bcsiALT - Hilbert Space Explorer

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Theorem bcsiALT 31763

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 6877	. . 3 ⊢ ((𝐴 ·_ih 𝐵) = 0 → (abs‘(𝐴 ·_ih 𝐵)) = (abs‘0))
2		abs0 15432	. . . 4 ⊢ (abs‘0) = 0
3		bcs.1	. . . . . 6 ⊢ 𝐴 ∈ ℋ
4		normge0 31710	. . . . . 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 31710	. . . . . 6 ⊢ (𝐵 ∈ ℋ → 0 ≤ (norm_ℎ‘𝐵))
8	6, 7	ax-mp 5	. . . . 5 ⊢ 0 ≤ (norm_ℎ‘𝐵)
9	3	normcli 31715	. . . . . 6 ⊢ (norm_ℎ‘𝐴) ∈ ℝ
10	6	normcli 31715	. . . . . 6 ⊢ (norm_ℎ‘𝐵) ∈ ℝ
11	9, 10	mulge0i 11844	. . . . 5 ⊢ ((0 ≤ (norm_ℎ‘𝐴) ∧ 0 ≤ (norm_ℎ‘𝐵)) → 0 ≤ ((norm_ℎ‘𝐴) · (norm_ℎ‘𝐵)))
12	5, 8, 11	mp2an 705	. . . 4 ⊢ 0 ≤ ((norm_ℎ‘𝐴) · (norm_ℎ‘𝐵))
13	2, 12	eqbrtri 5126	. . 3 ⊢ (abs‘0) ≤ ((norm_ℎ‘𝐴) · (norm_ℎ‘𝐵))
14	1, 13	eqbrtrdi 5144	. 2 ⊢ ((𝐴 ·_ih 𝐵) = 0 → (abs‘(𝐴 ·_ih 𝐵)) ≤ ((norm_ℎ‘𝐴) · (norm_ℎ‘𝐵)))
15		df-ne 2957	. . . 4 ⊢ ((𝐴 ·_ih 𝐵) ≠ 0 ↔ ¬ (𝐴 ·_ih 𝐵) = 0)
16	6, 3	his1i 31684	. . . . . . . 8 ⊢ (𝐵 ·_ih 𝐴) = (∗‘(𝐴 ·_ih 𝐵))
17	16	oveq2i 7423	. . . . . . 7 ⊢ (((𝐴 ·_ih 𝐵) / (abs‘(𝐴 ·_ih 𝐵))) · (𝐵 ·_ih 𝐴)) = (((𝐴 ·_ih 𝐵) / (abs‘(𝐴 ·_ih 𝐵))) · (∗‘(𝐴 ·_ih 𝐵)))
18	17	oveq2i 7423	. . . . . 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 31665	. . . . . . 7 ⊢ (𝐴 ·_ih 𝐵) ∈ ℂ
20		abslem2 15487	. . . . . . 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 703	. . . . . 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 2809	. . . . 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 15546	. . . . . . . 8 ⊢ ((abs‘(𝐴 ·_ih 𝐵)) = 0 ↔ (𝐴 ·_ih 𝐵) = 0)
24	23	necon3bii 3008	. . . . . . 7 ⊢ ((abs‘(𝐴 ·_ih 𝐵)) ≠ 0 ↔ (𝐴 ·_ih 𝐵) ≠ 0)
25	19	abscli 15543	. . . . . . . . . 10 ⊢ (abs‘(𝐴 ·_ih 𝐵)) ∈ ℝ
26	25	recni 11304	. . . . . . . . 9 ⊢ (abs‘(𝐴 ·_ih 𝐵)) ∈ ℂ
27	19, 26	divclzi 12033	. . . . . . . 8 ⊢ ((abs‘(𝐴 ·_ih 𝐵)) ≠ 0 → ((𝐴 ·_ih 𝐵) / (abs‘(𝐴 ·_ih 𝐵))) ∈ ℂ)
28	19, 26	divreczi 12036	. . . . . . . . . 10 ⊢ ((abs‘(𝐴 ·_ih 𝐵)) ≠ 0 → ((𝐴 ·_ih 𝐵) / (abs‘(𝐴 ·_ih 𝐵))) = ((𝐴 ·_ih 𝐵) · (1 / (abs‘(𝐴 ·_ih 𝐵)))))
