Theorem 2sqreulem1 27397

Description: Lemma 1 for 2sqreu 27407. (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 27389	. . 3 ⊢ ((𝑃 ∈ ℙ ∧ (𝑃 mod 4) = 1) → ∃𝑥 ∈ ℕ0 ∃𝑦 ∈ ℕ0 𝑃 = ((𝑥↑2) + (𝑦↑2)))
2		simpll 765	. . . . . . . . 9 ⊢ (((𝑥 ∈ ℕ0 ∧ 𝑦 ∈ ℕ0) ∧ 𝑃 = ((𝑥↑2) + (𝑦↑2))) → 𝑥 ∈ ℕ0)
3	2	adantl 480	. . . . . . . 8 ⊢ ((𝑥 ≤ 𝑦 ∧ ((𝑥 ∈ ℕ0 ∧ 𝑦 ∈ ℕ0) ∧ 𝑃 = ((𝑥↑2) + (𝑦↑2)))) → 𝑥 ∈ ℕ0)
4		breq1 5146	. . . . . . . . . . 11 ⊢ (𝑎 = 𝑥 → (𝑎 ≤ 𝑏 ↔ 𝑥 ≤ 𝑏))
5		oveq1 7423	. . . . . . . . . . . . 13 ⊢ (𝑎 = 𝑥 → (𝑎↑2) = (𝑥↑2))
6	5	oveq1d 7431	. . . . . . . . . . . 12 ⊢ (𝑎 = 𝑥 → ((𝑎↑2) + (𝑏↑2)) = ((𝑥↑2) + (𝑏↑2)))
7	6	eqeq1d 2727	. . . . . . . . . . 11 ⊢ (𝑎 = 𝑥 → (((𝑎↑2) + (𝑏↑2)) = 𝑃 ↔ ((𝑥↑2) + (𝑏↑2)) = 𝑃))
8	4, 7	anbi12d 630	. . . . . . . . . 10 ⊢ (𝑎 = 𝑥 → ((𝑎 ≤ 𝑏 ∧ ((𝑎↑2) + (𝑏↑2)) = 𝑃) ↔ (𝑥 ≤ 𝑏 ∧ ((𝑥↑2) + (𝑏↑2)) = 𝑃)))
9	8	reubidv 3382	. . . . . . . . 9 ⊢ (𝑎 = 𝑥 → (∃!𝑏 ∈ ℕ0 (𝑎 ≤ 𝑏 ∧ ((𝑎↑2) + (𝑏↑2)) = 𝑃) ↔ ∃!𝑏 ∈ ℕ0 (𝑥 ≤ 𝑏 ∧ ((𝑥↑2) + (𝑏↑2)) = 𝑃)))
10	9	adantl 480	. . . . . . . 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 483	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 ∈ ℕ0 ∧ 𝑦 ∈ ℕ0) → 𝑦 ∈ ℕ0)
12	11	adantr 479	. . . . . . . . . . . . . 14 ⊢ (((𝑥 ∈ ℕ0 ∧ 𝑦 ∈ ℕ0) ∧ 𝑥 ≤ 𝑦) → 𝑦 ∈ ℕ0)
13		breq2 5147	. . . . . . . . . . . . . . . . 17 ⊢ (𝑏 = 𝑦 → (𝑥 ≤ 𝑏 ↔ 𝑥 ≤ 𝑦))
14		oveq1 7423	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑏 = 𝑦 → (𝑏↑2) = (𝑦↑2))
15	14	oveq2d 7432	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑏 = 𝑦 → ((𝑥↑2) + (𝑏↑2)) = ((𝑥↑2) + (𝑦↑2)))
16	15	eqeq1d 2727	. . . . . . . . . . . . . . . . 17 ⊢ (𝑏 = 𝑦 → (((𝑥↑2) + (𝑏↑2)) = ((𝑥↑2) + (𝑦↑2)) ↔ ((𝑥↑2) + (𝑦↑2)) = ((𝑥↑2) + (𝑦↑2))))
17	13, 16	anbi12d 630	. . . . . . . . . . . . . . . 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 2020	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑏 = 𝑦 → (𝑏 = 𝑐 ↔ 𝑦 = 𝑐))
19	18	imbi2d 339	. . . . . . . . . . . . . . . . 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 3168	. . . . . . . . . . . . . . . 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 630	. . . . . . . . . . . . . . 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 480	. . . . . . . . . . . . . 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 483	. . . . . . . . . . . . . . 15 ⊢ (((𝑥 ∈ ℕ0 ∧ 𝑦 ∈ ℕ0) ∧ 𝑥 ≤ 𝑦) → 𝑥 ≤ 𝑦)
24		eqidd 2726	. . . . . . . . . . . . . . 15 ⊢ (((𝑥 ∈ ℕ0 ∧ 𝑦 ∈ ℕ0) ∧ 𝑥 ≤ 𝑦) → ((𝑥↑2) + (𝑦↑2)) = ((𝑥↑2) + (𝑦↑2)))
25		nn0re 12511	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑐 ∈ ℕ0 → 𝑐 ∈ ℝ)
26	25	resqcld 14121	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑐 ∈ ℕ0 → (𝑐↑2) ∈ ℝ)
