Theorem 2sqreulem1 27415

Description: Lemma 1 for 2sqreu 27425. (Contributed by AV, 4-Jun-2023.)

Assertion

Ref	Expression
2sqreulem1	⊢ ((𝑃 ∈ ℙ ∧ (𝑃 mod 4) = 1) → ∃!𝑎 ∈ ℕ0 ∃!𝑏 ∈ ℕ0 (𝑎 ≤ 𝑏 ∧ ((𝑎↑2) + (𝑏↑2)) = 𝑃))

Distinct variable group:   𝑃,𝑎,𝑏

Proof of Theorem 2sqreulem1

Dummy variables 𝑐 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		2sqnn0 27407	. . 3 ⊢ ((𝑃 ∈ ℙ ∧ (𝑃 mod 4) = 1) → ∃𝑥 ∈ ℕ0 ∃𝑦 ∈ ℕ0 𝑃 = ((𝑥↑2) + (𝑦↑2)))
2		simpll 767	. . . . . . . . 9 ⊢ (((𝑥 ∈ ℕ0 ∧ 𝑦 ∈ ℕ0) ∧ 𝑃 = ((𝑥↑2) + (𝑦↑2))) → 𝑥 ∈ ℕ0)
3	2	adantl 481	. . . . . . . 8 ⊢ ((𝑥 ≤ 𝑦 ∧ ((𝑥 ∈ ℕ0 ∧ 𝑦 ∈ ℕ0) ∧ 𝑃 = ((𝑥↑2) + (𝑦↑2)))) → 𝑥 ∈ ℕ0)
4		breq1 5100	. . . . . . . . . . 11 ⊢ (𝑎 = 𝑥 → (𝑎 ≤ 𝑏 ↔ 𝑥 ≤ 𝑏))
5		oveq1 7365	. . . . . . . . . . . . 13 ⊢ (𝑎 = 𝑥 → (𝑎↑2) = (𝑥↑2))
6	5	oveq1d 7373	. . . . . . . . . . . 12 ⊢ (𝑎 = 𝑥 → ((𝑎↑2) + (𝑏↑2)) = ((𝑥↑2) + (𝑏↑2)))
7	6	eqeq1d 2737	. . . . . . . . . . 11 ⊢ (𝑎 = 𝑥 → (((𝑎↑2) + (𝑏↑2)) = 𝑃 ↔ ((𝑥↑2) + (𝑏↑2)) = 𝑃))
8	4, 7	anbi12d 633	. . . . . . . . . 10 ⊢ (𝑎 = 𝑥 → ((𝑎 ≤ 𝑏 ∧ ((𝑎↑2) + (𝑏↑2)) = 𝑃) ↔ (𝑥 ≤ 𝑏 ∧ ((𝑥↑2) + (𝑏↑2)) = 𝑃)))
9	8	reubidv 3365	. . . . . . . . 9 ⊢ (𝑎 = 𝑥 → (∃!𝑏 ∈ ℕ0 (𝑎 ≤ 𝑏 ∧ ((𝑎↑2) + (𝑏↑2)) = 𝑃) ↔ ∃!𝑏 ∈ ℕ0 (𝑥 ≤ 𝑏 ∧ ((𝑥↑2) + (𝑏↑2)) = 𝑃)))
10	9	adantl 481	. . . . . . . 8 ⊢ (((𝑥 ≤ 𝑦 ∧ ((𝑥 ∈ ℕ0 ∧ 𝑦 ∈ ℕ0) ∧ 𝑃 = ((𝑥↑2) + (𝑦↑2)))) ∧ 𝑎 = 𝑥) → (∃!𝑏 ∈ ℕ0 (𝑎 ≤ 𝑏 ∧ ((𝑎↑2) + (𝑏↑2)) = 𝑃) ↔ ∃!𝑏 ∈ ℕ0 (𝑥 ≤ 𝑏 ∧ ((𝑥↑2) + (𝑏↑2)) = 𝑃)))
11		simpr 484	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 ∈ ℕ0 ∧ 𝑦 ∈ ℕ0) → 𝑦 ∈ ℕ0)
12	11	adantr 480	. . . . . . . . . . . . . 14 ⊢ (((𝑥 ∈ ℕ0 ∧ 𝑦 ∈ ℕ0) ∧ 𝑥 ≤ 𝑦) → 𝑦 ∈ ℕ0)
13		breq2 5101	. . . . . . . . . . . . . . . . 17 ⊢ (𝑏 = 𝑦 → (𝑥 ≤ 𝑏 ↔ 𝑥 ≤ 𝑦))
14		oveq1 7365	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑏 = 𝑦 → (𝑏↑2) = (𝑦↑2))
15	14	oveq2d 7374	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑏 = 𝑦 → ((𝑥↑2) + (𝑏↑2)) = ((𝑥↑2) + (𝑦↑2)))
16	15	eqeq1d 2737	. . . . . . . . . . . . . . . . 17 ⊢ (𝑏 = 𝑦 → (((𝑥↑2) + (𝑏↑2)) = ((𝑥↑2) + (𝑦↑2)) ↔ ((𝑥↑2) + (𝑦↑2)) = ((𝑥↑2) + (𝑦↑2))))
17	13, 16	anbi12d 633	. . . . . . . . . . . . . . . 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 2027	. . . . . . . . . . . . . . . . . 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 3158	. . . . . . . . . . . . . . . 16 ⊢ (𝑏 = 𝑦 → (∀𝑐 ∈ ℕ0 ((𝑥 ≤ 𝑐 ∧ ((𝑥↑2) + (𝑐↑2)) = ((𝑥↑2) + (𝑦↑2))) → 𝑏 = 𝑐) ↔ ∀𝑐 ∈ ℕ0 ((𝑥 ≤ 𝑐 ∧ ((𝑥↑2) + (𝑐↑2)) = ((𝑥↑2) + (𝑦↑2))) → 𝑦 = 𝑐)))
21	17, 20	anbi12d 633	. . . . . . . . . . . . . . 15 ⊢ (𝑏 = 𝑦 → (((𝑥 ≤ 𝑏 ∧ ((𝑥↑2) + (𝑏↑2)) = ((𝑥↑2) + (𝑦↑2))) ∧ ∀𝑐 ∈ ℕ0 ((𝑥 ≤ 𝑐 ∧ ((𝑥↑2) + (𝑐↑2)) = ((𝑥↑2) + (𝑦↑2))) → 𝑏 = 𝑐)) ↔ ((𝑥 ≤ 𝑦 ∧ ((𝑥↑2) + (𝑦↑2)) = ((𝑥↑2) + (𝑦↑2))) ∧ ∀𝑐 ∈ ℕ0 ((𝑥 ≤ 𝑐 ∧ ((𝑥↑2) + (𝑐↑2)) = ((𝑥↑2) + (𝑦↑2))) → 𝑦 = 𝑐))))
22	21	adantl 481	. . . . . . . . . . . . . 14 ⊢ ((((𝑥 ∈ ℕ0 ∧ 𝑦 ∈ ℕ0) ∧ 𝑥 ≤ 𝑦) ∧ 𝑏 = 𝑦) → (((𝑥 ≤ 𝑏 ∧ ((𝑥↑2) + (𝑏↑2)) = ((𝑥↑2) + (𝑦↑2))) ∧ ∀𝑐 ∈ ℕ0 ((𝑥 ≤ 𝑐 ∧ ((𝑥↑2) + (𝑐↑2)) = ((𝑥↑2) + (𝑦↑2))) → 𝑏 = 𝑐)) ↔ ((𝑥 ≤ 𝑦 ∧ ((𝑥↑2) + (𝑦↑2)) = ((𝑥↑2) + (𝑦↑2))) ∧ ∀𝑐 ∈ ℕ0 ((𝑥 ≤ 𝑐 ∧ ((𝑥↑2) + (𝑐↑2)) = ((𝑥↑2) + (𝑦↑2))) → 𝑦 = 𝑐))))
23		simpr 484	. . . . . . . . . . . . . . 15 ⊢ (((𝑥 ∈ ℕ0 ∧ 𝑦 ∈ ℕ0) ∧ 𝑥 ≤ 𝑦) → 𝑥 ≤ 𝑦)
24		eqidd 2736	. . . . . . . . . . . . . . 15 ⊢ (((𝑥 ∈ ℕ0 ∧ 𝑦 ∈ ℕ0) ∧ 𝑥 ≤ 𝑦) → ((𝑥↑2) + (𝑦↑2)) = ((𝑥↑2) + (𝑦↑2)))
25		nn0re 12412	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑐 ∈ ℕ0 → 𝑐 ∈ ℝ)
