2sqnn - Metamath Proof Explorer

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Theorem 2sqnn 27418

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 27417	. 2 ⊢ ((𝑃 ∈ ℙ ∧ (𝑃 mod 4) = 1) → ∃𝑎 ∈ ℕ₀ ∃𝑏 ∈ ℕ₀ 𝑃 = ((𝑎↑2) + (𝑏↑2)))
2		elnn0 12415	. . . . . . 7 ⊢ (𝑎 ∈ ℕ₀ ↔ (𝑎 ∈ ℕ ∨ 𝑎 = 0))
3		elnn0 12415	. . . . . . . . 9 ⊢ (𝑏 ∈ ℕ₀ ↔ (𝑏 ∈ ℕ ∨ 𝑏 = 0))
4		oveq1 7375	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 = 𝑎 → (𝑥↑2) = (𝑎↑2))
5	4	oveq1d 7383	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = 𝑎 → ((𝑥↑2) + (𝑦↑2)) = ((𝑎↑2) + (𝑦↑2)))
6	5	eqeq2d 2748	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = 𝑎 → (𝑃 = ((𝑥↑2) + (𝑦↑2)) ↔ 𝑃 = ((𝑎↑2) + (𝑦↑2))))
7		oveq1 7375	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 = 𝑏 → (𝑦↑2) = (𝑏↑2))
8	7	oveq2d 7384	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 = 𝑏 → ((𝑎↑2) + (𝑦↑2)) = ((𝑎↑2) + (𝑏↑2)))
9	8	eqeq2d 2748	. . . . . . . . . . . . . . 15 ⊢ (𝑦 = 𝑏 → (𝑃 = ((𝑎↑2) + (𝑦↑2)) ↔ 𝑃 = ((𝑎↑2) + (𝑏↑2))))
10	6, 9	rspc2ev 3591	. . . . . . . . . . . . . 14 ⊢ ((𝑎 ∈ ℕ ∧ 𝑏 ∈ ℕ ∧ 𝑃 = ((𝑎↑2) + (𝑏↑2))) → ∃𝑥 ∈ ℕ ∃𝑦 ∈ ℕ 𝑃 = ((𝑥↑2) + (𝑦↑2)))
11	10	3expia 1122	. . . . . . . . . . . . 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 14128	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑎 = 0 → (𝑎↑2) = 0)
15	14	adantr 480	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑎 = 0 ∧ 𝑏 ∈ ℕ) → (𝑎↑2) = 0)
16	15	oveq1d 7383	. . . . . . . . . . . . . . . 16 ⊢ ((𝑎 = 0 ∧ 𝑏 ∈ ℕ) → ((𝑎↑2) + (𝑏↑2)) = (0 + (𝑏↑2)))
17		nncn 12165	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑏 ∈ ℕ → 𝑏 ∈ ℂ)
18	17	sqcld 14079	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑏 ∈ ℕ → (𝑏↑2) ∈ ℂ)
19	18	addlidd 11346	. . . . . . . . . . . . . . . . 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 2772	. . . . . . . . . . . . . . 15 ⊢ ((𝑎 = 0 ∧ 𝑏 ∈ ℕ) → ((𝑎↑2) + (𝑏↑2)) = (𝑏↑2))
22	21	eqeq2d 2748	. . . . . . . . . . . . . 14 ⊢ ((𝑎 = 0 ∧ 𝑏 ∈ ℕ) → (𝑃 = ((𝑎↑2) + (𝑏↑2)) ↔ 𝑃 = (𝑏↑2)))
23		eleq1 2825	. . . . . . . . . . . . . . . . . 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 12521	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑏 ∈ ℕ → 𝑏 ∈ ℤ)
