2sqreultlem - Metamath Proof Explorer

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Theorem 2sqreultlem 27432

Description: Lemma for 2sqreult 27443. (Contributed by AV, 8-Jun-2023.) (Proposed by GL, 8-Jun-2023.)

**Assertion**
Ref	Expression
2sqreultlem	⊢ ((𝑃 ∈ ℙ ∧ (𝑃 mod 4) = 1) → ∃!𝑎 ∈ ℕ₀ ∃!𝑏 ∈ ℕ₀ (𝑎 < 𝑏 ∧ ((𝑎↑2) + (𝑏↑2)) = 𝑃))

Distinct variable group: 𝑃,𝑎,𝑏

**Proof of Theorem 2sqreultlem**
Step	Hyp	Ref	Expression
1		2sqreulem1 27431	. 2 ⊢ ((𝑃 ∈ ℙ ∧ (𝑃 mod 4) = 1) → ∃!𝑎 ∈ ℕ₀ ∃!𝑏 ∈ ℕ₀ (𝑎 ≤ 𝑏 ∧ ((𝑎↑2) + (𝑏↑2)) = 𝑃))
2		oveq1 7367	. . . . . . . . . . . . . . 15 ⊢ (𝑏 = 𝑎 → (𝑏↑2) = (𝑎↑2))
3	2	oveq2d 7376	. . . . . . . . . . . . . 14 ⊢ (𝑏 = 𝑎 → ((𝑎↑2) + (𝑏↑2)) = ((𝑎↑2) + (𝑎↑2)))
4	3	adantr 482	. . . . . . . . . . . . 13 ⊢ ((𝑏 = 𝑎 ∧ (((𝑃 ∈ ℙ ∧ (𝑃 mod 4) = 1) ∧ 𝑎 ∈ ℕ₀) ∧ 𝑏 ∈ ℕ₀)) → ((𝑎↑2) + (𝑏↑2)) = ((𝑎↑2) + (𝑎↑2)))
5		nn0cn 12442	. . . . . . . . . . . . . . . . 17 ⊢ (𝑎 ∈ ℕ₀ → 𝑎 ∈ ℂ)
6	5	sqcld 14101	. . . . . . . . . . . . . . . 16 ⊢ (𝑎 ∈ ℕ₀ → (𝑎↑2) ∈ ℂ)
7		2times 12307	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑎↑2) ∈ ℂ → (2 · (𝑎↑2)) = ((𝑎↑2) + (𝑎↑2)))
8	7	eqcomd 2747	. . . . . . . . . . . . . . . 16 ⊢ ((𝑎↑2) ∈ ℂ → ((𝑎↑2) + (𝑎↑2)) = (2 · (𝑎↑2)))
9	6, 8	syl 17	. . . . . . . . . . . . . . 15 ⊢ (𝑎 ∈ ℕ₀ → ((𝑎↑2) + (𝑎↑2)) = (2 · (𝑎↑2)))
10	9	adantl 483	. . . . . . . . . . . . . 14 ⊢ (((𝑃 ∈ ℙ ∧ (𝑃 mod 4) = 1) ∧ 𝑎 ∈ ℕ₀) → ((𝑎↑2) + (𝑎↑2)) = (2 · (𝑎↑2)))
11	10	ad2antrl 735	. . . . . . . . . . . . 13 ⊢ ((𝑏 = 𝑎 ∧ (((𝑃 ∈ ℙ ∧ (𝑃 mod 4) = 1) ∧ 𝑎 ∈ ℕ₀) ∧ 𝑏 ∈ ℕ₀)) → ((𝑎↑2) + (𝑎↑2)) = (2 · (𝑎↑2)))
12	4, 11	eqtrd 2776	. . . . . . . . . . . 12 ⊢ ((𝑏 = 𝑎 ∧ (((𝑃 ∈ ℙ ∧ (𝑃 mod 4) = 1) ∧ 𝑎 ∈ ℕ₀) ∧ 𝑏 ∈ ℕ₀)) → ((𝑎↑2) + (𝑏↑2)) = (2 · (𝑎↑2)))
13	12	eqeq1d 2743	. . . . . . . . . . 11 ⊢ ((𝑏 = 𝑎 ∧ (((𝑃 ∈ ℙ ∧ (𝑃 mod 4) = 1) ∧ 𝑎 ∈ ℕ₀) ∧ 𝑏 ∈ ℕ₀)) → (((𝑎↑2) + (𝑏↑2)) = 𝑃 ↔ (2 · (𝑎↑2)) = 𝑃))
14		oveq1 7367	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑃 = (2 · (𝑎↑2)) → (𝑃 mod 4) = ((2 · (𝑎↑2)) mod 4))
15	14	eqeq1d 2743	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑃 = (2 · (𝑎↑2)) → ((𝑃 mod 4) = 1 ↔ ((2 · (𝑎↑2)) mod 4) = 1))
