2sqreunnltlem - Metamath Proof Explorer

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Theorem 2sqreunnltlem 27394

Description: Lemma for 2sqreunnlt 27404. (Contributed by AV, 4-Jun-2023.) Specialization to different integers, proposed by GL. (Revised by AV, 11-Jun-2023.)

**Assertion**
Ref	Expression
2sqreunnltlem	⊢ ((𝑃 ∈ ℙ ∧ (𝑃 mod 4) = 1) → ∃!𝑎 ∈ ℕ ∃!𝑏 ∈ ℕ (𝑎 < 𝑏 ∧ ((𝑎↑2) + (𝑏↑2)) = 𝑃))

Distinct variable group: 𝑃,𝑎,𝑏

**Proof of Theorem 2sqreunnltlem**
Step	Hyp	Ref	Expression
1		2sqreunnlem1 27393	. 2 ⊢ ((𝑃 ∈ ℙ ∧ (𝑃 mod 4) = 1) → ∃!𝑎 ∈ ℕ ∃!𝑏 ∈ ℕ (𝑎 ≤ 𝑏 ∧ ((𝑎↑2) + (𝑏↑2)) = 𝑃))
2		oveq1 7376	. . . . . . . . . . . . . . 15 ⊢ (𝑏 = 𝑎 → (𝑏↑2) = (𝑎↑2))
3	2	oveq2d 7385	. . . . . . . . . . . . . 14 ⊢ (𝑏 = 𝑎 → ((𝑎↑2) + (𝑏↑2)) = ((𝑎↑2) + (𝑎↑2)))
4	3	adantr 480	. . . . . . . . . . . . 13 ⊢ ((𝑏 = 𝑎 ∧ (((𝑃 ∈ ℙ ∧ (𝑃 mod 4) = 1) ∧ 𝑎 ∈ ℕ) ∧ 𝑏 ∈ ℕ)) → ((𝑎↑2) + (𝑏↑2)) = ((𝑎↑2) + (𝑎↑2)))
5		nncn 12170	. . . . . . . . . . . . . . . . 17 ⊢ (𝑎 ∈ ℕ → 𝑎 ∈ ℂ)
6	5	sqcld 14085	. . . . . . . . . . . . . . . 16 ⊢ (𝑎 ∈ ℕ → (𝑎↑2) ∈ ℂ)
7		2times 12293	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑎↑2) ∈ ℂ → (2 · (𝑎↑2)) = ((𝑎↑2) + (𝑎↑2)))
8	7	eqcomd 2735	. . . . . . . . . . . . . . . 16 ⊢ ((𝑎↑2) ∈ ℂ → ((𝑎↑2) + (𝑎↑2)) = (2 · (𝑎↑2)))
9	6, 8	syl 17	. . . . . . . . . . . . . . 15 ⊢ (𝑎 ∈ ℕ → ((𝑎↑2) + (𝑎↑2)) = (2 · (𝑎↑2)))
10	9	adantl 481	. . . . . . . . . . . . . 14 ⊢ (((𝑃 ∈ ℙ ∧ (𝑃 mod 4) = 1) ∧ 𝑎 ∈ ℕ) → ((𝑎↑2) + (𝑎↑2)) = (2 · (𝑎↑2)))
11	10	ad2antrl 728	. . . . . . . . . . . . 13 ⊢ ((𝑏 = 𝑎 ∧ (((𝑃 ∈ ℙ ∧ (𝑃 mod 4) = 1) ∧ 𝑎 ∈ ℕ) ∧ 𝑏 ∈ ℕ)) → ((𝑎↑2) + (𝑎↑2)) = (2 · (𝑎↑2)))
12	4, 11	eqtrd 2764	. . . . . . . . . . . 12 ⊢ ((𝑏 = 𝑎 ∧ (((𝑃 ∈ ℙ ∧ (𝑃 mod 4) = 1) ∧ 𝑎 ∈ ℕ) ∧ 𝑏 ∈ ℕ)) → ((𝑎↑2) + (𝑏↑2)) = (2 · (𝑎↑2)))
13	12	eqeq1d 2731	. . . . . . . . . . 11 ⊢ ((𝑏 = 𝑎 ∧ (((𝑃 ∈ ℙ ∧ (𝑃 mod 4) = 1) ∧ 𝑎 ∈ ℕ) ∧ 𝑏 ∈ ℕ)) → (((𝑎↑2) + (𝑏↑2)) = 𝑃 ↔ (2 · (𝑎↑2)) = 𝑃))
14		oveq1 7376	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑃 = (2 · (𝑎↑2)) → (𝑃 mod 4) = ((2 · (𝑎↑2)) mod 4))
15	14	eqeq1d 2731	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑃 = (2 · (𝑎↑2)) → ((𝑃 mod 4) = 1 ↔ ((2 · (𝑎↑2)) mod 4) = 1))
