Theorem 2sqreulem1 27773

Description: Lemma 1 for 2sqreu 27783. (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 27765	. . 3 ⊢ ((𝑃 ∈ ℙ ∧ (𝑃 mod 4) = 1) → ∃𝑥 ∈ ℕ0 ∃𝑦 ∈ ℕ0 𝑃 = ((𝑥↑2) + (𝑦↑2)))
2		simpll 779	. . . . . . . . 9 ⊢ (((𝑥 ∈ ℕ0 ∧ 𝑦 ∈ ℕ0) ∧ 𝑃 = ((𝑥↑2) + (𝑦↑2))) → 𝑥 ∈ ℕ0)
3	2	adantl 487	. . . . . . . 8 ⊢ ((𝑥 ≤ 𝑦 ∧ ((𝑥 ∈ ℕ0 ∧ 𝑦 ∈ ℕ0) ∧ 𝑃 = ((𝑥↑2) + (𝑦↑2)))) → 𝑥 ∈ ℕ0)
4		breq1 5105	. . . . . . . . . . 11 ⊢ (𝑎 = 𝑥 → (𝑎 ≤ 𝑏 ↔ 𝑥 ≤ 𝑏))
5		oveq1 7427	. . . . . . . . . . . . 13 ⊢ (𝑎 = 𝑥 → (𝑎↑2) = (𝑥↑2))
6	5	oveq1d 7435	. . . . . . . . . . . 12 ⊢ (𝑎 = 𝑥 → ((𝑎↑2) + (𝑏↑2)) = ((𝑥↑2) + (𝑏↑2)))
7	6	eqeq1d 2762	. . . . . . . . . . 11 ⊢ (𝑎 = 𝑥 → (((𝑎↑2) + (𝑏↑2)) = 𝑃 ↔ ((𝑥↑2) + (𝑏↑2)) = 𝑃))
8	4, 7	anbi12d 644	. . . . . . . . . 10 ⊢ (𝑎 = 𝑥 → ((𝑎 ≤ 𝑏 ∧ ((𝑎↑2) + (𝑏↑2)) = 𝑃) ↔ (𝑥 ≤ 𝑏 ∧ ((𝑥↑2) + (𝑏↑2)) = 𝑃)))
9	8	reubidv 3381	. . . . . . . . 9 ⊢ (𝑎 = 𝑥 → (∃!𝑏 ∈ ℕ0 (𝑎 ≤ 𝑏 ∧ ((𝑎↑2) + (𝑏↑2)) = 𝑃) ↔ ∃!𝑏 ∈ ℕ0 (𝑥 ≤ 𝑏 ∧ ((𝑥↑2) + (𝑏↑2)) = 𝑃)))
10	9	adantl 487	. . . . . . . 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 490	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 ∈ ℕ0 ∧ 𝑦 ∈ ℕ0) → 𝑦 ∈ ℕ0)
12	11	adantr 486	. . . . . . . . . . . . . 14 ⊢ (((𝑥 ∈ ℕ0 ∧ 𝑦 ∈ ℕ0) ∧ 𝑥 ≤ 𝑦) → 𝑦 ∈ ℕ0)
13		breq2 5106	. . . . . . . . . . . . . . . . 17 ⊢ (𝑏 = 𝑦 → (𝑥 ≤ 𝑏 ↔ 𝑥 ≤ 𝑦))
14		oveq1 7427	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑏 = 𝑦 → (𝑏↑2) = (𝑦↑2))
15	14	oveq2d 7436	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑏 = 𝑦 → ((𝑥↑2) + (𝑏↑2)) = ((𝑥↑2) + (𝑦↑2)))
16	15	eqeq1d 2762	. . . . . . . . . . . . . . . . 17 ⊢ (𝑏 = 𝑦 → (((𝑥↑2) + (𝑏↑2)) = ((𝑥↑2) + (𝑦↑2)) ↔ ((𝑥↑2) + (𝑦↑2)) = ((𝑥↑2) + (𝑦↑2))))
17	13, 16	anbi12d 644	. . . . . . . . . . . . . . . 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 2057	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑏 = 𝑦 → (𝑏 = 𝑐 ↔ 𝑦 = 𝑐))
19	18	imbi2d 343	. . . . . . . . . . . . . . . . 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 3185	. . . . . . . . . . . . . . . 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 644	. . . . . . . . . . . . . . 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 487	. . . . . . . . . . . . . 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 490	. . . . . . . . . . . . . . 15 ⊢ (((𝑥 ∈ ℕ0 ∧ 𝑦 ∈ ℕ0) ∧ 𝑥 ≤ 𝑦) → 𝑥 ≤ 𝑦)
24		eqidd 2761	. . . . . . . . . . . . . . 15 ⊢ (((𝑥 ∈ ℕ0 ∧ 𝑦 ∈ ℕ0) ∧ 𝑥 ≤ 𝑦) → ((𝑥↑2) + (𝑦↑2)) = ((𝑥↑2) + (𝑦↑2)))
25		nn0re 12615	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑐 ∈ ℕ0 → 𝑐 ∈ ℝ)
26	25	resqcld 14268	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑐 ∈ ℕ0 → (𝑐↑2) ∈ ℝ)