26	25	resqcld 14050	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑐 ∈ ℕ0 → (𝑐↑2) ∈ ℝ)
27	26	adantl 481	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝑥 ∈ ℕ0 ∧ 𝑦 ∈ ℕ0) ∧ 𝑥 ≤ 𝑦) ∧ 𝑐 ∈ ℕ0) → (𝑐↑2) ∈ ℝ)
28		nn0re 12412	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑦 ∈ ℕ0 → 𝑦 ∈ ℝ)
29	28	resqcld 14050	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑦 ∈ ℕ0 → (𝑦↑2) ∈ ℝ)
30	29	adantl 481	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑥 ∈ ℕ0 ∧ 𝑦 ∈ ℕ0) → (𝑦↑2) ∈ ℝ)
31	30	ad2antrr 727	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝑥 ∈ ℕ0 ∧ 𝑦 ∈ ℕ0) ∧ 𝑥 ≤ 𝑦) ∧ 𝑐 ∈ ℕ0) → (𝑦↑2) ∈ ℝ)
32		nn0re 12412	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑥 ∈ ℕ0 → 𝑥 ∈ ℝ)
33	32	resqcld 14050	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑥 ∈ ℕ0 → (𝑥↑2) ∈ ℝ)
34	33	adantr 480	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑥 ∈ ℕ0 ∧ 𝑦 ∈ ℕ0) → (𝑥↑2) ∈ ℝ)
35	34	ad2antrr 727	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝑥 ∈ ℕ0 ∧ 𝑦 ∈ ℕ0) ∧ 𝑥 ≤ 𝑦) ∧ 𝑐 ∈ ℕ0) → (𝑥↑2) ∈ ℝ)
36		readdcan 11309	. . . . . . . . . . . . . . . . . . 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 1374	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝑥 ∈ ℕ0 ∧ 𝑦 ∈ ℕ0) ∧ 𝑥 ≤ 𝑦) ∧ 𝑐 ∈ ℕ0) → (((𝑥↑2) + (𝑐↑2)) = ((𝑥↑2) + (𝑦↑2)) ↔ (𝑐↑2) = (𝑦↑2)))
38	28	ad4antlr 734	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((((𝑥 ∈ ℕ0 ∧ 𝑦 ∈ ℕ0) ∧ 𝑥 ≤ 𝑦) ∧ 𝑐 ∈ ℕ0) ∧ (𝑐↑2) = (𝑦↑2)) → 𝑦 ∈ ℝ)
39	25	ad2antlr 728	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((((𝑥 ∈ ℕ0 ∧ 𝑦 ∈ ℕ0) ∧ 𝑥 ≤ 𝑦) ∧ 𝑐 ∈ ℕ0) ∧ (𝑐↑2) = (𝑦↑2)) → 𝑐 ∈ ℝ)
40		nn0ge0 12428	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑦 ∈ ℕ0 → 0 ≤ 𝑦)
41	40	ad4antlr 734	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((((𝑥 ∈ ℕ0 ∧ 𝑦 ∈ ℕ0) ∧ 𝑥 ≤ 𝑦) ∧ 𝑐 ∈ ℕ0) ∧ (𝑐↑2) = (𝑦↑2)) → 0 ≤ 𝑦)
42		nn0ge0 12428	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑐 ∈ ℕ0 → 0 ≤ 𝑐)
43	42	ad2antlr 728	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((((𝑥 ∈ ℕ0 ∧ 𝑦 ∈ ℕ0) ∧ 𝑥 ≤ 𝑦) ∧ 𝑐 ∈ ℕ0) ∧ (𝑐↑2) = (𝑦↑2)) → 0 ≤ 𝑐)
44		simpr 484	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((((𝑥 ∈ ℕ0 ∧ 𝑦 ∈ ℕ0) ∧ 𝑥 ≤ 𝑦) ∧ 𝑐 ∈ ℕ0) ∧ (𝑐↑2) = (𝑦↑2)) → (𝑐↑2) = (𝑦↑2))
45	44	eqcomd 2741	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((((𝑥 ∈ ℕ0 ∧ 𝑦 ∈ ℕ0) ∧ 𝑥 ≤ 𝑦) ∧ 𝑐 ∈ ℕ0) ∧ (𝑐↑2) = (𝑦↑2)) → (𝑦↑2) = (𝑐↑2))
46	38, 39, 41, 43, 45	sq11d 14183	. . . . . . . . . . . . . . . . . . 19 ⊢ (((((𝑥 ∈ ℕ0 ∧ 𝑦 ∈ ℕ0) ∧ 𝑥 ≤ 𝑦) ∧ 𝑐 ∈ ℕ0) ∧ (𝑐↑2) = (𝑦↑2)) → 𝑦 = 𝑐)
47	46	ex 412	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝑥 ∈ ℕ0 ∧ 𝑦 ∈ ℕ0) ∧ 𝑥 ≤ 𝑦) ∧ 𝑐 ∈ ℕ0) → ((𝑐↑2) = (𝑦↑2) → 𝑦 = 𝑐))
48	37, 47	sylbid 240	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝑥 ∈ ℕ0 ∧ 𝑦 ∈ ℕ0) ∧ 𝑥 ≤ 𝑦) ∧ 𝑐 ∈ ℕ0) → (((𝑥↑2) + (𝑐↑2)) = ((𝑥↑2) + (𝑦↑2)) → 𝑦 = 𝑐))
49	48	adantld 490	. . . . . . . . . . . . . . . 16 ⊢ ((((𝑥 ∈ ℕ0 ∧ 𝑦 ∈ ℕ0) ∧ 𝑥 ≤ 𝑦) ∧ 𝑐 ∈ ℕ0) → ((𝑥 ≤ 𝑐 ∧ ((𝑥↑2) + (𝑐↑2)) = ((𝑥↑2) + (𝑦↑2))) → 𝑦 = 𝑐))
50	49	ralrimiva 3127	. . . . . . . . . . . . . . 15 ⊢ (((𝑥 ∈ ℕ0 ∧ 𝑦 ∈ ℕ0) ∧ 𝑥 ≤ 𝑦) → ∀𝑐 ∈ ℕ0 ((𝑥 ≤ 𝑐 ∧ ((𝑥↑2) + (𝑐↑2)) = ((𝑥↑2) + (𝑦↑2))) → 𝑦 = 𝑐))
51	23, 24, 50	jca31 514	. . . . . . . . . . . . . 14 ⊢ (((𝑥 ∈ ℕ0 ∧ 𝑦 ∈ ℕ0) ∧ 𝑥 ≤ 𝑦) → ((𝑥 ≤ 𝑦 ∧ ((𝑥↑2) + (𝑦↑2)) = ((𝑥↑2) + (𝑦↑2))) ∧ ∀𝑐 ∈ ℕ0 ((𝑥 ≤ 𝑐 ∧ ((𝑥↑2) + (𝑐↑2)) = ((𝑥↑2) + (𝑦↑2))) → 𝑦 = 𝑐)))
