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Theorem 2sqnn 26274

Description: All primes of the form 4𝑘 + 1 are sums of squares of two positive integers. (Contributed by AV, 11-Jun-2023.)

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

Distinct variable group: 𝑥,𝑃,𝑦

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

Step	Hyp	Ref	Expression
1		2sqnn0 26273	. 2 ⊢ ((𝑃 ∈ ℙ ∧ (𝑃 mod 4) = 1) → ∃𝑎 ∈ ℕ₀ ∃𝑏 ∈ ℕ₀ 𝑃 = ((𝑎↑2) + (𝑏↑2)))
2		elnn0 12057	. . . . . . 7 ⊢ (𝑎 ∈ ℕ₀ ↔ (𝑎 ∈ ℕ ∨ 𝑎 = 0))
3		elnn0 12057	. . . . . . . . 9 ⊢ (𝑏 ∈ ℕ₀ ↔ (𝑏 ∈ ℕ ∨ 𝑏 = 0))
4		oveq1 7198	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 = 𝑎 → (𝑥↑2) = (𝑎↑2))
5	4	oveq1d 7206	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = 𝑎 → ((𝑥↑2) + (𝑦↑2)) = ((𝑎↑2) + (𝑦↑2)))
6	5	eqeq2d 2747	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = 𝑎 → (𝑃 = ((𝑥↑2) + (𝑦↑2)) ↔ 𝑃 = ((𝑎↑2) + (𝑦↑2))))
7		oveq1 7198	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 = 𝑏 → (𝑦↑2) = (𝑏↑2))
8	7	oveq2d 7207	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 = 𝑏 → ((𝑎↑2) + (𝑦↑2)) = ((𝑎↑2) + (𝑏↑2)))
9	8	eqeq2d 2747	. . . . . . . . . . . . . . 15 ⊢ (𝑦 = 𝑏 → (𝑃 = ((𝑎↑2) + (𝑦↑2)) ↔ 𝑃 = ((𝑎↑2) + (𝑏↑2))))
10	6, 9	rspc2ev 3539	. . . . . . . . . . . . . 14 ⊢ ((𝑎 ∈ ℕ ∧ 𝑏 ∈ ℕ ∧ 𝑃 = ((𝑎↑2) + (𝑏↑2))) → ∃𝑥 ∈ ℕ ∃𝑦 ∈ ℕ 𝑃 = ((𝑥↑2) + (𝑦↑2)))
11	10	3expia 1123	. . . . . . . . . . . . 13 ⊢ ((𝑎 ∈ ℕ ∧ 𝑏 ∈ ℕ) → (𝑃 = ((𝑎↑2) + (𝑏↑2)) → ∃𝑥 ∈ ℕ ∃𝑦 ∈ ℕ 𝑃 = ((𝑥↑2) + (𝑦↑2))))
12	11	a1d 25	. . . . . . . . . . . 12 ⊢ ((𝑎 ∈ ℕ ∧ 𝑏 ∈ ℕ) → (𝑃 ∈ ℙ → (𝑃 = ((𝑎↑2) + (𝑏↑2)) → ∃𝑥 ∈ ℕ ∃𝑦 ∈ ℕ 𝑃 = ((𝑥↑2) + (𝑦↑2)))))
13	12	expcom 417	. . . . . . . . . . 11 ⊢ (𝑏 ∈ ℕ → (𝑎 ∈ ℕ → (𝑃 ∈ ℙ → (𝑃 = ((𝑎↑2) + (𝑏↑2)) → ∃𝑥 ∈ ℕ ∃𝑦 ∈ ℕ 𝑃 = ((𝑥↑2) + (𝑦↑2))))))
14		sq0i 13727	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑎 = 0 → (𝑎↑2) = 0)
