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Theorem 2sqreunnlem1 26932

Description: Lemma 1 for 2sqreunn 26940. (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 26922	. . 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 483	. . . . . . . 8 ⊢ ((𝑥 ≤ 𝑦 ∧ ((𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ) ∧ 𝑃 = ((𝑥↑2) + (𝑦↑2)))) → 𝑥 ∈ ℕ)
4		breq1 5150	. . . . . . . . . . 11 ⊢ (𝑎 = 𝑥 → (𝑎 ≤ 𝑏 ↔ 𝑥 ≤ 𝑏))
5		oveq1 7411	. . . . . . . . . . . . 13 ⊢ (𝑎 = 𝑥 → (𝑎↑2) = (𝑥↑2))
6	5	oveq1d 7419	. . . . . . . . . . . 12 ⊢ (𝑎 = 𝑥 → ((𝑎↑2) + (𝑏↑2)) = ((𝑥↑2) + (𝑏↑2)))
7	6	eqeq1d 2735	. . . . . . . . . . 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 3395	. . . . . . . . 9 ⊢ (𝑎 = 𝑥 → (∃!𝑏 ∈ ℕ (𝑎 ≤ 𝑏 ∧ ((𝑎↑2) + (𝑏↑2)) = 𝑃) ↔ ∃!𝑏 ∈ ℕ (𝑥 ≤ 𝑏 ∧ ((𝑥↑2) + (𝑏↑2)) = 𝑃)))
10	9	adantl 483	. . . . . . . 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 486	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ) → 𝑦 ∈ ℕ)
12	11	adantr 482	. . . . . . . . . . . . . 14 ⊢ (((𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ) ∧ 𝑥 ≤ 𝑦) → 𝑦 ∈ ℕ)
13		breq2 5151	. . . . . . . . . . . . . . . . 17 ⊢ (𝑏 = 𝑦 → (𝑥 ≤ 𝑏 ↔ 𝑥 ≤ 𝑦))
14		oveq1 7411	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑏 = 𝑦 → (𝑏↑2) = (𝑦↑2))
15	14	oveq2d 7420	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑏 = 𝑦 → ((𝑥↑2) + (𝑏↑2)) = ((𝑥↑2) + (𝑦↑2)))
16	15	eqeq1d 2735	. . . . . . . . . . . . . . . . 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 2029	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑏 = 𝑦 → (𝑏 = 𝑐 ↔ 𝑦 = 𝑐))
19	18	imbi2d 341	. . . . . . . . . . . . . . . . 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 3178	. . . . . . . . . . . . . . . 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 483	. . . . . . . . . . . . . 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 486	. . . . . . . . . . . . . . 15 ⊢ (((𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ) ∧ 𝑥 ≤ 𝑦) → 𝑥 ≤ 𝑦)
24		eqidd 2734	. . . . . . . . . . . . . . 15 ⊢ (((𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ) ∧ 𝑥 ≤ 𝑦) → ((𝑥↑2) + (𝑦↑2)) = ((𝑥↑2) + (𝑦↑2)))
25		nnre 12215	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑐 ∈ ℕ → 𝑐 ∈ ℝ)
26	25	resqcld 14086	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑐 ∈ ℕ → (𝑐↑2) ∈ ℝ)
27	26	adantl 483	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ) ∧ 𝑥 ≤ 𝑦) ∧ 𝑐 ∈ ℕ) → (𝑐↑2) ∈ ℝ)
28		nnre 12215	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑦 ∈ ℕ → 𝑦 ∈ ℝ)
29	28	resqcld 14086	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑦 ∈ ℕ → (𝑦↑2) ∈ ℝ)
30	29	adantl 483	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ) → (𝑦↑2) ∈ ℝ)
31	30	ad2antrr 725	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ) ∧ 𝑥 ≤ 𝑦) ∧ 𝑐 ∈ ℕ) → (𝑦↑2) ∈ ℝ)
32		nnre 12215	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑥 ∈ ℕ → 𝑥 ∈ ℝ)
33	32	resqcld 14086	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑥 ∈ ℕ → (𝑥↑2) ∈ ℝ)
34	33	adantr 482	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ) → (𝑥↑2) ∈ ℝ)