52	12, 22, 51	rspcedvd 3577	. . . . . . . . . . . . 13 ⊢ (((𝑥 ∈ ℕ0 ∧ 𝑦 ∈ ℕ0) ∧ 𝑥 ≤ 𝑦) → ∃𝑏 ∈ ℕ0 ((𝑥 ≤ 𝑏 ∧ ((𝑥↑2) + (𝑏↑2)) = ((𝑥↑2) + (𝑦↑2))) ∧ ∀𝑐 ∈ ℕ0 ((𝑥 ≤ 𝑐 ∧ ((𝑥↑2) + (𝑐↑2)) = ((𝑥↑2) + (𝑦↑2))) → 𝑏 = 𝑐)))
53		breq2 5101	. . . . . . . . . . . . . . 15 ⊢ (𝑏 = 𝑐 → (𝑥 ≤ 𝑏 ↔ 𝑥 ≤ 𝑐))
54		oveq1 7365	. . . . . . . . . . . . . . . . 17 ⊢ (𝑏 = 𝑐 → (𝑏↑2) = (𝑐↑2))
55	54	oveq2d 7374	. . . . . . . . . . . . . . . 16 ⊢ (𝑏 = 𝑐 → ((𝑥↑2) + (𝑏↑2)) = ((𝑥↑2) + (𝑐↑2)))
56	55	eqeq1d 2737	. . . . . . . . . . . . . . 15 ⊢ (𝑏 = 𝑐 → (((𝑥↑2) + (𝑏↑2)) = ((𝑥↑2) + (𝑦↑2)) ↔ ((𝑥↑2) + (𝑐↑2)) = ((𝑥↑2) + (𝑦↑2))))
57	53, 56	anbi12d 633	. . . . . . . . . . . . . 14 ⊢ (𝑏 = 𝑐 → ((𝑥 ≤ 𝑏 ∧ ((𝑥↑2) + (𝑏↑2)) = ((𝑥↑2) + (𝑦↑2))) ↔ (𝑥 ≤ 𝑐 ∧ ((𝑥↑2) + (𝑐↑2)) = ((𝑥↑2) + (𝑦↑2)))))
58	57	reu8 3690	. . . . . . . . . . . . 13 ⊢ (∃!𝑏 ∈ ℕ0 (𝑥 ≤ 𝑏 ∧ ((𝑥↑2) + (𝑏↑2)) = ((𝑥↑2) + (𝑦↑2))) ↔ ∃𝑏 ∈ ℕ0 ((𝑥 ≤ 𝑏 ∧ ((𝑥↑2) + (𝑏↑2)) = ((𝑥↑2) + (𝑦↑2))) ∧ ∀𝑐 ∈ ℕ0 ((𝑥 ≤ 𝑐 ∧ ((𝑥↑2) + (𝑐↑2)) = ((𝑥↑2) + (𝑦↑2))) → 𝑏 = 𝑐)))
59	52, 58	sylibr 234	. . . . . . . . . . . 12 ⊢ (((𝑥 ∈ ℕ0 ∧ 𝑦 ∈ ℕ0) ∧ 𝑥 ≤ 𝑦) → ∃!𝑏 ∈ ℕ0 (𝑥 ≤ 𝑏 ∧ ((𝑥↑2) + (𝑏↑2)) = ((𝑥↑2) + (𝑦↑2))))
60	59	ex 412	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ ℕ0 ∧ 𝑦 ∈ ℕ0) → (𝑥 ≤ 𝑦 → ∃!𝑏 ∈ ℕ0 (𝑥 ≤ 𝑏 ∧ ((𝑥↑2) + (𝑏↑2)) = ((𝑥↑2) + (𝑦↑2)))))
61	60	adantr 480	. . . . . . . . . 10 ⊢ (((𝑥 ∈ ℕ0 ∧ 𝑦 ∈ ℕ0) ∧ 𝑃 = ((𝑥↑2) + (𝑦↑2))) → (𝑥 ≤ 𝑦 → ∃!𝑏 ∈ ℕ0 (𝑥 ≤ 𝑏 ∧ ((𝑥↑2) + (𝑏↑2)) = ((𝑥↑2) + (𝑦↑2)))))
62	61	impcom 407	. . . . . . . . 9 ⊢ ((𝑥 ≤ 𝑦 ∧ ((𝑥 ∈ ℕ0 ∧ 𝑦 ∈ ℕ0) ∧ 𝑃 = ((𝑥↑2) + (𝑦↑2)))) → ∃!𝑏 ∈ ℕ0 (𝑥 ≤ 𝑏 ∧ ((𝑥↑2) + (𝑏↑2)) = ((𝑥↑2) + (𝑦↑2))))
63		eqeq2 2747	. . . . . . . . . . . . 13 ⊢ (𝑃 = ((𝑥↑2) + (𝑦↑2)) → (((𝑥↑2) + (𝑏↑2)) = 𝑃 ↔ ((𝑥↑2) + (𝑏↑2)) = ((𝑥↑2) + (𝑦↑2))))
64	63	anbi2d 631	. . . . . . . . . . . 12 ⊢ (𝑃 = ((𝑥↑2) + (𝑦↑2)) → ((𝑥 ≤ 𝑏 ∧ ((𝑥↑2) + (𝑏↑2)) = 𝑃) ↔ (𝑥 ≤ 𝑏 ∧ ((𝑥↑2) + (𝑏↑2)) = ((𝑥↑2) + (𝑦↑2)))))
65	64	reubidv 3365	. . . . . . . . . . 11 ⊢ (𝑃 = ((𝑥↑2) + (𝑦↑2)) → (∃!𝑏 ∈ ℕ0 (𝑥 ≤ 𝑏 ∧ ((𝑥↑2) + (𝑏↑2)) = 𝑃) ↔ ∃!𝑏 ∈ ℕ0 (𝑥 ≤ 𝑏 ∧ ((𝑥↑2) + (𝑏↑2)) = ((𝑥↑2) + (𝑦↑2)))))
66	65	adantl 481	. . . . . . . . . 10 ⊢ (((𝑥 ∈ ℕ0 ∧ 𝑦 ∈ ℕ0) ∧ 𝑃 = ((𝑥↑2) + (𝑦↑2))) → (∃!𝑏 ∈ ℕ0 (𝑥 ≤ 𝑏 ∧ ((𝑥↑2) + (𝑏↑2)) = 𝑃) ↔ ∃!𝑏 ∈ ℕ0 (𝑥 ≤ 𝑏 ∧ ((𝑥↑2) + (𝑏↑2)) = ((𝑥↑2) + (𝑦↑2)))))
67	66	adantl 481	. . . . . . . . 9 ⊢ ((𝑥 ≤ 𝑦 ∧ ((𝑥 ∈ ℕ0 ∧ 𝑦 ∈ ℕ0) ∧ 𝑃 = ((𝑥↑2) + (𝑦↑2)))) → (∃!𝑏 ∈ ℕ0 (𝑥 ≤ 𝑏 ∧ ((𝑥↑2) + (𝑏↑2)) = 𝑃) ↔ ∃!𝑏 ∈ ℕ0 (𝑥 ≤ 𝑏 ∧ ((𝑥↑2) + (𝑏↑2)) = ((𝑥↑2) + (𝑦↑2)))))
68	62, 67	mpbird 257	. . . . . . . 8 ⊢ ((𝑥 ≤ 𝑦 ∧ ((𝑥 ∈ ℕ0 ∧ 𝑦 ∈ ℕ0) ∧ 𝑃 = ((𝑥↑2) + (𝑦↑2)))) → ∃!𝑏 ∈ ℕ0 (𝑥 ≤ 𝑏 ∧ ((𝑥↑2) + (𝑏↑2)) = 𝑃))
69	3, 10, 68	rspcedvd 3577	. . . . . . 7 ⊢ ((𝑥 ≤ 𝑦 ∧ ((𝑥 ∈ ℕ0 ∧ 𝑦 ∈ ℕ0) ∧ 𝑃 = ((𝑥↑2) + (𝑦↑2)))) → ∃𝑎 ∈ ℕ0 ∃!𝑏 ∈ ℕ0 (𝑎 ≤ 𝑏 ∧ ((𝑎↑2) + (𝑏↑2)) = 𝑃))
70	11	adantr 480	. . . . . . . . 9 ⊢ (((𝑥 ∈ ℕ0 ∧ 𝑦 ∈ ℕ0) ∧ 𝑃 = ((𝑥↑2) + (𝑦↑2))) → 𝑦 ∈ ℕ0)
71	70	adantl 481	. . . . . . . 8 ⊢ ((¬ 𝑥 ≤ 𝑦 ∧ ((𝑥 ∈ ℕ0 ∧ 𝑦 ∈ ℕ0) ∧ 𝑃 = ((𝑥↑2) + (𝑦↑2)))) → 𝑦 ∈ ℕ0)
72		breq1 5100	. . . . . . . . . . 11 ⊢ (𝑎 = 𝑦 → (𝑎 ≤ 𝑏 ↔ 𝑦 ≤ 𝑏))
73		oveq1 7365	. . . . . . . . . . . . 13 ⊢ (𝑎 = 𝑦 → (𝑎↑2) = (𝑦↑2))
74	73	oveq1d 7373	. . . . . . . . . . . 12 ⊢ (𝑎 = 𝑦 → ((𝑎↑2) + (𝑏↑2)) = ((𝑦↑2) + (𝑏↑2)))
75	74	eqeq1d 2737	. . . . . . . . . . 11 ⊢ (𝑎 = 𝑦 → (((𝑎↑2) + (𝑏↑2)) = 𝑃 ↔ ((𝑦↑2) + (𝑏↑2)) = 𝑃))