27	26	adantl 487	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝑥 ∈ ℕ0 ∧ 𝑦 ∈ ℕ0) ∧ 𝑥 ≤ 𝑦) ∧ 𝑐 ∈ ℕ0) → (𝑐↑2) ∈ ℝ)
28		nn0re 12615	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑦 ∈ ℕ0 → 𝑦 ∈ ℝ)
29	28	resqcld 14268	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑦 ∈ ℕ0 → (𝑦↑2) ∈ ℝ)
30	29	adantl 487	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑥 ∈ ℕ0 ∧ 𝑦 ∈ ℕ0) → (𝑦↑2) ∈ ℝ)
31	30	ad2antrr 739	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝑥 ∈ ℕ0 ∧ 𝑦 ∈ ℕ0) ∧ 𝑥 ≤ 𝑦) ∧ 𝑐 ∈ ℕ0) → (𝑦↑2) ∈ ℝ)
32		nn0re 12615	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑥 ∈ ℕ0 → 𝑥 ∈ ℝ)
33	32	resqcld 14268	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑥 ∈ ℕ0 → (𝑥↑2) ∈ ℝ)
34	33	adantr 486	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑥 ∈ ℕ0 ∧ 𝑦 ∈ ℕ0) → (𝑥↑2) ∈ ℝ)
35	34	ad2antrr 739	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝑥 ∈ ℕ0 ∧ 𝑦 ∈ ℕ0) ∧ 𝑥 ≤ 𝑦) ∧ 𝑐 ∈ ℕ0) → (𝑥↑2) ∈ ℝ)
36		readdcan 11484	. . . . . . . . . . . . . . . . . . 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 1398	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝑥 ∈ ℕ0 ∧ 𝑦 ∈ ℕ0) ∧ 𝑥 ≤ 𝑦) ∧ 𝑐 ∈ ℕ0) → (((𝑥↑2) + (𝑐↑2)) = ((𝑥↑2) + (𝑦↑2)) ↔ (𝑐↑2) = (𝑦↑2)))
38	28	ad4antlr 746	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((((𝑥 ∈ ℕ0 ∧ 𝑦 ∈ ℕ0) ∧ 𝑥 ≤ 𝑦) ∧ 𝑐 ∈ ℕ0) ∧ (𝑐↑2) = (𝑦↑2)) → 𝑦 ∈ ℝ)
39	25	ad2antlr 740	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((((𝑥 ∈ ℕ0 ∧ 𝑦 ∈ ℕ0) ∧ 𝑥 ≤ 𝑦) ∧ 𝑐 ∈ ℕ0) ∧ (𝑐↑2) = (𝑦↑2)) → 𝑐 ∈ ℝ)
40		nn0ge0 12631	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑦 ∈ ℕ0 → 0 ≤ 𝑦)
41	40	ad4antlr 746	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((((𝑥 ∈ ℕ0 ∧ 𝑦 ∈ ℕ0) ∧ 𝑥 ≤ 𝑦) ∧ 𝑐 ∈ ℕ0) ∧ (𝑐↑2) = (𝑦↑2)) → 0 ≤ 𝑦)
42		nn0ge0 12631	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑐 ∈ ℕ0 → 0 ≤ 𝑐)
43	42	ad2antlr 740	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((((𝑥 ∈ ℕ0 ∧ 𝑦 ∈ ℕ0) ∧ 𝑥 ≤ 𝑦) ∧ 𝑐 ∈ ℕ0) ∧ (𝑐↑2) = (𝑦↑2)) → 0 ≤ 𝑐)
44		simpr 490	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((((𝑥 ∈ ℕ0 ∧ 𝑦 ∈ ℕ0) ∧ 𝑥 ≤ 𝑦) ∧ 𝑐 ∈ ℕ0) ∧ (𝑐↑2) = (𝑦↑2)) → (𝑐↑2) = (𝑦↑2))
45	44	eqcomd 2766	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((((𝑥 ∈ ℕ0 ∧ 𝑦 ∈ ℕ0) ∧ 𝑥 ≤ 𝑦) ∧ 𝑐 ∈ ℕ0) ∧ (𝑐↑2) = (𝑦↑2)) → (𝑦↑2) = (𝑐↑2))
46	38, 39, 41, 43, 45	sq11d 14402	. . . . . . . . . . . . . . . . . . 19 ⊢ (((((𝑥 ∈ ℕ0 ∧ 𝑦 ∈ ℕ0) ∧ 𝑥 ≤ 𝑦) ∧ 𝑐 ∈ ℕ0) ∧ (𝑐↑2) = (𝑦↑2)) → 𝑦 = 𝑐)
47	46	ex 418	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝑥 ∈ ℕ0 ∧ 𝑦 ∈ ℕ0) ∧ 𝑥 ≤ 𝑦) ∧ 𝑐 ∈ ℕ0) → ((𝑐↑2) = (𝑦↑2) → 𝑦 = 𝑐))
48	37, 47	sylbid 243	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝑥 ∈ ℕ0 ∧ 𝑦 ∈ ℕ0) ∧ 𝑥 ≤ 𝑦) ∧ 𝑐 ∈ ℕ0) → (((𝑥↑2) + (𝑐↑2)) = ((𝑥↑2) + (𝑦↑2)) → 𝑦 = 𝑐))
