bcsiALT - Hilbert Space Explorer

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Theorem bcsiALT 31161

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 6828	. . 3 ⊢ ((𝐴 ·_ih 𝐵) = 0 → (abs‘(𝐴 ·_ih 𝐵)) = (abs‘0))
2		abs0 15194	. . . 4 ⊢ (abs‘0) = 0
3		bcs.1	. . . . . 6 ⊢ 𝐴 ∈ ℋ
4		normge0 31108	. . . . . 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 31108	. . . . . 6 ⊢ (𝐵 ∈ ℋ → 0 ≤ (norm_ℎ‘𝐵))
8	6, 7	ax-mp 5	. . . . 5 ⊢ 0 ≤ (norm_ℎ‘𝐵)
9	3	normcli 31113	. . . . . 6 ⊢ (norm_ℎ‘𝐴) ∈ ℝ
10	6	normcli 31113	. . . . . 6 ⊢ (norm_ℎ‘𝐵) ∈ ℝ
11	9, 10	mulge0i 11671	. . . . 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 5114	. . 3 ⊢ (abs‘0) ≤ ((norm_ℎ‘𝐴) · (norm_ℎ‘𝐵))
14	1, 13	eqbrtrdi 5132	. 2 ⊢ ((𝐴 ·_ih 𝐵) = 0 → (abs‘(𝐴 ·_ih 𝐵)) ≤ ((norm_ℎ‘𝐴) · (norm_ℎ‘𝐵)))
15		df-ne 2930	. . . 4 ⊢ ((𝐴 ·_ih 𝐵) ≠ 0 ↔ ¬ (𝐴 ·_ih 𝐵) = 0)
16	6, 3	his1i 31082	. . . . . . . 8 ⊢ (𝐵 ·_ih 𝐴) = (∗‘(𝐴 ·_ih 𝐵))
17	16	oveq2i 7363	. . . . . . 7 ⊢ (((𝐴 ·_ih 𝐵) / (abs‘(𝐴 ·_ih 𝐵))) · (𝐵 ·_ih 𝐴)) = (((𝐴 ·_ih 𝐵) / (abs‘(𝐴 ·_ih 𝐵))) · (∗‘(𝐴 ·_ih 𝐵)))
18	17	oveq2i 7363	. . . . . 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 31063	. . . . . . 7 ⊢ (𝐴 ·_ih 𝐵) ∈ ℂ
20		abslem2 15249	. . . . . . 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 2781	. . . . 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 15308	. . . . . . . 8 ⊢ ((abs‘(𝐴 ·_ih 𝐵)) = 0 ↔ (𝐴 ·_ih 𝐵) = 0)
24	23	necon3bii 2981	. . . . . . 7 ⊢ ((abs‘(𝐴 ·_ih 𝐵)) ≠ 0 ↔ (𝐴 ·_ih 𝐵) ≠ 0)
25	19	abscli 15305	. . . . . . . . . 10 ⊢ (abs‘(𝐴 ·_ih 𝐵)) ∈ ℝ
26	25	recni 11133	. . . . . . . . 9 ⊢ (abs‘(𝐴 ·_ih 𝐵)) ∈ ℂ
27	19, 26	divclzi 11863	. . . . . . . 8 ⊢ ((abs‘(𝐴 ·_ih 𝐵)) ≠ 0 → ((𝐴 ·_ih 𝐵) / (abs‘(𝐴 ·_ih 𝐵))) ∈ ℂ)
28	19, 26	divreczi 11866	. . . . . . . . . 10 ⊢ ((abs‘(𝐴 ·_ih 𝐵)) ≠ 0 → ((𝐴 ·_ih 𝐵) / (abs‘(𝐴 ·_ih 𝐵))) = ((𝐴 ·_ih 𝐵) · (1 / (abs‘(𝐴 ·_ih 𝐵)))))
29	28	fveq2d 6832	. . . . . . . . 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 11853	. . . . . . . . . . 11 ⊢ ((abs‘(𝐴 ·_ih 𝐵)) ≠ 0 → (1 / (abs‘(𝐴 ·_ih 𝐵))) ∈ ℂ)
31		absmul 15203	. . . . . . . . . . 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 11892	. . . . . . . . . . . 12 ⊢ ((abs‘(𝐴 ·_ih 𝐵)) ≠ 0 → (1 / (abs‘(𝐴 ·_ih 𝐵))) ∈ ℝ)
34		0re 11121	. . . . . . . . . . . . . 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 15309	. . . . . . . . . . . . . . 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 12034	. . . . . . . . . . . . . 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 11208	. . . . . . . . . . . . 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 15332	. . . . . . . . . . 11 ⊢ ((abs‘(𝐴 ·_ih 𝐵)) ≠ 0 → (abs‘(1 / (abs‘(𝐴 ·_ih 𝐵)))) = (1 / (abs‘(𝐴 ·_ih 𝐵))))