35	34	ad2antrr 725	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ) ∧ 𝑥 ≤ 𝑦) ∧ 𝑐 ∈ ℕ) → (𝑥↑2) ∈ ℝ)
36		readdcan 11384	. . . . . . . . . . . . . . . . . . 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 1372	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ) ∧ 𝑥 ≤ 𝑦) ∧ 𝑐 ∈ ℕ) → (((𝑥↑2) + (𝑐↑2)) = ((𝑥↑2) + (𝑦↑2)) ↔ (𝑐↑2) = (𝑦↑2)))
38	28	ad4antlr 732	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((((𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ) ∧ 𝑥 ≤ 𝑦) ∧ 𝑐 ∈ ℕ) ∧ (𝑐↑2) = (𝑦↑2)) → 𝑦 ∈ ℝ)
39	25	ad2antlr 726	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((((𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ) ∧ 𝑥 ≤ 𝑦) ∧ 𝑐 ∈ ℕ) ∧ (𝑐↑2) = (𝑦↑2)) → 𝑐 ∈ ℝ)
40		nnnn0 12475	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑦 ∈ ℕ → 𝑦 ∈ ℕ₀)
41	40	nn0ge0d 12531	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑦 ∈ ℕ → 0 ≤ 𝑦)
42	41	ad4antlr 732	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((((𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ) ∧ 𝑥 ≤ 𝑦) ∧ 𝑐 ∈ ℕ) ∧ (𝑐↑2) = (𝑦↑2)) → 0 ≤ 𝑦)
43		nnnn0 12475	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑐 ∈ ℕ → 𝑐 ∈ ℕ₀)
44	43	nn0ge0d 12531	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑐 ∈ ℕ → 0 ≤ 𝑐)
45	44	ad2antlr 726	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((((𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ) ∧ 𝑥 ≤ 𝑦) ∧ 𝑐 ∈ ℕ) ∧ (𝑐↑2) = (𝑦↑2)) → 0 ≤ 𝑐)
46		simpr 486	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((((𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ) ∧ 𝑥 ≤ 𝑦) ∧ 𝑐 ∈ ℕ) ∧ (𝑐↑2) = (𝑦↑2)) → (𝑐↑2) = (𝑦↑2))
47	46	eqcomd 2739	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((((𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ) ∧ 𝑥 ≤ 𝑦) ∧ 𝑐 ∈ ℕ) ∧ (𝑐↑2) = (𝑦↑2)) → (𝑦↑2) = (𝑐↑2))
48	38, 39, 42, 45, 47	sq11d 14217	. . . . . . . . . . . . . . . . . . 19 ⊢ (((((𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ) ∧ 𝑥 ≤ 𝑦) ∧ 𝑐 ∈ ℕ) ∧ (𝑐↑2) = (𝑦↑2)) → 𝑦 = 𝑐)
49	48	ex 414	. . . . . . . . . . . . . . . . . 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 492	. . . . . . . . . . . . . . . 16 ⊢ ((((𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ) ∧ 𝑥 ≤ 𝑦) ∧ 𝑐 ∈ ℕ) → ((𝑥 ≤ 𝑐 ∧ ((𝑥↑2) + (𝑐↑2)) = ((𝑥↑2) + (𝑦↑2))) → 𝑦 = 𝑐))
52	51	ralrimiva 3147	. . . . . . . . . . . . . . 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 516	. . . . . . . . . . . . . 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 3614	. . . . . . . . . . . . 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 5151	. . . . . . . . . . . . . . 15 ⊢ (𝑏 = 𝑐 → (𝑥 ≤ 𝑏 ↔ 𝑥 ≤ 𝑐))
56		oveq1 7411	. . . . . . . . . . . . . . . . 17 ⊢ (𝑏 = 𝑐 → (𝑏↑2) = (𝑐↑2))
57	56	oveq2d 7420	. . . . . . . . . . . . . . . 16 ⊢ (𝑏 = 𝑐 → ((𝑥↑2) + (𝑏↑2)) = ((𝑥↑2) + (𝑐↑2)))
58	57	eqeq1d 2735	. . . . . . . . . . . . . . 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 3728	. . . . . . . . . . . . 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 414	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ) → (𝑥 ≤ 𝑦 → ∃!𝑏 ∈ ℕ (𝑥 ≤ 𝑏 ∧ ((𝑥↑2) + (𝑏↑2)) = ((𝑥↑2) + (𝑦↑2)))))
