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Theorem 2sqnn 27350

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 27349	. 2 ⊢ ((𝑃 ∈ ℙ ∧ (𝑃 mod 4) = 1) → ∃𝑎 ∈ ℕ₀ ∃𝑏 ∈ ℕ₀ 𝑃 = ((𝑎↑2) + (𝑏↑2)))
2		elnn0 12444	. . . . . . 7 ⊢ (𝑎 ∈ ℕ₀ ↔ (𝑎 ∈ ℕ ∨ 𝑎 = 0))
3		elnn0 12444	. . . . . . . . 9 ⊢ (𝑏 ∈ ℕ₀ ↔ (𝑏 ∈ ℕ ∨ 𝑏 = 0))
4		oveq1 7394	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 = 𝑎 → (𝑥↑2) = (𝑎↑2))
5	4	oveq1d 7402	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = 𝑎 → ((𝑥↑2) + (𝑦↑2)) = ((𝑎↑2) + (𝑦↑2)))
6	5	eqeq2d 2740	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = 𝑎 → (𝑃 = ((𝑥↑2) + (𝑦↑2)) ↔ 𝑃 = ((𝑎↑2) + (𝑦↑2))))
7		oveq1 7394	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 = 𝑏 → (𝑦↑2) = (𝑏↑2))
8	7	oveq2d 7403	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 = 𝑏 → ((𝑎↑2) + (𝑦↑2)) = ((𝑎↑2) + (𝑏↑2)))
9	8	eqeq2d 2740	. . . . . . . . . . . . . . 15 ⊢ (𝑦 = 𝑏 → (𝑃 = ((𝑎↑2) + (𝑦↑2)) ↔ 𝑃 = ((𝑎↑2) + (𝑏↑2))))
10	6, 9	rspc2ev 3601	. . . . . . . . . . . . . 14 ⊢ ((𝑎 ∈ ℕ ∧ 𝑏 ∈ ℕ ∧ 𝑃 = ((𝑎↑2) + (𝑏↑2))) → ∃𝑥 ∈ ℕ ∃𝑦 ∈ ℕ 𝑃 = ((𝑥↑2) + (𝑦↑2)))
11	10	3expia 1121	. . . . . . . . . . . . 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 413	. . . . . . . . . . 11 ⊢ (𝑏 ∈ ℕ → (𝑎 ∈ ℕ → (𝑃 ∈ ℙ → (𝑃 = ((𝑎↑2) + (𝑏↑2)) → ∃𝑥 ∈ ℕ ∃𝑦 ∈ ℕ 𝑃 = ((𝑥↑2) + (𝑦↑2))))))
14		sq0i 14158	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑎 = 0 → (𝑎↑2) = 0)
15	14	adantr 480	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑎 = 0 ∧ 𝑏 ∈ ℕ) → (𝑎↑2) = 0)
16	15	oveq1d 7402	. . . . . . . . . . . . . . . 16 ⊢ ((𝑎 = 0 ∧ 𝑏 ∈ ℕ) → ((𝑎↑2) + (𝑏↑2)) = (0 + (𝑏↑2)))
17		nncn 12194	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑏 ∈ ℕ → 𝑏 ∈ ℂ)
18	17	sqcld 14109	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑏 ∈ ℕ → (𝑏↑2) ∈ ℂ)
19	18	addlidd 11375	. . . . . . . . . . . . . . . . 17 ⊢ (𝑏 ∈ ℕ → (0 + (𝑏↑2)) = (𝑏↑2))
20	19	adantl 481	. . . . . . . . . . . . . . . 16 ⊢ ((𝑎 = 0 ∧ 𝑏 ∈ ℕ) → (0 + (𝑏↑2)) = (𝑏↑2))
21	16, 20	eqtrd 2764	. . . . . . . . . . . . . . 15 ⊢ ((𝑎 = 0 ∧ 𝑏 ∈ ℕ) → ((𝑎↑2) + (𝑏↑2)) = (𝑏↑2))
22	21	eqeq2d 2740	. . . . . . . . . . . . . 14 ⊢ ((𝑎 = 0 ∧ 𝑏 ∈ ℕ) → (𝑃 = ((𝑎↑2) + (𝑏↑2)) ↔ 𝑃 = (𝑏↑2)))
23		eleq1 2816	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑃 = (𝑏↑2) → (𝑃 ∈ ℙ ↔ (𝑏↑2) ∈ ℙ))
24	23	adantl 481	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑏 ∈ ℕ ∧ 𝑃 = (𝑏↑2)) → (𝑃 ∈ ℙ ↔ (𝑏↑2) ∈ ℙ))
25		nnz 12550	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑏 ∈ ℕ → 𝑏 ∈ ℤ)
26		sqnprm 16672	. . . . . . . . . . . . . . . . . . . 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 480	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑏 ∈ ℕ ∧ 𝑃 = (𝑏↑2)) → ((𝑏↑2) ∈ ℙ → ∃𝑥 ∈ ℕ ∃𝑦 ∈ ℕ 𝑃 = ((𝑥↑2) + (𝑦↑2))))