43	42	oveq2d 7368	. . . . . . . . . 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 2768	. . . . . . . . 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 11855	. . . . . . . . 9 ⊢ ((abs‘(𝐴 ·_ih 𝐵)) ≠ 0 → ((abs‘(𝐴 ·_ih 𝐵)) · (1 / (abs‘(𝐴 ·_ih 𝐵)))) = 1)
46	29, 44, 45	3eqtrd 2772	. . . . . . . 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 31101	. . . . . 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 5115	. . . 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 11133	. . . . . 6 ⊢ (norm_ℎ‘𝐵) ∈ ℂ
54	9	recni 11133	. . . . . 6 ⊢ (norm_ℎ‘𝐴) ∈ ℂ
55		normval 31106	. . . . . . . 8 ⊢ (𝐵 ∈ ℋ → (norm_ℎ‘𝐵) = (√‘(𝐵 ·_ih 𝐵)))
56	6, 55	ax-mp 5	. . . . . . 7 ⊢ (norm_ℎ‘𝐵) = (√‘(𝐵 ·_ih 𝐵))
57		normval 31106	. . . . . . . 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 7364	. . . . . 6 ⊢ ((norm_ℎ‘𝐵) · (norm_ℎ‘𝐴)) = ((√‘(𝐵 ·_ih 𝐵)) · (√‘(𝐴 ·_ih 𝐴)))
60	53, 54, 59	mulcomli 11128	. . . . 5 ⊢ ((norm_ℎ‘𝐴) · (norm_ℎ‘𝐵)) = ((√‘(𝐵 ·_ih 𝐵)) · (√‘(𝐴 ·_ih 𝐴)))
61	60	breq2i 5101	. . . 4 ⊢ ((abs‘(𝐴 ·_ih 𝐵)) ≤ ((norm_ℎ‘𝐴) · (norm_ℎ‘𝐵)) ↔ (abs‘(𝐴 ·_ih 𝐵)) ≤ ((√‘(𝐵 ·_ih 𝐵)) · (√‘(𝐴 ·_ih 𝐴))))
62		2pos 12235	. . . . 5 ⊢ 0 < 2
63		hiidge0 31080	. . . . . . . 8 ⊢ (𝐵 ∈ ℋ → 0 ≤ (𝐵 ·_ih 𝐵))
64		hiidrcl 31077	. . . . . . . . . 10 ⊢ (𝐵 ∈ ℋ → (𝐵 ·_ih 𝐵) ∈ ℝ)
65	6, 64	ax-mp 5	. . . . . . . . 9 ⊢ (𝐵 ·_ih 𝐵) ∈ ℝ
66	65	sqrtcli 15281	. . . . . . . 8 ⊢ (0 ≤ (𝐵 ·_ih 𝐵) → (√‘(𝐵 ·_ih 𝐵)) ∈ ℝ)
67	6, 63, 66	mp2b 10	. . . . . . 7 ⊢ (√‘(𝐵 ·_ih 𝐵)) ∈ ℝ
68		hiidge0 31080	. . . . . . . 8 ⊢ (𝐴 ∈ ℋ → 0 ≤ (𝐴 ·_ih 𝐴))
69		hiidrcl 31077	. . . . . . . . . 10 ⊢ (𝐴 ∈ ℋ → (𝐴 ·_ih 𝐴) ∈ ℝ)
70	3, 69	ax-mp 5	. . . . . . . . 9 ⊢ (𝐴 ·_ih 𝐴) ∈ ℝ
71	70	sqrtcli 15281	. . . . . . . 8 ⊢ (0 ≤ (𝐴 ·_ih 𝐴) → (√‘(𝐴 ·_ih 𝐴)) ∈ ℝ)
72	3, 68, 71	mp2b 10	. . . . . . 7 ⊢ (√‘(𝐴 ·_ih 𝐴)) ∈ ℝ
73	67, 72	remulcli 11135	. . . . . 6 ⊢ ((√‘(𝐵 ·_ih 𝐵)) · (√‘(𝐴 ·_ih 𝐴))) ∈ ℝ
74		2re 12206	. . . . . 6 ⊢ 2 ∈ ℝ
75	25, 73, 74	lemul2i 12052	. . . . 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 1541 ∈ wcel 2113 ≠ wne 2929 class class class wbr 5093 ‘cfv 6486 (class class class)co 7352 ℂcc 11011 ℝcr 11012 0cc0 11013 1c1 11014 + caddc 11016 · cmul 11018 < clt 11153 ≤ cle 11154 / cdiv 11781 2c2 12187 ∗ccj 15005 √csqrt 15142 abscabs 15143 ℋchba 30901 ·_ih csp 30904 norm_ℎcno 30905

