bcsiALT - Hilbert Space Explorer

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Theorem bcsiALT 30432

Description: Bunjakovaskij-Cauchy-Schwarz inequality. Remark 3.4 of [Beran] p. 98. (Contributed by NM, 11-Oct-1999.) (New usage is discouraged.) (Proof modification is discouraged.)

**Hypotheses**
Ref	Expression
bcs.1	⊢ 𝐴 ∈ ℋ
bcs.2	⊢ 𝐵 ∈ ℋ

**Assertion**
Ref	Expression
bcsiALT	⊢ (abs‘(𝐴 ·_ih 𝐵)) ≤ ((norm_ℎ‘𝐴) · (norm_ℎ‘𝐵))

**Proof of Theorem bcsiALT**
Step	Hyp	Ref	Expression
1		fveq2 6892	. . 3 ⊢ ((𝐴 ·_ih 𝐵) = 0 → (abs‘(𝐴 ·_ih 𝐵)) = (abs‘0))
2		abs0 15232	. . . 4 ⊢ (abs‘0) = 0
3		bcs.1	. . . . . 6 ⊢ 𝐴 ∈ ℋ
4		normge0 30379	. . . . . 6 ⊢ (𝐴 ∈ ℋ → 0 ≤ (norm_ℎ‘𝐴))
5	3, 4	ax-mp 5	. . . . 5 ⊢ 0 ≤ (norm_ℎ‘𝐴)
6		bcs.2	. . . . . 6 ⊢ 𝐵 ∈ ℋ
7		normge0 30379	. . . . . 6 ⊢ (𝐵 ∈ ℋ → 0 ≤ (norm_ℎ‘𝐵))
8	6, 7	ax-mp 5	. . . . 5 ⊢ 0 ≤ (norm_ℎ‘𝐵)
9	3	normcli 30384	. . . . . 6 ⊢ (norm_ℎ‘𝐴) ∈ ℝ
10	6	normcli 30384	. . . . . 6 ⊢ (norm_ℎ‘𝐵) ∈ ℝ
11	9, 10	mulge0i 11761	. . . . 5 ⊢ ((0 ≤ (norm_ℎ‘𝐴) ∧ 0 ≤ (norm_ℎ‘𝐵)) → 0 ≤ ((norm_ℎ‘𝐴) · (norm_ℎ‘𝐵)))
12	5, 8, 11	mp2an 691	. . . 4 ⊢ 0 ≤ ((norm_ℎ‘𝐴) · (norm_ℎ‘𝐵))
13	2, 12	eqbrtri 5170	. . 3 ⊢ (abs‘0) ≤ ((norm_ℎ‘𝐴) · (norm_ℎ‘𝐵))
14	1, 13	eqbrtrdi 5188	. 2 ⊢ ((𝐴 ·_ih 𝐵) = 0 → (abs‘(𝐴 ·_ih 𝐵)) ≤ ((norm_ℎ‘𝐴) · (norm_ℎ‘𝐵)))
15		df-ne 2942	. . . 4 ⊢ ((𝐴 ·_ih 𝐵) ≠ 0 ↔ ¬ (𝐴 ·_ih 𝐵) = 0)
16	6, 3	his1i 30353	. . . . . . . 8 ⊢ (𝐵 ·_ih 𝐴) = (∗‘(𝐴 ·_ih 𝐵))
17	16	oveq2i 7420	. . . . . . 7 ⊢ (((𝐴 ·_ih 𝐵) / (abs‘(𝐴 ·_ih 𝐵))) · (𝐵 ·_ih 𝐴)) = (((𝐴 ·_ih 𝐵) / (abs‘(𝐴 ·_ih 𝐵))) · (∗‘(𝐴 ·_ih 𝐵)))
18	17	oveq2i 7420	. . . . . 6 ⊢ (((∗‘((𝐴 ·_ih 𝐵) / (abs‘(𝐴 ·_ih 𝐵)))) · (𝐴 ·_ih 𝐵)) + (((𝐴 ·_ih 𝐵) / (abs‘(𝐴 ·_ih 𝐵))) · (𝐵 ·_ih 𝐴))) = (((∗‘((𝐴 ·_ih 𝐵) / (abs‘(𝐴 ·_ih 𝐵)))) · (𝐴 ·_ih 𝐵)) + (((𝐴 ·_ih 𝐵) / (abs‘(𝐴 ·_ih 𝐵))) · (∗‘(𝐴 ·_ih 𝐵))))
