bcsiALT - Hilbert Space Explorer

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Theorem bcsiALT 31108

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 6858	. . 3 ⊢ ((𝐴 ·_ih 𝐵) = 0 → (abs‘(𝐴 ·_ih 𝐵)) = (abs‘0))
2		abs0 15251	. . . 4 ⊢ (abs‘0) = 0
3		bcs.1	. . . . . 6 ⊢ 𝐴 ∈ ℋ
4		normge0 31055	. . . . . 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 31055	. . . . . 6 ⊢ (𝐵 ∈ ℋ → 0 ≤ (norm_ℎ‘𝐵))
8	6, 7	ax-mp 5	. . . . 5 ⊢ 0 ≤ (norm_ℎ‘𝐵)
9	3	normcli 31060	. . . . . 6 ⊢ (norm_ℎ‘𝐴) ∈ ℝ
10	6	normcli 31060	. . . . . 6 ⊢ (norm_ℎ‘𝐵) ∈ ℝ
11	9, 10	mulge0i 11725	. . . . 5 ⊢ ((0 ≤ (norm_ℎ‘𝐴) ∧ 0 ≤ (norm_ℎ‘𝐵)) → 0 ≤ ((norm_ℎ‘𝐴) · (norm_ℎ‘𝐵)))
12	5, 8, 11	mp2an 692	. . . 4 ⊢ 0 ≤ ((norm_ℎ‘𝐴) · (norm_ℎ‘𝐵))
13	2, 12	eqbrtri 5128	. . 3 ⊢ (abs‘0) ≤ ((norm_ℎ‘𝐴) · (norm_ℎ‘𝐵))
14	1, 13	eqbrtrdi 5146	. 2 ⊢ ((𝐴 ·_ih 𝐵) = 0 → (abs‘(𝐴 ·_ih 𝐵)) ≤ ((norm_ℎ‘𝐴) · (norm_ℎ‘𝐵)))
15		df-ne 2926	. . . 4 ⊢ ((𝐴 ·_ih 𝐵) ≠ 0 ↔ ¬ (𝐴 ·_ih 𝐵) = 0)
16	6, 3	his1i 31029	. . . . . . . 8 ⊢ (𝐵 ·_ih 𝐴) = (∗‘(𝐴 ·_ih 𝐵))
17	16	oveq2i 7398	. . . . . . 7 ⊢ (((𝐴 ·_ih 𝐵) / (abs‘(𝐴 ·_ih 𝐵))) · (𝐵 ·_ih 𝐴)) = (((𝐴 ·_ih 𝐵) / (abs‘(𝐴 ·_ih 𝐵))) · (∗‘(𝐴 ·_ih 𝐵)))
18	17	oveq2i 7398	. . . . . 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 31010	. . . . . . 7 ⊢ (𝐴 ·_ih 𝐵) ∈ ℂ
20		abslem2 15306	. . . . . . 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 690	. . . . . 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 2777	. . . . 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 15365	. . . . . . . 8 ⊢ ((abs‘(𝐴 ·_ih 𝐵)) = 0 ↔ (𝐴 ·_ih 𝐵) = 0)
24	23	necon3bii 2977	. . . . . . 7 ⊢ ((abs‘(𝐴 ·_ih 𝐵)) ≠ 0 ↔ (𝐴 ·_ih 𝐵) ≠ 0)
25	19	abscli 15362	. . . . . . . . . 10 ⊢ (abs‘(𝐴 ·_ih 𝐵)) ∈ ℝ
26	25	recni 11188	. . . . . . . . 9 ⊢ (abs‘(𝐴 ·_ih 𝐵)) ∈ ℂ
27	19, 26	divclzi 11917	. . . . . . . 8 ⊢ ((abs‘(𝐴 ·_ih 𝐵)) ≠ 0 → ((𝐴 ·_ih 𝐵) / (abs‘(𝐴 ·_ih 𝐵))) ∈ ℂ)
28	19, 26	divreczi 11920	. . . . . . . . . 10 ⊢ ((abs‘(𝐴 ·_ih 𝐵)) ≠ 0 → ((𝐴 ·_ih 𝐵) / (abs‘(𝐴 ·_ih 𝐵))) = ((𝐴 ·_ih 𝐵) · (1 / (abs‘(𝐴 ·_ih 𝐵)))))
29	28	fveq2d 6862	. . . . . . . . 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 11907	. . . . . . . . . . 11 ⊢ ((abs‘(𝐴 ·_ih 𝐵)) ≠ 0 → (1 / (abs‘(𝐴 ·_ih 𝐵))) ∈ ℂ)
31		absmul 15260	. . . . . . . . . . 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 587	. . . . . . . . . 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 11946	. . . . . . . . . . . 12 ⊢ ((abs‘(𝐴 ·_ih 𝐵)) ≠ 0 → (1 / (abs‘(𝐴 ·_ih 𝐵))) ∈ ℝ)