76	72, 75	anbi12d 633	. . . . . . . . . 10 ⊢ (𝑎 = 𝑦 → ((𝑎 ≤ 𝑏 ∧ ((𝑎↑2) + (𝑏↑2)) = 𝑃) ↔ (𝑦 ≤ 𝑏 ∧ ((𝑦↑2) + (𝑏↑2)) = 𝑃)))
77	76	reubidv 3365	. . . . . . . . 9 ⊢ (𝑎 = 𝑦 → (∃!𝑏 ∈ ℕ0 (𝑎 ≤ 𝑏 ∧ ((𝑎↑2) + (𝑏↑2)) = 𝑃) ↔ ∃!𝑏 ∈ ℕ0 (𝑦 ≤ 𝑏 ∧ ((𝑦↑2) + (𝑏↑2)) = 𝑃)))
78	77	adantl 481	. . . . . . . 8 ⊢ (((¬ 𝑥 ≤ 𝑦 ∧ ((𝑥 ∈ ℕ0 ∧ 𝑦 ∈ ℕ0) ∧ 𝑃 = ((𝑥↑2) + (𝑦↑2)))) ∧ 𝑎 = 𝑦) → (∃!𝑏 ∈ ℕ0 (𝑎 ≤ 𝑏 ∧ ((𝑎↑2) + (𝑏↑2)) = 𝑃) ↔ ∃!𝑏 ∈ ℕ0 (𝑦 ≤ 𝑏 ∧ ((𝑦↑2) + (𝑏↑2)) = 𝑃)))
79		simpll 767	. . . . . . . . . . . . . 14 ⊢ (((𝑥 ∈ ℕ0 ∧ 𝑦 ∈ ℕ0) ∧ ¬ 𝑥 ≤ 𝑦) → 𝑥 ∈ ℕ0)
80		breq2 5101	. . . . . . . . . . . . . . . . 17 ⊢ (𝑏 = 𝑥 → (𝑦 ≤ 𝑏 ↔ 𝑦 ≤ 𝑥))
81		oveq1 7365	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑏 = 𝑥 → (𝑏↑2) = (𝑥↑2))
82	81	oveq2d 7374	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑏 = 𝑥 → ((𝑦↑2) + (𝑏↑2)) = ((𝑦↑2) + (𝑥↑2)))
83	82	eqeq1d 2737	. . . . . . . . . . . . . . . . 17 ⊢ (𝑏 = 𝑥 → (((𝑦↑2) + (𝑏↑2)) = ((𝑥↑2) + (𝑦↑2)) ↔ ((𝑦↑2) + (𝑥↑2)) = ((𝑥↑2) + (𝑦↑2))))
84	80, 83	anbi12d 633	. . . . . . . . . . . . . . . 16 ⊢ (𝑏 = 𝑥 → ((𝑦 ≤ 𝑏 ∧ ((𝑦↑2) + (𝑏↑2)) = ((𝑥↑2) + (𝑦↑2))) ↔ (𝑦 ≤ 𝑥 ∧ ((𝑦↑2) + (𝑥↑2)) = ((𝑥↑2) + (𝑦↑2)))))
85		equequ1 2027	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑏 = 𝑥 → (𝑏 = 𝑐 ↔ 𝑥 = 𝑐))
86	85	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)))
87	86	ralbidv 3158	. . . . . . . . . . . . . . . 16 ⊢ (𝑏 = 𝑥 → (∀𝑐 ∈ ℕ0 ((𝑦 ≤ 𝑐 ∧ ((𝑦↑2) + (𝑐↑2)) = ((𝑥↑2) + (𝑦↑2))) → 𝑏 = 𝑐) ↔ ∀𝑐 ∈ ℕ0 ((𝑦 ≤ 𝑐 ∧ ((𝑦↑2) + (𝑐↑2)) = ((𝑥↑2) + (𝑦↑2))) → 𝑥 = 𝑐)))
88	84, 87	anbi12d 633	. . . . . . . . . . . . . . 15 ⊢ (𝑏 = 𝑥 → (((𝑦 ≤ 𝑏 ∧ ((𝑦↑2) + (𝑏↑2)) = ((𝑥↑2) + (𝑦↑2))) ∧ ∀𝑐 ∈ ℕ0 ((𝑦 ≤ 𝑐 ∧ ((𝑦↑2) + (𝑐↑2)) = ((𝑥↑2) + (𝑦↑2))) → 𝑏 = 𝑐)) ↔ ((𝑦 ≤ 𝑥 ∧ ((𝑦↑2) + (𝑥↑2)) = ((𝑥↑2) + (𝑦↑2))) ∧ ∀𝑐 ∈ ℕ0 ((𝑦 ≤ 𝑐 ∧ ((𝑦↑2) + (𝑐↑2)) = ((𝑥↑2) + (𝑦↑2))) → 𝑥 = 𝑐))))
89	88	adantl 481	. . . . . . . . . . . . . 14 ⊢ ((((𝑥 ∈ ℕ0 ∧ 𝑦 ∈ ℕ0) ∧ ¬ 𝑥 ≤ 𝑦) ∧ 𝑏 = 𝑥) → (((𝑦 ≤ 𝑏 ∧ ((𝑦↑2) + (𝑏↑2)) = ((𝑥↑2) + (𝑦↑2))) ∧ ∀𝑐 ∈ ℕ0 ((𝑦 ≤ 𝑐 ∧ ((𝑦↑2) + (𝑐↑2)) = ((𝑥↑2) + (𝑦↑2))) → 𝑏 = 𝑐)) ↔ ((𝑦 ≤ 𝑥 ∧ ((𝑦↑2) + (𝑥↑2)) = ((𝑥↑2) + (𝑦↑2))) ∧ ∀𝑐 ∈ ℕ0 ((𝑦 ≤ 𝑐 ∧ ((𝑦↑2) + (𝑐↑2)) = ((𝑥↑2) + (𝑦↑2))) → 𝑥 = 𝑐))))
90		ltnle 11214	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑦 ∈ ℝ ∧ 𝑥 ∈ ℝ) → (𝑦 < 𝑥 ↔ ¬ 𝑥 ≤ 𝑦))
91	28, 32, 90	syl2anr 598	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑥 ∈ ℕ0 ∧ 𝑦 ∈ ℕ0) → (𝑦 < 𝑥 ↔ ¬ 𝑥 ≤ 𝑦))
92	28	ad2antlr 728	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑥 ∈ ℕ0 ∧ 𝑦 ∈ ℕ0) ∧ 𝑦 < 𝑥) → 𝑦 ∈ ℝ)
93	32	ad2antrr 727	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑥 ∈ ℕ0 ∧ 𝑦 ∈ ℕ0) ∧ 𝑦 < 𝑥) → 𝑥 ∈ ℝ)
94		simpr 484	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑥 ∈ ℕ0 ∧ 𝑦 ∈ ℕ0) ∧ 𝑦 < 𝑥) → 𝑦 < 𝑥)
95	92, 93, 94	ltled 11283	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑥 ∈ ℕ0 ∧ 𝑦 ∈ ℕ0) ∧ 𝑦 < 𝑥) → 𝑦 ≤ 𝑥)
96	95	ex 412	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑥 ∈ ℕ0 ∧ 𝑦 ∈ ℕ0) → (𝑦 < 𝑥 → 𝑦 ≤ 𝑥))
97	91, 96	sylbird 260	. . . . . . . . . . . . . . . 16 ⊢ ((𝑥 ∈ ℕ0 ∧ 𝑦 ∈ ℕ0) → (¬ 𝑥 ≤ 𝑦 → 𝑦 ≤ 𝑥))
98	97	imp 406	. . . . . . . . . . . . . . 15 ⊢ (((𝑥 ∈ ℕ0 ∧ 𝑦 ∈ ℕ0) ∧ ¬ 𝑥 ≤ 𝑦) → 𝑦 ≤ 𝑥)
99	29	recnd 11162	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑦 ∈ ℕ0 → (𝑦↑2) ∈ ℂ)
100	99	adantl 481	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑥 ∈ ℕ0 ∧ 𝑦 ∈ ℕ0) → (𝑦↑2) ∈ ℂ)
