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Theorem norm-ii-i 27378

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 9918	. . . . . . . . . . 11 ⊢ 1 ∈ ℝ
2		ax-1cn 9873	. . . . . . . . . . . 12 ⊢ 1 ∈ ℂ
3	2	cjrebi 13762	. . . . . . . . . . 11 ⊢ (1 ∈ ℝ ↔ (∗‘1) = 1)
4	1, 3	mpbi 219	. . . . . . . . . 10 ⊢ (∗‘1) = 1
5	4	oveq1i 6559	. . . . . . . . 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 27322	. . . . . . . . . 10 ⊢ (𝐵 ·_ih 𝐴) ∈ ℂ
9	8	mulid2i 9922	. . . . . . . . 9 ⊢ (1 · (𝐵 ·_ih 𝐴)) = (𝐵 ·_ih 𝐴)
10	5, 9	eqtri 2632	. . . . . . . 8 ⊢ ((∗‘1) · (𝐵 ·_ih 𝐴)) = (𝐵 ·_ih 𝐴)
11	7, 6	hicli 27322	. . . . . . . . 9 ⊢ (𝐴 ·_ih 𝐵) ∈ ℂ
12	11	mulid2i 9922	. . . . . . . 8 ⊢ (1 · (𝐴 ·_ih 𝐵)) = (𝐴 ·_ih 𝐵)
13	10, 12	oveq12i 6561	. . . . . . 7 ⊢ (((∗‘1) · (𝐵 ·_ih 𝐴)) + (1 · (𝐴 ·_ih 𝐵))) = ((𝐵 ·_ih 𝐴) + (𝐴 ·_ih 𝐵))
14		abs1 13885	. . . . . . . 8 ⊢ (abs‘1) = 1
15	2, 6, 7, 14	normlem7 27357	. . . . . . 7 ⊢ (((∗‘1) · (𝐵 ·_ih 𝐴)) + (1 · (𝐴 ·_ih 𝐵))) ≤ (2 · ((√‘(𝐴 ·_ih 𝐴)) · (√‘(𝐵 ·_ih 𝐵))))
16	13, 15	eqbrtrri 4606	. . . . . 6 ⊢ ((𝐵 ·_ih 𝐴) + (𝐴 ·_ih 𝐵)) ≤ (2 · ((√‘(𝐴 ·_ih 𝐴)) · (√‘(𝐵 ·_ih 𝐵))))
17		eqid 2610	. . . . . . . . . 10 ⊢ -(((∗‘1) · (𝐵 ·_ih 𝐴)) + (1 · (𝐴 ·_ih 𝐵))) = -(((∗‘1) · (𝐵 ·_ih 𝐴)) + (1 · (𝐴 ·_ih 𝐵)))
18	2, 6, 7, 17	normlem2 27352	. . . . . . . . 9 ⊢ -(((∗‘1) · (𝐵 ·_ih 𝐴)) + (1 · (𝐴 ·_ih 𝐵))) ∈ ℝ
19	2	cjcli 13757	. . . . . . . . . . . 12 ⊢ (∗‘1) ∈ ℂ
20	19, 8	mulcli 9924	. . . . . . . . . . 11 ⊢ ((∗‘1) · (𝐵 ·_ih 𝐴)) ∈ ℂ
21	2, 11	mulcli 9924	. . . . . . . . . . 11 ⊢ (1 · (𝐴 ·_ih 𝐵)) ∈ ℂ
22	20, 21	addcli 9923	. . . . . . . . . 10 ⊢ (((∗‘1) · (𝐵 ·_ih 𝐴)) + (1 · (𝐴 ·_ih 𝐵))) ∈ ℂ
23	22	negrebi 10234	. . . . . . . . 9 ⊢ (-(((∗‘1) · (𝐵 ·_ih 𝐴)) + (1 · (𝐴 ·_ih 𝐵))) ∈ ℝ ↔ (((∗‘1) · (𝐵 ·_ih 𝐴)) + (1 · (𝐴 ·_ih 𝐵))) ∈ ℝ)
24	18, 23	mpbi 219	. . . . . . . 8 ⊢ (((∗‘1) · (𝐵 ·_ih 𝐴)) + (1 · (𝐴 ·_ih 𝐵))) ∈ ℝ
25	13, 24	eqeltrri 2685	. . . . . . 7 ⊢ ((𝐵 ·_ih 𝐴) + (𝐴 ·_ih 𝐵)) ∈ ℝ
26		2re 10967	. . . . . . . 8 ⊢ 2 ∈ ℝ
27		hiidge0 27339	. . . . . . . . . . 11 ⊢ (𝐴 ∈ ℋ → 0 ≤ (𝐴 ·_ih 𝐴))
28	7, 27	ax-mp 5	. . . . . . . . . 10 ⊢ 0 ≤ (𝐴 ·_ih 𝐴)
29		hiidrcl 27336	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ ℋ → (𝐴 ·_ih 𝐴) ∈ ℝ)
30	7, 29	ax-mp 5	. . . . . . . . . . 11 ⊢ (𝐴 ·_ih 𝐴) ∈ ℝ
31	30	sqrtcli 13959	. . . . . . . . . 10 ⊢ (0 ≤ (𝐴 ·_ih 𝐴) → (√‘(𝐴 ·_ih 𝐴)) ∈ ℝ)
32	28, 31	ax-mp 5	. . . . . . . . 9 ⊢ (√‘(𝐴 ·_ih 𝐴)) ∈ ℝ
33		hiidge0 27339	. . . . . . . . . . 11 ⊢ (𝐵 ∈ ℋ → 0 ≤ (𝐵 ·_ih 𝐵))
34	6, 33	ax-mp 5	. . . . . . . . . 10 ⊢ 0 ≤ (𝐵 ·_ih 𝐵)
35		hiidrcl 27336	. . . . . . . . . . . 12 ⊢ (𝐵 ∈ ℋ → (𝐵 ·_ih 𝐵) ∈ ℝ)
36	6, 35	ax-mp 5	. . . . . . . . . . 11 ⊢ (𝐵 ·_ih 𝐵) ∈ ℝ
37	36	sqrtcli 13959	. . . . . . . . . 10 ⊢ (0 ≤ (𝐵 ·_ih 𝐵) → (√‘(𝐵 ·_ih 𝐵)) ∈ ℝ)
38	34, 37	ax-mp 5	. . . . . . . . 9 ⊢ (√‘(𝐵 ·_ih 𝐵)) ∈ ℝ
39	32, 38	remulcli 9933	. . . . . . . 8 ⊢ ((√‘(𝐴 ·_ih 𝐴)) · (√‘(𝐵 ·_ih 𝐵))) ∈ ℝ
40	26, 39	remulcli 9933	. . . . . . 7 ⊢ (2 · ((√‘(𝐴 ·_ih 𝐴)) · (√‘(𝐵 ·_ih 𝐵)))) ∈ ℝ
41	30, 36	readdcli 9932	. . . . . . 7 ⊢ ((𝐴 ·_ih 𝐴) + (𝐵 ·_ih 𝐵)) ∈ ℝ
42	25, 40, 41	leadd2i 10463	. . . . . 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 219	. . . . 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 27358	. . . . . 6 ⊢ ((𝐴 +_ℎ 𝐵) ·_ih (𝐴 +_ℎ 𝐵)) = (((𝐴 ·_ih 𝐴) + (𝐵 ·_ih 𝐵)) + ((𝐴 ·_ih 𝐵) + (𝐵 ·_ih 𝐴)))
