norm-ii-i - Hilbert Space Explorer

	Hilbert Space Explorer		< Previous Next > Nearby theorems
Mirrors > Home > HSE Home > Th. List > norm-ii-i		Structured version Visualization version GIF version

Theorem norm-ii-i 27994

Description: Triangle inequality for norms. Theorem 3.3(ii) of [Beran] p. 97. (Contributed by NM, 11-Aug-1999.) (New usage is discouraged.)

**Hypotheses**
Ref	Expression
norm-ii.1	⊢ 𝐴 ∈ ℋ
norm-ii.2	⊢ 𝐵 ∈ ℋ

**Assertion**
Ref	Expression
norm-ii-i	⊢ (norm_ℎ‘(𝐴 +_ℎ 𝐵)) ≤ ((norm_ℎ‘𝐴) + (norm_ℎ‘𝐵))

**Proof of Theorem norm-ii-i**
Step	Hyp	Ref	Expression
1		1re 10039	. . . . . . . . . . 11 ⊢ 1 ∈ ℝ
2		ax-1cn 9994	. . . . . . . . . . . 12 ⊢ 1 ∈ ℂ
3	2	cjrebi 13914	. . . . . . . . . . 11 ⊢ (1 ∈ ℝ ↔ (∗‘1) = 1)
4	1, 3	mpbi 220	. . . . . . . . . 10 ⊢ (∗‘1) = 1
5	4	oveq1i 6660	. . . . . . . . 9 ⊢ ((∗‘1) · (𝐵 ·_ih 𝐴)) = (1 · (𝐵 ·_ih 𝐴))
6		norm-ii.2	. . . . . . . . . . 11 ⊢ 𝐵 ∈ ℋ
7		norm-ii.1	. . . . . . . . . . 11 ⊢ 𝐴 ∈ ℋ
8	6, 7	hicli 27938	. . . . . . . . . 10 ⊢ (𝐵 ·_ih 𝐴) ∈ ℂ
9	8	mulid2i 10043	. . . . . . . . 9 ⊢ (1 · (𝐵 ·_ih 𝐴)) = (𝐵 ·_ih 𝐴)
10	5, 9	eqtri 2644	. . . . . . . 8 ⊢ ((∗‘1) · (𝐵 ·_ih 𝐴)) = (𝐵 ·_ih 𝐴)
11	7, 6	hicli 27938	. . . . . . . . 9 ⊢ (𝐴 ·_ih 𝐵) ∈ ℂ
12	11	mulid2i 10043	. . . . . . . 8 ⊢ (1 · (𝐴 ·_ih 𝐵)) = (𝐴 ·_ih 𝐵)
13	10, 12	oveq12i 6662	. . . . . . 7 ⊢ (((∗‘1) · (𝐵 ·_ih 𝐴)) + (1 · (𝐴 ·_ih 𝐵))) = ((𝐵 ·_ih 𝐴) + (𝐴 ·_ih 𝐵))
14		abs1 14037	. . . . . . . 8 ⊢ (abs‘1) = 1
15	2, 6, 7, 14	normlem7 27973	. . . . . . 7 ⊢ (((∗‘1) · (𝐵 ·_ih 𝐴)) + (1 · (𝐴 ·_ih 𝐵))) ≤ (2 · ((√‘(𝐴 ·_ih 𝐴)) · (√‘(𝐵 ·_ih 𝐵))))
16	13, 15	eqbrtrri 4676	. . . . . 6 ⊢ ((𝐵 ·_ih 𝐴) + (𝐴 ·_ih 𝐵)) ≤ (2 · ((√‘(𝐴 ·_ih 𝐴)) · (√‘(𝐵 ·_ih 𝐵))))