15	14	adantr 484	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑎 = 0 ∧ 𝑏 ∈ ℕ) → (𝑎↑2) = 0)
16	15	oveq1d 7206	. . . . . . . . . . . . . . . 16 ⊢ ((𝑎 = 0 ∧ 𝑏 ∈ ℕ) → ((𝑎↑2) + (𝑏↑2)) = (0 + (𝑏↑2)))
17		nncn 11803	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑏 ∈ ℕ → 𝑏 ∈ ℂ)
18	17	sqcld 13679	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑏 ∈ ℕ → (𝑏↑2) ∈ ℂ)
19	18	addid2d 10998	. . . . . . . . . . . . . . . . 17 ⊢ (𝑏 ∈ ℕ → (0 + (𝑏↑2)) = (𝑏↑2))
20	19	adantl 485	. . . . . . . . . . . . . . . 16 ⊢ ((𝑎 = 0 ∧ 𝑏 ∈ ℕ) → (0 + (𝑏↑2)) = (𝑏↑2))
21	16, 20	eqtrd 2771	. . . . . . . . . . . . . . 15 ⊢ ((𝑎 = 0 ∧ 𝑏 ∈ ℕ) → ((𝑎↑2) + (𝑏↑2)) = (𝑏↑2))
22	21	eqeq2d 2747	. . . . . . . . . . . . . 14 ⊢ ((𝑎 = 0 ∧ 𝑏 ∈ ℕ) → (𝑃 = ((𝑎↑2) + (𝑏↑2)) ↔ 𝑃 = (𝑏↑2)))
23		eleq1 2818	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑃 = (𝑏↑2) → (𝑃 ∈ ℙ ↔ (𝑏↑2) ∈ ℙ))
24	23	adantl 485	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑏 ∈ ℕ ∧ 𝑃 = (𝑏↑2)) → (𝑃 ∈ ℙ ↔ (𝑏↑2) ∈ ℙ))
25		nnz 12164	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑏 ∈ ℕ → 𝑏 ∈ ℤ)
26		sqnprm 16222	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑏 ∈ ℤ → ¬ (𝑏↑2) ∈ ℙ)
27	25, 26	syl 17	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑏 ∈ ℕ → ¬ (𝑏↑2) ∈ ℙ)
28	27	pm2.21d 121	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑏 ∈ ℕ → ((𝑏↑2) ∈ ℙ → ∃𝑥 ∈ ℕ ∃𝑦 ∈ ℕ 𝑃 = ((𝑥↑2) + (𝑦↑2))))
29	28	adantr 484	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑏 ∈ ℕ ∧ 𝑃 = (𝑏↑2)) → ((𝑏↑2) ∈ ℙ → ∃𝑥 ∈ ℕ ∃𝑦 ∈ ℕ 𝑃 = ((𝑥↑2) + (𝑦↑2))))
30	24, 29	sylbid 243	. . . . . . . . . . . . . . . 16 ⊢ ((𝑏 ∈ ℕ ∧ 𝑃 = (𝑏↑2)) → (𝑃 ∈ ℙ → ∃𝑥 ∈ ℕ ∃𝑦 ∈ ℕ 𝑃 = ((𝑥↑2) + (𝑦↑2))))
31	30	ex 416	. . . . . . . . . . . . . . 15 ⊢ (𝑏 ∈ ℕ → (𝑃 = (𝑏↑2) → (𝑃 ∈ ℙ → ∃𝑥 ∈ ℕ ∃𝑦 ∈ ℕ 𝑃 = ((𝑥↑2) + (𝑦↑2)))))
32	31	adantl 485	. . . . . . . . . . . . . 14 ⊢ ((𝑎 = 0 ∧ 𝑏 ∈ ℕ) → (𝑃 = (𝑏↑2) → (𝑃 ∈ ℙ → ∃𝑥 ∈ ℕ ∃𝑦 ∈ ℕ 𝑃 = ((𝑥↑2) + (𝑦↑2)))))