34		0re 11176	. . . . . . . . . . . . . 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 15366	. . . . . . . . . . . . . . 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 12088	. . . . . . . . . . . . . 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 11262	. . . . . . . . . . . . 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 15389	. . . . . . . . . . 11 ⊢ ((abs‘(𝐴 ·_ih 𝐵)) ≠ 0 → (abs‘(1 / (abs‘(𝐴 ·_ih 𝐵)))) = (1 / (abs‘(𝐴 ·_ih 𝐵))))
43	42	oveq2d 7403	. . . . . . . . . 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 2764	. . . . . . . . 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 11909	. . . . . . . . 9 ⊢ ((abs‘(𝐴 ·_ih 𝐵)) ≠ 0 → ((abs‘(𝐴 ·_ih 𝐵)) · (1 / (abs‘(𝐴 ·_ih 𝐵)))) = 1)
46	29, 44, 45	3eqtrd 2768	. . . . . . . 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 31048	. . . . . 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 5129	. . . 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 11188	. . . . . 6 ⊢ (norm_ℎ‘𝐵) ∈ ℂ
54	9	recni 11188	. . . . . 6 ⊢ (norm_ℎ‘𝐴) ∈ ℂ
55		normval 31053	. . . . . . . 8 ⊢ (𝐵 ∈ ℋ → (norm_ℎ‘𝐵) = (√‘(𝐵 ·_ih 𝐵)))
56	6, 55	ax-mp 5	. . . . . . 7 ⊢ (norm_ℎ‘𝐵) = (√‘(𝐵 ·_ih 𝐵))
57		normval 31053	. . . . . . . 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 7399	. . . . . 6 ⊢ ((norm_ℎ‘𝐵) · (norm_ℎ‘𝐴)) = ((√‘(𝐵 ·_ih 𝐵)) · (√‘(𝐴 ·_ih 𝐴)))
60	53, 54, 59	mulcomli 11183	. . . . 5 ⊢ ((norm_ℎ‘𝐴) · (norm_ℎ‘𝐵)) = ((√‘(𝐵 ·_ih 𝐵)) · (√‘(𝐴 ·_ih 𝐴)))
61	60	breq2i 5115	. . . 4 ⊢ ((abs‘(𝐴 ·_ih 𝐵)) ≤ ((norm_ℎ‘𝐴) · (norm_ℎ‘𝐵)) ↔ (abs‘(𝐴 ·_ih 𝐵)) ≤ ((√‘(𝐵 ·_ih 𝐵)) · (√‘(𝐴 ·_ih 𝐴))))
62		2pos 12289	. . . . 5 ⊢ 0 < 2
63		hiidge0 31027	. . . . . . . 8 ⊢ (𝐵 ∈ ℋ → 0 ≤ (𝐵 ·_ih 𝐵))
64		hiidrcl 31024	. . . . . . . . . 10 ⊢ (𝐵 ∈ ℋ → (𝐵 ·_ih 𝐵) ∈ ℝ)
65	6, 64	ax-mp 5	. . . . . . . . 9 ⊢ (𝐵 ·_ih 𝐵) ∈ ℝ
66	65	sqrtcli 15338	. . . . . . . 8 ⊢ (0 ≤ (𝐵 ·_ih 𝐵) → (√‘(𝐵 ·_ih 𝐵)) ∈ ℝ)
67	6, 63, 66	mp2b 10	. . . . . . 7 ⊢ (√‘(𝐵 ·_ih 𝐵)) ∈ ℝ
68		hiidge0 31027	. . . . . . . 8 ⊢ (𝐴 ∈ ℋ → 0 ≤ (𝐴 ·_ih 𝐴))
69		hiidrcl 31024	. . . . . . . . . 10 ⊢ (𝐴 ∈ ℋ → (𝐴 ·_ih 𝐴) ∈ ℝ)
70	3, 69	ax-mp 5	. . . . . . . . 9 ⊢ (𝐴 ·_ih 𝐴) ∈ ℝ
71	70	sqrtcli 15338	. . . . . . . 8 ⊢ (0 ≤ (𝐴 ·_ih 𝐴) → (√‘(𝐴 ·_ih 𝐴)) ∈ ℝ)
72	3, 68, 71	mp2b 10	. . . . . . 7 ⊢ (√‘(𝐴 ·_ih 𝐴)) ∈ ℝ
73	67, 72	remulcli 11190	. . . . . 6 ⊢ ((√‘(𝐵 ·_ih 𝐵)) · (√‘(𝐴 ·_ih 𝐴))) ∈ ℝ
74		2re 12260	. . . . . 6 ⊢ 2 ∈ ℝ
75	25, 73, 74	lemul2i 12106	. . . . 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 1540 ∈ wcel 2109 ≠ wne 2925 class class class wbr 5107 ‘cfv 6511 (class class class)co 7387 ℂcc 11066 ℝcr 11067 0cc0 11068 1c1 11069 + caddc 11071 · cmul 11073 < clt 11208 ≤ cle 11209 / cdiv 11835 2c2 12241 ∗ccj 15062 √csqrt 15199 abscabs 15200 ℋchba 30848 ·_ih csp 30851 norm_ℎcno 30852