27	26	adantl 480	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝑥 ∈ ℕ0 ∧ 𝑦 ∈ ℕ0) ∧ 𝑥 ≤ 𝑦) ∧ 𝑐 ∈ ℕ0) → (𝑐↑2) ∈ ℝ)
28		nn0re 12511	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑦 ∈ ℕ0 → 𝑦 ∈ ℝ)
29	28	resqcld 14121	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑦 ∈ ℕ0 → (𝑦↑2) ∈ ℝ)
30	29	adantl 480	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑥 ∈ ℕ0 ∧ 𝑦 ∈ ℕ0) → (𝑦↑2) ∈ ℝ)
31	30	ad2antrr 724	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝑥 ∈ ℕ0 ∧ 𝑦 ∈ ℕ0) ∧ 𝑥 ≤ 𝑦) ∧ 𝑐 ∈ ℕ0) → (𝑦↑2) ∈ ℝ)
32		nn0re 12511	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑥 ∈ ℕ0 → 𝑥 ∈ ℝ)
33	32	resqcld 14121	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑥 ∈ ℕ0 → (𝑥↑2) ∈ ℝ)
34	33	adantr 479	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑥 ∈ ℕ0 ∧ 𝑦 ∈ ℕ0) → (𝑥↑2) ∈ ℝ)
35	34	ad2antrr 724	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝑥 ∈ ℕ0 ∧ 𝑦 ∈ ℕ0) ∧ 𝑥 ≤ 𝑦) ∧ 𝑐 ∈ ℕ0) → (𝑥↑2) ∈ ℝ)
36		readdcan 11418	. . . . . . . . . . . . . . . . . . 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 1368	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝑥 ∈ ℕ0 ∧ 𝑦 ∈ ℕ0) ∧ 𝑥 ≤ 𝑦) ∧ 𝑐 ∈ ℕ0) → (((𝑥↑2) + (𝑐↑2)) = ((𝑥↑2) + (𝑦↑2)) ↔ (𝑐↑2) = (𝑦↑2)))
38	28	ad4antlr 731	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((((𝑥 ∈ ℕ0 ∧ 𝑦 ∈ ℕ0) ∧ 𝑥 ≤ 𝑦) ∧ 𝑐 ∈ ℕ0) ∧ (𝑐↑2) = (𝑦↑2)) → 𝑦 ∈ ℝ)
39	25	ad2antlr 725	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((((𝑥 ∈ ℕ0 ∧ 𝑦 ∈ ℕ0) ∧ 𝑥 ≤ 𝑦) ∧ 𝑐 ∈ ℕ0) ∧ (𝑐↑2) = (𝑦↑2)) → 𝑐 ∈ ℝ)
40		nn0ge0 12527	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑦 ∈ ℕ0 → 0 ≤ 𝑦)
41	40	ad4antlr 731	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((((𝑥 ∈ ℕ0 ∧ 𝑦 ∈ ℕ0) ∧ 𝑥 ≤ 𝑦) ∧ 𝑐 ∈ ℕ0) ∧ (𝑐↑2) = (𝑦↑2)) → 0 ≤ 𝑦)
42		nn0ge0 12527	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑐 ∈ ℕ0 → 0 ≤ 𝑐)
43	42	ad2antlr 725	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((((𝑥 ∈ ℕ0 ∧ 𝑦 ∈ ℕ0) ∧ 𝑥 ≤ 𝑦) ∧ 𝑐 ∈ ℕ0) ∧ (𝑐↑2) = (𝑦↑2)) → 0 ≤ 𝑐)
44		simpr 483	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((((𝑥 ∈ ℕ0 ∧ 𝑦 ∈ ℕ0) ∧ 𝑥 ≤ 𝑦) ∧ 𝑐 ∈ ℕ0) ∧ (𝑐↑2) = (𝑦↑2)) → (𝑐↑2) = (𝑦↑2))
45	44	eqcomd 2731	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((((𝑥 ∈ ℕ0 ∧ 𝑦 ∈ ℕ0) ∧ 𝑥 ≤ 𝑦) ∧ 𝑐 ∈ ℕ0) ∧ (𝑐↑2) = (𝑦↑2)) → (𝑦↑2) = (𝑐↑2))
46	38, 39, 41, 43, 45	sq11d 14252	. . . . . . . . . . . . . . . . . . 19 ⊢ (((((𝑥 ∈ ℕ0 ∧ 𝑦 ∈ ℕ0) ∧ 𝑥 ≤ 𝑦) ∧ 𝑐 ∈ ℕ0) ∧ (𝑐↑2) = (𝑦↑2)) → 𝑦 = 𝑐)
47	46	ex 411	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝑥 ∈ ℕ0 ∧ 𝑦 ∈ ℕ0) ∧ 𝑥 ≤ 𝑦) ∧ 𝑐 ∈ ℕ0) → ((𝑐↑2) = (𝑦↑2) → 𝑦 = 𝑐))
48	37, 47	sylbid 239	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝑥 ∈ ℕ0 ∧ 𝑦 ∈ ℕ0) ∧ 𝑥 ≤ 𝑦) ∧ 𝑐 ∈ ℕ0) → (((𝑥↑2) + (𝑐↑2)) = ((𝑥↑2) + (𝑦↑2)) → 𝑦 = 𝑐))