17		eqid 2622	. . . . . . . . . 10 ⊢ -(((∗‘1) · (𝐵 ·_ih 𝐴)) + (1 · (𝐴 ·_ih 𝐵))) = -(((∗‘1) · (𝐵 ·_ih 𝐴)) + (1 · (𝐴 ·_ih 𝐵)))
18	2, 6, 7, 17	normlem2 27968	. . . . . . . . 9 ⊢ -(((∗‘1) · (𝐵 ·_ih 𝐴)) + (1 · (𝐴 ·_ih 𝐵))) ∈ ℝ
19	2	cjcli 13909	. . . . . . . . . . . 12 ⊢ (∗‘1) ∈ ℂ
20	19, 8	mulcli 10045	. . . . . . . . . . 11 ⊢ ((∗‘1) · (𝐵 ·_ih 𝐴)) ∈ ℂ
21	2, 11	mulcli 10045	. . . . . . . . . . 11 ⊢ (1 · (𝐴 ·_ih 𝐵)) ∈ ℂ
22	20, 21	addcli 10044	. . . . . . . . . 10 ⊢ (((∗‘1) · (𝐵 ·_ih 𝐴)) + (1 · (𝐴 ·_ih 𝐵))) ∈ ℂ
23	22	negrebi 10355	. . . . . . . . 9 ⊢ (-(((∗‘1) · (𝐵 ·_ih 𝐴)) + (1 · (𝐴 ·_ih 𝐵))) ∈ ℝ ↔ (((∗‘1) · (𝐵 ·_ih 𝐴)) + (1 · (𝐴 ·_ih 𝐵))) ∈ ℝ)
24	18, 23	mpbi 220	. . . . . . . 8 ⊢ (((∗‘1) · (𝐵 ·_ih 𝐴)) + (1 · (𝐴 ·_ih 𝐵))) ∈ ℝ
25	13, 24	eqeltrri 2698	. . . . . . 7 ⊢ ((𝐵 ·_ih 𝐴) + (𝐴 ·_ih 𝐵)) ∈ ℝ
26		2re 11090	. . . . . . . 8 ⊢ 2 ∈ ℝ
27		hiidge0 27955	. . . . . . . . . . 11 ⊢ (𝐴 ∈ ℋ → 0 ≤ (𝐴 ·_ih 𝐴))
28	7, 27	ax-mp 5	. . . . . . . . . 10 ⊢ 0 ≤ (𝐴 ·_ih 𝐴)
29		hiidrcl 27952	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ ℋ → (𝐴 ·_ih 𝐴) ∈ ℝ)
30	7, 29	ax-mp 5	. . . . . . . . . . 11 ⊢ (𝐴 ·_ih 𝐴) ∈ ℝ
31	30	sqrtcli 14111	. . . . . . . . . 10 ⊢ (0 ≤ (𝐴 ·_ih 𝐴) → (√‘(𝐴 ·_ih 𝐴)) ∈ ℝ)
32	28, 31	ax-mp 5	. . . . . . . . 9 ⊢ (√‘(𝐴 ·_ih 𝐴)) ∈ ℝ
33		hiidge0 27955	. . . . . . . . . . 11 ⊢ (𝐵 ∈ ℋ → 0 ≤ (𝐵 ·_ih 𝐵))
34	6, 33	ax-mp 5	. . . . . . . . . 10 ⊢ 0 ≤ (𝐵 ·_ih 𝐵)
35		hiidrcl 27952	. . . . . . . . . . . 12 ⊢ (𝐵 ∈ ℋ → (𝐵 ·_ih 𝐵) ∈ ℝ)
36	6, 35	ax-mp 5	. . . . . . . . . . 11 ⊢ (𝐵 ·_ih 𝐵) ∈ ℝ
37	36	sqrtcli 14111	. . . . . . . . . 10 ⊢ (0 ≤ (𝐵 ·_ih 𝐵) → (√‘(𝐵 ·_ih 𝐵)) ∈ ℝ)
38	34, 37	ax-mp 5	. . . . . . . . 9 ⊢ (√‘(𝐵 ·_ih 𝐵)) ∈ ℝ
39	32, 38	remulcli 10054	. . . . . . . 8 ⊢ ((√‘(𝐴 ·_ih 𝐴)) · (√‘(𝐵 ·_ih 𝐵))) ∈ ℝ
40	26, 39	remulcli 10054	. . . . . . 7 ⊢ (2 · ((√‘(𝐴 ·_ih 𝐴)) · (√‘(𝐵 ·_ih 𝐵)))) ∈ ℝ