45	11, 8	addcomi 10106	. . . . . . 7 ⊢ ((𝐴 ·_ih 𝐵) + (𝐵 ·_ih 𝐴)) = ((𝐵 ·_ih 𝐴) + (𝐴 ·_ih 𝐵))
46	45	oveq2i 6560	. . . . . 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 2632	. . . . 5 ⊢ ((𝐴 +_ℎ 𝐵) ·_ih (𝐴 +_ℎ 𝐵)) = (((𝐴 ·_ih 𝐴) + (𝐵 ·_ih 𝐵)) + ((𝐵 ·_ih 𝐴) + (𝐴 ·_ih 𝐵)))
48	32	recni 9931	. . . . . . 7 ⊢ (√‘(𝐴 ·_ih 𝐴)) ∈ ℂ
49	38	recni 9931	. . . . . . 7 ⊢ (√‘(𝐵 ·_ih 𝐵)) ∈ ℂ
50	48, 49	binom2i 12836	. . . . . 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 12806	. . . . . . 7 ⊢ ((√‘(𝐴 ·_ih 𝐴))↑2) ∈ ℂ
52		2cn 10968	. . . . . . . 8 ⊢ 2 ∈ ℂ
53	48, 49	mulcli 9924	. . . . . . . 8 ⊢ ((√‘(𝐴 ·_ih 𝐴)) · (√‘(𝐵 ·_ih 𝐵))) ∈ ℂ
54	52, 53	mulcli 9924	. . . . . . 7 ⊢ (2 · ((√‘(𝐴 ·_ih 𝐴)) · (√‘(𝐵 ·_ih 𝐵)))) ∈ ℂ
55	49	sqcli 12806	. . . . . . 7 ⊢ ((√‘(𝐵 ·_ih 𝐵))↑2) ∈ ℂ
56	51, 54, 55	add32i 10138	. . . . . 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 13963	. . . . . . . . 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 13963	. . . . . . . . 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 6561	. . . . . . 7 ⊢ (((√‘(𝐴 ·_ih 𝐴))↑2) + ((√‘(𝐵 ·_ih 𝐵))↑2)) = ((𝐴 ·_ih 𝐴) + (𝐵 ·_ih 𝐵))
62	61	oveq1i 6559	. . . . . 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 2636	. . . . 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 4613	. . . 4 ⊢ ((𝐴 +_ℎ 𝐵) ·_ih (𝐴 +_ℎ 𝐵)) ≤ (((√‘(𝐴 ·_ih 𝐴)) + (√‘(𝐵 ·_ih 𝐵)))↑2)
65	7, 6	hvaddcli 27259	. . . . . 6 ⊢ (𝐴 +_ℎ 𝐵) ∈ ℋ
66		hiidge0 27339	. . . . . 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 9932	. . . . . 6 ⊢ ((√‘(𝐴 ·_ih 𝐴)) + (√‘(𝐵 ·_ih 𝐵))) ∈ ℝ
69	68	sqge0i 12813	. . . . 5 ⊢ 0 ≤ (((√‘(𝐴 ·_ih 𝐴)) + (√‘(𝐵 ·_ih 𝐵)))↑2)
70		hiidrcl 27336	. . . . . . 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 12811	. . . . . 6 ⊢ (((√‘(𝐴 ·_ih 𝐴)) + (√‘(𝐵 ·_ih 𝐵)))↑2) ∈ ℝ
73	71, 72	sqrtlei 13976	. . . . 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 704	. . . 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 219	. . 3 ⊢ (√‘((𝐴 +_ℎ 𝐵) ·_ih (𝐴 +_ℎ 𝐵))) ≤ (√‘(((√‘(𝐴 ·_ih 𝐴)) + (√‘(𝐵 ·_ih 𝐵)))↑2))
76	30	sqrtge0i 13964	. . . . . 6 ⊢ (0 ≤ (𝐴 ·_ih 𝐴) → 0 ≤ (√‘(𝐴 ·_ih 𝐴)))
77	28, 76	ax-mp 5	. . . . 5 ⊢ 0 ≤ (√‘(𝐴 ·_ih 𝐴))
78	36	sqrtge0i 13964	. . . . . 6 ⊢ (0 ≤ (𝐵 ·_ih 𝐵) → 0 ≤ (√‘(𝐵 ·_ih 𝐵)))
79	34, 78	ax-mp 5	. . . . 5 ⊢ 0 ≤ (√‘(𝐵 ·_ih 𝐵))
80	32, 38	addge0i 10447	. . . . 5 ⊢ ((0 ≤ (√‘(𝐴 ·_ih 𝐴)) ∧ 0 ≤ (√‘(𝐵 ·_ih 𝐵))) → 0 ≤ ((√‘(𝐴 ·_ih 𝐴)) + (√‘(𝐵 ·_ih 𝐵))))
81	77, 79, 80	mp2an 704	. . . 4 ⊢ 0 ≤ ((√‘(𝐴 ·_ih 𝐴)) + (√‘(𝐵 ·_ih 𝐵)))
82	68	sqrtsqi 13962	. . . 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 4608	. 2 ⊢ (√‘((𝐴 +_ℎ 𝐵) ·_ih (𝐴 +_ℎ 𝐵))) ≤ ((√‘(𝐴 ·_ih 𝐴)) + (√‘(𝐵 ·_ih 𝐵)))
85		normval 27365	. . 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 27365	. . . 4 ⊢ (𝐴 ∈ ℋ → (norm_ℎ‘𝐴) = (√‘(𝐴 ·_ih 𝐴)))
88	7, 87	ax-mp 5	. . 3 ⊢ (norm_ℎ‘𝐴) = (√‘(𝐴 ·_ih 𝐴))
89		normval 27365	. . . 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 6561	. 2 ⊢ ((norm_ℎ‘𝐴) + (norm_ℎ‘𝐵)) = ((√‘(𝐴 ·_ih 𝐴)) + (√‘(𝐵 ·_ih 𝐵)))
92	84, 86, 91	3brtr4i 4613	1 ⊢ (norm_ℎ‘(𝐴 +_ℎ 𝐵)) ≤ ((norm_ℎ‘𝐴) + (norm_ℎ‘𝐵))