26		sqnprm 16641	. . . . . . . . . . . . . . . . . . . 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 860	. . . . . . . . . 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 14128	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑏 = 0 → (𝑏↑2) = 0)
38	37	adantr 480	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑏 = 0 ∧ 𝑎 ∈ ℕ) → (𝑏↑2) = 0)
39	38	oveq2d 7384	. . . . . . . . . . . . . . . 16 ⊢ ((𝑏 = 0 ∧ 𝑎 ∈ ℕ) → ((𝑎↑2) + (𝑏↑2)) = ((𝑎↑2) + 0))
40		nncn 12165	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑎 ∈ ℕ → 𝑎 ∈ ℂ)
41	40	sqcld 14079	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑎 ∈ ℕ → (𝑎↑2) ∈ ℂ)
42	41	addridd 11345	. . . . . . . . . . . . . . . . 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 2772	. . . . . . . . . . . . . . 15 ⊢ ((𝑏 = 0 ∧ 𝑎 ∈ ℕ) → ((𝑎↑2) + (𝑏↑2)) = (𝑎↑2))
45	44	eqeq2d 2748	. . . . . . . . . . . . . 14 ⊢ ((𝑏 = 0 ∧ 𝑎 ∈ ℕ) → (𝑃 = ((𝑎↑2) + (𝑏↑2)) ↔ 𝑃 = (𝑎↑2)))
46		eleq1 2825	. . . . . . . . . . . . . . . . . 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 12521	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑎 ∈ ℕ → 𝑎 ∈ ℤ)
49		sqnprm 16641	. . . . . . . . . . . . . . . . . . . 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 7388	. . . . . . . . . . . . . . . 16 ⊢ ((𝑏 = 0 ∧ 𝑎 = 0) → ((𝑎↑2) + (𝑏↑2)) = (0 + 0))
60		00id 11320	. . . . . . . . . . . . . . . 16 ⊢ (0 + 0) = 0
61	59, 60	eqtrdi 2788	. . . . . . . . . . . . . . 15 ⊢ ((𝑏 = 0 ∧ 𝑎 = 0) → ((𝑎↑2) + (𝑏↑2)) = 0)
62	61	eqeq2d 2748	. . . . . . . . . . . . . 14 ⊢ ((𝑏 = 0 ∧ 𝑎 = 0) → (𝑃 = ((𝑎↑2) + (𝑏↑2)) ↔ 𝑃 = 0))
63		eleq1 2825	. . . . . . . . . . . . . . 15 ⊢ (𝑃 = 0 → (𝑃 ∈ ℙ ↔ 0 ∈ ℙ))
64		0nprm 16617	. . . . . . . . . . . . . . . 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 860	. . . . . . . . . 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 858	. . . . . . . . 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 3194	. 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 848 = wceq 1542 ∈ wcel 2114 ∃wrex 3062 (class class class)co 7368 0cc0 11038 1c1 11039 + caddc 11041 ℕcn 12157 2c2 12212 4c4 12214 ℕ₀cn0 12413 ℤcz 12500 mod cmo 13801 ↑cexp 13996 ℙcprime 16610