63	62	adantr 482	. . . . . . . . . 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 409	. . . . . . . . 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 2745	. . . . . . . . . . . . 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 3395	. . . . . . . . . . 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 483	. . . . . . . . . 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 483	. . . . . . . . 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 3614	. . . . . . 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 482	. . . . . . . . 9 ⊢ (((𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ) ∧ 𝑃 = ((𝑥↑2) + (𝑦↑2))) → 𝑦 ∈ ℕ)
73	72	adantl 483	. . . . . . . 8 ⊢ ((¬ 𝑥 ≤ 𝑦 ∧ ((𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ) ∧ 𝑃 = ((𝑥↑2) + (𝑦↑2)))) → 𝑦 ∈ ℕ)
74		breq1 5150	. . . . . . . . . . 11 ⊢ (𝑎 = 𝑦 → (𝑎 ≤ 𝑏 ↔ 𝑦 ≤ 𝑏))
75		oveq1 7411	. . . . . . . . . . . . 13 ⊢ (𝑎 = 𝑦 → (𝑎↑2) = (𝑦↑2))
76	75	oveq1d 7419	. . . . . . . . . . . 12 ⊢ (𝑎 = 𝑦 → ((𝑎↑2) + (𝑏↑2)) = ((𝑦↑2) + (𝑏↑2)))
77	76	eqeq1d 2735	. . . . . . . . . . 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 3395	. . . . . . . . 9 ⊢ (𝑎 = 𝑦 → (∃!𝑏 ∈ ℕ (𝑎 ≤ 𝑏 ∧ ((𝑎↑2) + (𝑏↑2)) = 𝑃) ↔ ∃!𝑏 ∈ ℕ (𝑦 ≤ 𝑏 ∧ ((𝑦↑2) + (𝑏↑2)) = 𝑃)))
80	79	adantl 483	. . . . . . . 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 5151	. . . . . . . . . . . . . . . . 17 ⊢ (𝑏 = 𝑥 → (𝑦 ≤ 𝑏 ↔ 𝑦 ≤ 𝑥))
83		oveq1 7411	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑏 = 𝑥 → (𝑏↑2) = (𝑥↑2))
84	83	oveq2d 7420	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑏 = 𝑥 → ((𝑦↑2) + (𝑏↑2)) = ((𝑦↑2) + (𝑥↑2)))
85	84	eqeq1d 2735	. . . . . . . . . . . . . . . . 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 2029	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑏 = 𝑥 → (𝑏 = 𝑐 ↔ 𝑥 = 𝑐))
88	87	imbi2d 341	. . . . . . . . . . . . . . . . 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 3178	. . . . . . . . . . . . . . . 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 483	. . . . . . . . . . . . . 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 11289	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑦 ∈ ℝ ∧ 𝑥 ∈ ℝ) → (𝑦 < 𝑥 ↔ ¬ 𝑥 ≤ 𝑦))
93	28, 32, 92	syl2anr 598	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ) → (𝑦 < 𝑥 ↔ ¬ 𝑥 ≤ 𝑦))
94	28	ad2antlr 726	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ) ∧ 𝑦 < 𝑥) → 𝑦 ∈ ℝ)
95	32	ad2antrr 725	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ) ∧ 𝑦 < 𝑥) → 𝑥 ∈ ℝ)
96		simpr 486	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ) ∧ 𝑦 < 𝑥) → 𝑦 < 𝑥)
97	94, 95, 96	ltled 11358	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ) ∧ 𝑦 < 𝑥) → 𝑦 ≤ 𝑥)
98	97	ex 414	. . . . . . . . . . . . . . . . 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 408	. . . . . . . . . . . . . . 15 ⊢ (((𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ) ∧ ¬ 𝑥 ≤ 𝑦) → 𝑦 ≤ 𝑥)
101	29	recnd 11238	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑦 ∈ ℕ → (𝑦↑2) ∈ ℂ)
102	101	adantl 483	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ) → (𝑦↑2) ∈ ℂ)
