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Theorem 2sqnn0 27416

Description: All primes of the form 4𝑘 + 1 are sums of squares of two nonnegative integers. (Contributed by AV, 3-Jun-2023.)

**Assertion**
Ref	Expression
2sqnn0	⊢ ((𝑃 ∈ ℙ ∧ (𝑃 mod 4) = 1) → ∃𝑥 ∈ ℕ₀ ∃𝑦 ∈ ℕ₀ 𝑃 = ((𝑥↑2) + (𝑦↑2)))

Distinct variable group: 𝑥,𝑃,𝑦

Proof of Theorem 2sqnn0 Dummy variables 𝑎 𝑏 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		2sq 27408	. 2 ⊢ ((𝑃 ∈ ℙ ∧ (𝑃 mod 4) = 1) → ∃𝑎 ∈ ℤ ∃𝑏 ∈ ℤ 𝑃 = ((𝑎↑2) + (𝑏↑2)))
2		elnn0z 12604	. . . . . . . . 9 ⊢ (𝑎 ∈ ℕ₀ ↔ (𝑎 ∈ ℤ ∧ 0 ≤ 𝑎))
3	2	biimpri 227	. . . . . . . 8 ⊢ ((𝑎 ∈ ℤ ∧ 0 ≤ 𝑎) → 𝑎 ∈ ℕ₀)
4		elznn0 12606	. . . . . . . . . 10 ⊢ (𝑎 ∈ ℤ ↔ (𝑎 ∈ ℝ ∧ (𝑎 ∈ ℕ₀ ∨ -𝑎 ∈ ℕ₀)))
5		nn0ge0 12530	. . . . . . . . . . . . . 14 ⊢ (𝑎 ∈ ℕ₀ → 0 ≤ 𝑎)
6	5	pm2.24d 151	. . . . . . . . . . . . 13 ⊢ (𝑎 ∈ ℕ₀ → (¬ 0 ≤ 𝑎 → -𝑎 ∈ ℕ₀))
7	6	a1i 11	. . . . . . . . . . . 12 ⊢ (𝑎 ∈ ℝ → (𝑎 ∈ ℕ₀ → (¬ 0 ≤ 𝑎 → -𝑎 ∈ ℕ₀)))
8		ax-1 6	. . . . . . . . . . . . 13 ⊢ (-𝑎 ∈ ℕ₀ → (¬ 0 ≤ 𝑎 → -𝑎 ∈ ℕ₀))
9	8	a1i 11	. . . . . . . . . . . 12 ⊢ (𝑎 ∈ ℝ → (-𝑎 ∈ ℕ₀ → (¬ 0 ≤ 𝑎 → -𝑎 ∈ ℕ₀)))
10	7, 9	jaod 857	. . . . . . . . . . 11 ⊢ (𝑎 ∈ ℝ → ((𝑎 ∈ ℕ₀ ∨ -𝑎 ∈ ℕ₀) → (¬ 0 ≤ 𝑎 → -𝑎 ∈ ℕ₀)))
11	10	imp 405	. . . . . . . . . 10 ⊢ ((𝑎 ∈ ℝ ∧ (𝑎 ∈ ℕ₀ ∨ -𝑎 ∈ ℕ₀)) → (¬ 0 ≤ 𝑎 → -𝑎 ∈ ℕ₀))
12	4, 11	sylbi 216	. . . . . . . . 9 ⊢ (𝑎 ∈ ℤ → (¬ 0 ≤ 𝑎 → -𝑎 ∈ ℕ₀))
13	12	imp 405	. . . . . . . 8 ⊢ ((𝑎 ∈ ℤ ∧ ¬ 0 ≤ 𝑎) → -𝑎 ∈ ℕ₀)
14	3, 13	ifclda 4565	. . . . . . 7 ⊢ (𝑎 ∈ ℤ → if(0 ≤ 𝑎, 𝑎, -𝑎) ∈ ℕ₀)
15	14	adantr 479	. . . . . 6 ⊢ ((𝑎 ∈ ℤ ∧ 𝑏 ∈ ℤ) → if(0 ≤ 𝑎, 𝑎, -𝑎) ∈ ℕ₀)
16	15	adantr 479	. . . . 5 ⊢ (((𝑎 ∈ ℤ ∧ 𝑏 ∈ ℤ) ∧ 𝑃 = ((𝑎↑2) + (𝑏↑2))) → if(0 ≤ 𝑎, 𝑎, -𝑎) ∈ ℕ₀)
17		elnn0z 12604	. . . . . . . . 9 ⊢ (𝑏 ∈ ℕ₀ ↔ (𝑏 ∈ ℤ ∧ 0 ≤ 𝑏))
18	17	biimpri 227	. . . . . . . 8 ⊢ ((𝑏 ∈ ℤ ∧ 0 ≤ 𝑏) → 𝑏 ∈ ℕ₀)