49	48	adantld 489	. . . . . . . . . . . . . . . 16 ⊢ ((((𝑥 ∈ ℕ0 ∧ 𝑦 ∈ ℕ0) ∧ 𝑥 ≤ 𝑦) ∧ 𝑐 ∈ ℕ0) → ((𝑥 ≤ 𝑐 ∧ ((𝑥↑2) + (𝑐↑2)) = ((𝑥↑2) + (𝑦↑2))) → 𝑦 = 𝑐))
50	49	ralrimiva 3136	. . . . . . . . . . . . . . 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 513	. . . . . . . . . . . . . 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 3603	. . . . . . . . . . . . 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 5147	. . . . . . . . . . . . . . 15 ⊢ (𝑏 = 𝑐 → (𝑥 ≤ 𝑏 ↔ 𝑥 ≤ 𝑐))
54		oveq1 7423	. . . . . . . . . . . . . . . . 17 ⊢ (𝑏 = 𝑐 → (𝑏↑2) = (𝑐↑2))
55	54	oveq2d 7432	. . . . . . . . . . . . . . . 16 ⊢ (𝑏 = 𝑐 → ((𝑥↑2) + (𝑏↑2)) = ((𝑥↑2) + (𝑐↑2)))
56	55	eqeq1d 2727	. . . . . . . . . . . . . . 15 ⊢ (𝑏 = 𝑐 → (((𝑥↑2) + (𝑏↑2)) = ((𝑥↑2) + (𝑦↑2)) ↔ ((𝑥↑2) + (𝑐↑2)) = ((𝑥↑2) + (𝑦↑2))))
57	53, 56	anbi12d 630	. . . . . . . . . . . . . 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 3720	. . . . . . . . . . . . 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 233	. . . . . . . . . . . 12 ⊢ (((𝑥 ∈ ℕ0 ∧ 𝑦 ∈ ℕ0) ∧ 𝑥 ≤ 𝑦) → ∃!𝑏 ∈ ℕ0 (𝑥 ≤ 𝑏 ∧ ((𝑥↑2) + (𝑏↑2)) = ((𝑥↑2) + (𝑦↑2))))
60	59	ex 411	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ ℕ0 ∧ 𝑦 ∈ ℕ0) → (𝑥 ≤ 𝑦 → ∃!𝑏 ∈ ℕ0 (𝑥 ≤ 𝑏 ∧ ((𝑥↑2) + (𝑏↑2)) = ((𝑥↑2) + (𝑦↑2)))))
61	60	adantr 479	. . . . . . . . . 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 406	. . . . . . . . 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 2737	. . . . . . . . . . . . 13 ⊢ (𝑃 = ((𝑥↑2) + (𝑦↑2)) → (((𝑥↑2) + (𝑏↑2)) = 𝑃 ↔ ((𝑥↑2) + (𝑏↑2)) = ((𝑥↑2) + (𝑦↑2))))
64	63	anbi2d 628	. . . . . . . . . . . 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 3382	. . . . . . . . . . 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 480	. . . . . . . . . 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 480	. . . . . . . . 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 256	. . . . . . . 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 3603	. . . . . . 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 479	. . . . . . . . 9 ⊢ (((𝑥 ∈ ℕ0 ∧ 𝑦 ∈ ℕ0) ∧ 𝑃 = ((𝑥↑2) + (𝑦↑2))) → 𝑦 ∈ ℕ0)
71	70	adantl 480	. . . . . . . 8 ⊢ ((¬ 𝑥 ≤ 𝑦 ∧ ((𝑥 ∈ ℕ0 ∧ 𝑦 ∈ ℕ0) ∧ 𝑃 = ((𝑥↑2) + (𝑦↑2)))) → 𝑦 ∈ ℕ0)
72		breq1 5146	. . . . . . . . . . 11 ⊢ (𝑎 = 𝑦 → (𝑎 ≤ 𝑏 ↔ 𝑦 ≤ 𝑏))
73		oveq1 7423	. . . . . . . . . . . . 13 ⊢ (𝑎 = 𝑦 → (𝑎↑2) = (𝑦↑2))
74	73	oveq1d 7431	. . . . . . . . . . . 12 ⊢ (𝑎 = 𝑦 → ((𝑎↑2) + (𝑏↑2)) = ((𝑦↑2) + (𝑏↑2)))
75	74	eqeq1d 2727	. . . . . . . . . . 11 ⊢ (𝑎 = 𝑦 → (((𝑎↑2) + (𝑏↑2)) = 𝑃 ↔ ((𝑦↑2) + (𝑏↑2)) = 𝑃))