19	3, 6	hicli 30334	. . . . . . 7 ⊢ (𝐴 ·_ih 𝐵) ∈ ℂ
20		abslem2 15286	. . . . . . 7 ⊢ (((𝐴 ·_ih 𝐵) ∈ ℂ ∧ (𝐴 ·_ih 𝐵) ≠ 0) → (((∗‘((𝐴 ·_ih 𝐵) / (abs‘(𝐴 ·_ih 𝐵)))) · (𝐴 ·_ih 𝐵)) + (((𝐴 ·_ih 𝐵) / (abs‘(𝐴 ·_ih 𝐵))) · (∗‘(𝐴 ·_ih 𝐵)))) = (2 · (abs‘(𝐴 ·_ih 𝐵))))
21	19, 20	mpan 689	. . . . . 6 ⊢ ((𝐴 ·_ih 𝐵) ≠ 0 → (((∗‘((𝐴 ·_ih 𝐵) / (abs‘(𝐴 ·_ih 𝐵)))) · (𝐴 ·_ih 𝐵)) + (((𝐴 ·_ih 𝐵) / (abs‘(𝐴 ·_ih 𝐵))) · (∗‘(𝐴 ·_ih 𝐵)))) = (2 · (abs‘(𝐴 ·_ih 𝐵))))
22	18, 21	eqtr2id 2786	. . . . 5 ⊢ ((𝐴 ·_ih 𝐵) ≠ 0 → (2 · (abs‘(𝐴 ·_ih 𝐵))) = (((∗‘((𝐴 ·_ih 𝐵) / (abs‘(𝐴 ·_ih 𝐵)))) · (𝐴 ·_ih 𝐵)) + (((𝐴 ·_ih 𝐵) / (abs‘(𝐴 ·_ih 𝐵))) · (𝐵 ·_ih 𝐴))))
23	19	abs00i 15345	. . . . . . . 8 ⊢ ((abs‘(𝐴 ·_ih 𝐵)) = 0 ↔ (𝐴 ·_ih 𝐵) = 0)
24	23	necon3bii 2994	. . . . . . 7 ⊢ ((abs‘(𝐴 ·_ih 𝐵)) ≠ 0 ↔ (𝐴 ·_ih 𝐵) ≠ 0)
25	19	abscli 15342	. . . . . . . . . 10 ⊢ (abs‘(𝐴 ·_ih 𝐵)) ∈ ℝ
26	25	recni 11228	. . . . . . . . 9 ⊢ (abs‘(𝐴 ·_ih 𝐵)) ∈ ℂ
27	19, 26	divclzi 11949	. . . . . . . 8 ⊢ ((abs‘(𝐴 ·_ih 𝐵)) ≠ 0 → ((𝐴 ·_ih 𝐵) / (abs‘(𝐴 ·_ih 𝐵))) ∈ ℂ)
28	19, 26	divreczi 11952	. . . . . . . . . 10 ⊢ ((abs‘(𝐴 ·_ih 𝐵)) ≠ 0 → ((𝐴 ·_ih 𝐵) / (abs‘(𝐴 ·_ih 𝐵))) = ((𝐴 ·_ih 𝐵) · (1 / (abs‘(𝐴 ·_ih 𝐵)))))
29	28	fveq2d 6896	. . . . . . . . 9 ⊢ ((abs‘(𝐴 ·_ih 𝐵)) ≠ 0 → (abs‘((𝐴 ·_ih 𝐵) / (abs‘(𝐴 ·_ih 𝐵)))) = (abs‘((𝐴 ·_ih 𝐵) · (1 / (abs‘(𝐴 ·_ih 𝐵))))))
30	26	recclzi 11939	. . . . . . . . . . 11 ⊢ ((abs‘(𝐴 ·_ih 𝐵)) ≠ 0 → (1 / (abs‘(𝐴 ·_ih 𝐵))) ∈ ℂ)
31		absmul 15241	. . . . . . . . . . 11 ⊢ (((𝐴 ·_ih 𝐵) ∈ ℂ ∧ (1 / (abs‘(𝐴 ·_ih 𝐵))) ∈ ℂ) → (abs‘((𝐴 ·_ih 𝐵) · (1 / (abs‘(𝐴 ·_ih 𝐵))))) = ((abs‘(𝐴 ·_ih 𝐵)) · (abs‘(1 / (abs‘(𝐴 ·_ih 𝐵))))))