49	48	adantld 496	. . . . . . . . . . . . . . . 16 ⊢ ((((𝑥 ∈ ℕ0 ∧ 𝑦 ∈ ℕ0) ∧ 𝑥 ≤ 𝑦) ∧ 𝑐 ∈ ℕ0) → ((𝑥 ≤ 𝑐 ∧ ((𝑥↑2) + (𝑐↑2)) = ((𝑥↑2) + (𝑦↑2))) → 𝑦 = 𝑐))
50	49	ralrimiva 3154	. . . . . . . . . . . . . . 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 524	. . . . . . . . . . . . . 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 3578	. . . . . . . . . . . . 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 5106	. . . . . . . . . . . . . . 15 ⊢ (𝑏 = 𝑐 → (𝑥 ≤ 𝑏 ↔ 𝑥 ≤ 𝑐))
54		oveq1 7427	. . . . . . . . . . . . . . . . 17 ⊢ (𝑏 = 𝑐 → (𝑏↑2) = (𝑐↑2))
55	54	oveq2d 7436	. . . . . . . . . . . . . . . 16 ⊢ (𝑏 = 𝑐 → ((𝑥↑2) + (𝑏↑2)) = ((𝑥↑2) + (𝑐↑2)))
56	55	eqeq1d 2762	. . . . . . . . . . . . . . 15 ⊢ (𝑏 = 𝑐 → (((𝑥↑2) + (𝑏↑2)) = ((𝑥↑2) + (𝑦↑2)) ↔ ((𝑥↑2) + (𝑐↑2)) = ((𝑥↑2) + (𝑦↑2))))
57	53, 56	anbi12d 644	. . . . . . . . . . . . . 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 237	. . . . . . . . . . . 12 ⊢ (((𝑥 ∈ ℕ0 ∧ 𝑦 ∈ ℕ0) ∧ 𝑥 ≤ 𝑦) → ∃!𝑏 ∈ ℕ0 (𝑥 ≤ 𝑏 ∧ ((𝑥↑2) + (𝑏↑2)) = ((𝑥↑2) + (𝑦↑2))))
60	59	ex 418	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ ℕ0 ∧ 𝑦 ∈ ℕ0) → (𝑥 ≤ 𝑦 → ∃!𝑏 ∈ ℕ0 (𝑥 ≤ 𝑏 ∧ ((𝑥↑2) + (𝑏↑2)) = ((𝑥↑2) + (𝑦↑2)))))
61	60	adantr 486	. . . . . . . . . 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 413	. . . . . . . . 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 2772	. . . . . . . . . . . . 13 ⊢ (𝑃 = ((𝑥↑2) + (𝑦↑2)) → (((𝑥↑2) + (𝑏↑2)) = 𝑃 ↔ ((𝑥↑2) + (𝑏↑2)) = ((𝑥↑2) + (𝑦↑2))))
64	63	anbi2d 642	. . . . . . . . . . . 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 3381	. . . . . . . . . . 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 487	. . . . . . . . . 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 487	. . . . . . . . 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 260	. . . . . . . 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 3578	. . . . . . 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 486	. . . . . . . . 9 ⊢ (((𝑥 ∈ ℕ0 ∧ 𝑦 ∈ ℕ0) ∧ 𝑃 = ((𝑥↑2) + (𝑦↑2))) → 𝑦 ∈ ℕ0)
71	70	adantl 487	. . . . . . . 8 ⊢ ((¬ 𝑥 ≤ 𝑦 ∧ ((𝑥 ∈ ℕ0 ∧ 𝑦 ∈ ℕ0) ∧ 𝑃 = ((𝑥↑2) + (𝑦↑2)))) → 𝑦 ∈ ℕ0)
72		breq1 5105	. . . . . . . . . . 11 ⊢ (𝑎 = 𝑦 → (𝑎 ≤ 𝑏 ↔ 𝑦 ≤ 𝑏))
73		oveq1 7427	. . . . . . . . . . . . 13 ⊢ (𝑎 = 𝑦 → (𝑎↑2) = (𝑦↑2))
74	73	oveq1d 7435	. . . . . . . . . . . 12 ⊢ (𝑎 = 𝑦 → ((𝑎↑2) + (𝑏↑2)) = ((𝑦↑2) + (𝑏↑2)))
75	74	eqeq1d 2762	. . . . . . . . . . 11 ⊢ (𝑎 = 𝑦 → (((𝑎↑2) + (𝑏↑2)) = 𝑃 ↔ ((𝑦↑2) + (𝑏↑2)) = 𝑃))
76	72, 75	anbi12d 644	. . . . . . . . . 10 ⊢ (𝑎 = 𝑦 → ((𝑎 ≤ 𝑏 ∧ ((𝑎↑2) + (𝑏↑2)) = 𝑃) ↔ (𝑦 ≤ 𝑏 ∧ ((𝑦↑2) + (𝑏↑2)) = 𝑃)))
