2sqreunnltlem - Metamath Proof Explorer

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Theorem 2sqreunnltlem 27186

Description: Lemma for 2sqreunnlt 27196. (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 27185	. 2 ⊢ ((𝑃 ∈ ℙ ∧ (𝑃 mod 4) = 1) → ∃!𝑎 ∈ ℕ ∃!𝑏 ∈ ℕ (𝑎 ≤ 𝑏 ∧ ((𝑎↑2) + (𝑏↑2)) = 𝑃))
2		oveq1 7419	. . . . . . . . . . . . . . 15 ⊢ (𝑏 = 𝑎 → (𝑏↑2) = (𝑎↑2))
3	2	oveq2d 7428	. . . . . . . . . . . . . 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 12225	. . . . . . . . . . . . . . . . 17 ⊢ (𝑎 ∈ ℕ → 𝑎 ∈ ℂ)
6	5	sqcld 14114	. . . . . . . . . . . . . . . 16 ⊢ (𝑎 ∈ ℕ → (𝑎↑2) ∈ ℂ)
7		2times 12353	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑎↑2) ∈ ℂ → (2 · (𝑎↑2)) = ((𝑎↑2) + (𝑎↑2)))
8	7	eqcomd 2737	. . . . . . . . . . . . . . . 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 725	. . . . . . . . . . . . 13 ⊢ ((𝑏 = 𝑎 ∧ (((𝑃 ∈ ℙ ∧ (𝑃 mod 4) = 1) ∧ 𝑎 ∈ ℕ) ∧ 𝑏 ∈ ℕ)) → ((𝑎↑2) + (𝑎↑2)) = (2 · (𝑎↑2)))
12	4, 11	eqtrd 2771	. . . . . . . . . . . 12 ⊢ ((𝑏 = 𝑎 ∧ (((𝑃 ∈ ℙ ∧ (𝑃 mod 4) = 1) ∧ 𝑎 ∈ ℕ) ∧ 𝑏 ∈ ℕ)) → ((𝑎↑2) + (𝑏↑2)) = (2 · (𝑎↑2)))
13	12	eqeq1d 2733	. . . . . . . . . . 11 ⊢ ((𝑏 = 𝑎 ∧ (((𝑃 ∈ ℙ ∧ (𝑃 mod 4) = 1) ∧ 𝑎 ∈ ℕ) ∧ 𝑏 ∈ ℕ)) → (((𝑎↑2) + (𝑏↑2)) = 𝑃 ↔ (2 · (𝑎↑2)) = 𝑃))
14		oveq1 7419	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑃 = (2 · (𝑎↑2)) → (𝑃 mod 4) = ((2 · (𝑎↑2)) mod 4))
15	14	eqeq1d 2733	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑃 = (2 · (𝑎↑2)) → ((𝑃 mod 4) = 1 ↔ ((2 · (𝑎↑2)) mod 4) = 1))
16		eleq1 2820	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑃 = (2 · (𝑎↑2)) → (𝑃 ∈ ℙ ↔ (2 · (𝑎↑2)) ∈ ℙ))
17	15, 16	anbi12d 630	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑃 = (2 · (𝑎↑2)) → (((𝑃 mod 4) = 1 ∧ 𝑃 ∈ ℙ) ↔ (((2 · (𝑎↑2)) mod 4) = 1 ∧ (2 · (𝑎↑2)) ∈ ℙ)))
18		nnz 12584	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑎 ∈ ℕ → 𝑎 ∈ ℤ)
19		2nn0 12494	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ 2 ∈ ℕ₀
20		zexpcl 14047	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑎 ∈ ℤ ∧ 2 ∈ ℕ₀) → (𝑎↑2) ∈ ℤ)
21	18, 19, 20	sylancl 585	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑎 ∈ ℕ → (𝑎↑2) ∈ ℤ)
22		2mulprm 16635	. . . . . . . . . . . . . . . . . . . . . 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 7420	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝑎↑2) = 1 → (2 · (𝑎↑2)) = (2 · 1))
25		2t1e2 12380	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (2 · 1) = 2
26	24, 25	eqtrdi 2787	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝑎↑2) = 1 → (2 · (𝑎↑2)) = 2)
27	26	oveq1d 7427	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝑎↑2) = 1 → ((2 · (𝑎↑2)) mod 4) = (2 mod 4))
28		2re 12291	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ 2 ∈ ℝ
29		4nn 12300	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ 4 ∈ ℕ
30		nnrp 12990	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (4 ∈ ℕ → 4 ∈ ℝ⁺)