16		eleq1 2816	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑃 = (2 · (𝑎↑2)) → (𝑃 ∈ ℙ ↔ (2 · (𝑎↑2)) ∈ ℙ))
17	15, 16	anbi12d 632	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑃 = (2 · (𝑎↑2)) → (((𝑃 mod 4) = 1 ∧ 𝑃 ∈ ℙ) ↔ (((2 · (𝑎↑2)) mod 4) = 1 ∧ (2 · (𝑎↑2)) ∈ ℙ)))
18		nnz 12526	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑎 ∈ ℕ → 𝑎 ∈ ℤ)
19		2nn0 12435	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ 2 ∈ ℕ₀
20		zexpcl 14017	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑎 ∈ ℤ ∧ 2 ∈ ℕ₀) → (𝑎↑2) ∈ ℤ)
21	18, 19, 20	sylancl 586	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑎 ∈ ℕ → (𝑎↑2) ∈ ℤ)
22		2mulprm 16639	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑎↑2) ∈ ℤ → ((2 · (𝑎↑2)) ∈ ℙ ↔ (𝑎↑2) = 1))
23	21, 22	syl 17	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑎 ∈ ℕ → ((2 · (𝑎↑2)) ∈ ℙ ↔ (𝑎↑2) = 1))
24		oveq2 7377	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝑎↑2) = 1 → (2 · (𝑎↑2)) = (2 · 1))
25		2t1e2 12320	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (2 · 1) = 2
26	24, 25	eqtrdi 2780	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝑎↑2) = 1 → (2 · (𝑎↑2)) = 2)
27	26	oveq1d 7384	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝑎↑2) = 1 → ((2 · (𝑎↑2)) mod 4) = (2 mod 4))
28		2re 12236	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ 2 ∈ ℝ
29		4nn 12245	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ 4 ∈ ℕ
30		nnrp 12939	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (4 ∈ ℕ → 4 ∈ ℝ⁺)
31	29, 30	ax-mp 5	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ 4 ∈ ℝ⁺
32		0le2 12264	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ 0 ≤ 2
33		2lt4 12332	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ 2 < 4
34		modid 13834	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((2 ∈ ℝ ∧ 4 ∈ ℝ⁺) ∧ (0 ≤ 2 ∧ 2 < 4)) → (2 mod 4) = 2)
35	28, 31, 32, 33, 34	mp4an 693	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (2 mod 4) = 2