41	30, 36	readdcli 10053	. . . . . . 7 ⊢ ((𝐴 ·_ih 𝐴) + (𝐵 ·_ih 𝐵)) ∈ ℝ
42	25, 40, 41	leadd2i 10584	. . . . . 6 ⊢ (((𝐵 ·_ih 𝐴) + (𝐴 ·_ih 𝐵)) ≤ (2 · ((√‘(𝐴 ·_ih 𝐴)) · (√‘(𝐵 ·_ih 𝐵)))) ↔ (((𝐴 ·_ih 𝐴) + (𝐵 ·_ih 𝐵)) + ((𝐵 ·_ih 𝐴) + (𝐴 ·_ih 𝐵))) ≤ (((𝐴 ·_ih 𝐴) + (𝐵 ·_ih 𝐵)) + (2 · ((√‘(𝐴 ·_ih 𝐴)) · (√‘(𝐵 ·_ih 𝐵))))))
43	16, 42	mpbi 220	. . . . 5 ⊢ (((𝐴 ·_ih 𝐴) + (𝐵 ·_ih 𝐵)) + ((𝐵 ·_ih 𝐴) + (𝐴 ·_ih 𝐵))) ≤ (((𝐴 ·_ih 𝐴) + (𝐵 ·_ih 𝐵)) + (2 · ((√‘(𝐴 ·_ih 𝐴)) · (√‘(𝐵 ·_ih 𝐵)))))
44	7, 6, 7, 6	normlem8 27974	. . . . . 6 ⊢ ((𝐴 +_ℎ 𝐵) ·_ih (𝐴 +_ℎ 𝐵)) = (((𝐴 ·_ih 𝐴) + (𝐵 ·_ih 𝐵)) + ((𝐴 ·_ih 𝐵) + (𝐵 ·_ih 𝐴)))
45	11, 8	addcomi 10227	. . . . . . 7 ⊢ ((𝐴 ·_ih 𝐵) + (𝐵 ·_ih 𝐴)) = ((𝐵 ·_ih 𝐴) + (𝐴 ·_ih 𝐵))
46	45	oveq2i 6661	. . . . . 6 ⊢ (((𝐴 ·_ih 𝐴) + (𝐵 ·_ih 𝐵)) + ((𝐴 ·_ih 𝐵) + (𝐵 ·_ih 𝐴))) = (((𝐴 ·_ih 𝐴) + (𝐵 ·_ih 𝐵)) + ((𝐵 ·_ih 𝐴) + (𝐴 ·_ih 𝐵)))
47	44, 46	eqtri 2644	. . . . 5 ⊢ ((𝐴 +_ℎ 𝐵) ·_ih (𝐴 +_ℎ 𝐵)) = (((𝐴 ·_ih 𝐴) + (𝐵 ·_ih 𝐵)) + ((𝐵 ·_ih 𝐴) + (𝐴 ·_ih 𝐵)))
48	32	recni 10052	. . . . . . 7 ⊢ (√‘(𝐴 ·_ih 𝐴)) ∈ ℂ
49	38	recni 10052	. . . . . . 7 ⊢ (√‘(𝐵 ·_ih 𝐵)) ∈ ℂ
50	48, 49	binom2i 12974	. . . . . 6 ⊢ (((√‘(𝐴 ·_ih 𝐴)) + (√‘(𝐵 ·_ih 𝐵)))↑2) = ((((√‘(𝐴 ·_ih 𝐴))↑2) + (2 · ((√‘(𝐴 ·_ih 𝐴)) · (√‘(𝐵 ·_ih 𝐵))))) + ((√‘(𝐵 ·_ih 𝐵))↑2))
51	48	sqcli 12944	. . . . . . 7 ⊢ ((√‘(𝐴 ·_ih 𝐴))↑2) ∈ ℂ
52		2cn 11091	. . . . . . . 8 ⊢ 2 ∈ ℂ
53	48, 49	mulcli 10045	. . . . . . . 8 ⊢ ((√‘(𝐴 ·_ih 𝐴)) · (√‘(𝐵 ·_ih 𝐵))) ∈ ℂ
54	52, 53	mulcli 10045	. . . . . . 7 ⊢ (2 · ((√‘(𝐴 ·_ih 𝐴)) · (√‘(𝐵 ·_ih 𝐵)))) ∈ ℂ
55	49	sqcli 12944	. . . . . . 7 ⊢ ((√‘(𝐵 ·_ih 𝐵))↑2) ∈ ℂ
56	51, 54, 55	add32i 10259	. . . . . 6 ⊢ ((((√‘(𝐴 ·_ih 𝐴))↑2) + (2 · ((√‘(𝐴 ·_ih 𝐴)) · (√‘(𝐵 ·_ih 𝐵))))) + ((√‘(𝐵 ·_ih 𝐵))↑2)) = ((((√‘(𝐴 ·_ih 𝐴))↑2) + ((√‘(𝐵 ·_ih 𝐵))↑2)) + (2 · ((√‘(𝐴 ·_ih 𝐴)) · (√‘(𝐵 ·_ih 𝐵)))))
