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Theorem 2sqnn 27603

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 27602	. 2 ⊢ ((𝑃 ∈ ℙ ∧ (𝑃 mod 4) = 1) → ∃𝑎 ∈ ℕ₀ ∃𝑏 ∈ ℕ₀ 𝑃 = ((𝑎↑2) + (𝑏↑2)))
2		elnn0 12501	. . . . . . 7 ⊢ (𝑎 ∈ ℕ₀ ↔ (𝑎 ∈ ℕ ∨ 𝑎 = 0))
3		elnn0 12501	. . . . . . . . 9 ⊢ (𝑏 ∈ ℕ₀ ↔ (𝑏 ∈ ℕ ∨ 𝑏 = 0))
4		oveq1 7417	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 = 𝑎 → (𝑥↑2) = (𝑎↑2))
5	4	oveq1d 7425	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = 𝑎 → ((𝑥↑2) + (𝑦↑2)) = ((𝑎↑2) + (𝑦↑2)))
6	5	eqeq2d 2774	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = 𝑎 → (𝑃 = ((𝑥↑2) + (𝑦↑2)) ↔ 𝑃 = ((𝑎↑2) + (𝑦↑2))))
7		oveq1 7417	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 = 𝑏 → (𝑦↑2) = (𝑏↑2))
8	7	oveq2d 7426	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 = 𝑏 → ((𝑎↑2) + (𝑦↑2)) = ((𝑎↑2) + (𝑏↑2)))
9	8	eqeq2d 2774	. . . . . . . . . . . . . . 15 ⊢ (𝑦 = 𝑏 → (𝑃 = ((𝑎↑2) + (𝑦↑2)) ↔ 𝑃 = ((𝑎↑2) + (𝑏↑2))))
10	6, 9	rspc2ev 3594	. . . . . . . . . . . . . 14 ⊢ ((𝑎 ∈ ℕ ∧ 𝑏 ∈ ℕ ∧ 𝑃 = ((𝑎↑2) + (𝑏↑2))) → ∃𝑥 ∈ ℕ ∃𝑦 ∈ ℕ 𝑃 = ((𝑥↑2) + (𝑦↑2)))
11	10	3expia 1139	. . . . . . . . . . . . 13 ⊢ ((𝑎 ∈ ℕ ∧ 𝑏 ∈ ℕ) → (𝑃 = ((𝑎↑2) + (𝑏↑2)) → ∃𝑥 ∈ ℕ ∃𝑦 ∈ ℕ 𝑃 = ((𝑥↑2) + (𝑦↑2))))
12	11	a1d 26	. . . . . . . . . . . 12 ⊢ ((𝑎 ∈ ℕ ∧ 𝑏 ∈ ℕ) → (𝑃 ∈ ℙ → (𝑃 = ((𝑎↑2) + (𝑏↑2)) → ∃𝑥 ∈ ℕ ∃𝑦 ∈ ℕ 𝑃 = ((𝑥↑2) + (𝑦↑2)))))
13	12	expcom 418	. . . . . . . . . . 11 ⊢ (𝑏 ∈ ℕ → (𝑎 ∈ ℕ → (𝑃 ∈ ℙ → (𝑃 = ((𝑎↑2) + (𝑏↑2)) → ∃𝑥 ∈ ℕ ∃𝑦 ∈ ℕ 𝑃 = ((𝑥↑2) + (𝑦↑2))))))
14		sq0i 14225	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑎 = 0 → (𝑎↑2) = 0)
15	14	adantr 485	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑎 = 0 ∧ 𝑏 ∈ ℕ) → (𝑎↑2) = 0)
16	15	oveq1d 7425	. . . . . . . . . . . . . . . 16 ⊢ ((𝑎 = 0 ∧ 𝑏 ∈ ℕ) → ((𝑎↑2) + (𝑏↑2)) = (0 + (𝑏↑2)))
17		nncn 12236	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑏 ∈ ℕ → 𝑏 ∈ ℂ)
18	17	sqcld 14176	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑏 ∈ ℕ → (𝑏↑2) ∈ ℂ)
19	18	addlidd 11406	. . . . . . . . . . . . . . . . 17 ⊢ (𝑏 ∈ ℕ → (0 + (𝑏↑2)) = (𝑏↑2))
20	19	adantl 486	. . . . . . . . . . . . . . . 16 ⊢ ((𝑎 = 0 ∧ 𝑏 ∈ ℕ) → (0 + (𝑏↑2)) = (𝑏↑2))
21	16, 20	eqtrd 2798	. . . . . . . . . . . . . . 15 ⊢ ((𝑎 = 0 ∧ 𝑏 ∈ ℕ) → ((𝑎↑2) + (𝑏↑2)) = (𝑏↑2))