76	72, 75	anbi12d 630	. . . . . . . . . 10 ⊢ (𝑎 = 𝑦 → ((𝑎 ≤ 𝑏 ∧ ((𝑎↑2) + (𝑏↑2)) = 𝑃) ↔ (𝑦 ≤ 𝑏 ∧ ((𝑦↑2) + (𝑏↑2)) = 𝑃)))
77	76	reubidv 3382	. . . . . . . . 9 ⊢ (𝑎 = 𝑦 → (∃!𝑏 ∈ ℕ0 (𝑎 ≤ 𝑏 ∧ ((𝑎↑2) + (𝑏↑2)) = 𝑃) ↔ ∃!𝑏 ∈ ℕ0 (𝑦 ≤ 𝑏 ∧ ((𝑦↑2) + (𝑏↑2)) = 𝑃)))
78	77	adantl 480	. . . . . . . 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 765	. . . . . . . . . . . . . 14 ⊢ (((𝑥 ∈ ℕ0 ∧ 𝑦 ∈ ℕ0) ∧ ¬ 𝑥 ≤ 𝑦) → 𝑥 ∈ ℕ0)
80		breq2 5147	. . . . . . . . . . . . . . . . 17 ⊢ (𝑏 = 𝑥 → (𝑦 ≤ 𝑏 ↔ 𝑦 ≤ 𝑥))
81		oveq1 7423	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑏 = 𝑥 → (𝑏↑2) = (𝑥↑2))
82	81	oveq2d 7432	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑏 = 𝑥 → ((𝑦↑2) + (𝑏↑2)) = ((𝑦↑2) + (𝑥↑2)))
83	82	eqeq1d 2727	. . . . . . . . . . . . . . . . 17 ⊢ (𝑏 = 𝑥 → (((𝑦↑2) + (𝑏↑2)) = ((𝑥↑2) + (𝑦↑2)) ↔ ((𝑦↑2) + (𝑥↑2)) = ((𝑥↑2) + (𝑦↑2))))
84	80, 83	anbi12d 630	. . . . . . . . . . . . . . . 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 2020	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑏 = 𝑥 → (𝑏 = 𝑐 ↔ 𝑥 = 𝑐))
86	85	imbi2d 339	. . . . . . . . . . . . . . . . 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 3168	. . . . . . . . . . . . . . . 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 630	. . . . . . . . . . . . . . 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 480	. . . . . . . . . . . . . 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 11323	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑦 ∈ ℝ ∧ 𝑥 ∈ ℝ) → (𝑦 < 𝑥 ↔ ¬ 𝑥 ≤ 𝑦))
91	28, 32, 90	syl2anr 595	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑥 ∈ ℕ0 ∧ 𝑦 ∈ ℕ0) → (𝑦 < 𝑥 ↔ ¬ 𝑥 ≤ 𝑦))
92	28	ad2antlr 725	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑥 ∈ ℕ0 ∧ 𝑦 ∈ ℕ0) ∧ 𝑦 < 𝑥) → 𝑦 ∈ ℝ)
93	32	ad2antrr 724	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑥 ∈ ℕ0 ∧ 𝑦 ∈ ℕ0) ∧ 𝑦 < 𝑥) → 𝑥 ∈ ℝ)
94		simpr 483	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑥 ∈ ℕ0 ∧ 𝑦 ∈ ℕ0) ∧ 𝑦 < 𝑥) → 𝑦 < 𝑥)
95	92, 93, 94	ltled 11392	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑥 ∈ ℕ0 ∧ 𝑦 ∈ ℕ0) ∧ 𝑦 < 𝑥) → 𝑦 ≤ 𝑥)
96	95	ex 411	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑥 ∈ ℕ0 ∧ 𝑦 ∈ ℕ0) → (𝑦 < 𝑥 → 𝑦 ≤ 𝑥))
97	91, 96	sylbird 259	. . . . . . . . . . . . . . . 16 ⊢ ((𝑥 ∈ ℕ0 ∧ 𝑦 ∈ ℕ0) → (¬ 𝑥 ≤ 𝑦 → 𝑦 ≤ 𝑥))
98	97	imp 405	. . . . . . . . . . . . . . 15 ⊢ (((𝑥 ∈ ℕ0 ∧ 𝑦 ∈ ℕ0) ∧ ¬ 𝑥 ≤ 𝑦) → 𝑦 ≤ 𝑥)
99	29	recnd 11272	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑦 ∈ ℕ0 → (𝑦↑2) ∈ ℂ)
100	99	adantl 480	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑥 ∈ ℕ0 ∧ 𝑦 ∈ ℕ0) → (𝑦↑2) ∈ ℂ)
101	33	recnd 11272	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 ∈ ℕ0 → (𝑥↑2) ∈ ℂ)