77	76	reubidv 3381	. . . . . . . . 9 ⊢ (𝑎 = 𝑦 → (∃!𝑏 ∈ ℕ0 (𝑎 ≤ 𝑏 ∧ ((𝑎↑2) + (𝑏↑2)) = 𝑃) ↔ ∃!𝑏 ∈ ℕ0 (𝑦 ≤ 𝑏 ∧ ((𝑦↑2) + (𝑏↑2)) = 𝑃)))
78	77	adantl 487	. . . . . . . 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 779	. . . . . . . . . . . . . 14 ⊢ (((𝑥 ∈ ℕ0 ∧ 𝑦 ∈ ℕ0) ∧ ¬ 𝑥 ≤ 𝑦) → 𝑥 ∈ ℕ0)
80		breq2 5106	. . . . . . . . . . . . . . . . 17 ⊢ (𝑏 = 𝑥 → (𝑦 ≤ 𝑏 ↔ 𝑦 ≤ 𝑥))
81		oveq1 7427	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑏 = 𝑥 → (𝑏↑2) = (𝑥↑2))
82	81	oveq2d 7436	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑏 = 𝑥 → ((𝑦↑2) + (𝑏↑2)) = ((𝑦↑2) + (𝑥↑2)))
83	82	eqeq1d 2762	. . . . . . . . . . . . . . . . 17 ⊢ (𝑏 = 𝑥 → (((𝑦↑2) + (𝑏↑2)) = ((𝑥↑2) + (𝑦↑2)) ↔ ((𝑦↑2) + (𝑥↑2)) = ((𝑥↑2) + (𝑦↑2))))
84	80, 83	anbi12d 644	. . . . . . . . . . . . . . . 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 2057	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑏 = 𝑥 → (𝑏 = 𝑐 ↔ 𝑥 = 𝑐))
86	85	imbi2d 343	. . . . . . . . . . . . . . . . 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 3185	. . . . . . . . . . . . . . . 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 644	. . . . . . . . . . . . . . 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 487	. . . . . . . . . . . . . 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 11389	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑦 ∈ ℝ ∧ 𝑥 ∈ ℝ) → (𝑦 < 𝑥 ↔ ¬ 𝑥 ≤ 𝑦))
91	28, 32, 90	syl2anr 609	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑥 ∈ ℕ0 ∧ 𝑦 ∈ ℕ0) → (𝑦 < 𝑥 ↔ ¬ 𝑥 ≤ 𝑦))
92	28	ad2antlr 740	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑥 ∈ ℕ0 ∧ 𝑦 ∈ ℕ0) ∧ 𝑦 < 𝑥) → 𝑦 ∈ ℝ)
93	32	ad2antrr 739	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑥 ∈ ℕ0 ∧ 𝑦 ∈ ℕ0) ∧ 𝑦 < 𝑥) → 𝑥 ∈ ℝ)
94		simpr 490	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑥 ∈ ℕ0 ∧ 𝑦 ∈ ℕ0) ∧ 𝑦 < 𝑥) → 𝑦 < 𝑥)
95	92, 93, 94	ltled 11458	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑥 ∈ ℕ0 ∧ 𝑦 ∈ ℕ0) ∧ 𝑦 < 𝑥) → 𝑦 ≤ 𝑥)
96	95	ex 418	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑥 ∈ ℕ0 ∧ 𝑦 ∈ ℕ0) → (𝑦 < 𝑥 → 𝑦 ≤ 𝑥))
97	91, 96	sylbird 263	. . . . . . . . . . . . . . . 16 ⊢ ((𝑥 ∈ ℕ0 ∧ 𝑦 ∈ ℕ0) → (¬ 𝑥 ≤ 𝑦 → 𝑦 ≤ 𝑥))
98	97	imp 412	. . . . . . . . . . . . . . 15 ⊢ (((𝑥 ∈ ℕ0 ∧ 𝑦 ∈ ℕ0) ∧ ¬ 𝑥 ≤ 𝑦) → 𝑦 ≤ 𝑥)
99	29	recnd 11337	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑦 ∈ ℕ0 → (𝑦↑2) ∈ ℂ)
100	99	adantl 487	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑥 ∈ ℕ0 ∧ 𝑦 ∈ ℕ0) → (𝑦↑2) ∈ ℂ)
101	33	recnd 11337	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 ∈ ℕ0 → (𝑥↑2) ∈ ℂ)
102	101	adantr 486	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑥 ∈ ℕ0 ∧ 𝑦 ∈ ℕ0) → (𝑥↑2) ∈ ℂ)
