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Theorem 2sqnn0 27568

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 27560	. 2 ⊢ ((𝑃 ∈ ℙ ∧ (𝑃 mod 4) = 1) → ∃𝑎 ∈ ℤ ∃𝑏 ∈ ℤ 𝑃 = ((𝑎↑2) + (𝑏↑2)))
2		oveq1 7418	. . . . . . 7 ⊢ (𝑥 = if(0 ≤ 𝑎, 𝑎, -𝑎) → (𝑥↑2) = (if(0 ≤ 𝑎, 𝑎, -𝑎)↑2))
3	2	oveq1d 7426	. . . . . 6 ⊢ (𝑥 = if(0 ≤ 𝑎, 𝑎, -𝑎) → ((𝑥↑2) + (𝑦↑2)) = ((if(0 ≤ 𝑎, 𝑎, -𝑎)↑2) + (𝑦↑2)))
4	3	eqeq2d 2780	. . . . 5 ⊢ (𝑥 = if(0 ≤ 𝑎, 𝑎, -𝑎) → (𝑃 = ((𝑥↑2) + (𝑦↑2)) ↔ 𝑃 = ((if(0 ≤ 𝑎, 𝑎, -𝑎)↑2) + (𝑦↑2))))
5		oveq1 7418	. . . . . . 7 ⊢ (𝑦 = if(0 ≤ 𝑏, 𝑏, -𝑏) → (𝑦↑2) = (if(0 ≤ 𝑏, 𝑏, -𝑏)↑2))
6	5	oveq2d 7427	. . . . . 6 ⊢ (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)))
7	6	eqeq2d 2780	. . . . 5 ⊢ (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))))
8		elnn0z 12604	. . . . . . . . 9 ⊢ (𝑎 ∈ ℕ₀ ↔ (𝑎 ∈ ℤ ∧ 0 ≤ 𝑎))
9	8	biimpri 231	. . . . . . . 8 ⊢ ((𝑎 ∈ ℤ ∧ 0 ≤ 𝑎) → 𝑎 ∈ ℕ₀)
10		elznn0 12606	. . . . . . . . . 10 ⊢ (𝑎 ∈ ℤ ↔ (𝑎 ∈ ℝ ∧ (𝑎 ∈ ℕ₀ ∨ -𝑎 ∈ ℕ₀)))
11		nn0ge0 12529	. . . . . . . . . . . . . 14 ⊢ (𝑎 ∈ ℕ₀ → 0 ≤ 𝑎)
12	11	pm2.24d 152	. . . . . . . . . . . . 13 ⊢ (𝑎 ∈ ℕ₀ → (¬ 0 ≤ 𝑎 → -𝑎 ∈ ℕ₀))
13	12	a1i 11	. . . . . . . . . . . 12 ⊢ (𝑎 ∈ ℝ → (𝑎 ∈ ℕ₀ → (¬ 0 ≤ 𝑎 → -𝑎 ∈ ℕ₀)))
14		ax1w 13	. . . . . . . . . . . 12 ⊢ (𝑎 ∈ ℝ → (-𝑎 ∈ ℕ₀ → (¬ 0 ≤ 𝑎 → -𝑎 ∈ ℕ₀)))
15	13, 14	jaod 872	. . . . . . . . . . 11 ⊢ (𝑎 ∈ ℝ → ((𝑎 ∈ ℕ₀ ∨ -𝑎 ∈ ℕ₀) → (¬ 0 ≤ 𝑎 → -𝑎 ∈ ℕ₀)))
16	15	imp 411	. . . . . . . . . 10 ⊢ ((𝑎 ∈ ℝ ∧ (𝑎 ∈ ℕ₀ ∨ -𝑎 ∈ ℕ₀)) → (¬ 0 ≤ 𝑎 → -𝑎 ∈ ℕ₀))
17	10, 16	sylbi 220	. . . . . . . . 9 ⊢ (𝑎 ∈ ℤ → (¬ 0 ≤ 𝑎 → -𝑎 ∈ ℕ₀))
18	17	imp 411	. . . . . . . 8 ⊢ ((𝑎 ∈ ℤ ∧ ¬ 0 ≤ 𝑎) → -𝑎 ∈ ℕ₀)
19	9, 18	ifclda 4528	. . . . . . 7 ⊢ (𝑎 ∈ ℤ → if(0 ≤ 𝑎, 𝑎, -𝑎) ∈ ℕ₀)
20	19	adantr 485	. . . . . 6 ⊢ ((𝑎 ∈ ℤ ∧ 𝑏 ∈ ℤ) → if(0 ≤ 𝑎, 𝑎, -𝑎) ∈ ℕ₀)
21	20	adantr 485	. . . . 5 ⊢ (((𝑎 ∈ ℤ ∧ 𝑏 ∈ ℤ) ∧ 𝑃 = ((𝑎↑2) + (𝑏↑2))) → if(0 ≤ 𝑎, 𝑎, -𝑎) ∈ ℕ₀)
22		elnn0z 12604	. . . . . . . 8 ⊢ (𝑏 ∈ ℕ₀ ↔ (𝑏 ∈ ℤ ∧ 0 ≤ 𝑏))
23	22	biimpri 231	. . . . . . 7 ⊢ ((𝑏 ∈ ℤ ∧ 0 ≤ 𝑏) → 𝑏 ∈ ℕ₀)