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2709 ax-rep 5226 ax-sep 5243 ax-nul 5253 ax-pow 5312 ax-pr 5379 ax-un 7690 ax-cnex 11094 ax-resscn 11095 ax-1cn 11096 ax-icn 11097 ax-addcl 11098 ax-addrcl 11099 ax-mulcl 11100 ax-mulrcl 11101 ax-mulcom 11102 ax-addass 11103 ax-mulass 11104 ax-distr 11105 ax-i2m1 11106 ax-1ne0 11107 ax-1rid 11108 ax-rnegex 11109 ax-rrecex 11110 ax-cnre 11111 ax-pre-lttri 11112 ax-pre-lttrn 11113 ax-pre-ltadd 11114 ax-pre-mulgt0 11115 ax-pre-sup 11116 ax-addf 11117 ax-mulf 11118

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2540 df-eu 2570 df-clab 2716 df-cleq 2729 df-clel 2812 df-nfc 2886 df-ne 2934 df-nel 3038 df-ral 3053 df-rex 3063 df-rmo 3352 df-reu 3353 df-rab 3402 df-v 3444 df-sbc 3743 df-csb 3852 df-dif 3906 df-un 3908 df-in 3910 df-ss 3920 df-pss 3923 df-nul 4288 df-if 4482 df-pw 4558 df-sn 4583 df-pr 4585 df-tp 4587 df-op 4589 df-uni 4866 df-int 4905 df-iun 4950 df-iin 4951 df-br 5101 df-opab 5163 df-mpt 5182 df-tr 5208 df-id 5527 df-eprel 5532 df-po 5540 df-so 5541 df-fr 5585 df-se 5586 df-we 5587 df-xp 5638 df-rel 5639 df-cnv 5640 df-co 5641 df-dm 5642 df-rn 5643 df-res 5644 df-ima 5645 df-pred 6267 df-ord 6328 df-on 6329 df-lim 6330 df-suc 6331 df-iota 6456 df-fun 6502 df-fn 6503 df-f 6504 df-f1 6505 df-fo 6506 df-f1o 6507 df-fv 6508 df-isom 6509 df-riota 7325 df-ov 7371 df-oprab 7372 df-mpo 7373 df-of 7632 df-ofr 7633 df-om 7819 df-1st 7943 df-2nd 7944 df-supp 8113 df-tpos 8178 df-frecs 8233 df-wrecs 8264 df-recs 8313 df-rdg 8351 df-1o 8407 df-2o 8408 df-oadd 8411 df-er 8645 df-ec 8647 df-qs 8651 df-map 8777 df-pm 8778 df-ixp 8848 df-en 8896 df-dom 8897 df-sdom 8898 df-fin 8899 df-fsupp 9277 df-sup 9357 df-inf 9358 df-oi 9427 df-dju 9825 df-card 9863 df-pnf 11180 df-mnf 11181 df-xr 11182 df-ltxr 11183 df-le 11184 df-sub 11378 df-neg 11379 df-div 11807 df-nn 12158 df-2 12220 df-3 12221 df-4 12222 df-5 12223 df-6 12224 df-7 12225 df-8 12226 df-9 12227 df-n0 12414 df-xnn0 12487 df-z 12501 df-dec 12620 df-uz 12764 df-q 12874 df-rp 12918 df-fz 13436 df-fzo 13583 df-fl 13724 df-mod 13802 df-seq 13937 df-exp 13997 df-hash 14266 df-cj 15034 df-re 15035 df-im 15036 df-sqrt 15170 df-abs 15171 df-dvds 16192 df-gcd 16434 df-prm 16611 df-phi 16705 df-pc 16777 df-gz 16870 df-struct 17086 df-sets 17103 df-slot 17121 df-ndx 17133 df-base 17149 df-ress 17170 df-plusg 17202 df-mulr 17203 df-starv 17204 df-sca 17205 df-vsca 17206 df-ip 17207 df-tset 17208 df-ple 17209 df-ds 17211 df-unif 17212 df-hom 17213 df-cco 17214 df-0g 17373 df-gsum 17374 df-prds 17379 df-pws 17381 df-imas 17441 df-qus 17442 df-mre 17517 df-mrc 17518 df-acs 17520 df-mgm 18577 df-sgrp 18656 df-mnd 18672 df-mhm 18720 df-submnd 18721 df-grp 18878 df-minusg 18879 df-sbg 18880 df-mulg 19010 df-subg 19065 df-nsg 19066 df-eqg 19067 df-ghm 19154 df-cntz 19258 df-cmn 19723 df-abl 19724 df-mgp 20088 df-rng 20100 df-ur 20129 df-srg 20134 df-ring 20182 df-cring 20183 df-oppr 20285 df-dvdsr 20305 df-unit 20306 df-invr 20336 df-dvr 20349 df-rhm 20420 df-nzr 20458 df-subrng 20491 df-subrg 20515 df-rlreg 20639 df-domn 20640 df-idom 20641 df-drng 20676 df-field 20677 df-lmod 20825 df-lss 20895 df-lsp 20935 df-sra 21137 df-rgmod 21138 df-lidl 21175 df-rsp 21176 df-2idl 21217 df-cnfld 21322 df-zring 21414 df-zrh 21470 df-zn 21473 df-assa 21820 df-asp 21821 df-ascl 21822 df-psr 21877 df-mvr 21878 df-mpl 21879 df-opsr 21881 df-evls 22041 df-evl 22042 df-psr1 22132 df-vr1 22133 df-ply1 22134 df-coe1 22135 df-evl1 22272 df-mdeg 26028 df-deg1 26029 df-mon1 26104 df-uc1p 26105 df-q1p 26106 df-r1p 26107 df-lgs 27274

This theorem is referenced by: 2sqreunnlem1 27428

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