32	19, 30, 31	sylancr 588	. . . . . . . . . 10 ⊢ ((abs‘(𝐴 ·_ih 𝐵)) ≠ 0 → (abs‘((𝐴 ·_ih 𝐵) · (1 / (abs‘(𝐴 ·_ih 𝐵))))) = ((abs‘(𝐴 ·_ih 𝐵)) · (abs‘(1 / (abs‘(𝐴 ·_ih 𝐵))))))
33	25	rerecclzi 11978	. . . . . . . . . . . 12 ⊢ ((abs‘(𝐴 ·_ih 𝐵)) ≠ 0 → (1 / (abs‘(𝐴 ·_ih 𝐵))) ∈ ℝ)
34		0re 11216	. . . . . . . . . . . . . 14 ⊢ 0 ∈ ℝ
35	33, 34	jctil 521	. . . . . . . . . . . . 13 ⊢ ((abs‘(𝐴 ·_ih 𝐵)) ≠ 0 → (0 ∈ ℝ ∧ (1 / (abs‘(𝐴 ·_ih 𝐵))) ∈ ℝ))
36	19	absgt0i 15346	. . . . . . . . . . . . . . 15 ⊢ ((𝐴 ·_ih 𝐵) ≠ 0 ↔ 0 < (abs‘(𝐴 ·_ih 𝐵)))
37	24, 36	bitri 275	. . . . . . . . . . . . . 14 ⊢ ((abs‘(𝐴 ·_ih 𝐵)) ≠ 0 ↔ 0 < (abs‘(𝐴 ·_ih 𝐵)))
38	25	recgt0i 12119	. . . . . . . . . . . . . 14 ⊢ (0 < (abs‘(𝐴 ·_ih 𝐵)) → 0 < (1 / (abs‘(𝐴 ·_ih 𝐵))))
39	37, 38	sylbi 216	. . . . . . . . . . . . 13 ⊢ ((abs‘(𝐴 ·_ih 𝐵)) ≠ 0 → 0 < (1 / (abs‘(𝐴 ·_ih 𝐵))))
40		ltle 11302	. . . . . . . . . . . . 13 ⊢ ((0 ∈ ℝ ∧ (1 / (abs‘(𝐴 ·_ih 𝐵))) ∈ ℝ) → (0 < (1 / (abs‘(𝐴 ·_ih 𝐵))) → 0 ≤ (1 / (abs‘(𝐴 ·_ih 𝐵)))))
41	35, 39, 40	sylc 65	. . . . . . . . . . . 12 ⊢ ((abs‘(𝐴 ·_ih 𝐵)) ≠ 0 → 0 ≤ (1 / (abs‘(𝐴 ·_ih 𝐵))))
42	33, 41	absidd 15369	. . . . . . . . . . 11 ⊢ ((abs‘(𝐴 ·_ih 𝐵)) ≠ 0 → (abs‘(1 / (abs‘(𝐴 ·_ih 𝐵)))) = (1 / (abs‘(𝐴 ·_ih 𝐵))))
43	42	oveq2d 7425	. . . . . . . . . 10 ⊢ ((abs‘(𝐴 ·_ih 𝐵)) ≠ 0 → ((abs‘(𝐴 ·_ih 𝐵)) · (abs‘(1 / (abs‘(𝐴 ·_ih 𝐵))))) = ((abs‘(𝐴 ·_ih 𝐵)) · (1 / (abs‘(𝐴 ·_ih 𝐵)))))
44	32, 43	eqtrd 2773	. . . . . . . . 9 ⊢ ((abs‘(𝐴 ·_ih 𝐵)) ≠ 0 → (abs‘((𝐴 ·_ih 𝐵) · (1 / (abs‘(𝐴 ·_ih 𝐵))))) = ((abs‘(𝐴 ·_ih 𝐵)) · (1 / (abs‘(𝐴 ·_ih 𝐵)))))
45	26	recidzi 11941	. . . . . . . . 9 ⊢ ((abs‘(𝐴 ·_ih 𝐵)) ≠ 0 → ((abs‘(𝐴 ·_ih 𝐵)) · (1 / (abs‘(𝐴 ·_ih 𝐵)))) = 1)