24		elznn0 12606	. . . . . . . . 9 ⊢ (𝑏 ∈ ℤ ↔ (𝑏 ∈ ℝ ∧ (𝑏 ∈ ℕ₀ ∨ -𝑏 ∈ ℕ₀)))
25		nn0ge0 12529	. . . . . . . . . . . . 13 ⊢ (𝑏 ∈ ℕ₀ → 0 ≤ 𝑏)
26	25	pm2.24d 152	. . . . . . . . . . . 12 ⊢ (𝑏 ∈ ℕ₀ → (¬ 0 ≤ 𝑏 → -𝑏 ∈ ℕ₀))
27	26	a1i 11	. . . . . . . . . . 11 ⊢ (𝑏 ∈ ℝ → (𝑏 ∈ ℕ₀ → (¬ 0 ≤ 𝑏 → -𝑏 ∈ ℕ₀)))
28		ax1w 13	. . . . . . . . . . 11 ⊢ (𝑏 ∈ ℝ → (-𝑏 ∈ ℕ₀ → (¬ 0 ≤ 𝑏 → -𝑏 ∈ ℕ₀)))
29	27, 28	jaod 872	. . . . . . . . . 10 ⊢ (𝑏 ∈ ℝ → ((𝑏 ∈ ℕ₀ ∨ -𝑏 ∈ ℕ₀) → (¬ 0 ≤ 𝑏 → -𝑏 ∈ ℕ₀)))
30	29	imp 411	. . . . . . . . 9 ⊢ ((𝑏 ∈ ℝ ∧ (𝑏 ∈ ℕ₀ ∨ -𝑏 ∈ ℕ₀)) → (¬ 0 ≤ 𝑏 → -𝑏 ∈ ℕ₀))
31	24, 30	sylbi 220	. . . . . . . 8 ⊢ (𝑏 ∈ ℤ → (¬ 0 ≤ 𝑏 → -𝑏 ∈ ℕ₀))
32	31	imp 411	. . . . . . 7 ⊢ ((𝑏 ∈ ℤ ∧ ¬ 0 ≤ 𝑏) → -𝑏 ∈ ℕ₀)
33	23, 32	ifclda 4528	. . . . . 6 ⊢ (𝑏 ∈ ℤ → if(0 ≤ 𝑏, 𝑏, -𝑏) ∈ ℕ₀)
34	33	ad2antlr 739	. . . . 5 ⊢ (((𝑎 ∈ ℤ ∧ 𝑏 ∈ ℤ) ∧ 𝑃 = ((𝑎↑2) + (𝑏↑2))) → if(0 ≤ 𝑏, 𝑏, -𝑏) ∈ ℕ₀)
35		elznn0nn 12605	. . . . . . . . . 10 ⊢ (𝑎 ∈ ℤ ↔ (𝑎 ∈ ℕ₀ ∨ (𝑎 ∈ ℝ ∧ -𝑎 ∈ ℕ)))
36	11	iftrued 4500	. . . . . . . . . . . . 13 ⊢ (𝑎 ∈ ℕ₀ → if(0 ≤ 𝑎, 𝑎, -𝑎) = 𝑎)
37	36	eqcomd 2775	. . . . . . . . . . . 12 ⊢ (𝑎 ∈ ℕ₀ → 𝑎 = if(0 ≤ 𝑎, 𝑎, -𝑎))
38	37	oveq1d 7426	. . . . . . . . . . 11 ⊢ (𝑎 ∈ ℕ₀ → (𝑎↑2) = (if(0 ≤ 𝑎, 𝑎, -𝑎)↑2))
39		elnnz 12601	. . . . . . . . . . . . . . . 16 ⊢ (-𝑎 ∈ ℕ ↔ (-𝑎 ∈ ℤ ∧ 0 < -𝑎))
40		lt0neg1 11720	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑎 ∈ ℝ → (𝑎 < 0 ↔ 0 < -𝑎))
41		id 23	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑎 ∈ ℝ → 𝑎 ∈ ℝ)
42		0red 11211	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑎 ∈ ℝ → 0 ∈ ℝ)
43	41, 42	ltnled 11357	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑎 ∈ ℝ → (𝑎 < 0 ↔ ¬ 0 ≤ 𝑎))
44	43	biimpd 232	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑎 ∈ ℝ → (𝑎 < 0 → ¬ 0 ≤ 𝑎))
45	40, 44	sylbird 263	. . . . . . . . . . . . . . . . 17 ⊢ (𝑎 ∈ ℝ → (0 < -𝑎 → ¬ 0 ≤ 𝑎))
46	45	com12 33	. . . . . . . . . . . . . . . 16 ⊢ (0 < -𝑎 → (𝑎 ∈ ℝ → ¬ 0 ≤ 𝑎))