19		elznn0 12606	. . . . . . . . . 10 ⊢ (𝑏 ∈ ℤ ↔ (𝑏 ∈ ℝ ∧ (𝑏 ∈ ℕ₀ ∨ -𝑏 ∈ ℕ₀)))
20		nn0ge0 12530	. . . . . . . . . . . . . 14 ⊢ (𝑏 ∈ ℕ₀ → 0 ≤ 𝑏)
21	20	pm2.24d 151	. . . . . . . . . . . . 13 ⊢ (𝑏 ∈ ℕ₀ → (¬ 0 ≤ 𝑏 → -𝑏 ∈ ℕ₀))
22	21	a1i 11	. . . . . . . . . . . 12 ⊢ (𝑏 ∈ ℝ → (𝑏 ∈ ℕ₀ → (¬ 0 ≤ 𝑏 → -𝑏 ∈ ℕ₀)))
23		ax-1 6	. . . . . . . . . . . . 13 ⊢ (-𝑏 ∈ ℕ₀ → (¬ 0 ≤ 𝑏 → -𝑏 ∈ ℕ₀))
24	23	a1i 11	. . . . . . . . . . . 12 ⊢ (𝑏 ∈ ℝ → (-𝑏 ∈ ℕ₀ → (¬ 0 ≤ 𝑏 → -𝑏 ∈ ℕ₀)))
25	22, 24	jaod 857	. . . . . . . . . . 11 ⊢ (𝑏 ∈ ℝ → ((𝑏 ∈ ℕ₀ ∨ -𝑏 ∈ ℕ₀) → (¬ 0 ≤ 𝑏 → -𝑏 ∈ ℕ₀)))
26	25	imp 405	. . . . . . . . . 10 ⊢ ((𝑏 ∈ ℝ ∧ (𝑏 ∈ ℕ₀ ∨ -𝑏 ∈ ℕ₀)) → (¬ 0 ≤ 𝑏 → -𝑏 ∈ ℕ₀))
27	19, 26	sylbi 216	. . . . . . . . 9 ⊢ (𝑏 ∈ ℤ → (¬ 0 ≤ 𝑏 → -𝑏 ∈ ℕ₀))
28	27	imp 405	. . . . . . . 8 ⊢ ((𝑏 ∈ ℤ ∧ ¬ 0 ≤ 𝑏) → -𝑏 ∈ ℕ₀)
29	18, 28	ifclda 4565	. . . . . . 7 ⊢ (𝑏 ∈ ℤ → if(0 ≤ 𝑏, 𝑏, -𝑏) ∈ ℕ₀)
30	29	adantl 480	. . . . . 6 ⊢ ((𝑎 ∈ ℤ ∧ 𝑏 ∈ ℤ) → if(0 ≤ 𝑏, 𝑏, -𝑏) ∈ ℕ₀)
31	30	adantr 479	. . . . 5 ⊢ (((𝑎 ∈ ℤ ∧ 𝑏 ∈ ℤ) ∧ 𝑃 = ((𝑎↑2) + (𝑏↑2))) → if(0 ≤ 𝑏, 𝑏, -𝑏) ∈ ℕ₀)
32		elznn0nn 12605	. . . . . . . . . 10 ⊢ (𝑎 ∈ ℤ ↔ (𝑎 ∈ ℕ₀ ∨ (𝑎 ∈ ℝ ∧ -𝑎 ∈ ℕ)))
33	5	iftrued 4538	. . . . . . . . . . . . 13 ⊢ (𝑎 ∈ ℕ₀ → if(0 ≤ 𝑎, 𝑎, -𝑎) = 𝑎)
34	33	eqcomd 2731	. . . . . . . . . . . 12 ⊢ (𝑎 ∈ ℕ₀ → 𝑎 = if(0 ≤ 𝑎, 𝑎, -𝑎))
35	34	oveq1d 7434	. . . . . . . . . . 11 ⊢ (𝑎 ∈ ℕ₀ → (𝑎↑2) = (if(0 ≤ 𝑎, 𝑎, -𝑎)↑2))
36		elnnz 12601	. . . . . . . . . . . . . . . 16 ⊢ (-𝑎 ∈ ℕ ↔ (-𝑎 ∈ ℤ ∧ 0 < -𝑎))
37		lt0neg1 11752	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑎 ∈ ℝ → (𝑎 < 0 ↔ 0 < -𝑎))
38		id 22	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑎 ∈ ℝ → 𝑎 ∈ ℝ)
39		0red 11249	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑎 ∈ ℝ → 0 ∈ ℝ)
40	38, 39	ltnled 11393	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑎 ∈ ℝ → (𝑎 < 0 ↔ ¬ 0 ≤ 𝑎))
