bcsiALT - Hilbert Space Explorer

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Theorem bcsiALT 31265

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 6834	. . 3 ⊢ ((𝐴 ·_ih 𝐵) = 0 → (abs‘(𝐴 ·_ih 𝐵)) = (abs‘0))
2		abs0 15238	. . . 4 ⊢ (abs‘0) = 0
3		bcs.1	. . . . . 6 ⊢ 𝐴 ∈ ℋ
4		normge0 31212	. . . . . 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 31212	. . . . . 6 ⊢ (𝐵 ∈ ℋ → 0 ≤ (norm_ℎ‘𝐵))
8	6, 7	ax-mp 5	. . . . 5 ⊢ 0 ≤ (norm_ℎ‘𝐵)
9	3	normcli 31217	. . . . . 6 ⊢ (norm_ℎ‘𝐴) ∈ ℝ
10	6	normcli 31217	. . . . . 6 ⊢ (norm_ℎ‘𝐵) ∈ ℝ
11	9, 10	mulge0i 11688	. . . . 5 ⊢ ((0 ≤ (norm_ℎ‘𝐴) ∧ 0 ≤ (norm_ℎ‘𝐵)) → 0 ≤ ((norm_ℎ‘𝐴) · (norm_ℎ‘𝐵)))
12	5, 8, 11	mp2an 693	. . . 4 ⊢ 0 ≤ ((norm_ℎ‘𝐴) · (norm_ℎ‘𝐵))
13	2, 12	eqbrtri 5107	. . 3 ⊢ (abs‘0) ≤ ((norm_ℎ‘𝐴) · (norm_ℎ‘𝐵))
14	1, 13	eqbrtrdi 5125	. 2 ⊢ ((𝐴 ·_ih 𝐵) = 0 → (abs‘(𝐴 ·_ih 𝐵)) ≤ ((norm_ℎ‘𝐴) · (norm_ℎ‘𝐵)))
15		df-ne 2934	. . . 4 ⊢ ((𝐴 ·_ih 𝐵) ≠ 0 ↔ ¬ (𝐴 ·_ih 𝐵) = 0)
16	6, 3	his1i 31186	. . . . . . . 8 ⊢ (𝐵 ·_ih 𝐴) = (∗‘(𝐴 ·_ih 𝐵))
17	16	oveq2i 7371	. . . . . . 7 ⊢ (((𝐴 ·_ih 𝐵) / (abs‘(𝐴 ·_ih 𝐵))) · (𝐵 ·_ih 𝐴)) = (((𝐴 ·_ih 𝐵) / (abs‘(𝐴 ·_ih 𝐵))) · (∗‘(𝐴 ·_ih 𝐵)))
18	17	oveq2i 7371	. . . . . 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 31167	. . . . . . 7 ⊢ (𝐴 ·_ih 𝐵) ∈ ℂ
20		abslem2 15293	. . . . . . 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 691	. . . . . 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 2785	. . . . 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 15352	. . . . . . . 8 ⊢ ((abs‘(𝐴 ·_ih 𝐵)) = 0 ↔ (𝐴 ·_ih 𝐵) = 0)
24	23	necon3bii 2985	. . . . . . 7 ⊢ ((abs‘(𝐴 ·_ih 𝐵)) ≠ 0 ↔ (𝐴 ·_ih 𝐵) ≠ 0)
25	19	abscli 15349	. . . . . . . . . 10 ⊢ (abs‘(𝐴 ·_ih 𝐵)) ∈ ℝ
26	25	recni 11150	. . . . . . . . 9 ⊢ (abs‘(𝐴 ·_ih 𝐵)) ∈ ℂ
27	19, 26	divclzi 11881	. . . . . . . 8 ⊢ ((abs‘(𝐴 ·_ih 𝐵)) ≠ 0 → ((𝐴 ·_ih 𝐵) / (abs‘(𝐴 ·_ih 𝐵))) ∈ ℂ)
28	19, 26	divreczi 11884	. . . . . . . . . 10 ⊢ ((abs‘(𝐴 ·_ih 𝐵)) ≠ 0 → ((𝐴 ·_ih 𝐵) / (abs‘(𝐴 ·_ih 𝐵))) = ((𝐴 ·_ih 𝐵) · (1 / (abs‘(𝐴 ·_ih 𝐵)))))
29	28	fveq2d 6838	. . . . . . . . 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 11871	. . . . . . . . . . 11 ⊢ ((abs‘(𝐴 ·_ih 𝐵)) ≠ 0 → (1 / (abs‘(𝐴 ·_ih 𝐵))) ∈ ℂ)
31		absmul 15247	. . . . . . . . . . 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 11910	. . . . . . . . . . . 12 ⊢ ((abs‘(𝐴 ·_ih 𝐵)) ≠ 0 → (1 / (abs‘(𝐴 ·_ih 𝐵))) ∈ ℝ)
34		0re 11137	. . . . . . . . . . . . . 14 ⊢ 0 ∈ ℝ
35	33, 34	jctil 519	. . . . . . . . . . . . 13 ⊢ ((abs‘(𝐴 ·_ih 𝐵)) ≠ 0 → (0 ∈ ℝ ∧ (1 / (abs‘(𝐴 ·_ih 𝐵))) ∈ ℝ))
36	19	absgt0i 15353	. . . . . . . . . . . . . . 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 12052	. . . . . . . . . . . . . 14 ⊢ (0 < (abs‘(𝐴 ·_ih 𝐵)) → 0 < (1 / (abs‘(𝐴 ·_ih 𝐵))))
39	37, 38	sylbi 217	. . . . . . . . . . . . 13 ⊢ ((abs‘(𝐴 ·_ih 𝐵)) ≠ 0 → 0 < (1 / (abs‘(𝐴 ·_ih 𝐵))))