103	33	recnd 11238	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 ∈ ℕ → (𝑥↑2) ∈ ℂ)
104	103	adantr 482	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ) → (𝑥↑2) ∈ ℂ)
105	102, 104	addcomd 11412	. . . . . . . . . . . . . . . 16 ⊢ ((𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ) → ((𝑦↑2) + (𝑥↑2)) = ((𝑥↑2) + (𝑦↑2)))
106	105	adantr 482	. . . . . . . . . . . . . . 15 ⊢ (((𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ) ∧ ¬ 𝑥 ≤ 𝑦) → ((𝑦↑2) + (𝑥↑2)) = ((𝑥↑2) + (𝑦↑2)))
107	34	recnd 11238	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ) → (𝑥↑2) ∈ ℂ)
108	107	adantr 482	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ) ∧ 𝑐 ∈ ℕ) → (𝑥↑2) ∈ ℂ)
109	30	recnd 11238	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ) → (𝑦↑2) ∈ ℂ)
110	109	adantr 482	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ) ∧ 𝑐 ∈ ℕ) → (𝑦↑2) ∈ ℂ)
111	108, 110	addcomd 11412	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ) ∧ 𝑐 ∈ ℕ) → ((𝑥↑2) + (𝑦↑2)) = ((𝑦↑2) + (𝑥↑2)))
112	111	eqeq2d 2744	. . . . . . . . . . . . . . . . . . . 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 483	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ) ∧ 𝑐 ∈ ℕ) → (𝑐↑2) ∈ ℝ)
114	33	ad2antrr 725	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ) ∧ 𝑐 ∈ ℕ) → (𝑥↑2) ∈ ℝ)
115	29	ad2antlr 726	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ) ∧ 𝑐 ∈ ℕ) → (𝑦↑2) ∈ ℝ)
116		readdcan 11384	. . . . . . . . . . . . . . . . . . . . 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 1372	. . . . . . . . . . . . . . . . . . . 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 726	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ) ∧ 𝑐 ∈ ℕ) ∧ (𝑐↑2) = (𝑥↑2)) → 𝑐 ∈ ℝ)
120	32	adantr 482	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ) → 𝑥 ∈ ℝ)
121	120	ad2antrr 725	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ) ∧ 𝑐 ∈ ℕ) ∧ (𝑐↑2) = (𝑥↑2)) → 𝑥 ∈ ℝ)
122	44	ad2antlr 726	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ) ∧ 𝑐 ∈ ℕ) ∧ (𝑐↑2) = (𝑥↑2)) → 0 ≤ 𝑐)
123		nnnn0 12475	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑥 ∈ ℕ → 𝑥 ∈ ℕ₀)
124	123	nn0ge0d 12531	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑥 ∈ ℕ → 0 ≤ 𝑥)
125	124	adantr 482	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ) → 0 ≤ 𝑥)
126	125	ad2antrr 725	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ) ∧ 𝑐 ∈ ℕ) ∧ (𝑐↑2) = (𝑥↑2)) → 0 ≤ 𝑥)
127		simpr 486	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ) ∧ 𝑐 ∈ ℕ) ∧ (𝑐↑2) = (𝑥↑2)) → (𝑐↑2) = (𝑥↑2))
128	119, 121, 122, 126, 127	sq11d 14217	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ) ∧ 𝑐 ∈ ℕ) ∧ (𝑐↑2) = (𝑥↑2)) → 𝑐 = 𝑥)
129	128	eqcomd 2739	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ) ∧ 𝑐 ∈ ℕ) ∧ (𝑐↑2) = (𝑥↑2)) → 𝑥 = 𝑐)
130	129	ex 414	. . . . . . . . . . . . . . . . . . 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 492	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ) ∧ 𝑐 ∈ ℕ) → ((𝑦 ≤ 𝑐 ∧ ((𝑦↑2) + (𝑐↑2)) = ((𝑥↑2) + (𝑦↑2))) → 𝑥 = 𝑐))
133	132	ralrimiva 3147	. . . . . . . . . . . . . . . 16 ⊢ ((𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ) → ∀𝑐 ∈ ℕ ((𝑦 ≤ 𝑐 ∧ ((𝑦↑2) + (𝑐↑2)) = ((𝑥↑2) + (𝑦↑2))) → 𝑥 = 𝑐))
