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Theorem norm-ii-i 31745

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 11312	. . . . . . . . . . 11 ⊢ 1 ∈ ℝ
2		ax-1cn 11262	. . . . . . . . . . . 12 ⊢ 1 ∈ ℂ
3	2	cjrebi 15347	. . . . . . . . . . 11 ⊢ (1 ∈ ℝ ↔ (∗‘1) = 1)
4	1, 3	mpbi 233	. . . . . . . . . 10 ⊢ (∗‘1) = 1
5	4	oveq1i 7431	. . . . . . . . 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 31689	. . . . . . . . . 10 ⊢ (𝐵 ·_ih 𝐴) ∈ ℂ
9	8	mullidi 11318	. . . . . . . . 9 ⊢ (1 · (𝐵 ·_ih 𝐴)) = (𝐵 ·_ih 𝐴)
10	5, 9	eqtri 2783	. . . . . . . 8 ⊢ ((∗‘1) · (𝐵 ·_ih 𝐴)) = (𝐵 ·_ih 𝐴)
11	7, 6	hicli 31689	. . . . . . . . 9 ⊢ (𝐴 ·_ih 𝐵) ∈ ℂ
12	11	mullidi 11318	. . . . . . . 8 ⊢ (1 · (𝐴 ·_ih 𝐵)) = (𝐴 ·_ih 𝐵)
13	10, 12	oveq12i 7433	. . . . . . 7 ⊢ (((∗‘1) · (𝐵 ·_ih 𝐴)) + (1 · (𝐴 ·_ih 𝐵))) = ((𝐵 ·_ih 𝐴) + (𝐴 ·_ih 𝐵))
14		abs1 15470	. . . . . . . 8 ⊢ (abs‘1) = 1
15	2, 6, 7, 14	normlem7 31724	. . . . . . 7 ⊢ (((∗‘1) · (𝐵 ·_ih 𝐴)) + (1 · (𝐴 ·_ih 𝐵))) ≤ (2 · ((√‘(𝐴 ·_ih 𝐴)) · (√‘(𝐵 ·_ih 𝐵))))
16	13, 15	eqbrtrri 5127	. . . . . 6 ⊢ ((𝐵 ·_ih 𝐴) + (𝐴 ·_ih 𝐵)) ≤ (2 · ((√‘(𝐴 ·_ih 𝐴)) · (√‘(𝐵 ·_ih 𝐵))))
17		eqid 2760	. . . . . . . . . 10 ⊢ -(((∗‘1) · (𝐵 ·_ih 𝐴)) + (1 · (𝐴 ·_ih 𝐵))) = -(((∗‘1) · (𝐵 ·_ih 𝐴)) + (1 · (𝐴 ·_ih 𝐵)))
18	2, 6, 7, 17	normlem2 31719	. . . . . . . . 9 ⊢ -(((∗‘1) · (𝐵 ·_ih 𝐴)) + (1 · (𝐴 ·_ih 𝐵))) ∈ ℝ
19	2	cjcli 15342	. . . . . . . . . . . 12 ⊢ (∗‘1) ∈ ℂ
20	19, 8	mulcli 11320	. . . . . . . . . . 11 ⊢ ((∗‘1) · (𝐵 ·_ih 𝐴)) ∈ ℂ
21	2, 11	mulcli 11320	. . . . . . . . . . 11 ⊢ (1 · (𝐴 ·_ih 𝐵)) ∈ ℂ
22	20, 21	addcli 11319	. . . . . . . . . 10 ⊢ (((∗‘1) · (𝐵 ·_ih 𝐴)) + (1 · (𝐴 ·_ih 𝐵))) ∈ ℂ
23	22	negrebi 11636	. . . . . . . . 9 ⊢ ( -(((∗‘1) · (𝐵 ·_ih 𝐴)) + (1 · (𝐴 ·_ih 𝐵))) ∈ ℝ ↔ (((∗‘1) · (𝐵 ·_ih 𝐴)) + (1 · (𝐴 ·_ih 𝐵))) ∈ ℝ)
24	18, 23	mpbi 233	. . . . . . . 8 ⊢ (((∗‘1) · (𝐵 ·_ih 𝐴)) + (1 · (𝐴 ·_ih 𝐵))) ∈ ℝ
25	13, 24	eqeltrri 2857	. . . . . . 7 ⊢ ((𝐵 ·_ih 𝐴) + (𝐴 ·_ih 𝐵)) ∈ ℝ
26		2re 12421	. . . . . . . 8 ⊢ 2 ∈ ℝ
27		hiidge0 31706	. . . . . . . . . . 11 ⊢ (𝐴 ∈ ℋ → 0 ≤ (𝐴 ·_ih 𝐴))
28	7, 27	ax-mp 5	. . . . . . . . . 10 ⊢ 0 ≤ (𝐴 ·_ih 𝐴)
29		hiidrcl 31703	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ ℋ → (𝐴 ·_ih 𝐴) ∈ ℝ)