41	40	biimpd 228	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑎 ∈ ℝ → (𝑎 < 0 → ¬ 0 ≤ 𝑎))
42	37, 41	sylbird 259	. . . . . . . . . . . . . . . . 17 ⊢ (𝑎 ∈ ℝ → (0 < -𝑎 → ¬ 0 ≤ 𝑎))
43	42	com12 32	. . . . . . . . . . . . . . . 16 ⊢ (0 < -𝑎 → (𝑎 ∈ ℝ → ¬ 0 ≤ 𝑎))
44	36, 43	simplbiim 503	. . . . . . . . . . . . . . 15 ⊢ (-𝑎 ∈ ℕ → (𝑎 ∈ ℝ → ¬ 0 ≤ 𝑎))
45	44	impcom 406	. . . . . . . . . . . . . 14 ⊢ ((𝑎 ∈ ℝ ∧ -𝑎 ∈ ℕ) → ¬ 0 ≤ 𝑎)
46	45	iffalsed 4541	. . . . . . . . . . . . 13 ⊢ ((𝑎 ∈ ℝ ∧ -𝑎 ∈ ℕ) → if(0 ≤ 𝑎, 𝑎, -𝑎) = -𝑎)
47	46	oveq1d 7434	. . . . . . . . . . . 12 ⊢ ((𝑎 ∈ ℝ ∧ -𝑎 ∈ ℕ) → (if(0 ≤ 𝑎, 𝑎, -𝑎)↑2) = (-𝑎↑2))
48		recn 11230	. . . . . . . . . . . . . 14 ⊢ (𝑎 ∈ ℝ → 𝑎 ∈ ℂ)
49		sqneg 14116	. . . . . . . . . . . . . 14 ⊢ (𝑎 ∈ ℂ → (-𝑎↑2) = (𝑎↑2))
50	48, 49	syl 17	. . . . . . . . . . . . 13 ⊢ (𝑎 ∈ ℝ → (-𝑎↑2) = (𝑎↑2))
51	50	adantr 479	. . . . . . . . . . . 12 ⊢ ((𝑎 ∈ ℝ ∧ -𝑎 ∈ ℕ) → (-𝑎↑2) = (𝑎↑2))
52	47, 51	eqtr2d 2766	. . . . . . . . . . 11 ⊢ ((𝑎 ∈ ℝ ∧ -𝑎 ∈ ℕ) → (𝑎↑2) = (if(0 ≤ 𝑎, 𝑎, -𝑎)↑2))
53	35, 52	jaoi 855	. . . . . . . . . 10 ⊢ ((𝑎 ∈ ℕ₀ ∨ (𝑎 ∈ ℝ ∧ -𝑎 ∈ ℕ)) → (𝑎↑2) = (if(0 ≤ 𝑎, 𝑎, -𝑎)↑2))
54	32, 53	sylbi 216	. . . . . . . . 9 ⊢ (𝑎 ∈ ℤ → (𝑎↑2) = (if(0 ≤ 𝑎, 𝑎, -𝑎)↑2))
55		elznn0nn 12605	. . . . . . . . . 10 ⊢ (𝑏 ∈ ℤ ↔ (𝑏 ∈ ℕ₀ ∨ (𝑏 ∈ ℝ ∧ -𝑏 ∈ ℕ)))
56	20	iftrued 4538	. . . . . . . . . . . . 13 ⊢ (𝑏 ∈ ℕ₀ → if(0 ≤ 𝑏, 𝑏, -𝑏) = 𝑏)
57	56	eqcomd 2731	. . . . . . . . . . . 12 ⊢ (𝑏 ∈ ℕ₀ → 𝑏 = if(0 ≤ 𝑏, 𝑏, -𝑏))
58	57	oveq1d 7434	. . . . . . . . . . 11 ⊢ (𝑏 ∈ ℕ₀ → (𝑏↑2) = (if(0 ≤ 𝑏, 𝑏, -𝑏)↑2))
59		elnnz 12601	. . . . . . . . . . . . . . . 16 ⊢ (-𝑏 ∈ ℕ ↔ (-𝑏 ∈ ℤ ∧ 0 < -𝑏))
60		lt0neg1 11752	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑏 ∈ ℝ → (𝑏 < 0 ↔ 0 < -𝑏))
61		id 22	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑏 ∈ ℝ → 𝑏 ∈ ℝ)