134	133	adantr 482	. . . . . . . . . . . . . . 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 516	. . . . . . . . . . . . . 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 3614	. . . . . . . . . . . . 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 5151	. . . . . . . . . . . . . . 15 ⊢ (𝑏 = 𝑐 → (𝑦 ≤ 𝑏 ↔ 𝑦 ≤ 𝑐))
138	56	oveq2d 7420	. . . . . . . . . . . . . . . 16 ⊢ (𝑏 = 𝑐 → ((𝑦↑2) + (𝑏↑2)) = ((𝑦↑2) + (𝑐↑2)))
139	138	eqeq1d 2735	. . . . . . . . . . . . . . 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 3728	. . . . . . . . . . . . 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 414	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ) → (¬ 𝑥 ≤ 𝑦 → ∃!𝑏 ∈ ℕ (𝑦 ≤ 𝑏 ∧ ((𝑦↑2) + (𝑏↑2)) = ((𝑥↑2) + (𝑦↑2)))))
144	143	adantr 482	. . . . . . . . . 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 409	. . . . . . . . 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 2745	. . . . . . . . . . . . 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 3395	. . . . . . . . . . 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 483	. . . . . . . . . 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 483	. . . . . . . . 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 3614	. . . . . . 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 414	. . . . 5 ⊢ ((𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ) → (𝑃 = ((𝑥↑2) + (𝑦↑2)) → ∃𝑎 ∈ ℕ ∃!𝑏 ∈ ℕ (𝑎 ≤ 𝑏 ∧ ((𝑎↑2) + (𝑏↑2)) = 𝑃)))
155	154	adantl 483	. . . 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 3212	. . 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 3381	. . . . 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 3147	. . 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 26920	. . . . 5 ⊢ (𝑃 ∈ ℙ → ∃*𝑎 ∈ ℕ₀ ∃𝑏 ∈ ℕ₀ (𝑎 ≤ 𝑏 ∧ ((𝑎↑2) + (𝑏↑2)) = 𝑃))
162		nnssnn0 12471	. . . . . 6 ⊢ ℕ ⊆ ℕ₀
163		nfcv 2904	. . . . . . 7 ⊢ Ⅎ𝑎ℕ
164		nfcv 2904	. . . . . . 7 ⊢ Ⅎ𝑎ℕ₀
165	163, 164	ssrmof 4048	. . . . . 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 4050	. . . . . . 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 3737	. . . . 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 482	. . 3 ⊢ ((𝑃 ∈ ℙ ∧ (𝑃 mod 4) = 1) → ∃*𝑎 ∈ ℕ ∃𝑏 ∈ ℕ (𝑎 ≤ 𝑏 ∧ ((𝑎↑2) + (𝑏↑2)) = 𝑃))
172		rmoim 3735	. . 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 3379	. 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 584	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 397 = wceq 1542 ∈ wcel 2107 ∀wral 3062 ∃wrex 3071 ∃!wreu 3375 ∃*wrmo 3376 ⊆ wss 3947 class class class wbr 5147 (class class class)co 7404 ℂcc 11104 ℝcr 11105 0cc0 11106 1c1 11107 + caddc 11109 < clt 11244 ≤ cle 11245 ℕcn 12208 2c2 12263 4c4 12265 ℕ₀cn0 12468 mod cmo 13830 ↑cexp 14023 ℙcprime 16604

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5284 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7720 ax-cnex 11162 ax-resscn 11163 ax-1cn 11164 ax-icn 11165 ax-addcl 11166 ax-addrcl 11167 ax-mulcl 11168 ax-mulrcl 11169 ax-mulcom 11170 ax-addass 11171 ax-mulass 11172 ax-distr 11173 ax-i2m1 11174 ax-1ne0 11175 ax-1rid 11176 ax-rnegex 11177 ax-rrecex 11178 ax-cnre 11179 ax-pre-lttri 11180 ax-pre-lttrn 11181 ax-pre-ltadd 11182 ax-pre-mulgt0 11183 ax-pre-sup 11184 ax-addf 11185 ax-mulf 11186