22	21	eqeq2d 2774	. . . . . . . . . . . . . 14 ⊢ ((𝑎 = 0 ∧ 𝑏 ∈ ℕ) → (𝑃 = ((𝑎↑2) + (𝑏↑2)) ↔ 𝑃 = (𝑏↑2)))
23		eleq1 2851	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑃 = (𝑏↑2) → (𝑃 ∈ ℙ ↔ (𝑏↑2) ∈ ℙ))
24	23	adantl 486	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑏 ∈ ℕ ∧ 𝑃 = (𝑏↑2)) → (𝑃 ∈ ℙ ↔ (𝑏↑2) ∈ ℙ))
25		nnz 12607	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑏 ∈ ℕ → 𝑏 ∈ ℤ)
26		sqnprm 16756	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑏 ∈ ℤ → ¬ (𝑏↑2) ∈ ℙ)
27	25, 26	syl 18	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑏 ∈ ℕ → ¬ (𝑏↑2) ∈ ℙ)
28	27	pm2.21d 122	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑏 ∈ ℕ → ((𝑏↑2) ∈ ℙ → ∃𝑥 ∈ ℕ ∃𝑦 ∈ ℕ 𝑃 = ((𝑥↑2) + (𝑦↑2))))
29	28	adantr 485	. . . . . . . . . . . . . . . . 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 417	. . . . . . . . . . . . . . 15 ⊢ (𝑏 ∈ ℕ → (𝑃 = (𝑏↑2) → (𝑃 ∈ ℙ → ∃𝑥 ∈ ℕ ∃𝑦 ∈ ℕ 𝑃 = ((𝑥↑2) + (𝑦↑2)))))
32	31	adantl 486	. . . . . . . . . . . . . 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 87	. . . . . . . . . . . 12 ⊢ ((𝑎 = 0 ∧ 𝑏 ∈ ℕ) → (𝑃 ∈ ℙ → (𝑃 = ((𝑎↑2) + (𝑏↑2)) → ∃𝑥 ∈ ℕ ∃𝑦 ∈ ℕ 𝑃 = ((𝑥↑2) + (𝑦↑2)))))
35	34	expcom 418	. . . . . . . . . . 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 872	. . . . . . . . . 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 14225	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑏 = 0 → (𝑏↑2) = 0)
38	37	adantr 485	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑏 = 0 ∧ 𝑎 ∈ ℕ) → (𝑏↑2) = 0)
39	38	oveq2d 7426	. . . . . . . . . . . . . . . 16 ⊢ ((𝑏 = 0 ∧ 𝑎 ∈ ℕ) → ((𝑎↑2) + (𝑏↑2)) = ((𝑎↑2) + 0))
40		nncn 12236	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑎 ∈ ℕ → 𝑎 ∈ ℂ)
41	40	sqcld 14176	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑎 ∈ ℕ → (𝑎↑2) ∈ ℂ)
42	41	addridd 11405	. . . . . . . . . . . . . . . . 17 ⊢ (𝑎 ∈ ℕ → ((𝑎↑2) + 0) = (𝑎↑2))
43	42	adantl 486	. . . . . . . . . . . . . . . 16 ⊢ ((𝑏 = 0 ∧ 𝑎 ∈ ℕ) → ((𝑎↑2) + 0) = (𝑎↑2))
44	39, 43	eqtrd 2798	. . . . . . . . . . . . . . 15 ⊢ ((𝑏 = 0 ∧ 𝑎 ∈ ℕ) → ((𝑎↑2) + (𝑏↑2)) = (𝑎↑2))
45	44	eqeq2d 2774	. . . . . . . . . . . . . 14 ⊢ ((𝑏 = 0 ∧ 𝑎 ∈ ℕ) → (𝑃 = ((𝑎↑2) + (𝑏↑2)) ↔ 𝑃 = (𝑎↑2)))
46		eleq1 2851	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑃 = (𝑎↑2) → (𝑃 ∈ ℙ ↔ (𝑎↑2) ∈ ℙ))