103	100, 102	addcomd 11512	. . . . . . . . . . . . . . . 16 ⊢ ((𝑥 ∈ ℕ0 ∧ 𝑦 ∈ ℕ0) → ((𝑦↑2) + (𝑥↑2)) = ((𝑥↑2) + (𝑦↑2)))
104	103	adantr 486	. . . . . . . . . . . . . . 15 ⊢ (((𝑥 ∈ ℕ0 ∧ 𝑦 ∈ ℕ0) ∧ ¬ 𝑥 ≤ 𝑦) → ((𝑦↑2) + (𝑥↑2)) = ((𝑥↑2) + (𝑦↑2)))
105	34	recnd 11337	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑥 ∈ ℕ0 ∧ 𝑦 ∈ ℕ0) → (𝑥↑2) ∈ ℂ)
106	105	adantr 486	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝑥 ∈ ℕ0 ∧ 𝑦 ∈ ℕ0) ∧ 𝑐 ∈ ℕ0) → (𝑥↑2) ∈ ℂ)
107	30	recnd 11337	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑥 ∈ ℕ0 ∧ 𝑦 ∈ ℕ0) → (𝑦↑2) ∈ ℂ)
108	107	adantr 486	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝑥 ∈ ℕ0 ∧ 𝑦 ∈ ℕ0) ∧ 𝑐 ∈ ℕ0) → (𝑦↑2) ∈ ℂ)
109	106, 108	addcomd 11512	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝑥 ∈ ℕ0 ∧ 𝑦 ∈ ℕ0) ∧ 𝑐 ∈ ℕ0) → ((𝑥↑2) + (𝑦↑2)) = ((𝑦↑2) + (𝑥↑2)))
110	109	eqeq2d 2771	. . . . . . . . . . . . . . . . . . . 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 487	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝑥 ∈ ℕ0 ∧ 𝑦 ∈ ℕ0) ∧ 𝑐 ∈ ℕ0) → (𝑐↑2) ∈ ℝ)
112	33	ad2antrr 739	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝑥 ∈ ℕ0 ∧ 𝑦 ∈ ℕ0) ∧ 𝑐 ∈ ℕ0) → (𝑥↑2) ∈ ℝ)
113	29	ad2antlr 740	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝑥 ∈ ℕ0 ∧ 𝑦 ∈ ℕ0) ∧ 𝑐 ∈ ℕ0) → (𝑦↑2) ∈ ℝ)
114		readdcan 11484	. . . . . . . . . . . . . . . . . . . . 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 1398	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝑥 ∈ ℕ0 ∧ 𝑦 ∈ ℕ0) ∧ 𝑐 ∈ ℕ0) → (((𝑦↑2) + (𝑐↑2)) = ((𝑦↑2) + (𝑥↑2)) ↔ (𝑐↑2) = (𝑥↑2)))
116	110, 115	bitrd 282	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑥 ∈ ℕ0 ∧ 𝑦 ∈ ℕ0) ∧ 𝑐 ∈ ℕ0) → (((𝑦↑2) + (𝑐↑2)) = ((𝑥↑2) + (𝑦↑2)) ↔ (𝑐↑2) = (𝑥↑2)))
117	25	ad2antlr 740	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((𝑥 ∈ ℕ0 ∧ 𝑦 ∈ ℕ0) ∧ 𝑐 ∈ ℕ0) ∧ (𝑐↑2) = (𝑥↑2)) → 𝑐 ∈ ℝ)
118	32	adantr 486	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑥 ∈ ℕ0 ∧ 𝑦 ∈ ℕ0) → 𝑥 ∈ ℝ)
119	118	ad2antrr 739	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((𝑥 ∈ ℕ0 ∧ 𝑦 ∈ ℕ0) ∧ 𝑐 ∈ ℕ0) ∧ (𝑐↑2) = (𝑥↑2)) → 𝑥 ∈ ℝ)
120	42	ad2antlr 740	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((𝑥 ∈ ℕ0 ∧ 𝑦 ∈ ℕ0) ∧ 𝑐 ∈ ℕ0) ∧ (𝑐↑2) = (𝑥↑2)) → 0 ≤ 𝑐)
121		nn0ge0 12631	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑥 ∈ ℕ0 → 0 ≤ 𝑥)
122	121	adantr 486	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑥 ∈ ℕ0 ∧ 𝑦 ∈ ℕ0) → 0 ≤ 𝑥)
123	122	ad2antrr 739	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((𝑥 ∈ ℕ0 ∧ 𝑦 ∈ ℕ0) ∧ 𝑐 ∈ ℕ0) ∧ (𝑐↑2) = (𝑥↑2)) → 0 ≤ 𝑥)