This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-rmo 3377 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-tp 4632 df-op 4634 df-uni 4908 df-int 4950 df-iun 4998 df-iin 4999 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5573 df-eprel 5579 df-po 5587 df-so 5588 df-fr 5630 df-se 5631 df-we 5632 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-pred 6297 df-ord 6364 df-on 6365 df-lim 6366 df-suc 6367 df-iota 6492 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-isom 6549 df-riota 7360 df-ov 7407 df-oprab 7408 df-mpo 7409 df-of 7665 df-ofr 7666 df-om 7851 df-1st 7970 df-2nd 7971 df-supp 8142 df-tpos 8206 df-frecs 8261 df-wrecs 8292 df-recs 8366 df-rdg 8405 df-1o 8461 df-2o 8462 df-oadd 8465 df-er 8699 df-ec 8701 df-qs 8705 df-map 8818 df-pm 8819 df-ixp 8888 df-en 8936 df-dom 8937 df-sdom 8938 df-fin 8939 df-fsupp 9358 df-sup 9433 df-inf 9434 df-oi 9501 df-dju 9892 df-card 9930 df-pnf 11246 df-mnf 11247 df-xr 11248 df-ltxr 11249 df-le 11250 df-sub 11442 df-neg 11443 df-div 11868 df-nn 12209 df-2 12271 df-3 12272 df-4 12273 df-5 12274 df-6 12275 df-7 12276 df-8 12277 df-9 12278 df-n0 12469 df-xnn0 12541 df-z 12555 df-dec 12674 df-uz 12819 df-q 12929 df-rp 12971 df-fz 13481 df-fzo 13624 df-fl 13753 df-mod 13831 df-seq 13963 df-exp 14024 df-hash 14287 df-cj 15042 df-re 15043 df-im 15044 df-sqrt 15178 df-abs 15179 df-dvds 16194 df-gcd 16432 df-prm 16605 df-phi 16695 df-pc 16766 df-gz 16859 df-struct 17076 df-sets 17093 df-slot 17111 df-ndx 17123 df-base 17141 df-ress 17170 df-plusg 17206 df-mulr 17207 df-starv 17208 df-sca 17209 df-vsca 17210 df-ip 17211 df-tset 17212 df-ple 17213 df-ds 17215 df-unif 17216 df-hom 17217 df-cco 17218 df-0g 17383 df-gsum 17384 df-prds 17389 df-pws 17391 df-imas 17450 df-qus 17451 df-mre 17526 df-mrc 17527 df-acs 17529 df-mgm 18557 df-sgrp 18606 df-mnd 18622 df-mhm 18667 df-submnd 18668 df-grp 18818 df-minusg 18819 df-sbg 18820 df-mulg 18945 df-subg 18997 df-nsg 18998 df-eqg 18999 df-ghm 19084 df-cntz 19175 df-cmn 19643 df-abl 19644 df-mgp 19980 df-ur 19997 df-srg 20001 df-ring 20049 df-cring 20050 df-oppr 20139 df-dvdsr 20160 df-unit 20161 df-invr 20191 df-dvr 20204 df-rnghom 20240 df-nzr 20281 df-drng 20306 df-field 20307 df-subrg 20349 df-lmod 20461 df-lss 20531 df-lsp 20571 df-sra 20773 df-rgmod 20774 df-lidl 20775 df-rsp 20776 df-2idl 20844 df-rlreg 20886 df-domn 20887 df-idom 20888 df-cnfld 20930 df-zring 21003 df-zrh 21037 df-zn 21040 df-assa 21392 df-asp 21393 df-ascl 21394 df-psr 21444 df-mvr 21445 df-mpl 21446 df-opsr 21448 df-evls 21617 df-evl 21618 df-psr1 21686 df-vr1 21687 df-ply1 21688 df-coe1 21689 df-evl1 21817 df-mdeg 25552 df-deg1 25553 df-mon1 25630 df-uc1p 25631 df-q1p 25632 df-r1p 25633 df-lgs 26778

This theorem is referenced by: 2sqreunnltlem 26933 2sqreunn 26940

W3C validator