33	22, 32	sylbid 243	. . . . . . . . . . . . 13 ⊢ ((𝑎 = 0 ∧ 𝑏 ∈ ℕ) → (𝑃 = ((𝑎↑2) + (𝑏↑2)) → (𝑃 ∈ ℙ → ∃𝑥 ∈ ℕ ∃𝑦 ∈ ℕ 𝑃 = ((𝑥↑2) + (𝑦↑2)))))
34	33	com23 86	. . . . . . . . . . . 12 ⊢ ((𝑎 = 0 ∧ 𝑏 ∈ ℕ) → (𝑃 ∈ ℙ → (𝑃 = ((𝑎↑2) + (𝑏↑2)) → ∃𝑥 ∈ ℕ ∃𝑦 ∈ ℕ 𝑃 = ((𝑥↑2) + (𝑦↑2)))))
35	34	expcom 417	. . . . . . . . . . 11 ⊢ (𝑏 ∈ ℕ → (𝑎 = 0 → (𝑃 ∈ ℙ → (𝑃 = ((𝑎↑2) + (𝑏↑2)) → ∃𝑥 ∈ ℕ ∃𝑦 ∈ ℕ 𝑃 = ((𝑥↑2) + (𝑦↑2))))))
36	13, 35	jaod 859	. . . . . . . . . 10 ⊢ (𝑏 ∈ ℕ → ((𝑎 ∈ ℕ ∨ 𝑎 = 0) → (𝑃 ∈ ℙ → (𝑃 = ((𝑎↑2) + (𝑏↑2)) → ∃𝑥 ∈ ℕ ∃𝑦 ∈ ℕ 𝑃 = ((𝑥↑2) + (𝑦↑2))))))
37		sq0i 13727	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑏 = 0 → (𝑏↑2) = 0)
38	37	adantr 484	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑏 = 0 ∧ 𝑎 ∈ ℕ) → (𝑏↑2) = 0)
39	38	oveq2d 7207	. . . . . . . . . . . . . . . 16 ⊢ ((𝑏 = 0 ∧ 𝑎 ∈ ℕ) → ((𝑎↑2) + (𝑏↑2)) = ((𝑎↑2) + 0))
40		nncn 11803	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑎 ∈ ℕ → 𝑎 ∈ ℂ)
41	40	sqcld 13679	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑎 ∈ ℕ → (𝑎↑2) ∈ ℂ)
42	41	addid1d 10997	. . . . . . . . . . . . . . . . 17 ⊢ (𝑎 ∈ ℕ → ((𝑎↑2) + 0) = (𝑎↑2))
43	42	adantl 485	. . . . . . . . . . . . . . . 16 ⊢ ((𝑏 = 0 ∧ 𝑎 ∈ ℕ) → ((𝑎↑2) + 0) = (𝑎↑2))
44	39, 43	eqtrd 2771	. . . . . . . . . . . . . . 15 ⊢ ((𝑏 = 0 ∧ 𝑎 ∈ ℕ) → ((𝑎↑2) + (𝑏↑2)) = (𝑎↑2))
45	44	eqeq2d 2747	. . . . . . . . . . . . . 14 ⊢ ((𝑏 = 0 ∧ 𝑎 ∈ ℕ) → (𝑃 = ((𝑎↑2) + (𝑏↑2)) ↔ 𝑃 = (𝑎↑2)))
46		eleq1 2818	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑃 = (𝑎↑2) → (𝑃 ∈ ℙ ↔ (𝑎↑2) ∈ ℙ))
47	46	adantl 485	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑎 ∈ ℕ ∧ 𝑃 = (𝑎↑2)) → (𝑃 ∈ ℙ ↔ (𝑎↑2) ∈ ℙ))
48		nnz 12164	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑎 ∈ ℕ → 𝑎 ∈ ℤ)
49		sqnprm 16222	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑎 ∈ ℤ → ¬ (𝑎↑2) ∈ ℙ)
50	48, 49	syl 17	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑎 ∈ ℕ → ¬ (𝑎↑2) ∈ ℙ)