36	27, 35	eqtrdi 2780	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑎↑2) = 1 → ((2 · (𝑎↑2)) mod 4) = 2)
37	36	eqeq1d 2731	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑎↑2) = 1 → (((2 · (𝑎↑2)) mod 4) = 1 ↔ 2 = 1))
38		1ne2 12365	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ 1 ≠ 2
39		eqcom 2736	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (2 = 1 ↔ 1 = 2)
40		eqneqall 2936	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (1 = 2 → (1 ≠ 2 → (𝑎 ≤ 𝑏 → 𝑏 ≠ 𝑎)))
41	40	com12 32	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (1 ≠ 2 → (1 = 2 → (𝑎 ≤ 𝑏 → 𝑏 ≠ 𝑎)))
42	39, 41	biimtrid 242	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (1 ≠ 2 → (2 = 1 → (𝑎 ≤ 𝑏 → 𝑏 ≠ 𝑎)))
43	38, 42	ax-mp 5	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (2 = 1 → (𝑎 ≤ 𝑏 → 𝑏 ≠ 𝑎))
44	37, 43	biimtrdi 253	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑎↑2) = 1 → (((2 · (𝑎↑2)) mod 4) = 1 → (𝑎 ≤ 𝑏 → 𝑏 ≠ 𝑎)))
45	23, 44	biimtrdi 253	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑎 ∈ ℕ → ((2 · (𝑎↑2)) ∈ ℙ → (((2 · (𝑎↑2)) mod 4) = 1 → (𝑎 ≤ 𝑏 → 𝑏 ≠ 𝑎))))
46	45	impcomd 411	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑎 ∈ ℕ → ((((2 · (𝑎↑2)) mod 4) = 1 ∧ (2 · (𝑎↑2)) ∈ ℙ) → (𝑎 ≤ 𝑏 → 𝑏 ≠ 𝑎)))
47	46	com12 32	. . . . . . . . . . . . . . . . . 18 ⊢ ((((2 · (𝑎↑2)) mod 4) = 1 ∧ (2 · (𝑎↑2)) ∈ ℙ) → (𝑎 ∈ ℕ → (𝑎 ≤ 𝑏 → 𝑏 ≠ 𝑎)))
48	17, 47	biimtrdi 253	. . . . . . . . . . . . . . . . 17 ⊢ (𝑃 = (2 · (𝑎↑2)) → (((𝑃 mod 4) = 1 ∧ 𝑃 ∈ ℙ) → (𝑎 ∈ ℕ → (𝑎 ≤ 𝑏 → 𝑏 ≠ 𝑎))))
49	48	expd 415	. . . . . . . . . . . . . . . 16 ⊢ (𝑃 = (2 · (𝑎↑2)) → ((𝑃 mod 4) = 1 → (𝑃 ∈ ℙ → (𝑎 ∈ ℕ → (𝑎 ≤ 𝑏 → 𝑏 ≠ 𝑎)))))
50	49	com34 91	. . . . . . . . . . . . . . 15 ⊢ (𝑃 = (2 · (𝑎↑2)) → ((𝑃 mod 4) = 1 → (𝑎 ∈ ℕ → (𝑃 ∈ ℙ → (𝑎 ≤ 𝑏 → 𝑏 ≠ 𝑎)))))
51	50	eqcoms 2737	. . . . . . . . . . . . . 14 ⊢ ((2 · (𝑎↑2)) = 𝑃 → ((𝑃 mod 4) = 1 → (𝑎 ∈ ℕ → (𝑃 ∈ ℙ → (𝑎 ≤ 𝑏 → 𝑏 ≠ 𝑎)))))
52	51	com14 96	. . . . . . . . . . . . 13 ⊢ (𝑃 ∈ ℙ → ((𝑃 mod 4) = 1 → (𝑎 ∈ ℕ → ((2 · (𝑎↑2)) = 𝑃 → (𝑎 ≤ 𝑏 → 𝑏 ≠ 𝑎)))))
53	52	imp31 417	. . . . . . . . . . . 12 ⊢ (((𝑃 ∈ ℙ ∧ (𝑃 mod 4) = 1) ∧ 𝑎 ∈ ℕ) → ((2 · (𝑎↑2)) = 𝑃 → (𝑎 ≤ 𝑏 → 𝑏 ≠ 𝑎)))