29	28	fveq2d 6881	. . . . . . . . 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 12023	. . . . . . . . . . 11 ⊢ ((abs‘(𝐴 ·_ih 𝐵)) ≠ 0 → (1 / (abs‘(𝐴 ·_ih 𝐵))) ∈ ℂ)
31		absmul 15441	. . . . . . . . . . 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 599	. . . . . . . . . 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 12062	. . . . . . . . . . . 12 ⊢ ((abs‘(𝐴 ·_ih 𝐵)) ≠ 0 → (1 / (abs‘(𝐴 ·_ih 𝐵))) ∈ ℝ)
34		0re 11291	. . . . . . . . . . . . . 14 ⊢ 0 ∈ ℝ
35	33, 34	jctil 529	. . . . . . . . . . . . 13 ⊢ ((abs‘(𝐴 ·_ih 𝐵)) ≠ 0 → (0 ∈ ℝ ∧ (1 / (abs‘(𝐴 ·_ih 𝐵))) ∈ ℝ))
36	19	absgt0i 15547	. . . . . . . . . . . . . . 15 ⊢ ((𝐴 ·_ih 𝐵) ≠ 0 ↔ 0 < (abs‘(𝐴 ·_ih 𝐵)))
37	24, 36	bitri 278	. . . . . . . . . . . . . 14 ⊢ ((abs‘(𝐴 ·_ih 𝐵)) ≠ 0 ↔ 0 < (abs‘(𝐴 ·_ih 𝐵)))
38	25	recgt0i 12203	. . . . . . . . . . . . . 14 ⊢ (0 < (abs‘(𝐴 ·_ih 𝐵)) → 0 < (1 / (abs‘(𝐴 ·_ih 𝐵))))
39	37, 38	sylbi 220	. . . . . . . . . . . . 13 ⊢ ((abs‘(𝐴 ·_ih 𝐵)) ≠ 0 → 0 < (1 / (abs‘(𝐴 ·_ih 𝐵))))
40		ltle 11379	. . . . . . . . . . . . 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 66	. . . . . . . . . . . 12 ⊢ ((abs‘(𝐴 ·_ih 𝐵)) ≠ 0 → 0 ≤ (1 / (abs‘(𝐴 ·_ih 𝐵))))
42	33, 41	absidd 15570	. . . . . . . . . . 11 ⊢ ((abs‘(𝐴 ·_ih 𝐵)) ≠ 0 → (abs‘(1 / (abs‘(𝐴 ·_ih 𝐵)))) = (1 / (abs‘(𝐴 ·_ih 𝐵))))
43	42	oveq2d 7428	. . . . . . . . . 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 2796	. . . . . . . . 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 12025	. . . . . . . . 9 ⊢ ((abs‘(𝐴 ·_ih 𝐵)) ≠ 0 → ((abs‘(𝐴 ·_ih 𝐵)) · (1 / (abs‘(𝐴 ·_ih 𝐵)))) = 1)
46	29, 44, 45	3eqtrd 2800	. . . . . . . 8 ⊢ ((abs‘(𝐴 ·_ih 𝐵)) ≠ 0 → (abs‘((𝐴 ·_ih 𝐵) / (abs‘(𝐴 ·_ih 𝐵)))) = 1)
47	27, 46	jca 521	. . . . . . 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 238	. . . . . 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 31703	. . . . . 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 18	. . . . 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 5127	. . . 4 ⊢ ((𝐴 ·_ih 𝐵) ≠ 0 → (2 · (abs‘(𝐴 ·_ih 𝐵))) ≤ (2 · ((√‘(𝐵 ·_ih 𝐵)) · (√‘(𝐴 ·_ih 𝐴)))))