51	50	pm2.21d 121	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑎 ∈ ℕ → ((𝑎↑2) ∈ ℙ → ∃𝑥 ∈ ℕ ∃𝑦 ∈ ℕ 𝑃 = ((𝑥↑2) + (𝑦↑2))))
52	51	adantr 484	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑎 ∈ ℕ ∧ 𝑃 = (𝑎↑2)) → ((𝑎↑2) ∈ ℙ → ∃𝑥 ∈ ℕ ∃𝑦 ∈ ℕ 𝑃 = ((𝑥↑2) + (𝑦↑2))))
53	47, 52	sylbid 243	. . . . . . . . . . . . . . . 16 ⊢ ((𝑎 ∈ ℕ ∧ 𝑃 = (𝑎↑2)) → (𝑃 ∈ ℙ → ∃𝑥 ∈ ℕ ∃𝑦 ∈ ℕ 𝑃 = ((𝑥↑2) + (𝑦↑2))))
54	53	ex 416	. . . . . . . . . . . . . . 15 ⊢ (𝑎 ∈ ℕ → (𝑃 = (𝑎↑2) → (𝑃 ∈ ℙ → ∃𝑥 ∈ ℕ ∃𝑦 ∈ ℕ 𝑃 = ((𝑥↑2) + (𝑦↑2)))))
55	54	adantl 485	. . . . . . . . . . . . . 14 ⊢ ((𝑏 = 0 ∧ 𝑎 ∈ ℕ) → (𝑃 = (𝑎↑2) → (𝑃 ∈ ℙ → ∃𝑥 ∈ ℕ ∃𝑦 ∈ ℕ 𝑃 = ((𝑥↑2) + (𝑦↑2)))))
56	45, 55	sylbid 243	. . . . . . . . . . . . 13 ⊢ ((𝑏 = 0 ∧ 𝑎 ∈ ℕ) → (𝑃 = ((𝑎↑2) + (𝑏↑2)) → (𝑃 ∈ ℙ → ∃𝑥 ∈ ℕ ∃𝑦 ∈ ℕ 𝑃 = ((𝑥↑2) + (𝑦↑2)))))
57	56	com23 86	. . . . . . . . . . . 12 ⊢ ((𝑏 = 0 ∧ 𝑎 ∈ ℕ) → (𝑃 ∈ ℙ → (𝑃 = ((𝑎↑2) + (𝑏↑2)) → ∃𝑥 ∈ ℕ ∃𝑦 ∈ ℕ 𝑃 = ((𝑥↑2) + (𝑦↑2)))))
58	57	ex 416	. . . . . . . . . . 11 ⊢ (𝑏 = 0 → (𝑎 ∈ ℕ → (𝑃 ∈ ℙ → (𝑃 = ((𝑎↑2) + (𝑏↑2)) → ∃𝑥 ∈ ℕ ∃𝑦 ∈ ℕ 𝑃 = ((𝑥↑2) + (𝑦↑2))))))
59	14, 37	oveqan12rd 7211	. . . . . . . . . . . . . . . 16 ⊢ ((𝑏 = 0 ∧ 𝑎 = 0) → ((𝑎↑2) + (𝑏↑2)) = (0 + 0))
60		00id 10972	. . . . . . . . . . . . . . . 16 ⊢ (0 + 0) = 0
61	59, 60	eqtrdi 2787	. . . . . . . . . . . . . . 15 ⊢ ((𝑏 = 0 ∧ 𝑎 = 0) → ((𝑎↑2) + (𝑏↑2)) = 0)
62	61	eqeq2d 2747	. . . . . . . . . . . . . 14 ⊢ ((𝑏 = 0 ∧ 𝑎 = 0) → (𝑃 = ((𝑎↑2) + (𝑏↑2)) ↔ 𝑃 = 0))
63		eleq1 2818	. . . . . . . . . . . . . . 15 ⊢ (𝑃 = 0 → (𝑃 ∈ ℙ ↔ 0 ∈ ℙ))
64		0nprm 16198	. . . . . . . . . . . . . . . 16 ⊢ ¬ 0 ∈ ℙ
65	64	pm2.21i 119	. . . . . . . . . . . . . . 15 ⊢ (0 ∈ ℙ → ∃𝑥 ∈ ℕ ∃𝑦 ∈ ℕ 𝑃 = ((𝑥↑2) + (𝑦↑2)))
66	63, 65	syl6bi 256	. . . . . . . . . . . . . 14 ⊢ (𝑃 = 0 → (𝑃 ∈ ℙ → ∃𝑥 ∈ ℕ ∃𝑦 ∈ ℕ 𝑃 = ((𝑥↑2) + (𝑦↑2))))
67	62, 66	syl6bi 256	. . . . . . . . . . . . 13 ⊢ ((𝑏 = 0 ∧ 𝑎 = 0) → (𝑃 = ((𝑎↑2) + (𝑏↑2)) → (𝑃 ∈ ℙ → ∃𝑥 ∈ ℕ ∃𝑦 ∈ ℕ 𝑃 = ((𝑥↑2) + (𝑦↑2)))))