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 1911 ax-6 1968 ax-7 2009 ax-8 2115 ax-9 2123 ax-10 2146 ax-11 2162 ax-12 2182 ax-ext 2705 ax-sep 5236 ax-nul 5246 ax-pow 5305 ax-pr 5372 ax-un 7674 ax-cnex 11069 ax-resscn 11070 ax-1cn 11071 ax-icn 11072 ax-addcl 11073 ax-addrcl 11074 ax-mulcl 11075 ax-mulrcl 11076 ax-mulcom 11077 ax-addass 11078 ax-mulass 11079 ax-distr 11080 ax-i2m1 11081 ax-1ne0 11082 ax-1rid 11083 ax-rnegex 11084 ax-rrecex 11085 ax-cnre 11086 ax-pre-lttri 11087 ax-pre-lttrn 11088 ax-pre-ltadd 11089 ax-pre-mulgt0 11090 ax-pre-sup 11091 ax-hfvadd 30982 ax-hv0cl 30985 ax-hfvmul 30987 ax-hvmulass 30989 ax-hvmul0 30992 ax-hfi 31061 ax-his1 31064 ax-his2 31065 ax-his3 31066 ax-his4 31067

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2068 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2725 df-clel 2808 df-nfc 2882 df-ne 2930 df-nel 3034 df-ral 3049 df-rex 3058 df-rmo 3347 df-reu 3348 df-rab 3397 df-v 3439 df-sbc 3738 df-csb 3847 df-dif 3901 df-un 3903 df-in 3905 df-ss 3915 df-pss 3918 df-nul 4283 df-if 4475 df-pw 4551 df-sn 4576 df-pr 4578 df-op 4582 df-uni 4859 df-iun 4943 df-br 5094 df-opab 5156 df-mpt 5175 df-tr 5201 df-id 5514 df-eprel 5519 df-po 5527 df-so 5528 df-fr 5572 df-we 5574 df-xp 5625 df-rel 5626 df-cnv 5627 df-co 5628 df-dm 5629 df-rn 5630 df-res 5631 df-ima 5632 df-pred 6253 df-ord 6314 df-on 6315 df-lim 6316 df-suc 6317 df-iota 6442 df-fun 6488 df-fn 6489 df-f 6490 df-f1 6491 df-fo 6492 df-f1o 6493 df-fv 6494 df-riota 7309 df-ov 7355 df-oprab 7356 df-mpo 7357 df-om 7803 df-2nd 7928 df-frecs 8217 df-wrecs 8248 df-recs 8297 df-rdg 8335 df-er 8628 df-en 8876 df-dom 8877 df-sdom 8878 df-sup 9333 df-pnf 11155 df-mnf 11156 df-xr 11157 df-ltxr 11158 df-le 11159 df-sub 11353 df-neg 11354 df-div 11782 df-nn 12133 df-2 12195 df-3 12196 df-4 12197 df-n0 12389 df-z 12476 df-uz 12739 df-rp 12893 df-seq 13911 df-exp 13971 df-cj 15008 df-re 15009 df-im 15010 df-sqrt 15144 df-abs 15145 df-hnorm 30950 df-hvsub 30953

This theorem is referenced by: (None)

W3C validator