47	39, 46	simplbiim 513	. . . . . . . . . . . . . . 15 ⊢ (-𝑎 ∈ ℕ → (𝑎 ∈ ℝ → ¬ 0 ≤ 𝑎))
48	47	impcom 412	. . . . . . . . . . . . . 14 ⊢ ((𝑎 ∈ ℝ ∧ -𝑎 ∈ ℕ) → ¬ 0 ≤ 𝑎)
49	48	iffalsed 4503	. . . . . . . . . . . . 13 ⊢ ((𝑎 ∈ ℝ ∧ -𝑎 ∈ ℕ) → if(0 ≤ 𝑎, 𝑎, -𝑎) = -𝑎)
50	49	oveq1d 7426	. . . . . . . . . . . 12 ⊢ ((𝑎 ∈ ℝ ∧ -𝑎 ∈ ℕ) → (if(0 ≤ 𝑎, 𝑎, -𝑎)↑2) = (-𝑎↑2))
51		recn 11190	. . . . . . . . . . . . . 14 ⊢ (𝑎 ∈ ℝ → 𝑎 ∈ ℂ)
52	51	sqnegd 14152	. . . . . . . . . . . . 13 ⊢ (𝑎 ∈ ℝ → (-𝑎↑2) = (𝑎↑2))
53	52	adantr 485	. . . . . . . . . . . 12 ⊢ ((𝑎 ∈ ℝ ∧ -𝑎 ∈ ℕ) → (-𝑎↑2) = (𝑎↑2))
54	50, 53	eqtr2d 2805	. . . . . . . . . . 11 ⊢ ((𝑎 ∈ ℝ ∧ -𝑎 ∈ ℕ) → (𝑎↑2) = (if(0 ≤ 𝑎, 𝑎, -𝑎)↑2))
55	38, 54	jaoi 870	. . . . . . . . . 10 ⊢ ((𝑎 ∈ ℕ₀ ∨ (𝑎 ∈ ℝ ∧ -𝑎 ∈ ℕ)) → (𝑎↑2) = (if(0 ≤ 𝑎, 𝑎, -𝑎)↑2))
56	35, 55	sylbi 220	. . . . . . . . 9 ⊢ (𝑎 ∈ ℤ → (𝑎↑2) = (if(0 ≤ 𝑎, 𝑎, -𝑎)↑2))
57		elznn0nn 12605	. . . . . . . . . 10 ⊢ (𝑏 ∈ ℤ ↔ (𝑏 ∈ ℕ₀ ∨ (𝑏 ∈ ℝ ∧ -𝑏 ∈ ℕ)))
58	25	iftrued 4500	. . . . . . . . . . . . 13 ⊢ (𝑏 ∈ ℕ₀ → if(0 ≤ 𝑏, 𝑏, -𝑏) = 𝑏)
59	58	eqcomd 2775	. . . . . . . . . . . 12 ⊢ (𝑏 ∈ ℕ₀ → 𝑏 = if(0 ≤ 𝑏, 𝑏, -𝑏))
60	59	oveq1d 7426	. . . . . . . . . . 11 ⊢ (𝑏 ∈ ℕ₀ → (𝑏↑2) = (if(0 ≤ 𝑏, 𝑏, -𝑏)↑2))
61		elnnz 12601	. . . . . . . . . . . . . . . 16 ⊢ (-𝑏 ∈ ℕ ↔ (-𝑏 ∈ ℤ ∧ 0 < -𝑏))
62		lt0neg1 11720	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑏 ∈ ℝ → (𝑏 < 0 ↔ 0 < -𝑏))
63		id 23	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑏 ∈ ℝ → 𝑏 ∈ ℝ)
64		0red 11211	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑏 ∈ ℝ → 0 ∈ ℝ)
65	63, 64	ltnled 11357	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑏 ∈ ℝ → (𝑏 < 0 ↔ ¬ 0 ≤ 𝑏))
66	65	biimpd 232	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑏 ∈ ℝ → (𝑏 < 0 → ¬ 0 ≤ 𝑏))
67	62, 66	sylbird 263	. . . . . . . . . . . . . . . . 17 ⊢ (𝑏 ∈ ℝ → (0 < -𝑏 → ¬ 0 ≤ 𝑏))
68	67	com12 33	. . . . . . . . . . . . . . . 16 ⊢ (0 < -𝑏 → (𝑏 ∈ ℝ → ¬ 0 ≤ 𝑏))
69	61, 68	simplbiim 513	. . . . . . . . . . . . . . 15 ⊢ (-𝑏 ∈ ℕ → (𝑏 ∈ ℝ → ¬ 0 ≤ 𝑏))