46	29, 44, 45	3eqtrd 2777	. . . . . . . 8 ⊢ ((abs‘(𝐴 ·_ih 𝐵)) ≠ 0 → (abs‘((𝐴 ·_ih 𝐵) / (abs‘(𝐴 ·_ih 𝐵)))) = 1)
47	27, 46	jca 513	. . . . . . 7 ⊢ ((abs‘(𝐴 ·_ih 𝐵)) ≠ 0 → (((𝐴 ·_ih 𝐵) / (abs‘(𝐴 ·_ih 𝐵))) ∈ ℂ ∧ (abs‘((𝐴 ·_ih 𝐵) / (abs‘(𝐴 ·_ih 𝐵)))) = 1))
48	24, 47	sylbir 234	. . . . . 6 ⊢ ((𝐴 ·_ih 𝐵) ≠ 0 → (((𝐴 ·_ih 𝐵) / (abs‘(𝐴 ·_ih 𝐵))) ∈ ℂ ∧ (abs‘((𝐴 ·_ih 𝐵) / (abs‘(𝐴 ·_ih 𝐵)))) = 1))
49	3, 6	normlem7tALT 30372	. . . . . 6 ⊢ ((((𝐴 ·_ih 𝐵) / (abs‘(𝐴 ·_ih 𝐵))) ∈ ℂ ∧ (abs‘((𝐴 ·_ih 𝐵) / (abs‘(𝐴 ·_ih 𝐵)))) = 1) → (((∗‘((𝐴 ·_ih 𝐵) / (abs‘(𝐴 ·_ih 𝐵)))) · (𝐴 ·_ih 𝐵)) + (((𝐴 ·_ih 𝐵) / (abs‘(𝐴 ·_ih 𝐵))) · (𝐵 ·_ih 𝐴))) ≤ (2 · ((√‘(𝐵 ·_ih 𝐵)) · (√‘(𝐴 ·_ih 𝐴)))))
50	48, 49	syl 17	. . . . 5 ⊢ ((𝐴 ·_ih 𝐵) ≠ 0 → (((∗‘((𝐴 ·_ih 𝐵) / (abs‘(𝐴 ·_ih 𝐵)))) · (𝐴 ·_ih 𝐵)) + (((𝐴 ·_ih 𝐵) / (abs‘(𝐴 ·_ih 𝐵))) · (𝐵 ·_ih 𝐴))) ≤ (2 · ((√‘(𝐵 ·_ih 𝐵)) · (√‘(𝐴 ·_ih 𝐴)))))
51	22, 50	eqbrtrd 5171	. . . 4 ⊢ ((𝐴 ·_ih 𝐵) ≠ 0 → (2 · (abs‘(𝐴 ·_ih 𝐵))) ≤ (2 · ((√‘(𝐵 ·_ih 𝐵)) · (√‘(𝐴 ·_ih 𝐴)))))
52	15, 51	sylbir 234	. . 3 ⊢ (¬ (𝐴 ·_ih 𝐵) = 0 → (2 · (abs‘(𝐴 ·_ih 𝐵))) ≤ (2 · ((√‘(𝐵 ·_ih 𝐵)) · (√‘(𝐴 ·_ih 𝐴)))))
53	10	recni 11228	. . . . . 6 ⊢ (norm_ℎ‘𝐵) ∈ ℂ
54	9	recni 11228	. . . . . 6 ⊢ (norm_ℎ‘𝐴) ∈ ℂ
55		normval 30377	. . . . . . . 8 ⊢ (𝐵 ∈ ℋ → (norm_ℎ‘𝐵) = (√‘(𝐵 ·_ih 𝐵)))
56	6, 55	ax-mp 5	. . . . . . 7 ⊢ (norm_ℎ‘𝐵) = (√‘(𝐵 ·_ih 𝐵))
57		normval 30377	. . . . . . . 8 ⊢ (𝐴 ∈ ℋ → (norm_ℎ‘𝐴) = (√‘(𝐴 ·_ih 𝐴)))
58	3, 57	ax-mp 5	. . . . . . 7 ⊢ (norm_ℎ‘𝐴) = (√‘(𝐴 ·_ih 𝐴))
59	56, 58	oveq12i 7421	. . . . . 6 ⊢ ((norm_ℎ‘𝐵) · (norm_ℎ‘𝐴)) = ((√‘(𝐵 ·_ih 𝐵)) · (√‘(𝐴 ·_ih 𝐴)))
60	53, 54, 59	mulcomli 11223	. . . . 5 ⊢ ((norm_ℎ‘𝐴) · (norm_ℎ‘𝐵)) = ((√‘(𝐵 ·_ih 𝐵)) · (√‘(𝐴 ·_ih 𝐴)))