Colors of variables: wff setvar class

Syntax hints: ↔ wb 195 = wceq 1475 ∈ wcel 1977 class class class wbr 4583 ‘cfv 5804 (class class class)co 6549 ℝcr 9814 0cc0 9815 1c1 9816 + caddc 9818 · cmul 9820 ≤ cle 9954 -cneg 10146 2c2 10947 ↑cexp 12722 ∗ccj 13684 √csqrt 13821 ℋchil 27160 +_ℎ cva 27161 ·_ih csp 27163 norm_ℎcno 27164

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 ax-pre-sup 9893 ax-hfvadd 27241 ax-hv0cl 27244 ax-hfvmul 27246 ax-hvmulass 27248 ax-hvmul0 27251 ax-hfi 27320 ax-his1 27323 ax-his2 27324 ax-his3 27325 ax-his4 27326

This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-2nd 7060 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-er 7629 df-en 7842 df-dom 7843 df-sdom 7844 df-sup 8231 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-div 10564 df-nn 10898 df-2 10956 df-3 10957 df-4 10958 df-n0 11170 df-z 11255 df-uz 11564 df-rp 11709 df-seq 12664 df-exp 12723 df-cj 13687 df-re 13688 df-im 13689 df-sqrt 13823 df-abs 13824 df-hnorm 27209 df-hvsub 27212

This theorem is referenced by: norm-ii 27379 norm3difi 27388

W3C validator