31	29, 30	ax-mp 5	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ 4 ∈ ℝ⁺
32		0le2 12319	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ 0 ≤ 2
33		2lt4 12392	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ 2 < 4
34		modid 13866	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((2 ∈ ℝ ∧ 4 ∈ ℝ⁺) ∧ (0 ≤ 2 ∧ 2 < 4)) → (2 mod 4) = 2)
35	28, 31, 32, 33, 34	mp4an 690	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (2 mod 4) = 2
36	27, 35	eqtrdi 2787	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑎↑2) = 1 → ((2 · (𝑎↑2)) mod 4) = 2)
37	36	eqeq1d 2733	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑎↑2) = 1 → (((2 · (𝑎↑2)) mod 4) = 1 ↔ 2 = 1))
38		1ne2 12425	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ 1 ≠ 2
39		eqcom 2738	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (2 = 1 ↔ 1 = 2)
40		eqneqall 2950	. . . . . . . . . . . . . . . . . . . . . . . . 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 241	. . . . . . . . . . . . . . . . . . . . . . 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	syl6bi 252	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑎↑2) = 1 → (((2 · (𝑎↑2)) mod 4) = 1 → (𝑎 ≤ 𝑏 → 𝑏 ≠ 𝑎)))
45	23, 44	syl6bi 252	. . . . . . . . . . . . . . . . . . . 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	syl6bi 252	. . . . . . . . . . . . . . . . 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 2739	. . . . . . . . . . . . . 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 725	. . . . . . . . . . 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 239	. . . . . . . . . 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 3024	. . . . . . . 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 12224	. . . . . . . . . . 11 ⊢ (𝑎 ∈ ℕ → 𝑎 ∈ ℝ)
61	60	adantl 481	. . . . . . . . . 10 ⊢ (((𝑃 ∈ ℙ ∧ (𝑃 mod 4) = 1) ∧ 𝑎 ∈ ℕ) → 𝑎 ∈ ℝ)
62		nnre 12224	. . . . . . . . . 10 ⊢ (𝑏 ∈ ℕ → 𝑏 ∈ ℝ)
63		ltlen 11320	. . . . . . . . . 10 ⊢ ((𝑎 ∈ ℝ ∧ 𝑏 ∈ ℝ) → (𝑎 < 𝑏 ↔ (𝑎 ≤ 𝑏 ∧ 𝑏 ≠ 𝑎)))
64	61, 62, 63	syl2an 595	. . . . . . . . 9 ⊢ ((((𝑃 ∈ ℙ ∧ (𝑃 mod 4) = 1) ∧ 𝑎 ∈ ℕ) ∧ 𝑏 ∈ ℕ) → (𝑎 < 𝑏 ↔ (𝑎 ≤ 𝑏 ∧ 𝑏 ≠ 𝑎)))
65	64	bibi2d 341	. . . . . . . 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 256	. . . . . 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 577	. . . 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 3391	. . 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 3391	. 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 231	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 205 ∧ wa 395 = wceq 1540 ∈ wcel 2105 ≠ wne 2939 ∃!wreu 3373 class class class wbr 5149 (class class class)co 7412 ℂcc 11111 ℝcr 11112 0cc0 11113 1c1 11114 + caddc 11116 · cmul 11118 < clt 11253 ≤ cle 11254 ℕcn 12217 2c2 12272 4c4 12274 ℕ₀cn0 12477 ℤcz 12563 ℝ⁺crp 12979 mod cmo 13839 ↑cexp 14032 ℙcprime 16613