102	101	adantr 479	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑥 ∈ ℕ0 ∧ 𝑦 ∈ ℕ0) → (𝑥↑2) ∈ ℂ)
103	100, 102	addcomd 11446	. . . . . . . . . . . . . . . 16 ⊢ ((𝑥 ∈ ℕ0 ∧ 𝑦 ∈ ℕ0) → ((𝑦↑2) + (𝑥↑2)) = ((𝑥↑2) + (𝑦↑2)))
104	103	adantr 479	. . . . . . . . . . . . . . 15 ⊢ (((𝑥 ∈ ℕ0 ∧ 𝑦 ∈ ℕ0) ∧ ¬ 𝑥 ≤ 𝑦) → ((𝑦↑2) + (𝑥↑2)) = ((𝑥↑2) + (𝑦↑2)))
105	34	recnd 11272	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑥 ∈ ℕ0 ∧ 𝑦 ∈ ℕ0) → (𝑥↑2) ∈ ℂ)
106	105	adantr 479	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝑥 ∈ ℕ0 ∧ 𝑦 ∈ ℕ0) ∧ 𝑐 ∈ ℕ0) → (𝑥↑2) ∈ ℂ)
107	30	recnd 11272	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑥 ∈ ℕ0 ∧ 𝑦 ∈ ℕ0) → (𝑦↑2) ∈ ℂ)
108	107	adantr 479	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝑥 ∈ ℕ0 ∧ 𝑦 ∈ ℕ0) ∧ 𝑐 ∈ ℕ0) → (𝑦↑2) ∈ ℂ)
109	106, 108	addcomd 11446	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝑥 ∈ ℕ0 ∧ 𝑦 ∈ ℕ0) ∧ 𝑐 ∈ ℕ0) → ((𝑥↑2) + (𝑦↑2)) = ((𝑦↑2) + (𝑥↑2)))
110	109	eqeq2d 2736	. . . . . . . . . . . . . . . . . . . 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 480	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝑥 ∈ ℕ0 ∧ 𝑦 ∈ ℕ0) ∧ 𝑐 ∈ ℕ0) → (𝑐↑2) ∈ ℝ)
112	33	ad2antrr 724	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝑥 ∈ ℕ0 ∧ 𝑦 ∈ ℕ0) ∧ 𝑐 ∈ ℕ0) → (𝑥↑2) ∈ ℝ)
113	29	ad2antlr 725	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝑥 ∈ ℕ0 ∧ 𝑦 ∈ ℕ0) ∧ 𝑐 ∈ ℕ0) → (𝑦↑2) ∈ ℝ)
114		readdcan 11418	. . . . . . . . . . . . . . . . . . . . 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 1368	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝑥 ∈ ℕ0 ∧ 𝑦 ∈ ℕ0) ∧ 𝑐 ∈ ℕ0) → (((𝑦↑2) + (𝑐↑2)) = ((𝑦↑2) + (𝑥↑2)) ↔ (𝑐↑2) = (𝑥↑2)))
116	110, 115	bitrd 278	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑥 ∈ ℕ0 ∧ 𝑦 ∈ ℕ0) ∧ 𝑐 ∈ ℕ0) → (((𝑦↑2) + (𝑐↑2)) = ((𝑥↑2) + (𝑦↑2)) ↔ (𝑐↑2) = (𝑥↑2)))
117	25	ad2antlr 725	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((𝑥 ∈ ℕ0 ∧ 𝑦 ∈ ℕ0) ∧ 𝑐 ∈ ℕ0) ∧ (𝑐↑2) = (𝑥↑2)) → 𝑐 ∈ ℝ)
118	32	adantr 479	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑥 ∈ ℕ0 ∧ 𝑦 ∈ ℕ0) → 𝑥 ∈ ℝ)
119	118	ad2antrr 724	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((𝑥 ∈ ℕ0 ∧ 𝑦 ∈ ℕ0) ∧ 𝑐 ∈ ℕ0) ∧ (𝑐↑2) = (𝑥↑2)) → 𝑥 ∈ ℝ)
120	42	ad2antlr 725	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((𝑥 ∈ ℕ0 ∧ 𝑦 ∈ ℕ0) ∧ 𝑐 ∈ ℕ0) ∧ (𝑐↑2) = (𝑥↑2)) → 0 ≤ 𝑐)
121		nn0ge0 12527	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑥 ∈ ℕ0 → 0 ≤ 𝑥)
122	121	adantr 479	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑥 ∈ ℕ0 ∧ 𝑦 ∈ ℕ0) → 0 ≤ 𝑥)
123	122	ad2antrr 724	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((𝑥 ∈ ℕ0 ∧ 𝑦 ∈ ℕ0) ∧ 𝑐 ∈ ℕ0) ∧ (𝑐↑2) = (𝑥↑2)) → 0 ≤ 𝑥)