62		0red 11249	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑏 ∈ ℝ → 0 ∈ ℝ)
63	61, 62	ltnled 11393	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑏 ∈ ℝ → (𝑏 < 0 ↔ ¬ 0 ≤ 𝑏))
64	63	biimpd 228	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑏 ∈ ℝ → (𝑏 < 0 → ¬ 0 ≤ 𝑏))
65	60, 64	sylbird 259	. . . . . . . . . . . . . . . . 17 ⊢ (𝑏 ∈ ℝ → (0 < -𝑏 → ¬ 0 ≤ 𝑏))
66	65	com12 32	. . . . . . . . . . . . . . . 16 ⊢ (0 < -𝑏 → (𝑏 ∈ ℝ → ¬ 0 ≤ 𝑏))
67	59, 66	simplbiim 503	. . . . . . . . . . . . . . 15 ⊢ (-𝑏 ∈ ℕ → (𝑏 ∈ ℝ → ¬ 0 ≤ 𝑏))
68	67	impcom 406	. . . . . . . . . . . . . 14 ⊢ ((𝑏 ∈ ℝ ∧ -𝑏 ∈ ℕ) → ¬ 0 ≤ 𝑏)
69	68	iffalsed 4541	. . . . . . . . . . . . 13 ⊢ ((𝑏 ∈ ℝ ∧ -𝑏 ∈ ℕ) → if(0 ≤ 𝑏, 𝑏, -𝑏) = -𝑏)
70	69	oveq1d 7434	. . . . . . . . . . . 12 ⊢ ((𝑏 ∈ ℝ ∧ -𝑏 ∈ ℕ) → (if(0 ≤ 𝑏, 𝑏, -𝑏)↑2) = (-𝑏↑2))
71		recn 11230	. . . . . . . . . . . . . 14 ⊢ (𝑏 ∈ ℝ → 𝑏 ∈ ℂ)
72		sqneg 14116	. . . . . . . . . . . . . 14 ⊢ (𝑏 ∈ ℂ → (-𝑏↑2) = (𝑏↑2))
73	71, 72	syl 17	. . . . . . . . . . . . 13 ⊢ (𝑏 ∈ ℝ → (-𝑏↑2) = (𝑏↑2))
74	73	adantr 479	. . . . . . . . . . . 12 ⊢ ((𝑏 ∈ ℝ ∧ -𝑏 ∈ ℕ) → (-𝑏↑2) = (𝑏↑2))
75	70, 74	eqtr2d 2766	. . . . . . . . . . 11 ⊢ ((𝑏 ∈ ℝ ∧ -𝑏 ∈ ℕ) → (𝑏↑2) = (if(0 ≤ 𝑏, 𝑏, -𝑏)↑2))
76	58, 75	jaoi 855	. . . . . . . . . 10 ⊢ ((𝑏 ∈ ℕ₀ ∨ (𝑏 ∈ ℝ ∧ -𝑏 ∈ ℕ)) → (𝑏↑2) = (if(0 ≤ 𝑏, 𝑏, -𝑏)↑2))
77	55, 76	sylbi 216	. . . . . . . . 9 ⊢ (𝑏 ∈ ℤ → (𝑏↑2) = (if(0 ≤ 𝑏, 𝑏, -𝑏)↑2))
78	54, 77	oveqan12d 7438	. . . . . . . 8 ⊢ ((𝑎 ∈ ℤ ∧ 𝑏 ∈ ℤ) → ((𝑎↑2) + (𝑏↑2)) = ((if(0 ≤ 𝑎, 𝑎, -𝑎)↑2) + (if(0 ≤ 𝑏, 𝑏, -𝑏)↑2)))
79	78	eqeq2d 2736	. . . . . . 7 ⊢ ((𝑎 ∈ ℤ ∧ 𝑏 ∈ ℤ) → (𝑃 = ((𝑎↑2) + (𝑏↑2)) ↔ 𝑃 = ((if(0 ≤ 𝑎, 𝑎, -𝑎)↑2) + (if(0 ≤ 𝑏, 𝑏, -𝑏)↑2))))
80	79	biimpd 228	. . . . . 6 ⊢ ((𝑎 ∈ ℤ ∧ 𝑏 ∈ ℤ) → (𝑃 = ((𝑎↑2) + (𝑏↑2)) → 𝑃 = ((if(0 ≤ 𝑎, 𝑎, -𝑎)↑2) + (if(0 ≤ 𝑏, 𝑏, -𝑏)↑2))))