47	46	adantl 486	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑎 ∈ ℕ ∧ 𝑃 = (𝑎↑2)) → (𝑃 ∈ ℙ ↔ (𝑎↑2) ∈ ℙ))
48		nnz 12607	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑎 ∈ ℕ → 𝑎 ∈ ℤ)
49		sqnprm 16756	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑎 ∈ ℤ → ¬ (𝑎↑2) ∈ ℙ)
50	48, 49	syl 18	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑎 ∈ ℕ → ¬ (𝑎↑2) ∈ ℙ)
51	50	pm2.21d 122	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑎 ∈ ℕ → ((𝑎↑2) ∈ ℙ → ∃𝑥 ∈ ℕ ∃𝑦 ∈ ℕ 𝑃 = ((𝑥↑2) + (𝑦↑2))))
52	51	adantr 485	. . . . . . . . . . . . . . . . 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 417	. . . . . . . . . . . . . . 15 ⊢ (𝑎 ∈ ℕ → (𝑃 = (𝑎↑2) → (𝑃 ∈ ℙ → ∃𝑥 ∈ ℕ ∃𝑦 ∈ ℕ 𝑃 = ((𝑥↑2) + (𝑦↑2)))))
55	54	adantl 486	. . . . . . . . . . . . . 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 87	. . . . . . . . . . . 12 ⊢ ((𝑏 = 0 ∧ 𝑎 ∈ ℕ) → (𝑃 ∈ ℙ → (𝑃 = ((𝑎↑2) + (𝑏↑2)) → ∃𝑥 ∈ ℕ ∃𝑦 ∈ ℕ 𝑃 = ((𝑥↑2) + (𝑦↑2)))))
58	57	ex 417	. . . . . . . . . . 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 7430	. . . . . . . . . . . . . . . 16 ⊢ ((𝑏 = 0 ∧ 𝑎 = 0) → ((𝑎↑2) + (𝑏↑2)) = (0 + 0))
60		00id 11380	. . . . . . . . . . . . . . . 16 ⊢ (0 + 0) = 0
61	59, 60	eqtrdi 2814	. . . . . . . . . . . . . . 15 ⊢ ((𝑏 = 0 ∧ 𝑎 = 0) → ((𝑎↑2) + (𝑏↑2)) = 0)
62	61	eqeq2d 2774	. . . . . . . . . . . . . 14 ⊢ ((𝑏 = 0 ∧ 𝑎 = 0) → (𝑃 = ((𝑎↑2) + (𝑏↑2)) ↔ 𝑃 = 0))
63		eleq1 2851	. . . . . . . . . . . . . . 15 ⊢ (𝑃 = 0 → (𝑃 ∈ ℙ ↔ 0 ∈ ℙ))
64		0nprm 16731	. . . . . . . . . . . . . . . 16 ⊢ ¬ 0 ∈ ℙ
65	64	pm2.21i 120	. . . . . . . . . . . . . . 15 ⊢ (0 ∈ ℙ → ∃𝑥 ∈ ℕ ∃𝑦 ∈ ℕ 𝑃 = ((𝑥↑2) + (𝑦↑2)))
66	63, 65	biimtrdi 256	. . . . . . . . . . . . . 14 ⊢ (𝑃 = 0 → (𝑃 ∈ ℙ → ∃𝑥 ∈ ℕ ∃𝑦 ∈ ℕ 𝑃 = ((𝑥↑2) + (𝑦↑2))))
67	62, 66	biimtrdi 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 87	. . . . . . . . . . . 12 ⊢ ((𝑏 = 0 ∧ 𝑎 = 0) → (𝑃 ∈ ℙ → (𝑃 = ((𝑎↑2) + (𝑏↑2)) → ∃𝑥 ∈ ℕ ∃𝑦 ∈ ℕ 𝑃 = ((𝑥↑2) + (𝑦↑2)))))
69	68	ex 417	. . . . . . . . . . 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 872	. . . . . . . . . 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 870	. . . . . . . . 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 33	. . . . . . 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 411	. . . . 5 ⊢ ((𝑎 ∈ ℕ₀ ∧ 𝑏 ∈ ℕ₀) → (𝑃 ∈ ℙ → (𝑃 = ((𝑎↑2) + (𝑏↑2)) → ∃𝑥 ∈ ℕ ∃𝑦 ∈ ℕ 𝑃 = ((𝑥↑2) + (𝑦↑2)))))