16		eleq1 2829	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑃 = (2 · (𝑎↑2)) → (𝑃 ∈ ℙ ↔ (2 · (𝑎↑2)) ∈ ℙ))
17	15, 16	anbi12d 639	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑃 = (2 · (𝑎↑2)) → (((𝑃 mod 4) = 1 ∧ 𝑃 ∈ ℙ) ↔ (((2 · (𝑎↑2)) mod 4) = 1 ∧ (2 · (𝑎↑2)) ∈ ℙ)))
18		nn0z 12543	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑎 ∈ ℕ₀ → 𝑎 ∈ ℤ)
19		2nn0 12449	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ 2 ∈ ℕ₀
20		zexpcl 14033	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑎 ∈ ℤ ∧ 2 ∈ ℕ₀) → (𝑎↑2) ∈ ℤ)
21	18, 19, 20	sylancl 593	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑎 ∈ ℕ₀ → (𝑎↑2) ∈ ℤ)
22		2mulprm 16657	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑎↑2) ∈ ℤ → ((2 · (𝑎↑2)) ∈ ℙ ↔ (𝑎↑2) = 1))
23	21, 22	syl 17	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑎 ∈ ℕ₀ → ((2 · (𝑎↑2)) ∈ ℙ ↔ (𝑎↑2) = 1))
24		oveq2 7368	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝑎↑2) = 1 → (2 · (𝑎↑2)) = (2 · 1))
25		2t1e2 12334	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (2 · 1) = 2
26	24, 25	eqtrdi 2792	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝑎↑2) = 1 → (2 · (𝑎↑2)) = 2)
27	26	oveq1d 7375	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝑎↑2) = 1 → ((2 · (𝑎↑2)) mod 4) = (2 mod 4))
28		2re 12250	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ 2 ∈ ℝ
29		4nn 12259	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ 4 ∈ ℕ
30		nnrp 12949	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (4 ∈ ℕ → 4 ∈ ℝ⁺)
31	29, 30	ax-mp 5	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ 4 ∈ ℝ⁺
32		0le2 12278	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ 0 ≤ 2
33		2lt4 12346	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ 2 < 4
34		modid 13850	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((2 ∈ ℝ ∧ 4 ∈ ℝ⁺) ∧ (0 ≤ 2 ∧ 2 < 4)) → (2 mod 4) = 2)
35	28, 31, 32, 33, 34	mp4an 700	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (2 mod 4) = 2
36	27, 35	eqtrdi 2792	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑎↑2) = 1 → ((2 · (𝑎↑2)) mod 4) = 2)