124		simpr 483	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((𝑥 ∈ ℕ0 ∧ 𝑦 ∈ ℕ0) ∧ 𝑐 ∈ ℕ0) ∧ (𝑐↑2) = (𝑥↑2)) → (𝑐↑2) = (𝑥↑2))
125	117, 119, 120, 123, 124	sq11d 14252	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((𝑥 ∈ ℕ0 ∧ 𝑦 ∈ ℕ0) ∧ 𝑐 ∈ ℕ0) ∧ (𝑐↑2) = (𝑥↑2)) → 𝑐 = 𝑥)
126	125	eqcomd 2731	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝑥 ∈ ℕ0 ∧ 𝑦 ∈ ℕ0) ∧ 𝑐 ∈ ℕ0) ∧ (𝑐↑2) = (𝑥↑2)) → 𝑥 = 𝑐)
127	126	ex 411	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑥 ∈ ℕ0 ∧ 𝑦 ∈ ℕ0) ∧ 𝑐 ∈ ℕ0) → ((𝑐↑2) = (𝑥↑2) → 𝑥 = 𝑐))
128	116, 127	sylbid 239	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑥 ∈ ℕ0 ∧ 𝑦 ∈ ℕ0) ∧ 𝑐 ∈ ℕ0) → (((𝑦↑2) + (𝑐↑2)) = ((𝑥↑2) + (𝑦↑2)) → 𝑥 = 𝑐))
129	128	adantld 489	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑥 ∈ ℕ0 ∧ 𝑦 ∈ ℕ0) ∧ 𝑐 ∈ ℕ0) → ((𝑦 ≤ 𝑐 ∧ ((𝑦↑2) + (𝑐↑2)) = ((𝑥↑2) + (𝑦↑2))) → 𝑥 = 𝑐))
130	129	ralrimiva 3136	. . . . . . . . . . . . . . . 16 ⊢ ((𝑥 ∈ ℕ0 ∧ 𝑦 ∈ ℕ0) → ∀𝑐 ∈ ℕ0 ((𝑦 ≤ 𝑐 ∧ ((𝑦↑2) + (𝑐↑2)) = ((𝑥↑2) + (𝑦↑2))) → 𝑥 = 𝑐))
131	130	adantr 479	. . . . . . . . . . . . . . 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 513	. . . . . . . . . . . . . 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 3603	. . . . . . . . . . . . 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 5147	. . . . . . . . . . . . . . 15 ⊢ (𝑏 = 𝑐 → (𝑦 ≤ 𝑏 ↔ 𝑦 ≤ 𝑐))
135	54	oveq2d 7432	. . . . . . . . . . . . . . . 16 ⊢ (𝑏 = 𝑐 → ((𝑦↑2) + (𝑏↑2)) = ((𝑦↑2) + (𝑐↑2)))
136	135	eqeq1d 2727	. . . . . . . . . . . . . . 15 ⊢ (𝑏 = 𝑐 → (((𝑦↑2) + (𝑏↑2)) = ((𝑥↑2) + (𝑦↑2)) ↔ ((𝑦↑2) + (𝑐↑2)) = ((𝑥↑2) + (𝑦↑2))))
137	134, 136	anbi12d 630	. . . . . . . . . . . . . 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 3720	. . . . . . . . . . . . 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 233	. . . . . . . . . . . 12 ⊢ (((𝑥 ∈ ℕ0 ∧ 𝑦 ∈ ℕ0) ∧ ¬ 𝑥 ≤ 𝑦) → ∃!𝑏 ∈ ℕ0 (𝑦 ≤ 𝑏 ∧ ((𝑦↑2) + (𝑏↑2)) = ((𝑥↑2) + (𝑦↑2))))
140	139	ex 411	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ ℕ0 ∧ 𝑦 ∈ ℕ0) → (¬ 𝑥 ≤ 𝑦 → ∃!𝑏 ∈ ℕ0 (𝑦 ≤ 𝑏 ∧ ((𝑦↑2) + (𝑏↑2)) = ((𝑥↑2) + (𝑦↑2)))))
141	140	adantr 479	. . . . . . . . . 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 406	. . . . . . . . 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 2737	. . . . . . . . . . . . 13 ⊢ (𝑃 = ((𝑥↑2) + (𝑦↑2)) → (((𝑦↑2) + (𝑏↑2)) = 𝑃 ↔ ((𝑦↑2) + (𝑏↑2)) = ((𝑥↑2) + (𝑦↑2))))