30	7, 29	ax-mp 5	. . . . . . . . . . 11 ⊢ (𝐴 ·_ih 𝐴) ∈ ℝ
31	30	sqrtcli 15545	. . . . . . . . . 10 ⊢ (0 ≤ (𝐴 ·_ih 𝐴) → (√‘(𝐴 ·_ih 𝐴)) ∈ ℝ)
32	28, 31	ax-mp 5	. . . . . . . . 9 ⊢ (√‘(𝐴 ·_ih 𝐴)) ∈ ℝ
33		hiidge0 31706	. . . . . . . . . . 11 ⊢ (𝐵 ∈ ℋ → 0 ≤ (𝐵 ·_ih 𝐵))
34	6, 33	ax-mp 5	. . . . . . . . . 10 ⊢ 0 ≤ (𝐵 ·_ih 𝐵)
35		hiidrcl 31703	. . . . . . . . . . . 12 ⊢ (𝐵 ∈ ℋ → (𝐵 ·_ih 𝐵) ∈ ℝ)
36	6, 35	ax-mp 5	. . . . . . . . . . 11 ⊢ (𝐵 ·_ih 𝐵) ∈ ℝ
37	36	sqrtcli 15545	. . . . . . . . . 10 ⊢ (0 ≤ (𝐵 ·_ih 𝐵) → (√‘(𝐵 ·_ih 𝐵)) ∈ ℝ)
38	34, 37	ax-mp 5	. . . . . . . . 9 ⊢ (√‘(𝐵 ·_ih 𝐵)) ∈ ℝ
39	32, 38	remulcli 11329	. . . . . . . 8 ⊢ ((√‘(𝐴 ·_ih 𝐴)) · (√‘(𝐵 ·_ih 𝐵))) ∈ ℝ
40	26, 39	remulcli 11329	. . . . . . 7 ⊢ (2 · ((√‘(𝐴 ·_ih 𝐴)) · (√‘(𝐵 ·_ih 𝐵)))) ∈ ℝ
41	30, 36	readdcli 11328	. . . . . . 7 ⊢ ((𝐴 ·_ih 𝐴) + (𝐵 ·_ih 𝐵)) ∈ ℝ
42	25, 40, 41	leadd2i 11876	. . . . . 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 233	. . . . 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 31725	. . . . . 6 ⊢ ((𝐴 +_ℎ 𝐵) ·_ih (𝐴 +_ℎ 𝐵)) = (((𝐴 ·_ih 𝐴) + (𝐵 ·_ih 𝐵)) + ((𝐴 ·_ih 𝐵) + (𝐵 ·_ih 𝐴)))
45	11, 8	addcomi 11505	. . . . . . 7 ⊢ ((𝐴 ·_ih 𝐵) + (𝐵 ·_ih 𝐴)) = ((𝐵 ·_ih 𝐴) + (𝐴 ·_ih 𝐵))
46	45	oveq2i 7432	. . . . . 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 2783	. . . . 5 ⊢ ((𝐴 +_ℎ 𝐵) ·_ih (𝐴 +_ℎ 𝐵)) = (((𝐴 ·_ih 𝐴) + (𝐵 ·_ih 𝐵)) + ((𝐵 ·_ih 𝐴) + (𝐴 ·_ih 𝐵)))
48	32	recni 11327	. . . . . . 7 ⊢ (√‘(𝐴 ·_ih 𝐴)) ∈ ℂ
49	38	recni 11327	. . . . . . 7 ⊢ (√‘(𝐵 ·_ih 𝐵)) ∈ ℂ
50	48, 49	binom2i 14360	. . . . . 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 14328	. . . . . . 7 ⊢ ((√‘(𝐴 ·_ih 𝐴))↑2) ∈ ℂ
52		2cn 12422	. . . . . . . 8 ⊢ 2 ∈ ℂ
53	48, 49	mulcli 11320	. . . . . . . 8 ⊢ ((√‘(𝐴 ·_ih 𝐴)) · (√‘(𝐵 ·_ih 𝐵))) ∈ ℂ
54	52, 53	mulcli 11320	. . . . . . 7 ⊢ (2 · ((√‘(𝐴 ·_ih 𝐴)) · (√‘(𝐵 ·_ih 𝐵)))) ∈ ℂ
55	49	sqcli 14328	. . . . . . 7 ⊢ ((√‘(𝐵 ·_ih 𝐵))↑2) ∈ ℂ
56	51, 54, 55	add32i 11538	. . . . . 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 15549	. . . . . . . . 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 15549	. . . . . . . . 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 7433	. . . . . . 7 ⊢ (((√‘(𝐴 ·_ih 𝐴))↑2) + ((√‘(𝐵 ·_ih 𝐵))↑2)) = ((𝐴 ·_ih 𝐴) + (𝐵 ·_ih 𝐵))