54	53	ad2antrl 728	. . . . . . . . . . 11 ⊢ ((𝑏 = 𝑎 ∧ (((𝑃 ∈ ℙ ∧ (𝑃 mod 4) = 1) ∧ 𝑎 ∈ ℕ) ∧ 𝑏 ∈ ℕ)) → ((2 · (𝑎↑2)) = 𝑃 → (𝑎 ≤ 𝑏 → 𝑏 ≠ 𝑎)))
55	13, 54	sylbid 240	. . . . . . . . . 10 ⊢ ((𝑏 = 𝑎 ∧ (((𝑃 ∈ ℙ ∧ (𝑃 mod 4) = 1) ∧ 𝑎 ∈ ℕ) ∧ 𝑏 ∈ ℕ)) → (((𝑎↑2) + (𝑏↑2)) = 𝑃 → (𝑎 ≤ 𝑏 → 𝑏 ≠ 𝑎)))
56	55	expimpd 453	. . . . . . . . 9 ⊢ (𝑏 = 𝑎 → (((((𝑃 ∈ ℙ ∧ (𝑃 mod 4) = 1) ∧ 𝑎 ∈ ℕ) ∧ 𝑏 ∈ ℕ) ∧ ((𝑎↑2) + (𝑏↑2)) = 𝑃) → (𝑎 ≤ 𝑏 → 𝑏 ≠ 𝑎)))
57		2a1 28	. . . . . . . . 9 ⊢ (𝑏 ≠ 𝑎 → (((((𝑃 ∈ ℙ ∧ (𝑃 mod 4) = 1) ∧ 𝑎 ∈ ℕ) ∧ 𝑏 ∈ ℕ) ∧ ((𝑎↑2) + (𝑏↑2)) = 𝑃) → (𝑎 ≤ 𝑏 → 𝑏 ≠ 𝑎)))
58	56, 57	pm2.61ine 3008	. . . . . . . 8 ⊢ (((((𝑃 ∈ ℙ ∧ (𝑃 mod 4) = 1) ∧ 𝑎 ∈ ℕ) ∧ 𝑏 ∈ ℕ) ∧ ((𝑎↑2) + (𝑏↑2)) = 𝑃) → (𝑎 ≤ 𝑏 → 𝑏 ≠ 𝑎))
59	58	pm4.71d 561	. . . . . . 7 ⊢ (((((𝑃 ∈ ℙ ∧ (𝑃 mod 4) = 1) ∧ 𝑎 ∈ ℕ) ∧ 𝑏 ∈ ℕ) ∧ ((𝑎↑2) + (𝑏↑2)) = 𝑃) → (𝑎 ≤ 𝑏 ↔ (𝑎 ≤ 𝑏 ∧ 𝑏 ≠ 𝑎)))
60		nnre 12169	. . . . . . . . . . 11 ⊢ (𝑎 ∈ ℕ → 𝑎 ∈ ℝ)
61	60	adantl 481	. . . . . . . . . 10 ⊢ (((𝑃 ∈ ℙ ∧ (𝑃 mod 4) = 1) ∧ 𝑎 ∈ ℕ) → 𝑎 ∈ ℝ)
62		nnre 12169	. . . . . . . . . 10 ⊢ (𝑏 ∈ ℕ → 𝑏 ∈ ℝ)
63		ltlen 11251	. . . . . . . . . 10 ⊢ ((𝑎 ∈ ℝ ∧ 𝑏 ∈ ℝ) → (𝑎 < 𝑏 ↔ (𝑎 ≤ 𝑏 ∧ 𝑏 ≠ 𝑎)))
64	61, 62, 63	syl2an 596	. . . . . . . . 9 ⊢ ((((𝑃 ∈ ℙ ∧ (𝑃 mod 4) = 1) ∧ 𝑎 ∈ ℕ) ∧ 𝑏 ∈ ℕ) → (𝑎 < 𝑏 ↔ (𝑎 ≤ 𝑏 ∧ 𝑏 ≠ 𝑎)))
65	64	bibi2d 342	. . . . . . . 8 ⊢ ((((𝑃 ∈ ℙ ∧ (𝑃 mod 4) = 1) ∧ 𝑎 ∈ ℕ) ∧ 𝑏 ∈ ℕ) → ((𝑎 ≤ 𝑏 ↔ 𝑎 < 𝑏) ↔ (𝑎 ≤ 𝑏 ↔ (𝑎 ≤ 𝑏 ∧ 𝑏 ≠ 𝑎))))
66	65	adantr 480	. . . . . . 7 ⊢ (((((𝑃 ∈ ℙ ∧ (𝑃 mod 4) = 1) ∧ 𝑎 ∈ ℕ) ∧ 𝑏 ∈ ℕ) ∧ ((𝑎↑2) + (𝑏↑2)) = 𝑃) → ((𝑎 ≤ 𝑏 ↔ 𝑎 < 𝑏) ↔ (𝑎 ≤ 𝑏 ↔ (𝑎 ≤ 𝑏 ∧ 𝑏 ≠ 𝑎))))
67	59, 66	mpbird 257	. . . . . 6 ⊢ (((((𝑃 ∈ ℙ ∧ (𝑃 mod 4) = 1) ∧ 𝑎 ∈ ℕ) ∧ 𝑏 ∈ ℕ) ∧ ((𝑎↑2) + (𝑏↑2)) = 𝑃) → (𝑎 ≤ 𝑏 ↔ 𝑎 < 𝑏))
68	67	ex 412	. . . . 5 ⊢ ((((𝑃 ∈ ℙ ∧ (𝑃 mod 4) = 1) ∧ 𝑎 ∈ ℕ) ∧ 𝑏 ∈ ℕ) → (((𝑎↑2) + (𝑏↑2)) = 𝑃 → (𝑎 ≤ 𝑏 ↔ 𝑎 < 𝑏)))
69	68	pm5.32rd 578	. . . 4 ⊢ ((((𝑃 ∈ ℙ ∧ (𝑃 mod 4) = 1) ∧ 𝑎 ∈ ℕ) ∧ 𝑏 ∈ ℕ) → ((𝑎 ≤ 𝑏 ∧ ((𝑎↑2) + (𝑏↑2)) = 𝑃) ↔ (𝑎 < 𝑏 ∧ ((𝑎↑2) + (𝑏↑2)) = 𝑃)))
70	69	reubidva 3367	. . 3 ⊢ (((𝑃 ∈ ℙ ∧ (𝑃 mod 4) = 1) ∧ 𝑎 ∈ ℕ) → (∃!𝑏 ∈ ℕ (𝑎 ≤ 𝑏 ∧ ((𝑎↑2) + (𝑏↑2)) = 𝑃) ↔ ∃!𝑏 ∈ ℕ (𝑎 < 𝑏 ∧ ((𝑎↑2) + (𝑏↑2)) = 𝑃)))