52	15, 51	sylbir 238	. . 3 ⊢ (¬ (𝐴 ·_ih 𝐵) = 0 → (2 · (abs‘(𝐴 ·_ih 𝐵))) ≤ (2 · ((√‘(𝐵 ·_ih 𝐵)) · (√‘(𝐴 ·_ih 𝐴)))))
53	10	recni 11304	. . . . . 6 ⊢ (norm_ℎ‘𝐵) ∈ ℂ
54	9	recni 11304	. . . . . 6 ⊢ (norm_ℎ‘𝐴) ∈ ℂ
55		normval 31708	. . . . . . . 8 ⊢ (𝐵 ∈ ℋ → (norm_ℎ‘𝐵) = (√‘(𝐵 ·_ih 𝐵)))
56	6, 55	ax-mp 5	. . . . . . 7 ⊢ (norm_ℎ‘𝐵) = (√‘(𝐵 ·_ih 𝐵))
57		normval 31708	. . . . . . . 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 7424	. . . . . 6 ⊢ ((norm_ℎ‘𝐵) · (norm_ℎ‘𝐴)) = ((√‘(𝐵 ·_ih 𝐵)) · (√‘(𝐴 ·_ih 𝐴)))
60	53, 54, 59	mulcomli 11299	. . . . 5 ⊢ ((norm_ℎ‘𝐴) · (norm_ℎ‘𝐵)) = ((√‘(𝐵 ·_ih 𝐵)) · (√‘(𝐴 ·_ih 𝐴)))
61	60	breq2i 5111	. . . 4 ⊢ ((abs‘(𝐴 ·_ih 𝐵)) ≤ ((norm_ℎ‘𝐴) · (norm_ℎ‘𝐵)) ↔ (abs‘(𝐴 ·_ih 𝐵)) ≤ ((√‘(𝐵 ·_ih 𝐵)) · (√‘(𝐴 ·_ih 𝐴))))
62		2pos 12428	. . . . 5 ⊢ 0 < 2
63		hiidge0 31682	. . . . . . . 8 ⊢ (𝐵 ∈ ℋ → 0 ≤ (𝐵 ·_ih 𝐵))
64		hiidrcl 31679	. . . . . . . . . 10 ⊢ (𝐵 ∈ ℋ → (𝐵 ·_ih 𝐵) ∈ ℝ)
65	6, 64	ax-mp 5	. . . . . . . . 9 ⊢ (𝐵 ·_ih 𝐵) ∈ ℝ
66	65	sqrtcli 15519	. . . . . . . 8 ⊢ (0 ≤ (𝐵 ·_ih 𝐵) → (√‘(𝐵 ·_ih 𝐵)) ∈ ℝ)
67	6, 63, 66	mp2b 10	. . . . . . 7 ⊢ (√‘(𝐵 ·_ih 𝐵)) ∈ ℝ
68		hiidge0 31682	. . . . . . . 8 ⊢ (𝐴 ∈ ℋ → 0 ≤ (𝐴 ·_ih 𝐴))
69		hiidrcl 31679	. . . . . . . . . 10 ⊢ (𝐴 ∈ ℋ → (𝐴 ·_ih 𝐴) ∈ ℝ)
70	3, 69	ax-mp 5	. . . . . . . . 9 ⊢ (𝐴 ·_ih 𝐴) ∈ ℝ
71	70	sqrtcli 15519	. . . . . . . 8 ⊢ (0 ≤ (𝐴 ·_ih 𝐴) → (√‘(𝐴 ·_ih 𝐴)) ∈ ℝ)
72	3, 68, 71	mp2b 10	. . . . . . 7 ⊢ (√‘(𝐴 ·_ih 𝐴)) ∈ ℝ
73	67, 72	remulcli 11306	. . . . . 6 ⊢ ((√‘(𝐵 ·_ih 𝐵)) · (√‘(𝐴 ·_ih 𝐴))) ∈ ℝ
74		2re 12398	. . . . . 6 ⊢ 2 ∈ ℝ
75	25, 73, 74	lemul2i 12221	. . . . 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 278	. . 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 237	. 2 ⊢ (¬ (𝐴 ·_ih 𝐵) = 0 → (abs‘(𝐴 ·_ih 𝐵)) ≤ ((norm_ℎ‘𝐴) · (norm_ℎ‘𝐵)))
79	14, 78	pm2.61i 184	1 ⊢ (abs‘(𝐴 ·_ih 𝐵)) ≤ ((norm_ℎ‘𝐴) · (norm_ℎ‘𝐵))