101	33	recnd 11162	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 ∈ ℕ0 → (𝑥↑2) ∈ ℂ)
102	101	adantr 480	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑥 ∈ ℕ0 ∧ 𝑦 ∈ ℕ0) → (𝑥↑2) ∈ ℂ)
103	100, 102	addcomd 11337	. . . . . . . . . . . . . . . 16 ⊢ ((𝑥 ∈ ℕ0 ∧ 𝑦 ∈ ℕ0) → ((𝑦↑2) + (𝑥↑2)) = ((𝑥↑2) + (𝑦↑2)))
104	103	adantr 480	. . . . . . . . . . . . . . 15 ⊢ (((𝑥 ∈ ℕ0 ∧ 𝑦 ∈ ℕ0) ∧ ¬ 𝑥 ≤ 𝑦) → ((𝑦↑2) + (𝑥↑2)) = ((𝑥↑2) + (𝑦↑2)))
105	34	recnd 11162	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑥 ∈ ℕ0 ∧ 𝑦 ∈ ℕ0) → (𝑥↑2) ∈ ℂ)
106	105	adantr 480	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝑥 ∈ ℕ0 ∧ 𝑦 ∈ ℕ0) ∧ 𝑐 ∈ ℕ0) → (𝑥↑2) ∈ ℂ)
107	30	recnd 11162	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑥 ∈ ℕ0 ∧ 𝑦 ∈ ℕ0) → (𝑦↑2) ∈ ℂ)
108	107	adantr 480	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝑥 ∈ ℕ0 ∧ 𝑦 ∈ ℕ0) ∧ 𝑐 ∈ ℕ0) → (𝑦↑2) ∈ ℂ)
109	106, 108	addcomd 11337	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝑥 ∈ ℕ0 ∧ 𝑦 ∈ ℕ0) ∧ 𝑐 ∈ ℕ0) → ((𝑥↑2) + (𝑦↑2)) = ((𝑦↑2) + (𝑥↑2)))
110	109	eqeq2d 2746	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝑥 ∈ ℕ0 ∧ 𝑦 ∈ ℕ0) ∧ 𝑐 ∈ ℕ0) → (((𝑦↑2) + (𝑐↑2)) = ((𝑥↑2) + (𝑦↑2)) ↔ ((𝑦↑2) + (𝑐↑2)) = ((𝑦↑2) + (𝑥↑2))))
111	26	adantl 481	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝑥 ∈ ℕ0 ∧ 𝑦 ∈ ℕ0) ∧ 𝑐 ∈ ℕ0) → (𝑐↑2) ∈ ℝ)
112	33	ad2antrr 727	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝑥 ∈ ℕ0 ∧ 𝑦 ∈ ℕ0) ∧ 𝑐 ∈ ℕ0) → (𝑥↑2) ∈ ℝ)
113	29	ad2antlr 728	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝑥 ∈ ℕ0 ∧ 𝑦 ∈ ℕ0) ∧ 𝑐 ∈ ℕ0) → (𝑦↑2) ∈ ℝ)
114		readdcan 11309	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝑐↑2) ∈ ℝ ∧ (𝑥↑2) ∈ ℝ ∧ (𝑦↑2) ∈ ℝ) → (((𝑦↑2) + (𝑐↑2)) = ((𝑦↑2) + (𝑥↑2)) ↔ (𝑐↑2) = (𝑥↑2)))
115	111, 112, 113, 114	syl3anc 1374	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝑥 ∈ ℕ0 ∧ 𝑦 ∈ ℕ0) ∧ 𝑐 ∈ ℕ0) → (((𝑦↑2) + (𝑐↑2)) = ((𝑦↑2) + (𝑥↑2)) ↔ (𝑐↑2) = (𝑥↑2)))
116	110, 115	bitrd 279	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑥 ∈ ℕ0 ∧ 𝑦 ∈ ℕ0) ∧ 𝑐 ∈ ℕ0) → (((𝑦↑2) + (𝑐↑2)) = ((𝑥↑2) + (𝑦↑2)) ↔ (𝑐↑2) = (𝑥↑2)))
117	25	ad2antlr 728	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((𝑥 ∈ ℕ0 ∧ 𝑦 ∈ ℕ0) ∧ 𝑐 ∈ ℕ0) ∧ (𝑐↑2) = (𝑥↑2)) → 𝑐 ∈ ℝ)
118	32	adantr 480	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑥 ∈ ℕ0 ∧ 𝑦 ∈ ℕ0) → 𝑥 ∈ ℝ)
119	118	ad2antrr 727	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((𝑥 ∈ ℕ0 ∧ 𝑦 ∈ ℕ0) ∧ 𝑐 ∈ ℕ0) ∧ (𝑐↑2) = (𝑥↑2)) → 𝑥 ∈ ℝ)
120	42	ad2antlr 728	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((𝑥 ∈ ℕ0 ∧ 𝑦 ∈ ℕ0) ∧ 𝑐 ∈ ℕ0) ∧ (𝑐↑2) = (𝑥↑2)) → 0 ≤ 𝑐)
121		nn0ge0 12428	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑥 ∈ ℕ0 → 0 ≤ 𝑥)
122	121	adantr 480	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑥 ∈ ℕ0 ∧ 𝑦 ∈ ℕ0) → 0 ≤ 𝑥)
123	122	ad2antrr 727	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((𝑥 ∈ ℕ0 ∧ 𝑦 ∈ ℕ0) ∧ 𝑐 ∈ ℕ0) ∧ (𝑐↑2) = (𝑥↑2)) → 0 ≤ 𝑥)
124		simpr 484	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((𝑥 ∈ ℕ0 ∧ 𝑦 ∈ ℕ0) ∧ 𝑐 ∈ ℕ0) ∧ (𝑐↑2) = (𝑥↑2)) → (𝑐↑2) = (𝑥↑2))
125	117, 119, 120, 123, 124	sq11d 14183	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((𝑥 ∈ ℕ0 ∧ 𝑦 ∈ ℕ0) ∧ 𝑐 ∈ ℕ0) ∧ (𝑐↑2) = (𝑥↑2)) → 𝑐 = 𝑥)
126	125	eqcomd 2741	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝑥 ∈ ℕ0 ∧ 𝑦 ∈ ℕ0) ∧ 𝑐 ∈ ℕ0) ∧ (𝑐↑2) = (𝑥↑2)) → 𝑥 = 𝑐)