30	24, 29	sylbid 240	. . . . . . . . . . . . . . . 16 ⊢ ((𝑏 ∈ ℕ ∧ 𝑃 = (𝑏↑2)) → (𝑃 ∈ ℙ → ∃𝑥 ∈ ℕ ∃𝑦 ∈ ℕ 𝑃 = ((𝑥↑2) + (𝑦↑2))))
31	30	ex 412	. . . . . . . . . . . . . . 15 ⊢ (𝑏 ∈ ℕ → (𝑃 = (𝑏↑2) → (𝑃 ∈ ℙ → ∃𝑥 ∈ ℕ ∃𝑦 ∈ ℕ 𝑃 = ((𝑥↑2) + (𝑦↑2)))))
32	31	adantl 481	. . . . . . . . . . . . . 14 ⊢ ((𝑎 = 0 ∧ 𝑏 ∈ ℕ) → (𝑃 = (𝑏↑2) → (𝑃 ∈ ℙ → ∃𝑥 ∈ ℕ ∃𝑦 ∈ ℕ 𝑃 = ((𝑥↑2) + (𝑦↑2)))))
33	22, 32	sylbid 240	. . . . . . . . . . . . 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 413	. . . . . . . . . . 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 14158	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑏 = 0 → (𝑏↑2) = 0)
38	37	adantr 480	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑏 = 0 ∧ 𝑎 ∈ ℕ) → (𝑏↑2) = 0)
39	38	oveq2d 7403	. . . . . . . . . . . . . . . 16 ⊢ ((𝑏 = 0 ∧ 𝑎 ∈ ℕ) → ((𝑎↑2) + (𝑏↑2)) = ((𝑎↑2) + 0))
40		nncn 12194	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑎 ∈ ℕ → 𝑎 ∈ ℂ)
41	40	sqcld 14109	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑎 ∈ ℕ → (𝑎↑2) ∈ ℂ)
42	41	addridd 11374	. . . . . . . . . . . . . . . . 17 ⊢ (𝑎 ∈ ℕ → ((𝑎↑2) + 0) = (𝑎↑2))
43	42	adantl 481	. . . . . . . . . . . . . . . 16 ⊢ ((𝑏 = 0 ∧ 𝑎 ∈ ℕ) → ((𝑎↑2) + 0) = (𝑎↑2))
44	39, 43	eqtrd 2764	. . . . . . . . . . . . . . 15 ⊢ ((𝑏 = 0 ∧ 𝑎 ∈ ℕ) → ((𝑎↑2) + (𝑏↑2)) = (𝑎↑2))
45	44	eqeq2d 2740	. . . . . . . . . . . . . 14 ⊢ ((𝑏 = 0 ∧ 𝑎 ∈ ℕ) → (𝑃 = ((𝑎↑2) + (𝑏↑2)) ↔ 𝑃 = (𝑎↑2)))
46		eleq1 2816	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑃 = (𝑎↑2) → (𝑃 ∈ ℙ ↔ (𝑎↑2) ∈ ℙ))
47	46	adantl 481	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑎 ∈ ℕ ∧ 𝑃 = (𝑎↑2)) → (𝑃 ∈ ℙ ↔ (𝑎↑2) ∈ ℙ))
48		nnz 12550	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑎 ∈ ℕ → 𝑎 ∈ ℤ)
49		sqnprm 16672	. . . . . . . . . . . . . . . . . . . 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 480	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑎 ∈ ℕ ∧ 𝑃 = (𝑎↑2)) → ((𝑎↑2) ∈ ℙ → ∃𝑥 ∈ ℕ ∃𝑦 ∈ ℕ 𝑃 = ((𝑥↑2) + (𝑦↑2))))
53	47, 52	sylbid 240	. . . . . . . . . . . . . . . 16 ⊢ ((𝑎 ∈ ℕ ∧ 𝑃 = (𝑎↑2)) → (𝑃 ∈ ℙ → ∃𝑥 ∈ ℕ ∃𝑦 ∈ ℕ 𝑃 = ((𝑥↑2) + (𝑦↑2))))
54	53	ex 412	. . . . . . . . . . . . . . 15 ⊢ (𝑎 ∈ ℕ → (𝑃 = (𝑎↑2) → (𝑃 ∈ ℙ → ∃𝑥 ∈ ℕ ∃𝑦 ∈ ℕ 𝑃 = ((𝑥↑2) + (𝑦↑2)))))
55	54	adantl 481	. . . . . . . . . . . . . 14 ⊢ ((𝑏 = 0 ∧ 𝑎 ∈ ℕ) → (𝑃 = (𝑎↑2) → (𝑃 ∈ ℙ → ∃𝑥 ∈ ℕ ∃𝑦 ∈ ℕ 𝑃 = ((𝑥↑2) + (𝑦↑2)))))
56	45, 55	sylbid 240	. . . . . . . . . . . . 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 412	. . . . . . . . . . 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 7407	. . . . . . . . . . . . . . . 16 ⊢ ((𝑏 = 0 ∧ 𝑎 = 0) → ((𝑎↑2) + (𝑏↑2)) = (0 + 0))