37	36	eqeq1d 2743	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑎↑2) = 1 → (((2 · (𝑎↑2)) mod 4) = 1 ↔ 2 = 1))
38		1ne2 12379	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ 1 ≠ 2
39		eqcom 2748	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (2 = 1 ↔ 1 = 2)
40		eqneqall 2947	. . . . . . . . . . . . . . . . . . . . . . . . 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 244	. . . . . . . . . . . . . . . . . . . . . . 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 255	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑎↑2) = 1 → (((2 · (𝑎↑2)) mod 4) = 1 → (𝑎 ≤ 𝑏 → 𝑏 ≠ 𝑎)))
45	23, 44	biimtrdi 255	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑎 ∈ ℕ₀ → ((2 · (𝑎↑2)) ∈ ℙ → (((2 · (𝑎↑2)) mod 4) = 1 → (𝑎 ≤ 𝑏 → 𝑏 ≠ 𝑎))))
46	45	impcomd 413	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑎 ∈ ℕ₀ → ((((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 ∈ ℕ₀ → (𝑎 ≤ 𝑏 → 𝑏 ≠ 𝑎)))
48	17, 47	biimtrdi 255	. . . . . . . . . . . . . . . . 17 ⊢ (𝑃 = (2 · (𝑎↑2)) → (((𝑃 mod 4) = 1 ∧ 𝑃 ∈ ℙ) → (𝑎 ∈ ℕ₀ → (𝑎 ≤ 𝑏 → 𝑏 ≠ 𝑎))))
49	48	expd 417	. . . . . . . . . . . . . . . 16 ⊢ (𝑃 = (2 · (𝑎↑2)) → ((𝑃 mod 4) = 1 → (𝑃 ∈ ℙ → (𝑎 ∈ ℕ₀ → (𝑎 ≤ 𝑏 → 𝑏 ≠ 𝑎)))))
50	49	com34 91	. . . . . . . . . . . . . . 15 ⊢ (𝑃 = (2 · (𝑎↑2)) → ((𝑃 mod 4) = 1 → (𝑎 ∈ ℕ₀ → (𝑃 ∈ ℙ → (𝑎 ≤ 𝑏 → 𝑏 ≠ 𝑎)))))
51	50	eqcoms 2749	. . . . . . . . . . . . . 14 ⊢ ((2 · (𝑎↑2)) = 𝑃 → ((𝑃 mod 4) = 1 → (𝑎 ∈ ℕ₀ → (𝑃 ∈ ℙ → (𝑎 ≤ 𝑏 → 𝑏 ≠ 𝑎)))))
52	51	com14 96	. . . . . . . . . . . . 13 ⊢ (𝑃 ∈ ℙ → ((𝑃 mod 4) = 1 → (𝑎 ∈ ℕ₀ → ((2 · (𝑎↑2)) = 𝑃 → (𝑎 ≤ 𝑏 → 𝑏 ≠ 𝑎)))))
53	52	imp31 419	. . . . . . . . . . . 12 ⊢ (((𝑃 ∈ ℙ ∧ (𝑃 mod 4) = 1) ∧ 𝑎 ∈ ℕ₀) → ((2 · (𝑎↑2)) = 𝑃 → (𝑎 ≤ 𝑏 → 𝑏 ≠ 𝑎)))
54	53	ad2antrl 735	. . . . . . . . . . 11 ⊢ ((𝑏 = 𝑎 ∧ (((𝑃 ∈ ℙ ∧ (𝑃 mod 4) = 1) ∧ 𝑎 ∈ ℕ₀) ∧ 𝑏 ∈ ℕ₀)) → ((2 · (𝑎↑2)) = 𝑃 → (𝑎 ≤ 𝑏 → 𝑏 ≠ 𝑎)))
55	13, 54	sylbid 242	. . . . . . . . . 10 ⊢ ((𝑏 = 𝑎 ∧ (((𝑃 ∈ ℙ ∧ (𝑃 mod 4) = 1) ∧ 𝑎 ∈ ℕ₀) ∧ 𝑏 ∈ ℕ₀)) → (((𝑎↑2) + (𝑏↑2)) = 𝑃 → (𝑎 ≤ 𝑏 → 𝑏 ≠ 𝑎)))
56	55	expimpd 455	. . . . . . . . 9 ⊢ (𝑏 = 𝑎 → (((((𝑃 ∈ ℙ ∧ (𝑃 mod 4) = 1) ∧ 𝑎 ∈ ℕ₀) ∧ 𝑏 ∈ ℕ₀) ∧ ((𝑎↑2) + (𝑏↑2)) = 𝑃) → (𝑎 ≤ 𝑏 → 𝑏 ≠ 𝑎)))