81	80	imp 405	. . . . 5 ⊢ (((𝑎 ∈ ℤ ∧ 𝑏 ∈ ℤ) ∧ 𝑃 = ((𝑎↑2) + (𝑏↑2))) → 𝑃 = ((if(0 ≤ 𝑎, 𝑎, -𝑎)↑2) + (if(0 ≤ 𝑏, 𝑏, -𝑏)↑2)))
82		oveq1 7426	. . . . . . . 8 ⊢ (𝑥 = if(0 ≤ 𝑎, 𝑎, -𝑎) → (𝑥↑2) = (if(0 ≤ 𝑎, 𝑎, -𝑎)↑2))
83	82	oveq1d 7434	. . . . . . 7 ⊢ (𝑥 = if(0 ≤ 𝑎, 𝑎, -𝑎) → ((𝑥↑2) + (𝑦↑2)) = ((if(0 ≤ 𝑎, 𝑎, -𝑎)↑2) + (𝑦↑2)))
84	83	eqeq2d 2736	. . . . . 6 ⊢ (𝑥 = if(0 ≤ 𝑎, 𝑎, -𝑎) → (𝑃 = ((𝑥↑2) + (𝑦↑2)) ↔ 𝑃 = ((if(0 ≤ 𝑎, 𝑎, -𝑎)↑2) + (𝑦↑2))))
85		oveq1 7426	. . . . . . . 8 ⊢ (𝑦 = if(0 ≤ 𝑏, 𝑏, -𝑏) → (𝑦↑2) = (if(0 ≤ 𝑏, 𝑏, -𝑏)↑2))
86	85	oveq2d 7435	. . . . . . 7 ⊢ (𝑦 = if(0 ≤ 𝑏, 𝑏, -𝑏) → ((if(0 ≤ 𝑎, 𝑎, -𝑎)↑2) + (𝑦↑2)) = ((if(0 ≤ 𝑎, 𝑎, -𝑎)↑2) + (if(0 ≤ 𝑏, 𝑏, -𝑏)↑2)))
87	86	eqeq2d 2736	. . . . . 6 ⊢ (𝑦 = if(0 ≤ 𝑏, 𝑏, -𝑏) → (𝑃 = ((if(0 ≤ 𝑎, 𝑎, -𝑎)↑2) + (𝑦↑2)) ↔ 𝑃 = ((if(0 ≤ 𝑎, 𝑎, -𝑎)↑2) + (if(0 ≤ 𝑏, 𝑏, -𝑏)↑2))))
88	84, 87	rspc2ev 3619	. . . . 5 ⊢ ((if(0 ≤ 𝑎, 𝑎, -𝑎) ∈ ℕ₀ ∧ if(0 ≤ 𝑏, 𝑏, -𝑏) ∈ ℕ₀ ∧ 𝑃 = ((if(0 ≤ 𝑎, 𝑎, -𝑎)↑2) + (if(0 ≤ 𝑏, 𝑏, -𝑏)↑2))) → ∃𝑥 ∈ ℕ₀ ∃𝑦 ∈ ℕ₀ 𝑃 = ((𝑥↑2) + (𝑦↑2)))
89	16, 31, 81, 88	syl3anc 1368	. . . 4 ⊢ (((𝑎 ∈ ℤ ∧ 𝑏 ∈ ℤ) ∧ 𝑃 = ((𝑎↑2) + (𝑏↑2))) → ∃𝑥 ∈ ℕ₀ ∃𝑦 ∈ ℕ₀ 𝑃 = ((𝑥↑2) + (𝑦↑2)))
90	89	ex 411	. . 3 ⊢ ((𝑎 ∈ ℤ ∧ 𝑏 ∈ ℤ) → (𝑃 = ((𝑎↑2) + (𝑏↑2)) → ∃𝑥 ∈ ℕ₀ ∃𝑦 ∈ ℕ₀ 𝑃 = ((𝑥↑2) + (𝑦↑2))))
91	90	rexlimivv 3189	. 2 ⊢ (∃𝑎 ∈ ℤ ∃𝑏 ∈ ℤ 𝑃 = ((𝑎↑2) + (𝑏↑2)) → ∃𝑥 ∈ ℕ₀ ∃𝑦 ∈ ℕ₀ 𝑃 = ((𝑥↑2) + (𝑦↑2)))
92	1, 91	syl 17	1 ⊢ ((𝑃 ∈ ℙ ∧ (𝑃 mod 4) = 1) → ∃𝑥 ∈ ℕ₀ ∃𝑦 ∈ ℕ₀ 𝑃 = ((𝑥↑2) + (𝑦↑2)))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 394 ∨ wo 845 = wceq 1533 ∈ wcel 2098 ∃wrex 3059 ifcif 4530 class class class wbr 5149 (class class class)co 7419 ℂcc 11138 ℝcr 11139 0cc0 11140 1c1 11141 + caddc 11143 < clt 11280 ≤ cle 11281 -cneg 11477 ℕcn 12245 2c2 12300 4c4 12302 ℕ₀cn0 12505 ℤcz 12591 mod cmo 13870 ↑cexp 14062 ℙcprime 16645