124		simpr 490	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((𝑥 ∈ ℕ0 ∧ 𝑦 ∈ ℕ0) ∧ 𝑐 ∈ ℕ0) ∧ (𝑐↑2) = (𝑥↑2)) → (𝑐↑2) = (𝑥↑2))
125	117, 119, 120, 123, 124	sq11d 14402	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((𝑥 ∈ ℕ0 ∧ 𝑦 ∈ ℕ0) ∧ 𝑐 ∈ ℕ0) ∧ (𝑐↑2) = (𝑥↑2)) → 𝑐 = 𝑥)
126	125	eqcomd 2766	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝑥 ∈ ℕ0 ∧ 𝑦 ∈ ℕ0) ∧ 𝑐 ∈ ℕ0) ∧ (𝑐↑2) = (𝑥↑2)) → 𝑥 = 𝑐)
127	126	ex 418	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑥 ∈ ℕ0 ∧ 𝑦 ∈ ℕ0) ∧ 𝑐 ∈ ℕ0) → ((𝑐↑2) = (𝑥↑2) → 𝑥 = 𝑐))
128	116, 127	sylbid 243	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑥 ∈ ℕ0 ∧ 𝑦 ∈ ℕ0) ∧ 𝑐 ∈ ℕ0) → (((𝑦↑2) + (𝑐↑2)) = ((𝑥↑2) + (𝑦↑2)) → 𝑥 = 𝑐))
129	128	adantld 496	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑥 ∈ ℕ0 ∧ 𝑦 ∈ ℕ0) ∧ 𝑐 ∈ ℕ0) → ((𝑦 ≤ 𝑐 ∧ ((𝑦↑2) + (𝑐↑2)) = ((𝑥↑2) + (𝑦↑2))) → 𝑥 = 𝑐))
130	129	ralrimiva 3154	. . . . . . . . . . . . . . . 16 ⊢ ((𝑥 ∈ ℕ0 ∧ 𝑦 ∈ ℕ0) → ∀𝑐 ∈ ℕ0 ((𝑦 ≤ 𝑐 ∧ ((𝑦↑2) + (𝑐↑2)) = ((𝑥↑2) + (𝑦↑2))) → 𝑥 = 𝑐))
131	130	adantr 486	. . . . . . . . . . . . . . 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 524	. . . . . . . . . . . . . 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 3578	. . . . . . . . . . . . 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 5106	. . . . . . . . . . . . . . 15 ⊢ (𝑏 = 𝑐 → (𝑦 ≤ 𝑏 ↔ 𝑦 ≤ 𝑐))
135	54	oveq2d 7436	. . . . . . . . . . . . . . . 16 ⊢ (𝑏 = 𝑐 → ((𝑦↑2) + (𝑏↑2)) = ((𝑦↑2) + (𝑐↑2)))
136	135	eqeq1d 2762	. . . . . . . . . . . . . . 15 ⊢ (𝑏 = 𝑐 → (((𝑦↑2) + (𝑏↑2)) = ((𝑥↑2) + (𝑦↑2)) ↔ ((𝑦↑2) + (𝑐↑2)) = ((𝑥↑2) + (𝑦↑2))))
137	134, 136	anbi12d 644	. . . . . . . . . . . . . 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 237	. . . . . . . . . . . 12 ⊢ (((𝑥 ∈ ℕ0 ∧ 𝑦 ∈ ℕ0) ∧ ¬ 𝑥 ≤ 𝑦) → ∃!𝑏 ∈ ℕ0 (𝑦 ≤ 𝑏 ∧ ((𝑦↑2) + (𝑏↑2)) = ((𝑥↑2) + (𝑦↑2))))
140	139	ex 418	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ ℕ0 ∧ 𝑦 ∈ ℕ0) → (¬ 𝑥 ≤ 𝑦 → ∃!𝑏 ∈ ℕ0 (𝑦 ≤ 𝑏 ∧ ((𝑦↑2) + (𝑏↑2)) = ((𝑥↑2) + (𝑦↑2)))))
141	140	adantr 486	. . . . . . . . . 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 413	. . . . . . . . 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 2772	. . . . . . . . . . . . 13 ⊢ (𝑃 = ((𝑥↑2) + (𝑦↑2)) → (((𝑦↑2) + (𝑏↑2)) = 𝑃 ↔ ((𝑦↑2) + (𝑏↑2)) = ((𝑥↑2) + (𝑦↑2))))
144	143	anbi2d 642	. . . . . . . . . . . 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 3381	. . . . . . . . . . 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 487	. . . . . . . . . 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 487	. . . . . . . . 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 260	. . . . . . . 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 3578	. . . . . . 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 824	. . . . . 6 ⊢ (((𝑥 ∈ ℕ0 ∧ 𝑦 ∈ ℕ0) ∧ 𝑃 = ((𝑥↑2) + (𝑦↑2))) → ∃𝑎 ∈ ℕ0 ∃!𝑏 ∈ ℕ0 (𝑎 ≤ 𝑏 ∧ ((𝑎↑2) + (𝑏↑2)) = 𝑃))