57		2a1 28	. . . . . . . . 9 ⊢ (𝑏 ≠ 𝑎 → (((((𝑃 ∈ ℙ ∧ (𝑃 mod 4) = 1) ∧ 𝑎 ∈ ℕ₀) ∧ 𝑏 ∈ ℕ₀) ∧ ((𝑎↑2) + (𝑏↑2)) = 𝑃) → (𝑎 ≤ 𝑏 → 𝑏 ≠ 𝑎)))
58	56, 57	pm2.61ine 3019	. . . . . . . 8 ⊢ (((((𝑃 ∈ ℙ ∧ (𝑃 mod 4) = 1) ∧ 𝑎 ∈ ℕ₀) ∧ 𝑏 ∈ ℕ₀) ∧ ((𝑎↑2) + (𝑏↑2)) = 𝑃) → (𝑎 ≤ 𝑏 → 𝑏 ≠ 𝑎))
59	58	pm4.71d 567	. . . . . . 7 ⊢ (((((𝑃 ∈ ℙ ∧ (𝑃 mod 4) = 1) ∧ 𝑎 ∈ ℕ₀) ∧ 𝑏 ∈ ℕ₀) ∧ ((𝑎↑2) + (𝑏↑2)) = 𝑃) → (𝑎 ≤ 𝑏 ↔ (𝑎 ≤ 𝑏 ∧ 𝑏 ≠ 𝑎)))
60		nn0re 12441	. . . . . . . . . . 11 ⊢ (𝑎 ∈ ℕ₀ → 𝑎 ∈ ℝ)
61	60	adantl 483	. . . . . . . . . 10 ⊢ (((𝑃 ∈ ℙ ∧ (𝑃 mod 4) = 1) ∧ 𝑎 ∈ ℕ₀) → 𝑎 ∈ ℝ)
62		nn0re 12441	. . . . . . . . . 10 ⊢ (𝑏 ∈ ℕ₀ → 𝑏 ∈ ℝ)
63		ltlen 11242	. . . . . . . . . 10 ⊢ ((𝑎 ∈ ℝ ∧ 𝑏 ∈ ℝ) → (𝑎 < 𝑏 ↔ (𝑎 ≤ 𝑏 ∧ 𝑏 ≠ 𝑎)))
64	61, 62, 63	syl2an 603	. . . . . . . . 9 ⊢ ((((𝑃 ∈ ℙ ∧ (𝑃 mod 4) = 1) ∧ 𝑎 ∈ ℕ₀) ∧ 𝑏 ∈ ℕ₀) → (𝑎 < 𝑏 ↔ (𝑎 ≤ 𝑏 ∧ 𝑏 ≠ 𝑎)))
65	64	bibi2d 344	. . . . . . . 8 ⊢ ((((𝑃 ∈ ℙ ∧ (𝑃 mod 4) = 1) ∧ 𝑎 ∈ ℕ₀) ∧ 𝑏 ∈ ℕ₀) → ((𝑎 ≤ 𝑏 ↔ 𝑎 < 𝑏) ↔ (𝑎 ≤ 𝑏 ↔ (𝑎 ≤ 𝑏 ∧ 𝑏 ≠ 𝑎))))
66	65	adantr 482	. . . . . . 7 ⊢ (((((𝑃 ∈ ℙ ∧ (𝑃 mod 4) = 1) ∧ 𝑎 ∈ ℕ₀) ∧ 𝑏 ∈ ℕ₀) ∧ ((𝑎↑2) + (𝑏↑2)) = 𝑃) → ((𝑎 ≤ 𝑏 ↔ 𝑎 < 𝑏) ↔ (𝑎 ≤ 𝑏 ↔ (𝑎 ≤ 𝑏 ∧ 𝑏 ≠ 𝑎))))
67	59, 66	mpbird 259	. . . . . 6 ⊢ (((((𝑃 ∈ ℙ ∧ (𝑃 mod 4) = 1) ∧ 𝑎 ∈ ℕ₀) ∧ 𝑏 ∈ ℕ₀) ∧ ((𝑎↑2) + (𝑏↑2)) = 𝑃) → (𝑎 ≤ 𝑏 ↔ 𝑎 < 𝑏))
68	67	ex 414	. . . . 5 ⊢ ((((𝑃 ∈ ℙ ∧ (𝑃 mod 4) = 1) ∧ 𝑎 ∈ ℕ₀) ∧ 𝑏 ∈ ℕ₀) → (((𝑎↑2) + (𝑏↑2)) = 𝑃 → (𝑎 ≤ 𝑏 ↔ 𝑎 < 𝑏)))
69	68	pm5.32rd 584	. . . 4 ⊢ ((((𝑃 ∈ ℙ ∧ (𝑃 mod 4) = 1) ∧ 𝑎 ∈ ℕ₀) ∧ 𝑏 ∈ ℕ₀) → ((𝑎 ≤ 𝑏 ∧ ((𝑎↑2) + (𝑏↑2)) = 𝑃) ↔ (𝑎 < 𝑏 ∧ ((𝑎↑2) + (𝑏↑2)) = 𝑃)))
70	69	reubidva 3360	. . 3 ⊢ (((𝑃 ∈ ℙ ∧ (𝑃 mod 4) = 1) ∧ 𝑎 ∈ ℕ₀) → (∃!𝑏 ∈ ℕ₀ (𝑎 ≤ 𝑏 ∧ ((𝑎↑2) + (𝑏↑2)) = 𝑃) ↔ ∃!𝑏 ∈ ℕ₀ (𝑎 < 𝑏 ∧ ((𝑎↑2) + (𝑏↑2)) = 𝑃)))
71	70	reubidva 3360	. 2 ⊢ ((𝑃 ∈ ℙ ∧ (𝑃 mod 4) = 1) → (∃!𝑎 ∈ ℕ₀ ∃!𝑏 ∈ ℕ₀ (𝑎 ≤ 𝑏 ∧ ((𝑎↑2) + (𝑏↑2)) = 𝑃) ↔ ∃!𝑎 ∈ ℕ₀ ∃!𝑏 ∈ ℕ₀ (𝑎 < 𝑏 ∧ ((𝑎↑2) + (𝑏↑2)) = 𝑃)))