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 5286 ax-sep 5300 ax-nul 5307 ax-pow 5365 ax-pr 5429 ax-un 7741 ax-cnex 11196 ax-resscn 11197 ax-1cn 11198 ax-icn 11199 ax-addcl 11200 ax-addrcl 11201 ax-mulcl 11202 ax-mulrcl 11203 ax-mulcom 11204 ax-addass 11205 ax-mulass 11206 ax-distr 11207 ax-i2m1 11208 ax-1ne0 11209 ax-1rid 11210 ax-rnegex 11211 ax-rrecex 11212 ax-cnre 11213 ax-pre-lttri 11214 ax-pre-lttrn 11215 ax-pre-ltadd 11216 ax-pre-mulgt0 11217 ax-pre-sup 11218 ax-addf 11219 ax-mulf 11220

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 2930 df-nel 3036 df-ral 3051 df-rex 3060 df-rmo 3363 df-reu 3364 df-rab 3419 df-v 3463 df-sbc 3774 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3964 df-nul 4323 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-tp 4635 df-op 4637 df-uni 4910 df-int 4951 df-iun 4999 df-iin 5000 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-id 5576 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5633 df-se 5634 df-we 5635 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-pred 6307 df-ord 6374 df-on 6375 df-lim 6376 df-suc 6377 df-iota 6501 df-fun 6551 df-fn 6552 df-f 6553 df-f1 6554 df-fo 6555 df-f1o 6556 df-fv 6557 df-isom 6558 df-riota 7375 df-ov 7422 df-oprab 7423 df-mpo 7424 df-of 7685 df-ofr 7686 df-om 7872 df-1st 7994 df-2nd 7995 df-supp 8166 df-tpos 8232 df-frecs 8287 df-wrecs 8318 df-recs 8392 df-rdg 8431 df-1o 8487 df-2o 8488 df-oadd 8491 df-er 8725 df-ec 8727 df-qs 8731 df-map 8847 df-pm 8848 df-ixp 8917 df-en 8965 df-dom 8966 df-sdom 8967 df-fin 8968 df-fsupp 9388 df-sup 9467 df-inf 9468 df-oi 9535 df-dju 9926 df-card 9964 df-pnf 11282 df-mnf 11283 df-xr 11284 df-ltxr 11285 df-le 11286 df-sub 11478 df-neg 11479 df-div 11904 df-nn 12246 df-2 12308 df-3 12309 df-4 12310 df-5 12311 df-6 12312 df-7 12313 