68	67	com23 86	. . . . . . . . . . . 12 ⊢ ((𝑏 = 0 ∧ 𝑎 = 0) → (𝑃 ∈ ℙ → (𝑃 = ((𝑎↑2) + (𝑏↑2)) → ∃𝑥 ∈ ℕ ∃𝑦 ∈ ℕ 𝑃 = ((𝑥↑2) + (𝑦↑2)))))
69	68	ex 416	. . . . . . . . . . 11 ⊢ (𝑏 = 0 → (𝑎 = 0 → (𝑃 ∈ ℙ → (𝑃 = ((𝑎↑2) + (𝑏↑2)) → ∃𝑥 ∈ ℕ ∃𝑦 ∈ ℕ 𝑃 = ((𝑥↑2) + (𝑦↑2))))))
70	58, 69	jaod 859	. . . . . . . . . 10 ⊢ (𝑏 = 0 → ((𝑎 ∈ ℕ ∨ 𝑎 = 0) → (𝑃 ∈ ℙ → (𝑃 = ((𝑎↑2) + (𝑏↑2)) → ∃𝑥 ∈ ℕ ∃𝑦 ∈ ℕ 𝑃 = ((𝑥↑2) + (𝑦↑2))))))
71	36, 70	jaoi 857	. . . . . . . . 9 ⊢ ((𝑏 ∈ ℕ ∨ 𝑏 = 0) → ((𝑎 ∈ ℕ ∨ 𝑎 = 0) → (𝑃 ∈ ℙ → (𝑃 = ((𝑎↑2) + (𝑏↑2)) → ∃𝑥 ∈ ℕ ∃𝑦 ∈ ℕ 𝑃 = ((𝑥↑2) + (𝑦↑2))))))
72	3, 71	sylbi 220	. . . . . . . 8 ⊢ (𝑏 ∈ ℕ₀ → ((𝑎 ∈ ℕ ∨ 𝑎 = 0) → (𝑃 ∈ ℙ → (𝑃 = ((𝑎↑2) + (𝑏↑2)) → ∃𝑥 ∈ ℕ ∃𝑦 ∈ ℕ 𝑃 = ((𝑥↑2) + (𝑦↑2))))))
73	72	com12 32	. . . . . . 7 ⊢ ((𝑎 ∈ ℕ ∨ 𝑎 = 0) → (𝑏 ∈ ℕ₀ → (𝑃 ∈ ℙ → (𝑃 = ((𝑎↑2) + (𝑏↑2)) → ∃𝑥 ∈ ℕ ∃𝑦 ∈ ℕ 𝑃 = ((𝑥↑2) + (𝑦↑2))))))
74	2, 73	sylbi 220	. . . . . 6 ⊢ (𝑎 ∈ ℕ₀ → (𝑏 ∈ ℕ₀ → (𝑃 ∈ ℙ → (𝑃 = ((𝑎↑2) + (𝑏↑2)) → ∃𝑥 ∈ ℕ ∃𝑦 ∈ ℕ 𝑃 = ((𝑥↑2) + (𝑦↑2))))))
75	74	imp 410	. . . . 5 ⊢ ((𝑎 ∈ ℕ₀ ∧ 𝑏 ∈ ℕ₀) → (𝑃 ∈ ℙ → (𝑃 = ((𝑎↑2) + (𝑏↑2)) → ∃𝑥 ∈ ℕ ∃𝑦 ∈ ℕ 𝑃 = ((𝑥↑2) + (𝑦↑2)))))
76	75	com12 32	. . . 4 ⊢ (𝑃 ∈ ℙ → ((𝑎 ∈ ℕ₀ ∧ 𝑏 ∈ ℕ₀) → (𝑃 = ((𝑎↑2) + (𝑏↑2)) → ∃𝑥 ∈ ℕ ∃𝑦 ∈ ℕ 𝑃 = ((𝑥↑2) + (𝑦↑2)))))
77	76	adantr 484	. . 3 ⊢ ((𝑃 ∈ ℙ ∧ (𝑃 mod 4) = 1) → ((𝑎 ∈ ℕ₀ ∧ 𝑏 ∈ ℕ₀) → (𝑃 = ((𝑎↑2) + (𝑏↑2)) → ∃𝑥 ∈ ℕ ∃𝑦 ∈ ℕ 𝑃 = ((𝑥↑2) + (𝑦↑2)))))
78	77	rexlimdvv 3202	. 2 ⊢ ((𝑃 ∈ ℙ ∧ (𝑃 mod 4) = 1) → (∃𝑎 ∈ ℕ₀ ∃𝑏 ∈ ℕ₀ 𝑃 = ((𝑎↑2) + (𝑏↑2)) → ∃𝑥 ∈ ℕ ∃𝑦 ∈ ℕ 𝑃 = ((𝑥↑2) + (𝑦↑2))))
79	1, 78	mpd 15	1 ⊢ ((𝑃 ∈ ℙ ∧ (𝑃 mod 4) = 1) → ∃𝑥 ∈ ℕ ∃𝑦 ∈ ℕ 𝑃 = ((𝑥↑2) + (𝑦↑2)))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 209 ∧ wa 399 ∨ wo 847 = wceq 1543 ∈ wcel 2112 ∃wrex 3052 (class class class)co 7191 0cc0 10694 1c1 10695 + caddc 10697 ℕcn 11795 2c2 11850 4c4 11852 ℕ₀cn0 12055 ℤcz 12141 mod cmo 13407 ↑cexp 13600 ℙcprime 16191