127	126	ex 412	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑥 ∈ ℕ0 ∧ 𝑦 ∈ ℕ0) ∧ 𝑐 ∈ ℕ0) → ((𝑐↑2) = (𝑥↑2) → 𝑥 = 𝑐))
128	116, 127	sylbid 240	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑥 ∈ ℕ0 ∧ 𝑦 ∈ ℕ0) ∧ 𝑐 ∈ ℕ0) → (((𝑦↑2) + (𝑐↑2)) = ((𝑥↑2) + (𝑦↑2)) → 𝑥 = 𝑐))
129	128	adantld 490	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑥 ∈ ℕ0 ∧ 𝑦 ∈ ℕ0) ∧ 𝑐 ∈ ℕ0) → ((𝑦 ≤ 𝑐 ∧ ((𝑦↑2) + (𝑐↑2)) = ((𝑥↑2) + (𝑦↑2))) → 𝑥 = 𝑐))
130	129	ralrimiva 3127	. . . . . . . . . . . . . . . 16 ⊢ ((𝑥 ∈ ℕ0 ∧ 𝑦 ∈ ℕ0) → ∀𝑐 ∈ ℕ0 ((𝑦 ≤ 𝑐 ∧ ((𝑦↑2) + (𝑐↑2)) = ((𝑥↑2) + (𝑦↑2))) → 𝑥 = 𝑐))
131	130	adantr 480	. . . . . . . . . . . . . . 15 ⊢ (((𝑥 ∈ ℕ0 ∧ 𝑦 ∈ ℕ0) ∧ ¬ 𝑥 ≤ 𝑦) → ∀𝑐 ∈ ℕ0 ((𝑦 ≤ 𝑐 ∧ ((𝑦↑2) + (𝑐↑2)) = ((𝑥↑2) + (𝑦↑2))) → 𝑥 = 𝑐))
132	98, 104, 131	jca31 514	. . . . . . . . . . . . . 14 ⊢ (((𝑥 ∈ ℕ0 ∧ 𝑦 ∈ ℕ0) ∧ ¬ 𝑥 ≤ 𝑦) → ((𝑦 ≤ 𝑥 ∧ ((𝑦↑2) + (𝑥↑2)) = ((𝑥↑2) + (𝑦↑2))) ∧ ∀𝑐 ∈ ℕ0 ((𝑦 ≤ 𝑐 ∧ ((𝑦↑2) + (𝑐↑2)) = ((𝑥↑2) + (𝑦↑2))) → 𝑥 = 𝑐)))
133	79, 89, 132	rspcedvd 3577	. . . . . . . . . . . . 13 ⊢ (((𝑥 ∈ ℕ0 ∧ 𝑦 ∈ ℕ0) ∧ ¬ 𝑥 ≤ 𝑦) → ∃𝑏 ∈ ℕ0 ((𝑦 ≤ 𝑏 ∧ ((𝑦↑2) + (𝑏↑2)) = ((𝑥↑2) + (𝑦↑2))) ∧ ∀𝑐 ∈ ℕ0 ((𝑦 ≤ 𝑐 ∧ ((𝑦↑2) + (𝑐↑2)) = ((𝑥↑2) + (𝑦↑2))) → 𝑏 = 𝑐)))
134		breq2 5101	. . . . . . . . . . . . . . 15 ⊢ (𝑏 = 𝑐 → (𝑦 ≤ 𝑏 ↔ 𝑦 ≤ 𝑐))
135	54	oveq2d 7374	. . . . . . . . . . . . . . . 16 ⊢ (𝑏 = 𝑐 → ((𝑦↑2) + (𝑏↑2)) = ((𝑦↑2) + (𝑐↑2)))
136	135	eqeq1d 2737	. . . . . . . . . . . . . . 15 ⊢ (𝑏 = 𝑐 → (((𝑦↑2) + (𝑏↑2)) = ((𝑥↑2) + (𝑦↑2)) ↔ ((𝑦↑2) + (𝑐↑2)) = ((𝑥↑2) + (𝑦↑2))))
137	134, 136	anbi12d 633	. . . . . . . . . . . . . 14 ⊢ (𝑏 = 𝑐 → ((𝑦 ≤ 𝑏 ∧ ((𝑦↑2) + (𝑏↑2)) = ((𝑥↑2) + (𝑦↑2))) ↔ (𝑦 ≤ 𝑐 ∧ ((𝑦↑2) + (𝑐↑2)) = ((𝑥↑2) + (𝑦↑2)))))
138	137	reu8 3690	. . . . . . . . . . . . 13 ⊢ (∃!𝑏 ∈ ℕ0 (𝑦 ≤ 𝑏 ∧ ((𝑦↑2) + (𝑏↑2)) = ((𝑥↑2) + (𝑦↑2))) ↔ ∃𝑏 ∈ ℕ0 ((𝑦 ≤ 𝑏 ∧ ((𝑦↑2) + (𝑏↑2)) = ((𝑥↑2) + (𝑦↑2))) ∧ ∀𝑐 ∈ ℕ0 ((𝑦 ≤ 𝑐 ∧ ((𝑦↑2) + (𝑐↑2)) = ((𝑥↑2) + (𝑦↑2))) → 𝑏 = 𝑐)))
139	133, 138	sylibr 234	. . . . . . . . . . . 12 ⊢ (((𝑥 ∈ ℕ0 ∧ 𝑦 ∈ ℕ0) ∧ ¬ 𝑥 ≤ 𝑦) → ∃!𝑏 ∈ ℕ0 (𝑦 ≤ 𝑏 ∧ ((𝑦↑2) + (𝑏↑2)) = ((𝑥↑2) + (𝑦↑2))))
140	139	ex 412	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ ℕ0 ∧ 𝑦 ∈ ℕ0) → (¬ 𝑥 ≤ 𝑦 → ∃!𝑏 ∈ ℕ0 (𝑦 ≤ 𝑏 ∧ ((𝑦↑2) + (𝑏↑2)) = ((𝑥↑2) + (𝑦↑2)))))
141	140	adantr 480	. . . . . . . . . 10 ⊢ (((𝑥 ∈ ℕ0 ∧ 𝑦 ∈ ℕ0) ∧ 𝑃 = ((𝑥↑2) + (𝑦↑2))) → (¬ 𝑥 ≤ 𝑦 → ∃!𝑏 ∈ ℕ0 (𝑦 ≤ 𝑏 ∧ ((𝑦↑2) + (𝑏↑2)) = ((𝑥↑2) + (𝑦↑2)))))
142	141	impcom 407	. . . . . . . . 9 ⊢ ((¬ 𝑥 ≤ 𝑦 ∧ ((𝑥 ∈ ℕ0 ∧ 𝑦 ∈ ℕ0) ∧ 𝑃 = ((𝑥↑2) + (𝑦↑2)))) → ∃!𝑏 ∈ ℕ0 (𝑦 ≤ 𝑏 ∧ ((𝑦↑2) + (𝑏↑2)) = ((𝑥↑2) + (𝑦↑2))))
143		eqeq2 2747	. . . . . . . . . . . . 13 ⊢ (𝑃 = ((𝑥↑2) + (𝑦↑2)) → (((𝑦↑2) + (𝑏↑2)) = 𝑃 ↔ ((𝑦↑2) + (𝑏↑2)) = ((𝑥↑2) + (𝑦↑2))))
144	143	anbi2d 631	. . . . . . . . . . . 12 ⊢ (𝑃 = ((𝑥↑2) + (𝑦↑2)) → ((𝑦 ≤ 𝑏 ∧ ((𝑦↑2) + (𝑏↑2)) = 𝑃) ↔ (𝑦 ≤ 𝑏 ∧ ((𝑦↑2) + (𝑏↑2)) = ((𝑥↑2) + (𝑦↑2)))))
145	144	reubidv 3365	. . . . . . . . . . 11 ⊢ (𝑃 = ((𝑥↑2) + (𝑦↑2)) → (∃!𝑏 ∈ ℕ0 (𝑦 ≤ 𝑏 ∧ ((𝑦↑2) + (𝑏↑2)) = 𝑃) ↔ ∃!𝑏 ∈ ℕ0 (𝑦 ≤ 𝑏 ∧ ((𝑦↑2) + (𝑏↑2)) = ((𝑥↑2) + (𝑦↑2)))))
146	145	adantl 481	. . . . . . . . . 10 ⊢ (((𝑥 ∈ ℕ0 ∧ 𝑦 ∈ ℕ0) ∧ 𝑃 = ((𝑥↑2) + (𝑦↑2))) → (∃!𝑏 ∈ ℕ0 (𝑦 ≤ 𝑏 ∧ ((𝑦↑2) + (𝑏↑2)) = 𝑃) ↔ ∃!𝑏 ∈ ℕ0 (𝑦 ≤ 𝑏 ∧ ((𝑦↑2) + (𝑏↑2)) = ((𝑥↑2) + (𝑦↑2)))))
147	146	adantl 481	. . . . . . . . 9 ⊢ ((¬ 𝑥 ≤ 𝑦 ∧ ((𝑥 ∈ ℕ0 ∧ 𝑦 ∈ ℕ0) ∧ 𝑃 = ((𝑥↑2) + (𝑦↑2)))) → (∃!𝑏 ∈ ℕ0 (𝑦 ≤ 𝑏 ∧ ((𝑦↑2) + (𝑏↑2)) = 𝑃) ↔ ∃!𝑏 ∈ ℕ0 (𝑦 ≤ 𝑏 ∧ ((𝑦↑2) + (𝑏↑2)) = ((𝑥↑2) + (𝑦↑2)))))