40		ltle 11225	. . . . . . . . . . . . 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 15376	. . . . . . . . . . 11 ⊢ ((abs‘(𝐴 ·_ih 𝐵)) ≠ 0 → (abs‘(1 / (abs‘(𝐴 ·_ih 𝐵)))) = (1 / (abs‘(𝐴 ·_ih 𝐵))))
43	42	oveq2d 7376	. . . . . . . . . 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 2772	. . . . . . . . 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 11873	. . . . . . . . 9 ⊢ ((abs‘(𝐴 ·_ih 𝐵)) ≠ 0 → ((abs‘(𝐴 ·_ih 𝐵)) · (1 / (abs‘(𝐴 ·_ih 𝐵)))) = 1)
46	29, 44, 45	3eqtrd 2776	. . . . . . . 8 ⊢ ((abs‘(𝐴 ·_ih 𝐵)) ≠ 0 → (abs‘((𝐴 ·_ih 𝐵) / (abs‘(𝐴 ·_ih 𝐵)))) = 1)
47	27, 46	jca 511	. . . . . . 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 235	. . . . . 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 31205	. . . . . 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 5108	. . . 4 ⊢ ((𝐴 ·_ih 𝐵) ≠ 0 → (2 · (abs‘(𝐴 ·_ih 𝐵))) ≤ (2 · ((√‘(𝐵 ·_ih 𝐵)) · (√‘(𝐴 ·_ih 𝐴)))))
52	15, 51	sylbir 235	. . 3 ⊢ (¬ (𝐴 ·_ih 𝐵) = 0 → (2 · (abs‘(𝐴 ·_ih 𝐵))) ≤ (2 · ((√‘(𝐵 ·_ih 𝐵)) · (√‘(𝐴 ·_ih 𝐴)))))
53	10	recni 11150	. . . . . 6 ⊢ (norm_ℎ‘𝐵) ∈ ℂ
54	9	recni 11150	. . . . . 6 ⊢ (norm_ℎ‘𝐴) ∈ ℂ
55		normval 31210	. . . . . . . 8 ⊢ (𝐵 ∈ ℋ → (norm_ℎ‘𝐵) = (√‘(𝐵 ·_ih 𝐵)))
56	6, 55	ax-mp 5	. . . . . . 7 ⊢ (norm_ℎ‘𝐵) = (√‘(𝐵 ·_ih 𝐵))
57		normval 31210	. . . . . . . 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 7372	. . . . . 6 ⊢ ((norm_ℎ‘𝐵) · (norm_ℎ‘𝐴)) = ((√‘(𝐵 ·_ih 𝐵)) · (√‘(𝐴 ·_ih 𝐴)))
60	53, 54, 59	mulcomli 11145	. . . . 5 ⊢ ((norm_ℎ‘𝐴) · (norm_ℎ‘𝐵)) = ((√‘(𝐵 ·_ih 𝐵)) · (√‘(𝐴 ·_ih 𝐴)))
61	60	breq2i 5094	. . . 4 ⊢ ((abs‘(𝐴 ·_ih 𝐵)) ≤ ((norm_ℎ‘𝐴) · (norm_ℎ‘𝐵)) ↔ (abs‘(𝐴 ·_ih 𝐵)) ≤ ((√‘(𝐵 ·_ih 𝐵)) · (√‘(𝐴 ·_ih 𝐴))))
62		2pos 12275	. . . . 5 ⊢ 0 < 2
63		hiidge0 31184	. . . . . . . 8 ⊢ (𝐵 ∈ ℋ → 0 ≤ (𝐵 ·_ih 𝐵))
64		hiidrcl 31181	. . . . . . . . . 10 ⊢ (𝐵 ∈ ℋ → (𝐵 ·_ih 𝐵) ∈ ℝ)
65	6, 64	ax-mp 5	. . . . . . . . 9 ⊢ (𝐵 ·_ih 𝐵) ∈ ℝ
66	65	sqrtcli 15325	. . . . . . . 8 ⊢ (0 ≤ (𝐵 ·_ih 𝐵) → (√‘(𝐵 ·_ih 𝐵)) ∈ ℝ)
67	6, 63, 66	mp2b 10	. . . . . . 7 ⊢ (√‘(𝐵 ·_ih 𝐵)) ∈ ℝ
68		hiidge0 31184	. . . . . . . 8 ⊢ (𝐴 ∈ ℋ → 0 ≤ (𝐴 ·_ih 𝐴))
69		hiidrcl 31181	. . . . . . . . . 10 ⊢ (𝐴 ∈ ℋ → (𝐴 ·_ih 𝐴) ∈ ℝ)
70	3, 69	ax-mp 5	. . . . . . . . 9 ⊢ (𝐴 ·_ih 𝐴) ∈ ℝ
71	70	sqrtcli 15325	. . . . . . . 8 ⊢ (0 ≤ (𝐴 ·_ih 𝐴) → (√‘(𝐴 ·_ih 𝐴)) ∈ ℝ)
72	3, 68, 71	mp2b 10	. . . . . . 7 ⊢ (√‘(𝐴 ·_ih 𝐴)) ∈ ℝ
73	67, 72	remulcli 11152	. . . . . 6 ⊢ ((√‘(𝐵 ·_ih 𝐵)) · (√‘(𝐴 ·_ih 𝐴))) ∈ ℝ
74		2re 12246	. . . . . 6 ⊢ 2 ∈ ℝ
75	25, 73, 74	lemul2i 12070	. . . . 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 234	. 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 206 ∧ wa 395 = wceq 1542 ∈ wcel 2114 ≠ wne 2933 class class class wbr 5086 ‘cfv 6492 (class class class)co 7360 ℂcc 11027 ℝcr 11028 0cc0 11029 1c1 11030 + caddc 11032 · cmul 11034 < clt 11170 ≤ cle 11171 / cdiv 11798 2c2 12227 ∗ccj 15049 √csqrt 15186 abscabs 15187 ℋchba 31005 ·_ih csp 31008 norm_ℎcno 31009