72	1, 71	mpbid 234	1 ⊢ ((𝑃 ∈ ℙ ∧ (𝑃 mod 4) = 1) → ∃!𝑎 ∈ ℕ₀ ∃!𝑏 ∈ ℕ₀ (𝑎 < 𝑏 ∧ ((𝑎↑2) + (𝑏↑2)) = 𝑃))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 208 ∧ wa 397 = wceq 1548 ∈ wcel 2121 ≠ wne 2936 ∃!wreu 3344 class class class wbr 5075 (class class class)co 7360 ℂcc 11031 ℝcr 11032 0cc0 11033 1c1 11034 + caddc 11036 · cmul 11038 < clt 11174 ≤ cle 11175 ℕcn 12169 2c2 12231 4c4 12233 ℕ₀cn0 12432 ℤcz 12519 ℝ⁺crp 12937 mod cmo 13823 ↑cexp 14018 ℙcprime 16635

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1975 ax-7 2016 ax-8 2123 ax-9 2131 ax-10 2154 ax-11 2170 ax-12 2191 ax-ext 2713 ax-rep 5202 ax-sep 5221 ax-nul 5231 ax-pow 5297 ax-pr 5365 ax-un 7682 ax-cnex 11089 ax-resscn 11090 ax-1cn 11091 ax-icn 11092 ax-addcl 11093 ax-addrcl 11094 ax-mulcl 11095 ax-mulrcl 11096 ax-mulcom 11097 ax-addass 11098 ax-mulass 11099 ax-distr 11100 ax-i2m1 11101 ax-1ne0 11102 ax-1rid 11103 ax-rnegex 11104 ax-rrecex 11105 ax-cnre 11106 ax-pre-lttri 11107 ax-pre-lttrn 11108 ax-pre-ltadd 11109 ax-pre-mulgt0 11110 ax-pre-sup 11111 ax-addf 11112 ax-mulf 11113

This theorem depends on definitions: df-bi 209 df-an 398 df-or 855 df-3or 1094 df-3an 1095 df-tru 1551 df-fal 1561 df-ex 1788 df-nf 1792 df-sb 2075 df-mo 2545 df-eu 2575 df-clab 2720 df-cleq 2733 df-clel 2816 df-nfc 2890 df-ne 2937 df-nel 3041 df-ral 3056 df-rex 3066 df-rmo 3346 df-reu 3347 df-rab 3394 df-v 3435 df-sbc 3726 df-csb 3834 df-dif 3888 df-un 3890 df-in 3892 df-ss 3902 df-pss 3905 df-nul 4265 df-if 4458 df-pw 4534 df-sn 4559 df-pr 4561 df-tp 4563 df-op 4565 df-uni 4842 df-int 4881 df-iun 4926 df-iin 4927 df-br 5076 df-opab 5138 df-mpt 5157 df-tr 5183 df-id 5516 df-eprel 5521 df-po 5529 df-so 5530 df-fr 5574 df-se 5575 df-we 5576 df-xp 5627 df-rel 5628 df-cnv 5629 df-co 5630 df-dm 5631 df-rn 5632 df-res 5633 df-ima 5634 df-pred 6256 df-ord 6317 df-on 6318 df-lim 6319 df-suc 6320 df-iota 6445 df-fun 6491 df-fn 6492 df-f 6493 df-f1 6494 df-fo 6495 df-f1o 6496 df-fv 6497 df-isom 6498 df-riota 7317 df-ov 7363 df-oprab 7364 df-mpo 7365 df-of 7624 df-ofr 7625 df-om 7811 df-1st 7935 df-2nd 7936 df-supp 8105 df-tpos 8170 df-frecs 8225 df-wrecs 8256 df-recs 8305 df-rdg 8343 df-1o 8399 df-2o 8400 df-oadd 8403 df-er 8637 df-ec 8639 df-qs 8643 df-map 8769 df-pm 8770 df-ixp 8840 df-en 8888 df-dom 