144	143	anbi2d 628	. . . . . . . . . . . 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 3382	. . . . . . . . . . 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 480	. . . . . . . . . 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 480	. . . . . . . . 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 256	. . . . . . . 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 3603	. . . . . . 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 810	. . . . . 6 ⊢ (((𝑥 ∈ ℕ0 ∧ 𝑦 ∈ ℕ0) ∧ 𝑃 = ((𝑥↑2) + (𝑦↑2))) → ∃𝑎 ∈ ℕ0 ∃!𝑏 ∈ ℕ0 (𝑎 ≤ 𝑏 ∧ ((𝑎↑2) + (𝑏↑2)) = 𝑃))
151	150	ex 411	. . . . 5 ⊢ ((𝑥 ∈ ℕ0 ∧ 𝑦 ∈ ℕ0) → (𝑃 = ((𝑥↑2) + (𝑦↑2)) → ∃𝑎 ∈ ℕ0 ∃!𝑏 ∈ ℕ0 (𝑎 ≤ 𝑏 ∧ ((𝑎↑2) + (𝑏↑2)) = 𝑃)))
152	151	adantl 480	. . . 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 3202	. . 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 3368	. . . . 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 3136	. . 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 27388	. . . 4 ⊢ (𝑃 ∈ ℙ → ∃*𝑎 ∈ ℕ0 ∃𝑏 ∈ ℕ0 (𝑎 ≤ 𝑏 ∧ ((𝑎↑2) + (𝑏↑2)) = 𝑃))
159	158	adantr 479	. . 3 ⊢ ((𝑃 ∈ ℙ ∧ (𝑃 mod 4) = 1) → ∃*𝑎 ∈ ℕ0 ∃𝑏 ∈ ℕ0 (𝑎 ≤ 𝑏 ∧ ((𝑎↑2) + (𝑏↑2)) = 𝑃))
160		rmoim 3727	. . 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 3366	. 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 581	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 205   ∧ wa 394   = wceq 1533   ∈ wcel 2098  ∀wral 3051  ∃wrex 3060  ∃!wreu 3362  ∃*wrmo 3363   class class class wbr 5143  (class class class)co 7416  ℂcc 11136  ℝcr 11137  0cc0 11138  1c1 11139   + caddc 11141   < clt 11278   ≤ cle 11279  2c2 12297  4c4 12299  ℕ₀cn0 12502   mod cmo 13866  ↑cexp 14058  ℙcprime 16641

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-rep 5280  ax-sep 5294  ax-nul 5301  ax-pow 5359  ax-pr 5423  ax-un 7738  ax-cnex 11194  ax-resscn 11195  ax-1cn 11196  ax-icn 11197  ax-addcl 11198  ax-addrcl 11199  ax-mulcl 11200  ax-mulrcl 11201  ax-mulcom 11202  ax-addass 11203  ax-mulass 11204  ax-distr 11205  ax-i2m1 11206  ax-1ne0 11207  ax-1rid 11208  ax-rnegex 11209  ax-rrecex 11210  ax-cnre 11211  ax-pre-lttri 11212  ax-pre-lttrn 11213  ax-pre-ltadd 11214  ax-pre-mulgt0 11215  ax-pre-sup 11216  ax-addf 11217  ax-mulf 11218

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2931  df-nel 3037  df-ral 3052  df-rex 3061  df-rmo 3364  df-reu 3365  df-rab 3420  df-v 3465  df-sbc 3769  df-csb 3885  df-dif 3942  df-un 3944  df-in 3946  df-ss 3956  df-pss 3959  df-nul 4319  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-tp 4629  df-op 4631  df-uni 4904  df-int 4945  df-iun 4993  df-iin 4994  df-br 5144  df-opab 5206  df-mpt 5227  df-tr 5261  df-id 5570  df-eprel 5576  df-po 5584  df-so 5585  df-fr 5627  df-se 5628  df-we 5629  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-pred 