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2702 ax-rep 5286 ax-sep 5300 ax-nul 5307 ax-pow 5364 ax-pr 5428 ax-un 7728 ax-cnex 11169 ax-resscn 11170 ax-1cn 11171 ax-icn 11172 ax-addcl 11173 ax-addrcl 11174 ax-mulcl 11175 ax-mulrcl 11176 ax-mulcom 11177 ax-addass 11178 ax-mulass 11179 ax-distr 11180 ax-i2m1 11181 ax-1ne0 11182 ax-1rid 11183 ax-rnegex 11184 ax-rrecex 11185 ax-cnre 11186 ax-pre-lttri 11187 ax-pre-lttrn 11188 ax-pre-ltadd 11189 ax-pre-mulgt0 11190 ax-pre-sup 11191 ax-addf 11192 ax-mulf 11193

This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-rmo 3375 df-reu 3376 df-rab 3432 df-v 3475 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-tp 4634 df-op 4636 df-uni 4910 df-int 4952 df-iun 5000 df-iin 5001 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-id 5575 df-eprel 5581 df-po 5589 df-so 5590 df-fr 5632 df-se 5633 df-we 5634 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-rn 5688 df-res 5689 df-ima 5690 df-pred 6301 df-ord 6368 df-on 6369 df-lim 6370 df-suc 6371 df-iota 6496 df-fun 6546 df-fn 6547 df-f 6548 df-f1 6549 df-fo 6550 df-f1o 6551 df-fv 6552 df-isom 6553 df-riota 7368 df-ov 7415 df-oprab 7416 df-mpo 7417 df-of 7673 df-ofr 7674 df-om 7859 df-1st 7978 df-2nd 7979 df-supp 8150 df-tpos 8214 df-frecs 8269 df-wrecs 8300 df-recs 8374 df-rdg 8413 df-1o 8469 df-2o 8470 df-oadd 8473 df-er 8706 df-ec 8708 df-qs 8712 df-map 8825 df-pm 8826 df-ixp 8895 df-en 8943 df-dom 8944 df-sdom 8945 df-fin 8946 df-fsupp 9365 df-sup 9440 df-inf 9441 df-oi 9508 df-dju 9899 df-card 9937 df-pnf 11255 df-mnf 11256 df-xr 11257 df-ltxr 11258 df-le 11259 df-sub 11451 df-neg 11452 df-div 11877 df-nn 12218 df-2 12280 df-3 12281 df-4 12282 df-5 12283 df-6 12284 df-7 12285 df-8 12286 df-9 12287 df-n0 12478 df-xnn0 12550 df-z 12564 df-dec 12683 df-uz 12828 df-q 12938 df-rp 12980 df-fz 13490 df-fzo 13633 df-fl 13762 df-mod 13840 df-seq 13972 df-exp 14033 df-hash 14296 df-cj 15051 df-re 15052 df-im 15053 df-sqrt 15187 df-abs 15188 df-dvds 16203 df-gcd 16441 df-prm 16614 df-phi 16704 df-pc 16775 df-gz 16868 df-struct 17085 df-sets 17102 df-slot 17120 df-ndx 17132 df-base 17150 df-ress 17179 df-plusg 17215 df-mulr 17216 df-starv 17217 df-sca 17218 df-vsca 17219 df-ip 17220 df-tset 17221 df-ple 17222 df-ds 17224 df-unif 17225 df-hom 17226 df-cco 17227 df-0g 17392 df-gsum 17393 df-prds 17398 df-pws 17400 df-imas 17459 df-qus 17460 df-mre 17535 df-mrc 17536 df-acs 17538 df-mgm 18566 df-sgrp 18645 df-mnd 18661 df-mhm 18706 df-submnd 18707 df-grp 18859 df-minusg 18860 df-sbg 18861 df-mulg 18988 df-subg 19040 df-nsg 19041 df-eqg 19042 df-ghm 19129 df-cntz 19223 df-cmn 19692 df-abl 19693 df-mgp 20030 df-rng 20048 df-ur 20077 df-srg 20082 df-ring 20130 df-cring 20131 df-oppr 20226 df-dvdsr 20249 df-unit 20250 df-invr 20280 df-dvr 20293 df-rhm 20364 df-nzr 20405 df-subrng 20435 df-subrg 20460 df-drng 20503 df-field 20504 df-lmod 20617 df-lss 20688 df-lsp 20728 df-sra 20931 df-rgmod 20932 df-lidl 20933 df-rsp 20934 df-2idl 21007 df-rlreg 21100 df-domn 21101 df-idom 21102 df-cnfld 21146 df-zring 21219 df-zrh 21273 df-zn 21276 df-assa 21628 df-asp 21629 df-ascl 21630 df-psr 21682 df-mvr 21683 df-mpl 21684 df-opsr 21686 df-evls 21855 df-evl 21856 df-psr1 21924 df-vr1 21925 df-ply1 21926 df-coe1 21927 df-evl1 22056 df-mdeg 25803 df-deg1 25804 df-mon1 25881 df-uc1p 25882 df-q1p 25883 df-r1p 25884 df-lgs 27031

This theorem is referenced by: 2sqreunnltblem 27187 2sqreunnlt 27196

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