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-sep 5251 ax-nul 5261 ax-pow 5320 ax-pr 5387 ax-un 7711 ax-cnex 11124 ax-resscn 11125 ax-1cn 11126 ax-icn 11127 ax-addcl 11128 ax-addrcl 11129 ax-mulcl 11130 ax-mulrcl 11131 ax-mulcom 11132 ax-addass 11133 ax-mulass 11134 ax-distr 11135 ax-i2m1 11136 ax-1ne0 11137 ax-1rid 11138 ax-rnegex 11139 ax-rrecex 11140 ax-cnre 11141 ax-pre-lttri 11142 ax-pre-lttrn 11143 ax-pre-ltadd 11144 ax-pre-mulgt0 11145 ax-pre-sup 11146 ax-hfvadd 30929 ax-hv0cl 30932 ax-hfvmul 30934 ax-hvmulass 30936 ax-hvmul0 30939 ax-hfi 31008 ax-his1 31011 ax-his2 31012 ax-his3 31013 ax-his4 31014

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 3354 df-reu 3355 df-rab 3406 df-v 3449 df-sbc 3754 df-csb 3863 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-pss 3934 df-nul 4297 df-if 4489 df-pw 4565 df-sn 4590 df-pr 4592 df-op 4596 df-uni 4872 df-iun 4957 df-br 5108 df-opab 5170 df-mpt 5189 df-tr 5215 df-id 5533 df-eprel 5538 df-po 5546 df-so 5547 df-fr 5591 df-we 5593 df-xp 5644 df-rel 5645 df-cnv 5646 df-co 5647 df-dm 5648 df-rn 5649 df-res 5650 df-ima 5651 df-pred 6274 df-ord 6335 df-on 6336 df-lim 6337 df-suc 6338 df-iota 6464 df-fun 6513 df-fn 6514 df-f 6515 df-f1 6516 df-fo 6517 df-f1o 6518 df-fv 6519 df-riota 7344 df-ov 7390 df-oprab 7391 df-mpo 7392 df-om 7843 df-2nd 7969 df-frecs 8260 df-wrecs 8291 df-recs 8340 df-rdg 8378 df-er 8671 df-en 8919 df-dom 8920 df-sdom 8921 df-sup 9393 df-pnf 11210 df-mnf 11211 df-xr 11212 df-ltxr 11213 df-le 11214 df-sub 11407 df-neg 11408 df-div 11836 df-nn 12187 df-2 12249 df-3 12250 df-4 12251 df-n0 12443 df-z 12530 df-uz 12794 df-rp 12952 df-seq 13967 df-exp 14027 df-cj 15065 df-re 15066 df-im 15067 df-sqrt 15201 df-abs 15202 df-hnorm 30897 df-hvsub 30900

This theorem is referenced by: (None)

W3C validator