70	69	impcom 412	. . . . . . . . . . . . . 14 ⊢ ((𝑏 ∈ ℝ ∧ -𝑏 ∈ ℕ) → ¬ 0 ≤ 𝑏)
71	70	iffalsed 4503	. . . . . . . . . . . . 13 ⊢ ((𝑏 ∈ ℝ ∧ -𝑏 ∈ ℕ) → if(0 ≤ 𝑏, 𝑏, -𝑏) = -𝑏)
72	71	oveq1d 7426	. . . . . . . . . . . 12 ⊢ ((𝑏 ∈ ℝ ∧ -𝑏 ∈ ℕ) → (if(0 ≤ 𝑏, 𝑏, -𝑏)↑2) = (-𝑏↑2))
73		recn 11190	. . . . . . . . . . . . . 14 ⊢ (𝑏 ∈ ℝ → 𝑏 ∈ ℂ)
74	73	sqnegd 14152	. . . . . . . . . . . . 13 ⊢ (𝑏 ∈ ℝ → (-𝑏↑2) = (𝑏↑2))
75	74	adantr 485	. . . . . . . . . . . 12 ⊢ ((𝑏 ∈ ℝ ∧ -𝑏 ∈ ℕ) → (-𝑏↑2) = (𝑏↑2))
76	72, 75	eqtr2d 2805	. . . . . . . . . . 11 ⊢ ((𝑏 ∈ ℝ ∧ -𝑏 ∈ ℕ) → (𝑏↑2) = (if(0 ≤ 𝑏, 𝑏, -𝑏)↑2))
77	60, 76	jaoi 870	. . . . . . . . . 10 ⊢ ((𝑏 ∈ ℕ₀ ∨ (𝑏 ∈ ℝ ∧ -𝑏 ∈ ℕ)) → (𝑏↑2) = (if(0 ≤ 𝑏, 𝑏, -𝑏)↑2))
78	57, 77	sylbi 220	. . . . . . . . 9 ⊢ (𝑏 ∈ ℤ → (𝑏↑2) = (if(0 ≤ 𝑏, 𝑏, -𝑏)↑2))
79	56, 78	oveqan12d 7430	. . . . . . . 8 ⊢ ((𝑎 ∈ ℤ ∧ 𝑏 ∈ ℤ) → ((𝑎↑2) + (𝑏↑2)) = ((if(0 ≤ 𝑎, 𝑎, -𝑎)↑2) + (if(0 ≤ 𝑏, 𝑏, -𝑏)↑2)))
80	79	eqeq2d 2780	. . . . . . 7 ⊢ ((𝑎 ∈ ℤ ∧ 𝑏 ∈ ℤ) → (𝑃 = ((𝑎↑2) + (𝑏↑2)) ↔ 𝑃 = ((if(0 ≤ 𝑎, 𝑎, -𝑎)↑2) + (if(0 ≤ 𝑏, 𝑏, -𝑏)↑2))))
81	80	biimpd 232	. . . . . 6 ⊢ ((𝑎 ∈ ℤ ∧ 𝑏 ∈ ℤ) → (𝑃 = ((𝑎↑2) + (𝑏↑2)) → 𝑃 = ((if(0 ≤ 𝑎, 𝑎, -𝑎)↑2) + (if(0 ≤ 𝑏, 𝑏, -𝑏)↑2))))
82	81	imp 411	. . . . 5 ⊢ (((𝑎 ∈ ℤ ∧ 𝑏 ∈ ℤ) ∧ 𝑃 = ((𝑎↑2) + (𝑏↑2))) → 𝑃 = ((if(0 ≤ 𝑎, 𝑎, -𝑎)↑2) + (if(0 ≤ 𝑏, 𝑏, -𝑏)↑2)))
83	4, 7, 21, 34, 82	2rspcedvdw 3604	. . . 4 ⊢ (((𝑎 ∈ ℤ ∧ 𝑏 ∈ ℤ) ∧ 𝑃 = ((𝑎↑2) + (𝑏↑2))) → ∃𝑥 ∈ ℕ₀ ∃𝑦 ∈ ℕ₀ 𝑃 = ((𝑥↑2) + (𝑦↑2)))
84	83	ex 417	. . 3 ⊢ ((𝑎 ∈ ℤ ∧ 𝑏 ∈ ℤ) → (𝑃 = ((𝑎↑2) + (𝑏↑2)) → ∃𝑥 ∈ ℕ₀ ∃𝑦 ∈ ℕ₀ 𝑃 = ((𝑥↑2) + (𝑦↑2))))
85	84	rexlimivv 3213	. 2 ⊢ (∃𝑎 ∈ ℤ ∃𝑏 ∈ ℤ 𝑃 = ((𝑎↑2) + (𝑏↑2)) → ∃𝑥 ∈ ℕ₀ ∃𝑦 ∈ ℕ₀ 𝑃 = ((𝑥↑2) + (𝑦↑2)))
86	1, 85	syl 18	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 400 ∨ wo 860 = wceq 1567 ∈ wcel 2149 ∃wrex 3095 ifcif 4492 class class class wbr 5113 (class class class)co 7411 ℝcr 11099 0cc0 11100 1c1 11101 + caddc 11103 < clt 11243 ≤ cle 11244 -cneg 11442 ℕcn 12233 2c2 12295 4c4 12297 ℕ₀cn0 12504 ℤcz 12591 mod cmo 13902 ↑cexp 14097 ℙcprime 16729

