2sqnn - Metamath Proof Explorer

	Metamath Proof Explorer		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > 2sqnn		Structured version Visualization version GIF version

Theorem 2sqnn 27407

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 27406	. 2 ⊢ ((𝑃 ∈ ℙ ∧ (𝑃 mod 4) = 1) → ∃𝑎 ∈ ℕ₀ ∃𝑏 ∈ ℕ₀ 𝑃 = ((𝑎↑2) + (𝑏↑2)))
2		elnn0 12508	. . . . . . 7 ⊢ (𝑎 ∈ ℕ₀ ↔ (𝑎 ∈ ℕ ∨ 𝑎 = 0))
3		elnn0 12508	. . . . . . . . 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 2747	. . . . . . . . . . . . . . 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 2747	. . . . . . . . . . . . . . 15 ⊢ (𝑦 = 𝑏 → (𝑃 = ((𝑎↑2) + (𝑦↑2)) ↔ 𝑃 = ((𝑎↑2) + (𝑏↑2))))
10	6, 9	rspc2ev 3619	. . . . . . . . . . . . . 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 14216	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑎 = 0 → (𝑎↑2) = 0)
15	14	adantr 480	. . . . . . . . . . . . . . . . 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 12253	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑏 ∈ ℕ → 𝑏 ∈ ℂ)
18	17	sqcld 14167	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑏 ∈ ℕ → (𝑏↑2) ∈ ℂ)
19	18	addlidd 11441	. . . . . . . . . . . . . . . . 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 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 2823	. . . . . . . . . . . . . . . . . 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 12614	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑏 ∈ ℕ → 𝑏 ∈ ℤ)
26		sqnprm 16726	. . . . . . . . . . . . . . . . . . . 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 14216	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑏 = 0 → (𝑏↑2) = 0)
38	37	adantr 480	. . . . . . . . . . . . . . . . 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 12253	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑎 ∈ ℕ → 𝑎 ∈ ℂ)
41	40	sqcld 14167	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑎 ∈ ℕ → (𝑎↑2) ∈ ℂ)
42	41	addridd 11440	. . . . . . . . . . . . . . . . 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 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 2823	. . . . . . . . . . . . . . . . . 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 12614	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑎 ∈ ℕ → 𝑎 ∈ ℤ)
49		sqnprm 16726	. . . . . . . . . . . . . . . . . . . 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 7430	. . . . . . . . . . . . . . . 16 ⊢ ((𝑏 = 0 ∧ 𝑎 = 0) → ((𝑎↑2) + (𝑏↑2)) = (0 + 0))
60		00id 11415	. . . . . . . . . . . . . . . 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 2823	. . . . . . . . . . . . . . 15 ⊢ (𝑃 = 0 → (𝑃 ∈ ℙ ↔ 0 ∈ ℙ))
64		0nprm 16702	. . . . . . . . . . . . . . . 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 3201	. 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 3061 (class class class)co 7410 0cc0 11134 1c1 11135 + caddc 11137 ℕcn 12245 2c2 12300 4c4 12302 ℕ₀cn0 12506 ℤcz 12593 mod cmo 13891 ↑cexp 14084 ℙcprime 16695

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 2708 ax-rep 5254 ax-sep 5271 ax-nul 5281 ax-pow 5340 ax-pr 5407 ax-un 7734 ax-cnex 11190 ax-resscn 11191 ax-1cn 11192 ax-icn 11193 ax-addcl 11194 ax-addrcl 11195 ax-mulcl 11196 ax-mulrcl 11197 ax-mulcom 11198 ax-addass 11199 ax-mulass 11200 ax-distr 11201 ax-i2m1 11202 ax-1ne0 11203 ax-1rid 11204 ax-rnegex 11205 ax-rrecex 11206 ax-cnre 11207 ax-pre-lttri 11208 ax-pre-lttrn 11209 ax-pre-ltadd 11210 ax-pre-mulgt0 11211 ax-pre-sup 11212 ax-addf 11213 ax-mulf 11214