151	150	ex 418	. . . . 5 ⊢ ((𝑥 ∈ ℕ0 ∧ 𝑦 ∈ ℕ0) → (𝑃 = ((𝑥↑2) + (𝑦↑2)) → ∃𝑎 ∈ ℕ0 ∃!𝑏 ∈ ℕ0 (𝑎 ≤ 𝑏 ∧ ((𝑎↑2) + (𝑏↑2)) = 𝑃)))
152	151	adantl 487	. . . 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 3219	. . 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 16	. 2 ⊢ ((𝑃 ∈ ℙ ∧ (𝑃 mod 4) = 1) → ∃𝑎 ∈ ℕ0 ∃!𝑏 ∈ ℕ0 (𝑎 ≤ 𝑏 ∧ ((𝑎↑2) + (𝑏↑2)) = 𝑃))
155		reurex 3369	. . . . 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 3154	. . 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 27764	. . . 4 ⊢ (𝑃 ∈ ℙ → ∃*𝑎 ∈ ℕ0 ∃𝑏 ∈ ℕ0 (𝑎 ≤ 𝑏 ∧ ((𝑎↑2) + (𝑏↑2)) = 𝑃))
159	158	adantr 486	. . 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 66	. 2 ⊢ ((𝑃 ∈ ℙ ∧ (𝑃 mod 4) = 1) → ∃*𝑎 ∈ ℕ0 ∃!𝑏 ∈ ℕ0 (𝑎 ≤ 𝑏 ∧ ((𝑎↑2) + (𝑏↑2)) = 𝑃))
162		reu5 3367	. 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 595	1 ⊢ ((𝑃 ∈ ℙ ∧ (𝑃 mod 4) = 1) → ∃!𝑎 ∈ ℕ0 ∃!𝑏 ∈ ℕ0 (𝑎 ≤ 𝑏 ∧ ((𝑎↑2) + (𝑏↑2)) = 𝑃))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1569   ∈ wcel 2144  ∀wral 3076  ∃wrex 3086  ∃!wreu 3363  ∃*wrmo 3364   class class class wbr 5102  (class class class)co 7420  ℂcc 11198  ℝcr 11199  0cc0 11200  1c1 11201   + caddc 11203   < clt 11343   ≤ cle 11344  2c2 12397  4c4 12399  ℕ₀cn0 12606   mod cmo 14009  ↑cexp 14204  ℙcprime 16846

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-rep 5231  ax-sep 5248  ax-nul 5259  ax-pow 5326  ax-pr 5390  ax-un 7751  ax-cnex 11256  ax-resscn 11257  ax-1cn 11258  ax-icn 11259  ax-addcl 11260  ax-addrcl 11261  ax-mulcl 11262  ax-mulrcl 11263  ax-mulcom 11264  ax-addass 11265  ax-mulass 11266  ax-distr 11267  ax-i2m1 11268  ax-1ne0 11269  ax-1rid 11270  ax-rnegex 11271  ax-rrecex 11272  ax-cnre 11273  ax-pre-lttri 11274  ax-pre-lttrn 11275  ax-pre-ltadd 11276  ax-pre-mulgt0 11277  ax-pre-sup 11278  ax-addf 11279  ax-mulf 11280

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-nel 3062  df-ral 3077  df-rex 3087  df-rmo 3365  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3847  df-ss 3896  df-pss 4014  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4584  df-pr 4586  df-tp 4588  df-op 4590  df-uni 4867  df-int 4907  df-iun 4952  df-iin 4953  df-br 5103  df-opab 5167  df-mpt 5186  df-tr 5212  df-id 5545  df-eprel 5550  df-po 5558  df-so 5559  df-fr 5603  df-se 5604  df-we 5605  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-res 5662  df-ima 5663  df-pred 6304  df-ord 6365  df-on 6366  df-lim 6367  df-suc 6368  