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1822 ax-4 1836 ax-5 1937 ax-6 1994 ax-7 2035 ax-8 2151 ax-9 2159 ax-10 2182 ax-11 2198 ax-12 2219 ax-ext 2741 ax-rep 5242 ax-sep 5261 ax-nul 5271 ax-pow 5337 ax-pr 5405 ax-un 7733 ax-cnex 11156 ax-resscn 11157 ax-1cn 11158 ax-icn 11159 ax-addcl 11160 ax-addrcl 11161 ax-mulcl 11162 ax-mulrcl 11163 ax-mulcom 11164 ax-addass 11165 ax-mulass 11166 ax-distr 11167 ax-i2m1 11168 ax-1ne0 11169 ax-1rid 11170 ax-rnegex 11171 ax-rrecex 11172 ax-cnre 11173 ax-pre-lttri 11174 ax-pre-lttrn 11175 ax-pre-ltadd 11176 ax-pre-mulgt0 11177 ax-pre-sup 11178 ax-addf 11179 ax-mulf 11180

This theorem depends on definitions: df-bi 210 df-an 401 df-or 861 df-3or 1102 df-3an 1103 df-tru 1570 df-fal 1580 df-ex 1807 df-nf 1811 df-sb 2098 df-mo 2573 df-eu 2603 df-clab 2748 df-cleq 2761 df-clel 2844 df-nfc 2918 df-ne 2965 df-nel 3071 df-ral 3086 df-rex 3096 df-rmo 3376 df-reu 3377 df-rab 3424 df-v 3465 df-sbc 3754 df-csb 3862 df-dif 3916 df-un 3918 df-in 3920 df-ss 3930 df-pss 3933 df-nul 4295 df-if 4493 df-pw 4569 df-sn 4595 df-pr 4597 df-tp 4599 df-op 4601 df-uni 4877 df-int 4917 df-iun 4962 df-iin 4963 df-br 5114 df-opab 5178 df-mpt 5197 df-tr 5223 df-id 5557 df-eprel 5562 df-po 5570 df-so 5571 df-fr 5615 df-se 5616 df-we 5617 df-xp 5668 df-rel 5669 df-cnv 5670 df-co 5671 df-dm 5672 df-rn 5673 df-res 5674 df-ima 5675 df-pred 6303 df-ord 6364 df-on 6365 df-lim 6366 df-suc 6367 df-iota 6493 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-isom 6546 df-riota 7368 df-ov 7414 df-oprab 7415 df-mpo 7416 df-of 7675 df-ofr 7676 df-om 7863 df-1st 7986 df-2nd 7987 df-supp 8157 df-tpos 8222 df-frecs 8278 df-wrecs 8309 df-recs 8358 df-rdg 8397 df-1o 8453 df-2o 8454 df-oadd 8457 df-er 8694 df-ec 8696 df-qs 8700 df-map 8826 df-pm 8827 df-ixp 8896 df-en 8944 df-dom 8945 df-sdom 8946 df-fin 8947 df-fsupp 9322 df-sup 9402 df-inf 9403 df-oi 9472 df-dju 