61	60	breq2i 5157	. . . 4 ⊢ ((abs‘(𝐴 ·_ih 𝐵)) ≤ ((norm_ℎ‘𝐴) · (norm_ℎ‘𝐵)) ↔ (abs‘(𝐴 ·_ih 𝐵)) ≤ ((√‘(𝐵 ·_ih 𝐵)) · (√‘(𝐴 ·_ih 𝐴))))
62		2pos 12315	. . . . 5 ⊢ 0 < 2
63		hiidge0 30351	. . . . . . . 8 ⊢ (𝐵 ∈ ℋ → 0 ≤ (𝐵 ·_ih 𝐵))
64		hiidrcl 30348	. . . . . . . . . 10 ⊢ (𝐵 ∈ ℋ → (𝐵 ·_ih 𝐵) ∈ ℝ)
65	6, 64	ax-mp 5	. . . . . . . . 9 ⊢ (𝐵 ·_ih 𝐵) ∈ ℝ
66	65	sqrtcli 15318	. . . . . . . 8 ⊢ (0 ≤ (𝐵 ·_ih 𝐵) → (√‘(𝐵 ·_ih 𝐵)) ∈ ℝ)
67	6, 63, 66	mp2b 10	. . . . . . 7 ⊢ (√‘(𝐵 ·_ih 𝐵)) ∈ ℝ
68		hiidge0 30351	. . . . . . . 8 ⊢ (𝐴 ∈ ℋ → 0 ≤ (𝐴 ·_ih 𝐴))
69		hiidrcl 30348	. . . . . . . . . 10 ⊢ (𝐴 ∈ ℋ → (𝐴 ·_ih 𝐴) ∈ ℝ)
70	3, 69	ax-mp 5	. . . . . . . . 9 ⊢ (𝐴 ·_ih 𝐴) ∈ ℝ
71	70	sqrtcli 15318	. . . . . . . 8 ⊢ (0 ≤ (𝐴 ·_ih 𝐴) → (√‘(𝐴 ·_ih 𝐴)) ∈ ℝ)
72	3, 68, 71	mp2b 10	. . . . . . 7 ⊢ (√‘(𝐴 ·_ih 𝐴)) ∈ ℝ
73	67, 72	remulcli 11230	. . . . . 6 ⊢ ((√‘(𝐵 ·_ih 𝐵)) · (√‘(𝐴 ·_ih 𝐴))) ∈ ℝ
74		2re 12286	. . . . . 6 ⊢ 2 ∈ ℝ
75	25, 73, 74	lemul2i 12137	. . . . 5 ⊢ (0 < 2 → ((abs‘(𝐴 ·_ih 𝐵)) ≤ ((√‘(𝐵 ·_ih 𝐵)) · (√‘(𝐴 ·_ih 𝐴))) ↔ (2 · (abs‘(𝐴 ·_ih 𝐵))) ≤ (2 · ((√‘(𝐵 ·_ih 𝐵)) · (√‘(𝐴 ·_ih 𝐴))))))
76	62, 75	ax-mp 5	. . . 4 ⊢ ((abs‘(𝐴 ·_ih 𝐵)) ≤ ((√‘(𝐵 ·_ih 𝐵)) · (√‘(𝐴 ·_ih 𝐴))) ↔ (2 · (abs‘(𝐴 ·_ih 𝐵))) ≤ (2 · ((√‘(𝐵 ·_ih 𝐵)) · (√‘(𝐴 ·_ih 𝐴)))))
77	61, 76	bitri 275	. . 3 ⊢ ((abs‘(𝐴 ·_ih 𝐵)) ≤ ((norm_ℎ‘𝐴) · (norm_ℎ‘𝐵)) ↔ (2 · (abs‘(𝐴 ·_ih 𝐵))) ≤ (2 · ((√‘(𝐵 ·_ih 𝐵)) · (√‘(𝐴 ·_ih 𝐴)))))
78	52, 77	sylibr 233	. 2 ⊢ (¬ (𝐴 ·_ih 𝐵) = 0 → (abs‘(𝐴 ·_ih 𝐵)) ≤ ((norm_ℎ‘𝐴) · (norm_ℎ‘𝐵)))
79	14, 78	pm2.61i 182	1 ⊢ (abs‘(𝐴 ·_ih 𝐵)) ≤ ((norm_ℎ‘𝐴) · (norm_ℎ‘𝐵))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 ↔ wb 205 ∧ wa 397 = wceq 1542 ∈ wcel 2107 ≠ wne 2941 class class class wbr 5149 ‘cfv 6544 (class class class)co 7409 ℂcc 11108 ℝcr 11109 0cc0 11110 1c1 11111 + caddc 11113 · cmul 11115 < clt 11248 ≤ cle 11249 / cdiv 11871 2c2 12267 ∗ccj 15043 √csqrt 15180 abscabs 15181 ℋchba 30172 ·_ih csp 30175 norm_ℎcno 30176