71	70	reubidva 3367	. 2 ⊢ ((𝑃 ∈ ℙ ∧ (𝑃 mod 4) = 1) → (∃!𝑎 ∈ ℕ ∃!𝑏 ∈ ℕ (𝑎 ≤ 𝑏 ∧ ((𝑎↑2) + (𝑏↑2)) = 𝑃) ↔ ∃!𝑎 ∈ ℕ ∃!𝑏 ∈ ℕ (𝑎 < 𝑏 ∧ ((𝑎↑2) + (𝑏↑2)) = 𝑃)))
72	1, 71	mpbid 232	1 ⊢ ((𝑃 ∈ ℙ ∧ (𝑃 mod 4) = 1) → ∃!𝑎 ∈ ℕ ∃!𝑏 ∈ ℕ (𝑎 < 𝑏 ∧ ((𝑎↑2) + (𝑏↑2)) = 𝑃))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1540 ∈ wcel 2109 ≠ wne 2925 ∃!wreu 3349 class class class wbr 5102 (class class class)co 7369 ℂcc 11042 ℝcr 11043 0cc0 11044 1c1 11045 + caddc 11047 · cmul 11049 < clt 11184 ≤ cle 11185 ℕcn 12162 2c2 12217 4c4 12219 ℕ₀cn0 12418 ℤcz 12505 ℝ⁺crp 12927 mod cmo 13807 ↑cexp 14002 ℙcprime 16617

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 5229 ax-sep 5246 ax-nul 5256 ax-pow 5315 ax-pr 5382 ax-un 7691 ax-cnex 11100 ax-resscn 11101 ax-1cn 11102 ax-icn 11103 ax-addcl 11104 ax-addrcl 11105 ax-mulcl 11106 ax-mulrcl 11107 ax-mulcom 11108 ax-addass 11109 ax-mulass 11110 ax-distr 11111 ax-i2m1 11112 ax-1ne0 11113 ax-1rid 11114 ax-rnegex 11115 ax-rrecex 11116 ax-cnre 11117 ax-pre-lttri 11118 ax-pre-lttrn 11119 ax-pre-ltadd 11120 ax-pre-mulgt0 11121 ax-pre-sup 11122 ax-addf 11123 ax-mulf 11124

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 3351 df-reu 3352 df-rab 3403 df-v 3446 df-sbc 3751 df-csb 3860 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-pss 3931 df-nul 4293 df-if 4485 df-pw 4561 df-sn 4586 df-pr 4588 df-tp 4590 df-op 4592 df-uni 4868 df-int 4907 df-iun 4953 df-iin 4954 df-br 5103 df-opab 5165 df-mpt 5184 df-tr 5210 df-id 5526 df-eprel 5531 df-po 5539 df-so 5540 df-fr 5584 df-se 5585 df-we 5586 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-pred 6262 df-ord 6323 df-on 6324 df-lim 6325 df-suc 6326 df-iota 6452 df-fun 6501 df-fn 6502 df-f 6503 df-f1 6504 df-fo 6505 df-f1o 6506 df-fv 6507 df-isom 6508 df-riota 7326 df-ov 7372 df-oprab 7373 df-mpo 7374 df-of 7633 df-ofr 7634 df-om 7823 df-1st 7947 df-2nd 7948 df-supp 8117 df-tpos 8182 df-frecs 8237 df-wrecs 8268 df-recs 8317 df-rdg 8355 df-1o 8411 df-2o 8412 df-oadd 8415 df-er 8648 df-ec 8650 df-qs 8654 df-map 8778 df-pm 8779 df-ixp 8848 df-en 8896 df-dom 8897 df-sdom 