57	30	sqsqrti 14115	. . . . . . . . 9 ⊢ (0 ≤ (𝐴 ·_ih 𝐴) → ((√‘(𝐴 ·_ih 𝐴))↑2) = (𝐴 ·_ih 𝐴))
58	28, 57	ax-mp 5	. . . . . . . 8 ⊢ ((√‘(𝐴 ·_ih 𝐴))↑2) = (𝐴 ·_ih 𝐴)
59	36	sqsqrti 14115	. . . . . . . . 9 ⊢ (0 ≤ (𝐵 ·_ih 𝐵) → ((√‘(𝐵 ·_ih 𝐵))↑2) = (𝐵 ·_ih 𝐵))
60	34, 59	ax-mp 5	. . . . . . . 8 ⊢ ((√‘(𝐵 ·_ih 𝐵))↑2) = (𝐵 ·_ih 𝐵)
61	58, 60	oveq12i 6662	. . . . . . 7 ⊢ (((√‘(𝐴 ·_ih 𝐴))↑2) + ((√‘(𝐵 ·_ih 𝐵))↑2)) = ((𝐴 ·_ih 𝐴) + (𝐵 ·_ih 𝐵))
62	61	oveq1i 6660	. . . . . 6 ⊢ ((((√‘(𝐴 ·_ih 𝐴))↑2) + ((√‘(𝐵 ·_ih 𝐵))↑2)) + (2 · ((√‘(𝐴 ·_ih 𝐴)) · (√‘(𝐵 ·_ih 𝐵))))) = (((𝐴 ·_ih 𝐴) + (𝐵 ·_ih 𝐵)) + (2 · ((√‘(𝐴 ·_ih 𝐴)) · (√‘(𝐵 ·_ih 𝐵)))))
63	50, 56, 62	3eqtri 2648	. . . . 5 ⊢ (((√‘(𝐴 ·_ih 𝐴)) + (√‘(𝐵 ·_ih 𝐵)))↑2) = (((𝐴 ·_ih 𝐴) + (𝐵 ·_ih 𝐵)) + (2 · ((√‘(𝐴 ·_ih 𝐴)) · (√‘(𝐵 ·_ih 𝐵)))))
64	43, 47, 63	3brtr4i 4683	. . . 4 ⊢ ((𝐴 +_ℎ 𝐵) ·_ih (𝐴 +_ℎ 𝐵)) ≤ (((√‘(𝐴 ·_ih 𝐴)) + (√‘(𝐵 ·_ih 𝐵)))↑2)
65	7, 6	hvaddcli 27875	. . . . . 6 ⊢ (𝐴 +_ℎ 𝐵) ∈ ℋ
66		hiidge0 27955	. . . . . 6 ⊢ ((𝐴 +_ℎ 𝐵) ∈ ℋ → 0 ≤ ((𝐴 +_ℎ 𝐵) ·_ih (𝐴 +_ℎ 𝐵)))
67	65, 66	ax-mp 5	. . . . 5 ⊢ 0 ≤ ((𝐴 +_ℎ 𝐵) ·_ih (𝐴 +_ℎ 𝐵))
68	32, 38	readdcli 10053	. . . . . 6 ⊢ ((√‘(𝐴 ·_ih 𝐴)) + (√‘(𝐵 ·_ih 𝐵))) ∈ ℝ
69	68	sqge0i 12951	. . . . 5 ⊢ 0 ≤ (((√‘(𝐴 ·_ih 𝐴)) + (√‘(𝐵 ·_ih 𝐵)))↑2)
70		hiidrcl 27952	. . . . . . 7 ⊢ ((𝐴 +_ℎ 𝐵) ∈ ℋ → ((𝐴 +_ℎ 𝐵) ·_ih (𝐴 +_ℎ 𝐵)) ∈ ℝ)
71	65, 70	ax-mp 5	. . . . . 6 ⊢ ((𝐴 +_ℎ 𝐵) ·_ih (𝐴 +_ℎ 𝐵)) ∈ ℝ
72	68	resqcli 12949	. . . . . 6 ⊢ (((√‘(𝐴 ·_ih 𝐴)) + (√‘(𝐵 ·_ih 𝐵)))↑2) ∈ ℝ
73	71, 72	sqrtlei 14128	. . . . 5 ⊢ ((0 ≤ ((𝐴 +_ℎ 𝐵) ·_ih (𝐴 +_ℎ 𝐵)) ∧ 0 ≤ (((√‘(𝐴 ·_ih 𝐴)) + (√‘(𝐵 ·_ih 𝐵)))↑2)) → (((𝐴 +_ℎ 𝐵) ·_ih (𝐴 +_ℎ 𝐵)) ≤ (((√‘(𝐴 ·_ih 𝐴)) + (√‘(𝐵 ·_ih 𝐵)))↑2) ↔ (√‘((𝐴 +_ℎ 𝐵) ·_ih (𝐴 +_ℎ 𝐵))) ≤ (√‘(((√‘(𝐴 ·_ih 𝐴)) + (√‘(𝐵 ·_ih 𝐵)))↑2))))
