 3904 df-ss 3914 df-pss 3917 df-nul 4281 df-if 4473 df-pw 4549 df-sn 4574 df-pr 4576 df-tp 4578 df-op 4580 df-uni 4857 df-int 4896 df-iun 4941 df-iin 4942 df-br 5090 df-opab 5152 df-mpt 5171 df-tr 5197 df-id 5509 df-eprel 5514 df-po 5522 df-so 5523 df-fr 5567 df-se 5568 df-we 5569 df-xp 5620 df-rel 5621 df-cnv 5622 df-co 5623 df-dm 5624 df-rn 5625 df-res 5626 df-ima 5627 df-pred 6248 df-ord 6309 df-on 6310 df-lim 6311 df-suc 6312 df-iota 6437 df-fun 6483 df-fn 6484 df-f 6485 df-f1 6486 df-fo 6487 df-f1o 6488 df-fv 6489 df-isom 6490 df-riota 7303 df-ov 7349 df-oprab 7350 df-mpo 7351 df-of 7610 df-ofr 7611 df-om 7797 df-1st 7921 df-2nd 7922 df-supp 8091 df-tpos 8156 df-frecs 8211 df-wrecs 8242 df-recs 8291 df-rdg 8329 df-1o 8385 df-2o 8386 df-oadd 8389 df-er 8622 df-ec 8624 df-qs 8628 df-map 8752 df-pm 8753 df-ixp 8822 df-en 8870 df-dom 8871 df-sdom 8872 df-fin 8873 df-fsupp 9246 df-sup 9326 df-inf 9327 df-oi 9396 df-dju 9794 df-card 9832 df-pnf 11148 df-mnf 11149 df-xr 11150 df-ltxr 11151 df-le 11152 df-sub 11346 df-neg 11347 df-div 11775 df-nn 12126 df-2 12188 df-3 12189 df-4 12190 df-5 12191 df-6 12192 df-7 12193 df-8 12194 df-9 12195 df-n0 12382 df-xnn0 12455 df-z 12469 df-dec 12589 df-uz 12733 df-q 12847 df-rp 12891 df-fz 13408 df-fzo 13555 df-fl 13696 df-mod 13774 df-seq 13909 df-exp 13969 df-hash 14238 df-cj 15006 df-re 15007 df-im 15008 df-sqrt 15142 df-abs 15143 df-dvds 16164 df-gcd 16406 df-prm 16583 df-phi 16677 df-pc 16749 df-gz 16842 df-struct 17058 df-sets 17075 df-slot 17093 df-ndx 17105 df-base 17121 df-ress 17142 df-plusg 17174 df-mulr 17175 df-starv 17176 df-sca 17177 df-vsca 17178 df-ip 17179 df-tset 17180 df-ple 17181 df-ds 17183 df-unif 17184 df-hom 17185 df-cco 17186 df-0g 17345 df-gsum 17346 df-prds 17351 df-pws 17353 df-imas 17412 df-qus 17413 df-mre 17488 df-mrc 17489 df-acs 17491 df-mgm 18548 df-sgrp 18627 df-mnd 18643 df-mhm 18691 df-submnd 18692 df-grp 18849 df-minusg 18850 df-sbg 18851 df-mulg 18981 df-subg 19036 df-nsg 19037 df-eqg 19038 df-ghm 19125 df-cntz 19229 df-cmn 19694 df-abl 19695 df-mgp 20059 df-rng 20071 df-ur 20100 df-srg 20105 df-ring 20153 df-cring 20154 df-oppr 20255 df-dvdsr 20275 df-unit 20276 df-invr 20306 df-dvr 20319 df-rhm 20390 df-nzr 20428 df-subrng 20461 df-subrg 20485 df-rlreg 20609 df-domn 20610 df-idom 20611 df-drng 20646 df-field 20647 df-lmod 20795 df-lss 20865 df-lsp 20905 df-sra 21107 df-rgmod 21108 df-lidl 21145 df-rsp 21146 df-2idl 21187 df-cnfld 21292 df-zring 21384 df-zrh 21440 df-zn 21443 df-assa 21790 df-asp 21791 df-ascl 21792 df-psr 21846 df-mvr 21847 df-mpl 21848 df-opsr 21850 df-evls 22009 df-evl 22010 df-psr1 22092 df-vr1 22093 df-ply1 22094 df-coe1 22095 df-evl1 22231 df-mdeg 25987 df-deg1 25988 df-mon1 26063 df-uc1p 26064 df-q1p 26065 df-r1p 26066 df-lgs 27233

This theorem is referenced by: 2sqreunnltlem 27388 2sqreunn 27395

W3C validator