8898 df-fin 8899 df-fsupp 9289 df-sup 9369 df-inf 9370 df-oi 9439 df-dju 9830 df-card 9868 df-pnf 11186 df-mnf 11187 df-xr 11188 df-ltxr 11189 df-le 11190 df-sub 11383 df-neg 11384 df-div 11812 df-nn 12163 df-2 12225 df-3 12226 df-4 12227 df-5 12228 df-6 12229 df-7 12230 df-8 12231 df-9 12232 df-n0 12419 df-xnn0 12492 df-z 12506 df-dec 12626 df-uz 12770 df-q 12884 df-rp 12928 df-fz 13445 df-fzo 13592 df-fl 13730 df-mod 13808 df-seq 13943 df-exp 14003 df-hash 14272 df-cj 15041 df-re 15042 df-im 15043 df-sqrt 15177 df-abs 15178 df-dvds 16199 df-gcd 16441 df-prm 16618 df-phi 16712 df-pc 16784 df-gz 16877 df-struct 17093 df-sets 17110 df-slot 17128 df-ndx 17140 df-base 17156 df-ress 17177 df-plusg 17209 df-mulr 17210 df-starv 17211 df-sca 17212 df-vsca 17213 df-ip 17214 df-tset 17215 df-ple 17216 df-ds 17218 df-unif 17219 df-hom 17220 df-cco 17221 df-0g 17380 df-gsum 17381 df-prds 17386 df-pws 17388 df-imas 17447 df-qus 17448 df-mre 17523 df-mrc 17524 df-acs 17526 df-mgm 18549 df-sgrp 18628 df-mnd 18644 df-mhm 18692 df-submnd 18693 df-grp 18850 df-minusg 18851 df-sbg 18852 df-mulg 18982 df-subg 19037 df-nsg 19038 df-eqg 19039 df-ghm 19127 df-cntz 19231 df-cmn 19696 df-abl 19697 df-mgp 20061 df-rng 20073 df-ur 20102 df-srg 20107 df-ring 20155 df-cring 20156 df-oppr 20257 df-dvdsr 20277 df-unit 20278 df-invr 20308 df-dvr 20321 df-rhm 20392 df-nzr 20433 df-subrng 20466 df-subrg 20490 df-rlreg 20614 df-domn 20615 df-idom 20616 df-drng 20651 df-field 20652 df-lmod 20800 df-lss 20870 df-lsp 20910 df-sra 21112 df-rgmod 21113 df-lidl 21150 df-rsp 21151 df-2idl 21192 df-cnfld 21297 df-zring 21389 df-zrh 21445 df-zn 21448 df-assa 21795 df-asp 21796 df-ascl 21797 df-psr 21851 df-mvr 21852 df-mpl 21853 df-opsr 21855 df-evls 22014 df-evl 22015 df-psr1 22097 df-vr1 22098 df-ply1 22099 df-coe1 22100 df-evl1 22236 df-mdeg 25993 df-deg1 25994 df-mon1 26069 df-uc1p 26070 df-q1p 26071 df-r1p 26072 df-lgs 27239

This theorem is referenced by: 2sqreunnltblem 27395 2sqreunnlt 27404

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