62	61	oveq1i 7431	. . . . . 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 2787	. . . . 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 5134	. . . 4 ⊢ ((𝐴 +_ℎ 𝐵) ·_ih (𝐴 +_ℎ 𝐵)) ≤ (((√‘(𝐴 ·_ih 𝐴)) + (√‘(𝐵 ·_ih 𝐵)))↑2)
65	7, 6	hvaddcli 31626	. . . . . 6 ⊢ (𝐴 +_ℎ 𝐵) ∈ ℋ
66		hiidge0 31706	. . . . . 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 11328	. . . . . 6 ⊢ ((√‘(𝐴 ·_ih 𝐴)) + (√‘(𝐵 ·_ih 𝐵))) ∈ ℝ
69	68	sqge0i 14335	. . . . 5 ⊢ 0 ≤ (((√‘(𝐴 ·_ih 𝐴)) + (√‘(𝐵 ·_ih 𝐵)))↑2)
70		hiidrcl 31703	. . . . . . 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 14333	. . . . . 6 ⊢ (((√‘(𝐴 ·_ih 𝐴)) + (√‘(𝐵 ·_ih 𝐵)))↑2) ∈ ℝ
73	71, 72	sqrtlei 15562	. . . . 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 705	. . . 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 233	. . 3 ⊢ (√‘((𝐴 +_ℎ 𝐵) ·_ih (𝐴 +_ℎ 𝐵))) ≤ (√‘(((√‘(𝐴 ·_ih 𝐴)) + (√‘(𝐵 ·_ih 𝐵)))↑2))
76	30	sqrtge0i 15550	. . . . . 6 ⊢ (0 ≤ (𝐴 ·_ih 𝐴) → 0 ≤ (√‘(𝐴 ·_ih 𝐴)))
77	28, 76	ax-mp 5	. . . . 5 ⊢ 0 ≤ (√‘(𝐴 ·_ih 𝐴))
78	36	sqrtge0i 15550	. . . . . 6 ⊢ (0 ≤ (𝐵 ·_ih 𝐵) → 0 ≤ (√‘(𝐵 ·_ih 𝐵)))
79	34, 78	ax-mp 5	. . . . 5 ⊢ 0 ≤ (√‘(𝐵 ·_ih 𝐵))
80	32, 38	addge0i 11860	. . . . 5 ⊢ ((0 ≤ (√‘(𝐴 ·_ih 𝐴)) ∧ 0 ≤ (√‘(𝐵 ·_ih 𝐵))) → 0 ≤ ((√‘(𝐴 ·_ih 𝐴)) + (√‘(𝐵 ·_ih 𝐵))))
81	77, 79, 80	mp2an 705	. . . 4 ⊢ 0 ≤ ((√‘(𝐴 ·_ih 𝐴)) + (√‘(𝐵 ·_ih 𝐵)))
82	68	sqrtsqi 15548	. . . 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 5129	. 2 ⊢ (√‘((𝐴 +_ℎ 𝐵) ·_ih (𝐴 +_ℎ 𝐵))) ≤ ((√‘(𝐴 ·_ih 𝐴)) + (√‘(𝐵 ·_ih 𝐵)))
85		normval 31732	. . 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 31732	. . . 4 ⊢ (𝐴 ∈ ℋ → (norm_ℎ‘𝐴) = (√‘(𝐴 ·_ih 𝐴)))
88	7, 87	ax-mp 5	. . 3 ⊢ (norm_ℎ‘𝐴) = (√‘(𝐴 ·_ih 𝐴))
89		normval 31732	. . . 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 7433	. 2 ⊢ ((norm_ℎ‘𝐴) + (norm_ℎ‘𝐵)) = ((√‘(𝐴 ·_ih 𝐴)) + (√‘(𝐵 ·_ih 𝐵)))
92	84, 86, 91	3brtr4i 5134	1 ⊢ (norm_ℎ‘(𝐴 +_ℎ 𝐵)) ≤ ((norm_ℎ‘𝐴) + (norm_ℎ‘𝐵))

Colors of variables: wff setvar class