Colors of variables: wff setvar class

This proof depends on syntax axioms: ¬ wn 3 ↔ wb 209 ∧ wa 401 = wceq 1570 ∈ wcel 2145 ≠ wne 2956 class class class wbr 5103 ‘cfv 6531 (class class class)co 7412 ℂcc 11179 ℝcr 11180 0cc0 11181 1c1 11182 + caddc 11184 · cmul 11186 < clt 11324 ≤ cle 11325 / cdiv 11954 2c2 12378 ∗ccj 15243 √csqrt 15380 abscabs 15381 ℋchba 31503 ·_ih csp 31506 norm_ℎcno 31507

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1943 ax-6 2000 ax-7 2041 ax-8 2147 ax-9 2155 ax-10 2178 ax-11 2194 ax-12 2213 ax-ext 2733 ax-sep 5249 ax-nul 5260 ax-pow 5327 ax-pr 5391 ax-un 7740 ax-cnex 11237 ax-resscn 11238 ax-1cn 11239 ax-icn 11240 ax-addcl 11241 ax-addrcl 11242 ax-mulcl 11243 ax-mulrcl 11244 ax-mulcom 11245 ax-addass 11246 ax-mulass 11247 ax-distr 11248 ax-i2m1 11249 ax-1ne0 11250 ax-1rid 11251 ax-rnegex 11252 ax-rrecex 11253 ax-cnre 11254 ax-pre-lttri 11255 ax-pre-lttrn 11256 ax-pre-ltadd 11257 ax-pre-mulgt0 11258 ax-pre-sup 11259 ax-hfvadd 31584 ax-hv0cl 31587 ax-hfvmul 31589 ax-hvmulass 31591 ax-hvmul0 31594 ax-hfi 31663 ax-his1 31666 ax-his2 31667 ax-his3 31668 ax-his4 31669

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2100 df-mo 2565 df-eu 2595 df-clab 2740 df-cleq 2753 df-clel 2836 df-nfc 2910 df-ne 2957 df-nel 3063 df-ral 3078 df-rex 3088 df-rmo 3366 df-reu 3367 df-rab 3414 df-v 3453 df-sbc 3740 df-csb 3848 df-dif 3902 df-un 3904 df-in 3906 df-ss 3916 df-pss 3919 df-nul 4280 df-if 4483 df-pw 4559 df-sn 4585 df-pr 4587 df-op 4591 df-uni 4868 df-iun 4953 df-br 5104 df-opab 5168 df-mpt 5187 df-tr 5213 df-id 5546 df-eprel 5551 df-po 5559 df-so 5560 df-fr 5604 df-we 5606 df-xp 5657 df-rel 5658 df-cnv 5659 df-co 5660 df-dm 5661 df-rn 5662 df-res 5663 df-ima 5664 df-pred 6297 df-ord 6358 df-on 6359 df-lim 6360 df-suc 6361 df-iota 6487 df-fun 6533 df-fn 6534 df-f 6535 df-f1 6536 df-fo 6537 df-f1o 6538 df-fv 6539 df-riota 7369 df-ov 7415 df-oprab 7416 df-mpo 7417 df-om 7867 df-2nd 7991 df-frecs 8283 df-wrecs 8314 df-recs 8363 df-rdg 8402 df-er 8701 df-en 8958 df-dom 8959 df-sdom 8960 df-sup 9418 df-pnf 11326 df-mnf 11327 df-xr 11328 df-ltxr 11329 df-le 11330 df-sub 11524 df-neg 11525 df-div 11955 df-nn 12317 df-2 12386 df-3 12387 df-4 12388 df-n0 12588 df-z 12675 df-uz 12947 df-rp 13102 df-seq 14125 df-exp 14185 df-cj 15246 df-re 15247 df-im 15248 df-sqrt 15382 df-abs 15383 df-hnorm 31552 df-hvsub 31555

This theorem is used by: (None)

W3C validator