76	75	com12 33	. . . 4 ⊢ (𝑃 ∈ ℙ → ((𝑎 ∈ ℕ₀ ∧ 𝑏 ∈ ℕ₀) → (𝑃 = ((𝑎↑2) + (𝑏↑2)) → ∃𝑥 ∈ ℕ ∃𝑦 ∈ ℕ 𝑃 = ((𝑥↑2) + (𝑦↑2)))))
77	76	adantr 485	. . 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 3221	. 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 16	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 400 ∨ wo 860 = wceq 1570 ∈ wcel 2143 ∃wrex 3089 (class class class)co 7410 0cc0 11095 1c1 11096 + caddc 11098 ℕcn 12228 2c2 12290 4c4 12292 ℕ₀cn0 12499 ℤcz 12586 mod cmo 13898 ↑cexp 14093 ℙcprime 16724

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1825 ax-4 1839 ax-5 1940 ax-6 1997 ax-7 2038 ax-8 2145 ax-9 2153 ax-10 2176 ax-11 2192 ax-12 2213 ax-ext 2735 ax-rep 5238 ax-sep 5257 ax-nul 5269 ax-pow 5336 ax-pr 5404 ax-un 7732 ax-cnex 11151 ax-resscn 11152 ax-1cn 11153 ax-icn 11154 ax-addcl 11155 ax-addrcl 11156 ax-mulcl 11157 ax-mulrcl 11158 ax-mulcom 11159 ax-addass 11160 ax-mulass 11161 ax-distr 11162 ax-i2m1 11163 ax-1ne0 11164 ax-1rid 11165 ax-rnegex 11166 ax-rrecex 11167 ax-cnre 11168 ax-pre-lttri 11169 ax-pre-lttrn 11170 ax-pre-ltadd 11171 ax-pre-mulgt0 11172 ax-pre-sup 11173 ax-addf 11174 ax-mulf 11175

This theorem depends on definitions: df-bi 210 df-an 401 df-or 861 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1810 df-nf 1814 df-sb 2097 df-mo 2567 df-eu 2597 df-clab 2742 df-cleq 2755 df-clel 2838 df-nfc 2912 df-ne 2959 df-nel 3065 df-ral 3080 df-rex 3090 df-rmo 3369 df-reu 3370 df-rab 3417 df-v 3457 df-sbc 3745 df-csb 3854 df-dif 3908 df-un 3910 df-in 3912 df-ss 3922 df-pss 3925 df-nul 4287 df-if 4488 df-pw 4564 df-sn 4590 df-pr 4592 df-tp 4594 df-op 4596 df-uni 4873 df-int 4913 df-iun 4958 df-iin 4959 df-br 5110 df-opab 5174 df-mpt 5193 df-tr 5219 df-id 5556 df-eprel 5561 df-po 5569 df-so 5570 df-fr 5614 df-se 5615 df-we 5616 df-xp 5667 df-rel 5668 df-cnv 5669 df-co 5670 df-dm 5671 df-rn 5672 df-res 5673 df-ima 5674 df-pred 6302 df-ord 6363 df-on 6364 df-lim 6365 df-suc 6366 df-iota 6492 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-isom 6545 df-riota 7367 df-ov 7413 df-oprab 7414 df-mpo 7415 df-of 7674 df-ofr 7675 df-om 7859 df-1st 7982 df-2nd 7983 df-supp 8153 df-tpos 