df-8 12314 df-9 12315 df-n0 12506 df-xnn0 12578 df-z 12592 df-dec 12711 df-uz 12856 df-q 12966 df-rp 13010 df-fz 13520 df-fzo 13663 df-fl 13793 df-mod 13871 df-seq 14003 df-exp 14063 df-hash 14326 df-cj 15082 df-re 15083 df-im 15084 df-sqrt 15218 df-abs 15219 df-dvds 16235 df-gcd 16473 df-prm 16646 df-phi 16738 df-pc 16809 df-gz 16902 df-struct 17119 df-sets 17136 df-slot 17154 df-ndx 17166 df-base 17184 df-ress 17213 df-plusg 17249 df-mulr 17250 df-starv 17251 df-sca 17252 df-vsca 17253 df-ip 17254 df-tset 17255 df-ple 17256 df-ds 17258 df-unif 17259 df-hom 17260 df-cco 17261 df-0g 17426 df-gsum 17427 df-prds 17432 df-pws 17434 df-imas 17493 df-qus 17494 df-mre 17569 df-mrc 17570 df-acs 17572 df-mgm 18603 df-sgrp 18682 df-mnd 18698 df-mhm 18743 df-submnd 18744 df-grp 18901 df-minusg 18902 df-sbg 18903 df-mulg 19032 df-subg 19086 df-nsg 19087 df-eqg 19088 df-ghm 19176 df-cntz 19280 df-cmn 19749 df-abl 19750 df-mgp 20087 df-rng 20105 df-ur 20134 df-srg 20139 df-ring 20187 df-cring 20188 df-oppr 20285 df-dvdsr 20308 df-unit 20309 df-invr 20339 df-dvr 20352 df-rhm 20423 df-nzr 20464 df-subrng 20495 df-subrg 20520 df-drng 20638 df-field 20639 df-lmod 20757 df-lss 20828 df-lsp 20868 df-sra 21070 df-rgmod 21071 df-lidl 21116 df-rsp 21117 df-2idl 21157 df-rlreg 21247 df-domn 21248 df-idom 21249 df-cnfld 21297 df-zring 21390 df-zrh 21446 df-zn 21449 df-assa 21804 df-asp 21805 df-ascl 21806 df-psr 21859 df-mvr 21860 df-mpl 21861 df-opsr 21863 df-evls 22040 df-evl 22041 df-psr1 22122 df-vr1 22123 df-ply1 22124 df-coe1 22125 df-evl1 22260 df-mdeg 26032 df-deg1 26033 df-mon1 26111 df-uc1p 26112 df-q1p 26113 df-r1p 26114 df-lgs 27273

This theorem is referenced by: 2sqnn 27417 2sqreulem1 27424

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