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5300 ax-nul 5307 ax-pow 5364 ax-pr 5428 ax-un 7725 ax-cnex 11166 ax-resscn 11167 ax-1cn 11168 ax-icn 11169 ax-addcl 11170 ax-addrcl 11171 ax-mulcl 11172 ax-mulrcl 11173 ax-mulcom 11174 ax-addass 11175 ax-mulass 11176 ax-distr 11177 ax-i2m1 11178 ax-1ne0 11179 ax-1rid 11180 ax-rnegex 11181 ax-rrecex 11182 ax-cnre 11183 ax-pre-lttri 11184 ax-pre-lttrn 11185 ax-pre-ltadd 11186 ax-pre-mulgt0 11187 ax-pre-sup 11188 ax-hfvadd 30253 ax-hv0cl 30256 ax-hfvmul 30258 ax-hvmulass 30260 ax-hvmul0 30263 ax-hfi 30332 ax-his1 30335 ax-his2 30336 ax-his3 30337 ax-his4 30338

This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-rmo 3377 df-reu 3378 df-rab 3434 df-v 3477 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-op 4636 df-uni 4910 df-iun 5000 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-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-riota 7365 df-ov 7412 df-oprab 7413 df-mpo 7414 df-om 7856 df-2nd 7976 df-frecs 8266 df-wrecs 8297 df-recs 8371 df-rdg 8410 df-er 8703 df-en 8940 df-dom 8941 df-sdom 8942 df-sup 9437 df-pnf 11250 df-mnf 11251 df-xr 11252 df-ltxr 11253 df-le 11254 df-sub 11446 df-neg 11447 df-div 11872 df-nn 12213 df-2 12275 df-3 12276 df-4 12277 df-n0 12473 df-z 12559 df-uz 12823 df-rp 12975 df-seq 13967 df-exp 14028 df-cj 15046 df-re 15047 df-im 15048 df-sqrt 15182 df-abs 15183 df-hnorm 30221 df-hvsub 30224

This theorem is referenced by: (None)

W3C validator