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2018 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2160 ax-12 2177 ax-ext 2708 ax-rep 5164 ax-sep 5177 ax-nul 5184 ax-pow 5243 ax-pr 5307 ax-un 7501 ax-cnex 10750 ax-resscn 10751 ax-1cn 10752 ax-icn 10753 ax-addcl 10754 ax-addrcl 10755 ax-mulcl 10756 ax-mulrcl 10757 ax-mulcom 10758 ax-addass 10759 ax-mulass 10760 ax-distr 10761 ax-i2m1 10762 ax-1ne0 10763 ax-1rid 10764 ax-rnegex 10765 ax-rrecex 10766 ax-cnre 10767 ax-pre-lttri 10768 ax-pre-lttrn 10769 ax-pre-ltadd 10770 ax-pre-mulgt0 10771 ax-pre-sup 10772 ax-addf 10773 ax-mulf 10774

This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2073 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2728 df-clel 2809 df-nfc 2879 df-ne 2933 df-nel 3037 df-ral 3056 df-rex 3057 df-reu 3058 df-rmo 3059 df-rab 3060 df-v 3400 df-sbc 3684 df-csb 3799 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-pss 3872 df-nul 4224 df-if 4426 df-pw 4501 df-sn 4528 df-pr 4530 df-tp 4532 df-op 4534 df-uni 4806 df-int 4846 df-iun 4892 df-iin 4893 df-br 5040 df-opab 5102 df-mpt 5121 df-tr 5147 df-id 5440 df-eprel 5445 df-po 5453 df-so 5454 df-fr 5494 df-se 5495 df-we 5496 df-xp 5542 df-rel 5543 df-cnv 5544 df-co 5545 df-dm 5546 df-rn 5547 df-res 5548 df-ima 5549 df-pred 6140 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6316 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-isom 6367 df-riota 7148 df-ov 7194 df-oprab 7195 df-mpo 7196 df-of 7447 df-ofr 7448 df-om 7623 df-1st 7739 df-2nd 7740 df-supp 7882 df-tpos 7946 df-wrecs 8025 df-recs 8086 df-rdg 8124 df-1o 8180 df-2o 8181 df-oadd 8184 df-er 8369 df-ec 8371 df-qs 8375 df-map 8488 df-pm 8489 df-ixp 8557 df-en 8605 df-dom 8606 df-sdom 8607 df-fin 8608 df-fsupp 8964 df-sup 9036 df-inf 9037 df-oi 9104 df-dju 9482 df-card 9520 df-pnf 10834 df-mnf 10835 df-xr 10836 df-ltxr 10837 df-le 10838 df-sub 