9887 df-card 9925 df-pnf 11245 df-mnf 11246 df-xr 11247 df-ltxr 11248 df-le 11249 df-sub 11443 df-neg 11444 df-div 11872 df-nn 12234 df-2 12303 df-3 12304 df-4 12305 df-5 12306 df-6 12307 df-7 12308 df-8 12309 df-9 12310 df-n0 12505 df-xnn0 12578 df-z 12592 df-dec 12712 df-uz 12863 df-q 12973 df-rp 13017 df-fz 13536 df-fzo 13683 df-fl 13825 df-mod 13903 df-seq 14038 df-exp 14098 df-hash 14367 df-cj 15150 df-re 15151 df-im 15152 df-sqrt 15286 df-abs 15287 df-dvds 16311 df-gcd 16553 df-prm 16730 df-phi 16825 df-pc 16897 df-gz 16990 df-struct 17207 df-sets 17224 df-slot 17242 df-ndx 17254 df-base 17270 df-ress 17291 df-plusg 17323 df-mulr 17324 df-starv 17325 df-sca 17326 df-vsca 17327 df-ip 17328 df-tset 17329 df-ple 17330 df-ds 17332 df-unif 17333 df-hom 17334 df-cco 17335 df-0g 17494 df-gsum 17495 df-prds 17500 df-pws 17502 df-imas 17562 df-qus 17563 df-mre 17638 df-mrc 17639 df-acs 17641 df-mgm 18698 df-sgrp 18777 df-mnd 18793 df-mhm 18841 df-submnd 18842 df-grp 19003 df-minusg 19004 df-sbg 19005 df-mulg 19134 df-subg 19189 df-nsg 19190 df-eqg 19191 df-ghm 19284 df-cntz 19387 df-cmn 19852 df-abl 19853 df-mgp 20217 df-rng 20231 df-ur 20264 df-srg 20269 df-ring 20317 df-cring 20318 df-oppr 20419 df-dvdsr 20439 df-unit 20440 df-invr 20470 df-dvr 20483 df-rhm 20554 df-nzr 20596 df-subrng 20631 df-subrg 20655 df-rlreg 20779 df-domn 20780 df-idom 20781 df-drng 20815 df-field 20816 df-lmod 20961 df-lss 21031 df-lsp 21071 df-sra 21272 df-rgmod 21273 df-lidl 21310 df-rsp 21311 df-2idl 21360 df-cnfld 21492 df-zring 21566 df-zrh 21622 df-zn 21625 df-assa 21972 df-asp 21973 df-ascl 21974 df-psr 22028 df-mvr 22029 df-mpl 22030 df-opsr 22032 df-evls 22194 df-evl 22195 df-psr1 22309 df-vr1 22310 df-ply1 22311 df-coe1 22312 df-evl1 22445 df-mdeg 26181 df-deg1 26182 df-mon1 26257 df-uc1p 26258 df-q1p 26259 df-r1p 26260 df-lgs 27425

This theorem is referenced by: 2sqnn 27569 2sqreulem1 27576

W3C validator