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545  df-fv 6546  df-isom 6547  df-riota 7377  df-ov 7423  df-oprab 7424  df-mpo 7425  df-of 7693  df-ofr 7694  df-om 7878  df-1st 8001  df-2nd 8002  df-supp 8178  df-tpos 8243  df-frecs 8299  df-wrecs 8330  df-recs 8379  df-rdg 8418  df-1o 8476  df-2o 8477  df-oadd 8480  df-er 8717  df-ec 8719  df-qs 8723  df-map 8849  df-pm 8850  df-ixp 8926  df-en 8974  df-dom 8975  df-sdom 8976  df-fin 8977  df-fsupp 9354  df-sup 9434  df-inf 9435  df-oi 9504  df-dju 9982  df-card 10020  df-pnf 11345  df-mnf 11346  df-xr 11347  df-ltxr 11348  df-le 11349  df-sub 11543  df-neg 11544  df-div 11974  df-nn 12336  df-2 12405  df-3 12406  df-4 12407  df-5 12408  df-6 12409  df-7 12410  df-8 12411  df-9 12412  df-n0 12607  df-xnn0 12680  df-z 12694  df-dec 12815  df-uz 12966  df-q 13076  df-rp 13121  df-fz 13640  df-fzo 13789  df-fl 13932  df-mod 14010  df-seq 14145  df-exp 14205  df-hash 14475  df-cj 15266  df-re 15267  df-im 15268  df-sqrt 15402  df-abs 15403  df-dvds 16423  df-gcd 16665  df-prm 16847  df-phi 16943  df-pc 17015  df-gz 17108  df-struct 17325  df-sets 17342  df-slot 17360  df-ndx 17372  df-base 17388  df-ress 17409  df-plusg 17441  df-mulr 17442  df-starv 17443  df-sca 17444  df-vsca 17445  df-ip 17446  df-tset 17447  df-ple 17448  df-ds 17450  df-unif 17451  df-hom 17452  df-cco 17453  df-0g 17612  df-gsum 17613  df-prds 17618  df-pws 17620  df-imas 17680  df-qus 17681  df-mre 17756  df-mrc 17757  df-acs 17759  df-mgm 18816  df-sgrp 18908  df-mnd 18924  df-mhm 18978  df-submnd 18979  df-grp 19147  df-minusg 19148  df-sbg 19149  df-mulg 19278  df-subg 19333  df-nsg 19334  df-eqg 19335  df-ghm 19428  df-cntz 19531  df-cmn 19996  df-abl 19997  df-mgp 20361  df-rng 20375  df-ur 20408  df-srg 20413  df-ring 20461  df-cring 20462  df-oppr 20567  df-dvdsr 20587  df-unit 20588  df-invr 20618  df-dvr 20631  df-rhm 20702  df-nzr 20763  df-subrng 20798  df-subrg 20822  df-rlreg 20946  df-domn 20947  df-idom 20948  df-drng 20982  df-field 20983  df-lmod 21137  df-lss 21207  df-lsp 21247  df-sra 21448  df-rgmod 21449  df-lidl 21486  df-rsp 21487  df-2idl 21543  df-cnfld 21679  df-zring 21753  df-zrh 21809  df-zn 21812  df-assa 22161  df-asp 22162  df-ascl 22163  df-psr 22217  df-mvr 22218  df-mpl 22219  df-opsr 22221  df-evls 22383  df-evl 22384  df-psr1 22498  df-vr1 22499  df-ply1 22500  df-coe1 22501  df-evl1 22634  df-mdeg 26373  df-deg1 26374  df-mon1 26449  df-uc1p 26450  df-q1p 26451  df-r1p 26452  df-lgs 27622

This theorem is used by:  2sqreultlem  27774  2sqreu  27783