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-sep 5231 ax-nul 5241 ax-pow 5302 ax-pr 5370 ax-un 7682 ax-cnex 11085 ax-resscn 11086 ax-1cn 11087 ax-icn 11088 ax-addcl 11089 ax-addrcl 11090 ax-mulcl 11091 ax-mulrcl 11092 ax-mulcom 11093 ax-addass 11094 ax-mulass 11095 ax-distr 11096 ax-i2m1 11097 ax-1ne0 11098 ax-1rid 11099 ax-rnegex 11100 ax-rrecex 11101 ax-cnre 11102 ax-pre-lttri 11103 ax-pre-lttrn 11104 ax-pre-ltadd 11105 ax-pre-mulgt0 11106 ax-pre-sup 11107 ax-hfvadd 31086 ax-hv0cl 31089 ax-hfvmul 31091 ax-hvmulass 31093 ax-hvmul0 31096 ax-hfi 31165 ax-his1 31168 ax-his2 31169 ax-his3 31170 ax-his4 31171

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 3343 df-reu 3344 df-rab 3391 df-v 3432 df-sbc 3730 df-csb 3839 df-dif 3893 df-un 3895 df-in 3897 df-ss 3907 df-pss 3910 df-nul 4275 df-if 4468 df-pw 4544 df-sn 4569 df-pr 4571 df-op 4575 df-uni 4852 df-iun 4936 df-br 5087 df-opab 5149 df-mpt 5168 df-tr 5194 df-id 5519 df-eprel 5524 df-po 5532 df-so 5533 df-fr 5577 df-we 5579 df-xp 5630 df-rel 5631 df-cnv 5632 df-co 5633 df-dm 5634 df-rn 5635 df-res 5636 df-ima 5637 df-pred 6259 df-ord 6320 df-on 6321 df-lim 6322 df-suc 6323 df-iota 6448 df-fun 6494 df-fn 6495 df-f 6496 df-f1 6497 df-fo 6498 df-f1o 6499 df-fv 6500 df-riota 7317 df-ov 7363 df-oprab 7364 df-mpo 7365 df-om 7811 df-2nd 7936 df-frecs 8224 df-wrecs 8255 df-recs 8304 df-rdg 8342 df-er 8636 df-en 8887 df-dom 8888 df-sdom 8889 df-sup 9348 df-pnf 11172 df-mnf 11173 df-xr 11174 df-ltxr 11175 df-le 11176 df-sub 11370 df-neg 11371 df-div 11799 df-nn 12166 df-2 12235 df-3 12236 df-4 12237 df-n0 12429 df-z 12516 df-uz 12780 df-rp 12934 df-seq 13955 df-exp 14015 df-cj 15052 df-re 15053 df-im 15054 df-sqrt 15188 df-abs 15189 df-hnorm 31054 df-hvsub 31057

This theorem is referenced by: (None)

W3C validator