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 2540 df-eu 2569 df-clab 2715 df-cleq 2728 df-clel 2810 df-nfc 2886 df-ne 2934 df-nel 3038 df-ral 3053 df-rex 3062 df-rmo 3364 df-reu 3365 df-rab 3421 df-v 3466 df-sbc 3771 df-csb 3880 df-dif 3934 df-un 3936 df-in 3938 df-ss 3948 df-pss 3951 df-nul 4314 df-if 4506 df-pw 4582 df-sn 4607 df-pr 4609 df-tp 4611 df-op 4613 df-uni 4889 df-int 4928 df-iun 4974 df-iin 4975 df-br 5125 df-opab 5187 df-mpt 5207 df-tr 5235 df-id 5553 df-eprel 5558 df-po 5566 df-so 5567 df-fr 5611 df-se 5612 df-we 5613 df-xp 5665 df-rel 5666 df-cnv 5667 df-co 5668 df-dm 5669 df-rn 5670 df-res 5671 df-ima 5672 df-pred 6295 df-ord 6360 df-on 6361 df-lim 6362 df-suc 6363 df-iota 6489 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 7676 df-ofr 7677 df-om 7867 df-1st 7993 df-2nd 7994 df-supp 8165 df-tpos 8230 df-frecs 8285 df-wrecs 8316 df-recs 8390 df-rdg 8429 df-1o 8485 df-2o 8486 df-oadd 8489 df-er 8724 df-ec 8726 df-qs 8730 df-map 8847 df-pm 8848 df-ixp 8917 df-en 8965 df-dom 8966 df-sdom 8967 df-fin 8968 df-fsupp 9379 df-sup 9459 df-inf 9460 df-oi 9529 df-dju 9920 df-card 9958 df-pnf 11276 df-mnf 11277 df-xr 11278 df-ltxr 11279 df-le 11280 df-sub 11473 df-neg 11474 df-div 11900 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 12507 df-xnn0 12580 df-z 12594 df-dec 12714 df-uz 12858 df-q 12970 df-rp 13014 df-fz 13530 df-fzo 13677 df-fl 13814 df-mod 13892 df-seq 14025 df-exp 14085 df-hash 14354 df-cj 15123 df-re 15124 df-im 15125 df-sqrt 15259 df-abs 15260 df-dvds 16278 df-gcd 16519 df-prm 16696 df-phi 16790 df-pc 16862 df-gz 16955 df-struct 17171 df-sets 17188 df-slot 17206 df-ndx 17218 df-base 17234 df-ress 17257 df-plusg 17289 df-mulr 17290 df-starv 17291 df-sca 17292 df-vsca 17293 df-ip 17294 df-tset 17295 df-ple 17296 df-ds 17298 df-unif 17299 df-hom 17300 df-cco 17301 df-0g 17460 df-gsum 17461 df-prds 17466 df-pws 17468 df-imas 17527 df-qus 17528 df-mre 17603 df-mrc 17604 df-acs 17606 df-mgm 18623 df-sgrp 18702 df-mnd 18718 df-mhm 18766 df-submnd 18767 df-grp 18924 df-minusg 18925 df-sbg 18926 df-mulg 19056 df-subg 19111 df-nsg 19112 df-eqg 19113 df-ghm 19201 df-cntz 19305 df-cmn 19768 df-abl 19769 df-mgp 20106 df-rng 20118 df-ur 20147 df-srg 20152 df-ring 20200 df-cring 20201 df-oppr 20302 df-dvdsr 20322 df-unit 20323 df-invr 20353 df-dvr 20366 df-rhm 20437 df-nzr 20478 df-subrng 20511 df-subrg 20535 df-rlreg 20659 df-domn 20660 df-idom 20661 df-drng 20696 df-field 20697 df-lmod 20824 df-lss 20894 df-lsp 20934 df-sra 21136 df-rgmod 21137 df-lidl 21174 df-rsp 21175 df-2idl 21216 df-cnfld 21321 df-zring 21413 df-zrh 21469 df-zn 21472 df-assa 21818 df-asp 21819 df-ascl 21820 df-psr 21874 df-mvr 21875 df-mpl 21876 df-opsr 21878 df-evls 22037 df-evl 22038 df-psr1 22120 df-vr1 22121 df-ply1 22122 df-coe1 22123 df-evl1 22259 df-mdeg 26017 df-deg1 26018 df-mon1 26093 df-uc1p 26094 df-q1p 26095 df-r1p 26096 df-lgs 27263

This theorem is referenced by: 2sqreunnlem1 27417

W3C validator