8889 df-sdom 8890 df-fin 8891 df-fsupp 9269 df-sup 9349 df-inf 9350 df-oi 9419 df-dju 9820 df-card 9858 df-pnf 11176 df-mnf 11177 df-xr 11178 df-ltxr 11179 df-le 11180 df-sub 11374 df-neg 11375 df-div 11803 df-nn 12170 df-2 12239 df-3 12240 df-4 12241 df-5 12242 df-6 12243 df-7 12244 df-8 12245 df-9 12246 df-n0 12433 df-xnn0 12506 df-z 12520 df-dec 12640 df-uz 12784 df-q 12894 df-rp 12938 df-fz 13457 df-fzo 13604 df-fl 13746 df-mod 13824 df-seq 13959 df-exp 14019 df-hash 14288 df-cj 15056 df-re 15057 df-im 15058 df-sqrt 15192 df-abs 15193 df-dvds 16217 df-gcd 16459 df-prm 16636 df-phi 16731 df-pc 16803 df-gz 16896 df-struct 17112 df-sets 17129 df-slot 17147 df-ndx 17159 df-base 17175 df-ress 17196 df-plusg 17228 df-mulr 17229 df-starv 17230 df-sca 17231 df-vsca 17232 df-ip 17233 df-tset 17234 df-ple 17235 df-ds 17237 df-unif 17238 df-hom 17239 df-cco 17240 df-0g 17399 df-gsum 17400 df-prds 17405 df-pws 17407 df-imas 17467 df-qus 17468 df-mre 17543 df-mrc 17544 df-acs 17546 df-mgm 18603 df-sgrp 18682 df-mnd 18698 df-mhm 18746 df-submnd 18747 df-grp 18907 df-minusg 18908 df-sbg 18909 df-mulg 19039 df-subg 19094 df-nsg 19095 df-eqg 19096 df-ghm 19183 df-cntz 19287 df-cmn 19752 df-abl 19753 df-mgp 20117 df-rng 20129 df-ur 20158 df-srg 20163 df-ring 20211 df-cring 20212 df-oppr 20312 df-dvdsr 20332 df-unit 20333 df-invr 20363 df-dvr 20376 df-rhm 20447 df-nzr 20489 df-subrng 20522 df-subrg 20546 df-rlreg 20670 df-domn 20671 df-idom 20672 df-drng 20707 df-field 20708 df-lmod 20856 df-lss 20926 df-lsp 20966 df-sra 21167 df-rgmod 21168 df-lidl 21205 df-rsp 21206 df-2idl 21247 df-cnfld 21352 df-zring 21426 df-zrh 21482 df-zn 21485 df-assa 21832 df-asp 21833 df-ascl 21834 df-psr 21888 df-mvr 21889 df-mpl 21890 df-opsr 21892 df-evls 22054 df-evl 22055 df-psr1 22169 df-vr1 22170 df-ply1 22171 df-coe1 22172 df-evl1 22306 df-mdeg 26042 df-deg1 26043 df-mon1 26118 df-uc1p 26119 df-q1p 26120 df-r1p 26121 df-lgs 27280

This theorem is referenced by: 2sqreultblem 27433 2sqreult 27443

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