148	142, 147	mpbird 257	. . . . . . . 8 ⊢ ((¬ 𝑥 ≤ 𝑦 ∧ ((𝑥 ∈ ℕ0 ∧ 𝑦 ∈ ℕ0) ∧ 𝑃 = ((𝑥↑2) + (𝑦↑2)))) → ∃!𝑏 ∈ ℕ0 (𝑦 ≤ 𝑏 ∧ ((𝑦↑2) + (𝑏↑2)) = 𝑃))
149	71, 78, 148	rspcedvd 3577	. . . . . . 7 ⊢ ((¬ 𝑥 ≤ 𝑦 ∧ ((𝑥 ∈ ℕ0 ∧ 𝑦 ∈ ℕ0) ∧ 𝑃 = ((𝑥↑2) + (𝑦↑2)))) → ∃𝑎 ∈ ℕ0 ∃!𝑏 ∈ ℕ0 (𝑎 ≤ 𝑏 ∧ ((𝑎↑2) + (𝑏↑2)) = 𝑃))
150	69, 149	pm2.61ian 812	. . . . . 6 ⊢ (((𝑥 ∈ ℕ0 ∧ 𝑦 ∈ ℕ0) ∧ 𝑃 = ((𝑥↑2) + (𝑦↑2))) → ∃𝑎 ∈ ℕ0 ∃!𝑏 ∈ ℕ0 (𝑎 ≤ 𝑏 ∧ ((𝑎↑2) + (𝑏↑2)) = 𝑃))
151	150	ex 412	. . . . 5 ⊢ ((𝑥 ∈ ℕ0 ∧ 𝑦 ∈ ℕ0) → (𝑃 = ((𝑥↑2) + (𝑦↑2)) → ∃𝑎 ∈ ℕ0 ∃!𝑏 ∈ ℕ0 (𝑎 ≤ 𝑏 ∧ ((𝑎↑2) + (𝑏↑2)) = 𝑃)))
152	151	adantl 481	. . . 4 ⊢ (((𝑃 ∈ ℙ ∧ (𝑃 mod 4) = 1) ∧ (𝑥 ∈ ℕ0 ∧ 𝑦 ∈ ℕ0)) → (𝑃 = ((𝑥↑2) + (𝑦↑2)) → ∃𝑎 ∈ ℕ0 ∃!𝑏 ∈ ℕ0 (𝑎 ≤ 𝑏 ∧ ((𝑎↑2) + (𝑏↑2)) = 𝑃)))
153	152	rexlimdvva 3192	. . 3 ⊢ ((𝑃 ∈ ℙ ∧ (𝑃 mod 4) = 1) → (∃𝑥 ∈ ℕ0 ∃𝑦 ∈ ℕ0 𝑃 = ((𝑥↑2) + (𝑦↑2)) → ∃𝑎 ∈ ℕ0 ∃!𝑏 ∈ ℕ0 (𝑎 ≤ 𝑏 ∧ ((𝑎↑2) + (𝑏↑2)) = 𝑃)))
154	1, 153	mpd 15	. 2 ⊢ ((𝑃 ∈ ℙ ∧ (𝑃 mod 4) = 1) → ∃𝑎 ∈ ℕ0 ∃!𝑏 ∈ ℕ0 (𝑎 ≤ 𝑏 ∧ ((𝑎↑2) + (𝑏↑2)) = 𝑃))
155		reurex 3353	. . . . 5 ⊢ (∃!𝑏 ∈ ℕ0 (𝑎 ≤ 𝑏 ∧ ((𝑎↑2) + (𝑏↑2)) = 𝑃) → ∃𝑏 ∈ ℕ0 (𝑎 ≤ 𝑏 ∧ ((𝑎↑2) + (𝑏↑2)) = 𝑃))
156	155	a1i 11	. . . 4 ⊢ (((𝑃 ∈ ℙ ∧ (𝑃 mod 4) = 1) ∧ 𝑎 ∈ ℕ0) → (∃!𝑏 ∈ ℕ0 (𝑎 ≤ 𝑏 ∧ ((𝑎↑2) + (𝑏↑2)) = 𝑃) → ∃𝑏 ∈ ℕ0 (𝑎 ≤ 𝑏 ∧ ((𝑎↑2) + (𝑏↑2)) = 𝑃)))
157	156	ralrimiva 3127	. . 3 ⊢ ((𝑃 ∈ ℙ ∧ (𝑃 mod 4) = 1) → ∀𝑎 ∈ ℕ0 (∃!𝑏 ∈ ℕ0 (𝑎 ≤ 𝑏 ∧ ((𝑎↑2) + (𝑏↑2)) = 𝑃) → ∃𝑏 ∈ ℕ0 (𝑎 ≤ 𝑏 ∧ ((𝑎↑2) + (𝑏↑2)) = 𝑃)))
158		2sqmo 27406	. . . 4 ⊢ (𝑃 ∈ ℙ → ∃*𝑎 ∈ ℕ0 ∃𝑏 ∈ ℕ0 (𝑎 ≤ 𝑏 ∧ ((𝑎↑2) + (𝑏↑2)) = 𝑃))
159	158	adantr 480	. . 3 ⊢ ((𝑃 ∈ ℙ ∧ (𝑃 mod 4) = 1) → ∃*𝑎 ∈ ℕ0 ∃𝑏 ∈ ℕ0 (𝑎 ≤ 𝑏 ∧ ((𝑎↑2) + (𝑏↑2)) = 𝑃))
160		rmoim 3697	. . 3 ⊢ (∀𝑎 ∈ ℕ0 (∃!𝑏 ∈ ℕ0 (𝑎 ≤ 𝑏 ∧ ((𝑎↑2) + (𝑏↑2)) = 𝑃) → ∃𝑏 ∈ ℕ0 (𝑎 ≤ 𝑏 ∧ ((𝑎↑2) + (𝑏↑2)) = 𝑃)) → (∃*𝑎 ∈ ℕ0 ∃𝑏 ∈ ℕ0 (𝑎 ≤ 𝑏 ∧ ((𝑎↑2) + (𝑏↑2)) = 𝑃) → ∃*𝑎 ∈ ℕ0 ∃!𝑏 ∈ ℕ0 (𝑎 ≤ 𝑏 ∧ ((𝑎↑2) + (𝑏↑2)) = 𝑃)))
161	157, 159, 160	sylc 65	. 2 ⊢ ((𝑃 ∈ ℙ ∧ (𝑃 mod 4) = 1) → ∃*𝑎 ∈ ℕ0 ∃!𝑏 ∈ ℕ0 (𝑎 ≤ 𝑏 ∧ ((𝑎↑2) + (𝑏↑2)) = 𝑃))
162		reu5 3351	. 2 ⊢ (∃!𝑎 ∈ ℕ0 ∃!𝑏 ∈ ℕ0 (𝑎 ≤ 𝑏 ∧ ((𝑎↑2) + (𝑏↑2)) = 𝑃) ↔ (∃𝑎 ∈ ℕ0 ∃!𝑏 ∈ ℕ0 (𝑎 ≤ 𝑏 ∧ ((𝑎↑2) + (𝑏↑2)) = 𝑃) ∧ ∃*𝑎 ∈ ℕ0 ∃!𝑏 ∈ ℕ0 (𝑎 ≤ 𝑏 ∧ ((𝑎↑2) + (𝑏↑2)) = 𝑃)))
163	154, 161, 162	sylanbrc 584	1 ⊢ ((𝑃 ∈ ℙ ∧ (𝑃 mod 4) = 1) → ∃!𝑎 ∈ ℕ0 ∃!𝑏 ∈ ℕ0 (𝑎 ≤ 𝑏 ∧ ((𝑎↑2) + (𝑏↑2)) = 𝑃))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1542   ∈ wcel 2114  ∀wral 3050  ∃wrex 3059  ∃!wreu 3347  ∃*wrmo 3348   class class class wbr 5097  (class class class)co 7358  ℂcc 11026  ℝcr 11027  0cc0 11028  1c1 11029   + caddc 11031   < clt 11168   ≤ cle 11169  2c2 12202  4c4 12204  ℕ₀cn0 12403   mod cmo 13791  ↑cexp 13986  ℙcprime 16600

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2183  ax-ext 2707  ax-rep 5223  ax-sep 5240  ax-nul 5250  ax-pow 5309  ax-pr 5376  ax-un 7680  ax-cnex 11084  ax-resscn 11085  ax-1cn 11086  ax-icn 11087  ax-addcl 11088  ax-addrcl 11089  ax-mulcl 11090  ax-mulrcl 11091  ax-mulcom 11092  ax-addass 11093  ax-mulass 11094  ax-distr 11095  ax-i2m1 11096  ax-1ne0 11097  ax-1rid 11098  ax-rnegex 11099  ax-rrecex 11100  ax-cnre 11101  ax-pre-lttri 11102  ax-pre-lttrn 11103  ax-pre-ltadd 11104  ax-pre-mulgt0 11105  ax-pre-sup 11106  ax-addf 11107  ax-mulf 11108