74	67, 69, 73	mp2an 708	. . . 4 ⊢ (((𝐴 +_ℎ 𝐵) ·_ih (𝐴 +_ℎ 𝐵)) ≤ (((√‘(𝐴 ·_ih 𝐴)) + (√‘(𝐵 ·_ih 𝐵)))↑2) ↔ (√‘((𝐴 +_ℎ 𝐵) ·_ih (𝐴 +_ℎ 𝐵))) ≤ (√‘(((√‘(𝐴 ·_ih 𝐴)) + (√‘(𝐵 ·_ih 𝐵)))↑2)))
75	64, 74	mpbi 220	. . 3 ⊢ (√‘((𝐴 +_ℎ 𝐵) ·_ih (𝐴 +_ℎ 𝐵))) ≤ (√‘(((√‘(𝐴 ·_ih 𝐴)) + (√‘(𝐵 ·_ih 𝐵)))↑2))
76	30	sqrtge0i 14116	. . . . . 6 ⊢ (0 ≤ (𝐴 ·_ih 𝐴) → 0 ≤ (√‘(𝐴 ·_ih 𝐴)))
77	28, 76	ax-mp 5	. . . . 5 ⊢ 0 ≤ (√‘(𝐴 ·_ih 𝐴))
78	36	sqrtge0i 14116	. . . . . 6 ⊢ (0 ≤ (𝐵 ·_ih 𝐵) → 0 ≤ (√‘(𝐵 ·_ih 𝐵)))
79	34, 78	ax-mp 5	. . . . 5 ⊢ 0 ≤ (√‘(𝐵 ·_ih 𝐵))
80	32, 38	addge0i 10568	. . . . 5 ⊢ ((0 ≤ (√‘(𝐴 ·_ih 𝐴)) ∧ 0 ≤ (√‘(𝐵 ·_ih 𝐵))) → 0 ≤ ((√‘(𝐴 ·_ih 𝐴)) + (√‘(𝐵 ·_ih 𝐵))))
81	77, 79, 80	mp2an 708	. . . 4 ⊢ 0 ≤ ((√‘(𝐴 ·_ih 𝐴)) + (√‘(𝐵 ·_ih 𝐵)))
82	68	sqrtsqi 14114	. . . 4 ⊢ (0 ≤ ((√‘(𝐴 ·_ih 𝐴)) + (√‘(𝐵 ·_ih 𝐵))) → (√‘(((√‘(𝐴 ·_ih 𝐴)) + (√‘(𝐵 ·_ih 𝐵)))↑2)) = ((√‘(𝐴 ·_ih 𝐴)) + (√‘(𝐵 ·_ih 𝐵))))
83	81, 82	ax-mp 5	. . 3 ⊢ (√‘(((√‘(𝐴 ·_ih 𝐴)) + (√‘(𝐵 ·_ih 𝐵)))↑2)) = ((√‘(𝐴 ·_ih 𝐴)) + (√‘(𝐵 ·_ih 𝐵)))
84	75, 83	breqtri 4678	. 2 ⊢ (√‘((𝐴 +_ℎ 𝐵) ·_ih (𝐴 +_ℎ 𝐵))) ≤ ((√‘(𝐴 ·_ih 𝐴)) + (√‘(𝐵 ·_ih 𝐵)))
85		normval 27981	. . 3 ⊢ ((𝐴 +_ℎ 𝐵) ∈ ℋ → (norm_ℎ‘(𝐴 +_ℎ 𝐵)) = (√‘((𝐴 +_ℎ 𝐵) ·_ih (𝐴 +_ℎ 𝐵))))
86	65, 85	ax-mp 5	. 2 ⊢ (norm_ℎ‘(𝐴 +_ℎ 𝐵)) = (√‘((𝐴 +_ℎ 𝐵) ·_ih (𝐴 +_ℎ 𝐵)))
87		normval 27981	. . . 4 ⊢ (𝐴 ∈ ℋ → (norm_ℎ‘𝐴) = (√‘(𝐴 ·_ih 𝐴)))
88	7, 87	ax-mp 5	. . 3 ⊢ (norm_ℎ‘𝐴) = (√‘(𝐴 ·_ih 𝐴))
89		normval 27981	. . . 4 ⊢ (𝐵 ∈ ℋ → (norm_ℎ‘𝐵) = (√‘(𝐵 ·_ih 𝐵)))
90	6, 89	ax-mp 5	. . 3 ⊢ (norm_ℎ‘𝐵) = (√‘(𝐵 ·_ih 𝐵))
91	88, 90	oveq12i 6662	. 2 ⊢ ((norm_ℎ‘𝐴) + (norm_ℎ‘𝐵)) = ((√‘(𝐴 ·_ih 𝐴)) + (√‘(𝐵 ·_ih 𝐵)))
92	84, 86, 91	3brtr4i 4683	1 ⊢ (norm_ℎ‘(𝐴 +_ℎ 𝐵)) ≤ ((norm_ℎ‘𝐴) + (norm_ℎ‘𝐵))