60		00id 11349	. . . . . . . . . . . . . . . 16 ⊢ (0 + 0) = 0
61	59, 60	eqtrdi 2780	. . . . . . . . . . . . . . 15 ⊢ ((𝑏 = 0 ∧ 𝑎 = 0) → ((𝑎↑2) + (𝑏↑2)) = 0)
62	61	eqeq2d 2740	. . . . . . . . . . . . . 14 ⊢ ((𝑏 = 0 ∧ 𝑎 = 0) → (𝑃 = ((𝑎↑2) + (𝑏↑2)) ↔ 𝑃 = 0))
63		eleq1 2816	. . . . . . . . . . . . . . 15 ⊢ (𝑃 = 0 → (𝑃 ∈ ℙ ↔ 0 ∈ ℙ))
64		0nprm 16648	. . . . . . . . . . . . . . . 16 ⊢ ¬ 0 ∈ ℙ
65	64	pm2.21i 119	. . . . . . . . . . . . . . 15 ⊢ (0 ∈ ℙ → ∃𝑥 ∈ ℕ ∃𝑦 ∈ ℕ 𝑃 = ((𝑥↑2) + (𝑦↑2)))
66	63, 65	biimtrdi 253	. . . . . . . . . . . . . 14 ⊢ (𝑃 = 0 → (𝑃 ∈ ℙ → ∃𝑥 ∈ ℕ ∃𝑦 ∈ ℕ 𝑃 = ((𝑥↑2) + (𝑦↑2))))
67	62, 66	biimtrdi 253	. . . . . . . . . . . . 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 412	. . . . . . . . . . 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 217	. . . . . . . 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 217	. . . . . 6 ⊢ (𝑎 ∈ ℕ₀ → (𝑏 ∈ ℕ₀ → (𝑃 ∈ ℙ → (𝑃 = ((𝑎↑2) + (𝑏↑2)) → ∃𝑥 ∈ ℕ ∃𝑦 ∈ ℕ 𝑃 = ((𝑥↑2) + (𝑦↑2))))))
75	74	imp 406	. . . . 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 480	. . 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 3193	. 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 206 ∧ wa 395 ∨ wo 847 = wceq 1540 ∈ wcel 2109 ∃wrex 3053 (class class class)co 7387 0cc0 11068 1c1 11069 + caddc 11071 ℕcn 12186 2c2 12241 4c4 12243 ℕ₀cn0 12442 ℤcz 12529 mod cmo 13831 ↑cexp 14026 ℙcprime 16641

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2701 ax-rep 5234 ax-sep 5251 ax-nul 5261 ax-pow 5320 ax-pr 5387 ax-un 7711 ax-cnex 11124 ax-resscn 11125 ax-1cn 11126 ax-icn 11127 ax-addcl 11128 ax-addrcl 11129 ax-mulcl 11130 ax-mulrcl 11131 ax-mulcom 11132 ax-addass 11133 ax-mulass 11134 ax-distr 11135 ax-i2m1 11136 ax-1ne0 11137 ax-1rid 11138 ax-rnegex 11139 ax-rrecex 11140 ax-cnre 11141 ax-pre-lttri 11142 ax-pre-lttrn 11143 ax-pre-ltadd 11144 ax-pre-mulgt0 11145 ax-pre-sup 11146 ax-addf 11147 ax-mulf 11148

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2533 df-eu 2562 df-clab 2708 df-cleq 2721 df-clel 2803 df-nfc 2878 df-ne 2926 df-nel 3030 df-ral 3045 df-rex 3054 df-rmo 3354 df-reu 3355 df-rab 3406 df-v 3449 df-sbc 3754 df-csb 3863 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-pss 3934 df-nul 4297 df-if 4489 df-pw 4565 df-sn 4590 df-pr 4592 df-tp 4594 df-op 4596 df-uni 4872 df-int 4911 df-iun 4957 df-iin 4958 df-br 5108 df-opab 5170 df-mpt 5189 df-tr 5215 df-id 5533 df-eprel 5538 df-po 5546 df-so 5547 df-fr 5591 df-se 5592 df-we 5593 df-xp 5644 df-rel 5645 df-cnv 5646 df-co 