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2538  df-eu 2568  df-clab 2714  df-cleq 2727  df-clel 2810  df-nfc 2884  df-ne 2932  df-nel 3036  df-ral 3051  df-rex 3060  df-rmo 3349  df-reu 3350  df-rab 3399  df-v 3441  df-sbc 3740  df-csb 3849  df-dif 3903  df-un 3905  df-in 3907  df-ss 3917  df-pss 3920  df-nul 4285  df-if 4479  df-pw 4555  df-sn 4580  df-pr 4582  df-tp 4584  df-op 4586  df-uni 4863  df-int 4902  df-iun 4947  df-iin 4948  df-br 5098  df-opab 5160  df-mpt 5179  df-tr 5205  df-id 5518  df-eprel 5523  df-po 5531  df-so 5532  df-fr 5576  df-se 5577  df-we 5578  df-xp 5629  df-rel 5630  df-cnv 5631  df-co 5632  df-dm 5633  df-rn 5634  df-res 5635  df-ima 5636  df-pred 6258  df-ord 6319  df-on 6320  df-lim 6321  df-suc 6322  df-iota 6447  df-fun 6493  df-fn 6494  df-f 6495  df-f1 6496  df-fo 6497  df-f1o 6498  df-fv 6499  df-isom 6500  df-riota 7315  df-ov 7361  df-oprab 7362  df-mpo 7363  df-of 7622  df-ofr 7623  df-om 7809  df-1st 7933  df-2nd 7934  df-supp 8103  df-tpos 8168  df-frecs 8223  df-wrecs 8254  df-recs 8303  df-rdg 8341  df-1o 8397  df-2o 8398  df-oadd 8401  df-er 8635  df-ec 8637  df-qs 8641  df-map 8767  df-pm 8768  df-ixp 8838  df-en 8886  df-dom 8887  df-sdom 8888  df-fin 8889  df-fsupp 9267  df-sup 9347  df-inf 9348  df-oi 9417  df-dju 9815  df-card 9853  df-pnf 11170  df-mnf 11171  df-xr 11172  df-ltxr 11173  df-le 11174  df-sub 11368  df-neg 11369  df-div 11797  df-nn 12148  df-2 12210  df-3 12211  df-4 12212  df-5 12213  df-6 12214  df-7 12215  df-8 12216  df-9 12217  df-n0 12404  df-xnn0 12477  df-z 12491  df-dec 12610  df-uz 12754  df-q 12864  df-rp 12908  df-fz 13426  df-fzo 13573  df-fl 13714  df-mod 13792  df-seq 13927  df-exp 13987  df-hash 14256  df-cj 15024  df-re 15025  df-im 15026  df-sqrt 15160  df-abs 15161  df-dvds 16182  df-gcd 16424  df-prm 16601  df-phi 16695  df-pc 16767  df-gz 16860  df-struct 17076  df-sets 17093  df-slot 17111  df-ndx 17123  df-base 17139  df-ress 17160  df-plusg 17192  df-mulr 17193  df-starv 17194  df-sca 17195  df-vsca 17196  df-ip 17197  df-tset 17198  df-ple 17199  df-ds 17201  df-unif 17202  df-hom 17203  df-cco 17204  df-0g 17363  df-gsum 17364  df-prds 17369  df-pws 17371  df-imas 17431  df-qus 17432  df-mre 17507  df-mrc 17508  df-acs 17510  df-mgm 18567  df-sgrp 18646  df-mnd 18662  df-mhm 18710  df-submnd 18711  df-grp 18868  df-minusg 18869  df-sbg 18870  df-mulg 19000  df-subg 19055  df-nsg 19056  df-eqg 19057  df-ghm 19144  df-cntz 19248  df-cmn 19713  df-abl 19714  df-mgp 20078  df-rng 20090  df-ur 20119  df-srg 20124  df-ring 20172  df-cring 20173  df-oppr 20275  df-dvdsr 20295  df-unit 20296  df-invr 20326  df-dvr 20339  df-rhm 20410  df-nzr 20448  df-subrng 20481  df-subrg 20505  df-rlreg 20629  df-domn 20630  df-idom 20631  df-drng 20666  df-field 20667  df-lmod 20815  df-lss 20885  df-lsp 20925  df-sra 21127  df-rgmod 21128  df-lidl 21165  df-rsp 21166  df-2idl 21207  df-cnfld 21312  df-zring 21404  df-zrh 21460  df-zn 21463  df-assa 21810  df-asp 21811  df-ascl 21812  df-psr 21867  df-mvr 21868  df-mpl 21869  df-opsr 21871  df-evls 22031  df-evl 22032  df-psr1 22122  df-vr1 22123  df-ply1 22124  df-coe1 22125  df-evl1 22262  df-mdeg 26018  df-deg1 26019  df-mon1 26094  df-uc1p 26095  df-q1p 26096  df-r1p 26097  df-lgs 27264

This theorem is referenced by:  2sqreultlem  27416  2sqreu  27425