Colors of variables: wff setvar class

Syntax hints: ↔ wb 196 = wceq 1483 ∈ wcel 1990 class class class wbr 4653 ‘cfv 5888 (class class class)co 6650 ℝcr 9935 0cc0 9936 1c1 9937 + caddc 9939 · cmul 9941 ≤ cle 10075 -cneg 10267 2c2 11070 ↑cexp 12860 ∗ccj 13836 √csqrt 13973 ℋchil 27776 +_ℎ cva 27777 ·_ih csp 27779 norm_ℎcno 27780

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013 ax-pre-sup 10014 ax-hfvadd 27857 ax-hv0cl 27860 ax-hfvmul 27862 ax-hvmulass 27864 ax-hvmul0 27867 ax-hfi 27936 ax-his1 27939 ax-his2 27940 ax-his3 27941 ax-his4 27942

This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-om 7066 df-2nd 7169 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-er 7742 df-en 7956 df-dom 7957 df-sdom 7958 df-sup 8348 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-div 10685 df-nn 11021 df-2 11079 df-3 11080 df-4 11081 df-n0 11293 df-z 11378 df-uz 11688 df-rp 11833 df-seq 12802 df-exp 12861 df-cj 13839 df-re 13840 df-im 13841 df-sqrt 13975 df-abs 13976 df-hnorm 27825 df-hvsub 27828

This theorem is referenced by: norm-ii 27995 norm3difi 28004

W3C validator