8218 df-frecs 8274 df-wrecs 8305 df-recs 8354 df-rdg 8393 df-1o 8449 df-2o 8450 df-oadd 8453 df-er 8690 df-ec 8692 df-qs 8696 df-map 8822 df-pm 8823 df-ixp 8892 df-en 8940 df-dom 8941 df-sdom 8942 df-fin 8943 df-fsupp 9318 df-sup 9398 df-inf 9399 df-oi 9468 df-dju 9883 df-card 9921 df-pnf 11240 df-mnf 11241 df-xr 11242 df-ltxr 11243 df-le 11244 df-sub 11438 df-neg 11439 df-div 11867 df-nn 12229 df-2 12298 df-3 12299 df-4 12300 df-5 12301 df-6 12302 df-7 12303 df-8 12304 df-9 12305 df-n0 12500 df-xnn0 12573 df-z 12587 df-dec 12707 df-uz 12858 df-q 12968 df-rp 13012 df-fz 13531 df-fzo 13679 df-fl 13821 df-mod 13899 df-seq 14034 df-exp 14094 df-hash 14363 df-cj 15146 df-re 15147 df-im 15148 df-sqrt 15282 df-abs 15283 df-dvds 16306 df-gcd 16548 df-prm 16725 df-phi 16820 df-pc 16892 df-gz 16985 df-struct 17202 df-sets 17219 df-slot 17237 df-ndx 17249 df-base 17265 df-ress 17286 df-plusg 17318 df-mulr 17319 df-starv 17320 df-sca 17321 df-vsca 17322 df-ip 17323 df-tset 17324 df-ple 17325 df-ds 17327 df-unif 17328 df-hom 17329 df-cco 17330 df-0g 17489 df-gsum 17490 df-prds 17495 df-pws 17497 df-imas 17557 df-qus 17558 df-mre 17633 df-mrc 17634 df-acs 17636 df-mgm 18693 df-sgrp 18772 df-mnd 18788 df-mhm 18836 df-submnd 18837 df-grp 18998 df-minusg 18999 df-sbg 19000 df-mulg 19129 df-subg 19184 df-nsg 19185 df-eqg 19186 df-ghm 19279 df-cntz 19382 df-cmn 19847 df-abl 19848 df-mgp 20212 df-rng 20226 df-ur 20259 df-srg 20264 df-ring 20312 df-cring 20313 df-oppr 20415 df-dvdsr 20435 df-unit 20436 df-invr 20466 df-dvr 20479 df-rhm 20550 df-nzr 20610 df-subrng 20645 df-subrg 20669 df-rlreg 20793 df-domn 20794 df-idom 20795 df-drng 20829 df-field 20830 df-lmod 20983 df-lss 21053 df-lsp 21093 df-sra 21294 df-rgmod 21295 df-lidl 21332 df-rsp 21333 df-2idl 21389 df-cnfld 21523 df-zring 21597 df-zrh 21653 df-zn 21656 df-assa 22003 df-asp 22004 df-ascl 22005 df-psr 22059 df-mvr 22060 df-mpl 22061 df-opsr 22063 df-evls 22225 df-evl 22226 df-psr1 22340 df-vr1 22341 df-ply1 22342 df-coe1 22343 df-evl1 22476 df-mdeg 26212 df-deg1 26213 df-mon1 26288 df-uc1p 26289 df-q1p 26290 df-r1p 26291 df-lgs 27459

This theorem is referenced by: 2sqreunnlem1 27613

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