6300  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-isom 6552  df-riota 7372  df-ov 7419  df-oprab 7420  df-mpo 7421  df-of 7682  df-ofr 7683  df-om 7869  df-1st 7991  df-2nd 7992  df-supp 8164  df-tpos 8230  df-frecs 8285  df-wrecs 8316  df-recs 8390  df-rdg 8429  df-1o 8485  df-2o 8486  df-oadd 8489  df-er 8723  df-ec 8725  df-qs 8729  df-map 8845  df-pm 8846  df-ixp 8915  df-en 8963  df-dom 8964  df-sdom 8965  df-fin 8966  df-fsupp 9386  df-sup 9465  df-inf 9466  df-oi 9533  df-dju 9924  df-card 9962  df-pnf 11280  df-mnf 11281  df-xr 11282  df-ltxr 11283  df-le 11284  df-sub 11476  df-neg 11477  df-div 11902  df-nn 12243  df-2 12305  df-3 12306  df-4 12307  df-5 12308  df-6 12309  df-7 12310  df-8 12311  df-9 12312  df-n0 12503  df-xnn0 12575  df-z 12589  df-dec 12708  df-uz 12853  df-q 12963  df-rp 13007  df-fz 13517  df-fzo 13660  df-fl 13789  df-mod 13867  df-seq 13999  df-exp 14059  df-hash 14322  df-cj 15078  df-re 15079  df-im 15080  df-sqrt 15214  df-abs 15215  df-dvds 16231  df-gcd 16469  df-prm 16642  df-phi 16734  df-pc 16805  df-gz 16898  df-struct 17115  df-sets 17132  df-slot 17150  df-ndx 17162  df-base 17180  df-ress 17209  df-plusg 17245  df-mulr 17246  df-starv 17247  df-sca 17248  df-vsca 17249  df-ip 17250  df-tset 17251  df-ple 17252  df-ds 17254  df-unif 17255  df-hom 17256  df-cco 17257  df-0g 17422  df-gsum 17423  df-prds 17428  df-pws 17430  df-imas 17489  df-qus 17490  df-mre 17565  df-mrc 17566  df-acs 17568  df-mgm 18599  df-sgrp 18678  df-mnd 18694  df-mhm 18739  df-submnd 18740  df-grp 18897  df-minusg 18898  df-sbg 18899  df-mulg 19028  df-subg 19082  df-nsg 19083  df-eqg 19084  df-ghm 19172  df-cntz 19272  df-cmn 19741  df-abl 19742  df-mgp 20079  df-rng 20097  df-ur 20126  df-srg 20131  df-ring 20179  df-cring 20180  df-oppr 20277  df-dvdsr 20300  df-unit 20301  df-invr 20331  df-dvr 20344  df-rhm 20415  df-nzr 20456  df-subrng 20487  df-subrg 20512  df-drng 20630  df-field 20631  df-lmod 20749  df-lss 20820  df-lsp 20860  df-sra 21062  df-rgmod 21063  df-lidl 21108  df-rsp 21109  df-2idl 21148  df-rlreg 21234  df-domn 21235  df-idom 21236  df-cnfld 21284  df-zring 21377  df-zrh 21433  df-zn 21436  df-assa 21791  df-asp 21792  df-ascl 21793  df-psr 21846  df-mvr 21847  df-mpl 21848  df-opsr 21850  df-evls 22025  df-evl 22026  df-psr1 22107  df-vr1 22108  df-ply1 22109  df-coe1 22110  df-evl1 22244  df-mdeg 26006  df-deg1 26007  df-mon1 26084  df-uc1p 26085  df-q1p 26086  df-r1p 26087  df-lgs 27246

This theorem is referenced by:  2sqreultlem  27398  2sqreu  27407