5647 df-dm 5648 df-rn 5649 df-res 5650 df-ima 5651 df-pred 6274 df-ord 6335 df-on 6336 df-lim 6337 df-suc 6338 df-iota 6464 df-fun 6513 df-fn 6514 df-f 6515 df-f1 6516 df-fo 6517 df-f1o 6518 df-fv 6519 df-isom 6520 df-riota 7344 df-ov 7390 df-oprab 7391 df-mpo 7392 df-of 7653 df-ofr 7654 df-om 7843 df-1st 7968 df-2nd 7969 df-supp 8140 df-tpos 8205 df-frecs 8260 df-wrecs 8291 df-recs 8340 df-rdg 8378 df-1o 8434 df-2o 8435 df-oadd 8438 df-er 8671 df-ec 8673 df-qs 8677 df-map 8801 df-pm 8802 df-ixp 8871 df-en 8919 df-dom 8920 df-sdom 8921 df-fin 8922 df-fsupp 9313 df-sup 9393 df-inf 9394 df-oi 9463 df-dju 9854 df-card 9892 df-pnf 11210 df-mnf 11211 df-xr 11212 df-ltxr 11213 df-le 11214 df-sub 11407 df-neg 11408 df-div 11836 df-nn 12187 df-2 12249 df-3 12250 df-4 12251 df-5 12252 df-6 12253 df-7 12254 df-8 12255 df-9 12256 df-n0 12443 df-xnn0 12516 df-z 12530 df-dec 12650 df-uz 12794 df-q 12908 df-rp 12952 df-fz 13469 df-fzo 13616 df-fl 13754 df-mod 13832 df-seq 13967 df-exp 14027 df-hash 14296 df-cj 15065 df-re 15066 df-im 15067 df-sqrt 15201 df-abs 15202 df-dvds 16223 df-gcd 16465 df-prm 16642 df-phi 16736 df-pc 16808 df-gz 16901 df-struct 17117 df-sets 17134 df-slot 17152 df-ndx 17164 df-base 17180 df-ress 17201 df-plusg 17233 df-mulr 17234 df-starv 17235 df-sca 17236 df-vsca 17237 df-ip 17238 df-tset 17239 df-ple 17240 df-ds 17242 df-unif 17243 df-hom 17244 df-cco 17245 df-0g 17404 df-gsum 17405 df-prds 17410 df-pws 17412 df-imas 17471 df-qus 17472 df-mre 17547 df-mrc 17548 df-acs 17550 df-mgm 18567 df-sgrp 18646 df-mnd 18662 df-mhm 18710 df-submnd 18711 df-grp 18868 df-minusg 18869 df-sbg 18870 df-mulg 19000 df-subg 19055 df-nsg 19056 df-eqg 19057 df-ghm 19145 df-cntz 19249 df-cmn 19712 df-abl 19713 df-mgp 20050 df-rng 20062 df-ur 20091 df-srg 20096 df-ring 20144 df-cring 20145 df-oppr 20246 df-dvdsr 20266 df-unit 20267 df-invr 20297 df-dvr 20310 df-rhm 20381 df-nzr 20422 df-subrng 20455 df-subrg 20479 df-rlreg 20603 df-domn 20604 df-idom 20605 df-drng 20640 df-field 20641 df-lmod 20768 df-lss 20838 df-lsp 20878 df-sra 21080 df-rgmod 21081 df-lidl 21118 df-rsp 21119 df-2idl 21160 df-cnfld 21265 df-zring 21357 df-zrh 21413 df-zn 21416 df-assa 21762 df-asp 21763 df-ascl 21764 df-psr 21818 df-mvr 21819 df-mpl 21820 df-opsr 21822 df-evls 21981 df-evl 21982 df-psr1 22064 df-vr1 22065 df-ply1 22066 df-coe1 22067 df-evl1 22203 df-mdeg 25960 df-deg1 25961 df-mon1 26036 df-uc1p 26037 df-q1p 26038 df-r1p 26039 df-lgs 27206

This theorem is referenced by: 2sqreunnlem1 27360

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