11029 df-neg 11030 df-div 11455 df-nn 11796 df-2 11858 df-3 11859 df-4 11860 df-5 11861 df-6 11862 df-7 11863 df-8 11864 df-9 11865 df-n0 12056 df-xnn0 12128 df-z 12142 df-dec 12259 df-uz 12404 df-q 12510 df-rp 12552 df-fz 13061 df-fzo 13204 df-fl 13332 df-mod 13408 df-seq 13540 df-exp 13601 df-hash 13862 df-cj 14627 df-re 14628 df-im 14629 df-sqrt 14763 df-abs 14764 df-dvds 15779 df-gcd 16017 df-prm 16192 df-phi 16282 df-pc 16353 df-gz 16446 df-struct 16668 df-ndx 16669 df-slot 16670 df-base 16672 df-sets 16673 df-ress 16674 df-plusg 16762 df-mulr 16763 df-starv 16764 df-sca 16765 df-vsca 16766 df-ip 16767 df-tset 16768 df-ple 16769 df-ds 16771 df-unif 16772 df-hom 16773 df-cco 16774 df-0g 16900 df-gsum 16901 df-prds 16906 df-pws 16908 df-imas 16967 df-qus 16968 df-mre 17043 df-mrc 17044 df-acs 17046 df-mgm 18068 df-sgrp 18117 df-mnd 18128 df-mhm 18172 df-submnd 18173 df-grp 18322 df-minusg 18323 df-sbg 18324 df-mulg 18443 df-subg 18494 df-nsg 18495 df-eqg 18496 df-ghm 18574 df-cntz 18665 df-cmn 19126 df-abl 19127 df-mgp 19459 df-ur 19471 df-srg 19475 df-ring 19518 df-cring 19519 df-oppr 19595 df-dvdsr 19613 df-unit 19614 df-invr 19644 df-dvr 19655 df-rnghom 19689 df-drng 19723 df-field 19724 df-subrg 19752 df-lmod 19855 df-lss 19923 df-lsp 19963 df-sra 20163 df-rgmod 20164 df-lidl 20165 df-rsp 20166 df-2idl 20224 df-nzr 20250 df-rlreg 20275 df-domn 20276 df-idom 20277 df-cnfld 20318 df-zring 20390 df-zrh 20424 df-zn 20427 df-assa 20769 df-asp 20770 df-ascl 20771 df-psr 20822 df-mvr 20823 df-mpl 20824 df-opsr 20826 df-evls 20986 df-evl 20987 df-psr1 21055 df-vr1 21056 df-ply1 21057 df-coe1 21058 df-evl1 21186 df-mdeg 24904 df-deg1 24905 df-mon1 24982 df-uc1p 24983 df-q1p 24984 df-r1p 24985 df-lgs 26130

This theorem is referenced by: 2sqreunnlem1 26284

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