This proof depends on syntax axioms: ↔ wb 209 = wceq 1569 ∈ wcel 2144 class class class wbr 5102 ‘cfv 6539 (class class class)co 7421 ℝcr 11203 0cc0 11204 1c1 11205 + caddc 11207 · cmul 11209 ≤ cle 11348 -cneg 11546 2c2 12401 ↑cexp 14208 ∗ccj 15269 √csqrt 15406 ℋchba 31527 +_ℎ cva 31528 ·_ih csp 31530 norm_ℎcno 31531

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1827 ax-4 1841 ax-5 1942 ax-6 1999 ax-7 2040 ax-8 2146 ax-9 2154 ax-10 2177 ax-11 2193 ax-12 2212 ax-ext 2732 ax-sep 5248 ax-nul 5259 ax-pow 5326 ax-pr 5390 ax-un 7752 ax-cnex 11260 ax-resscn 11261 ax-1cn 11262 ax-icn 11263 ax-addcl 11264 ax-addrcl 11265 ax-mulcl 11266 ax-mulrcl 11267 ax-mulcom 11268 ax-addass 11269 ax-mulass 11270 ax-distr 11271 ax-i2m1 11272 ax-1ne0 11273 ax-1rid 11274 ax-rnegex 11275 ax-rrecex 11276 ax-cnre 11277 ax-pre-lttri 11278 ax-pre-lttrn 11279 ax-pre-ltadd 11280 ax-pre-mulgt0 11281 ax-pre-sup 11282 ax-hfvadd 31608 ax-hv0cl 31611 ax-hfvmul 31613 ax-hvmulass 31615 ax-hvmul0 31618 ax-hfi 31687 ax-his1 31690 ax-his2 31691 ax-his3 31692 ax-his4 31693

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1572 df-fal 1582 df-ex 1812 df-nf 1816 df-sb 2099 df-mo 2564 df-eu 2594 df-clab 2739 df-cleq 2752 df-clel 2835 df-nfc 2909 df-ne 2956 df-nel 3062 df-ral 3077 df-rex 3087 df-rmo 3365 df-reu 3366 df-rab 3413 df-v 3452 df-sbc 3739 df-csb 3847 df-ss 3896 df-pss 4014 df-dif 4047 df-in 4091 df-un 4151 df-nul 4279 df-if 4482 df-pw 4558 df-sn 4585 df-pr 4587 df-op 4835 df-uni 4867 df-iun 4952 df-br 5103 df-opab 5167 df-mpt 5186 df-tr 5212 df-id 5545 df-eprel 5550 df-po 5558 df-so 5559 df-fr 5603 df-we 5605 df-xp 5656 df-rel 5657 df-cnv 5658 df-co 5659 df-dm 5660 df-rn 5661 df-res 5662 df-ima 5663 df-pred 6304 df-ord 6365 df-on 6366 df-lim 6367 df-suc 6368 df-iota 6494 df-fun 6541 df-fn 6542 df-f 6543 df-f1 6544 df-fo 6545 df-f1o 6546 df-fv 6547 df-riota 7378 df-ov 7424 df-oprab 7425 df-mpo 7426 df-om 7879 df-2nd 8003 df-frecs 8303 df-wrecs 8334 df-recs 8383 df-rdg 8422 df-er 8721 df-en 8978 df-dom 8979 df-sdom 8980 df-sup 9438 df-pnf 11349 df-mnf 11350 df-xr 11351 df-ltxr 11352 df-le 11353 df-sub 11547 df-neg 11548 df-div 11978 df-nn 12340 df-2 12409 df-3 12410 df-4 12411 df-n0 12611 df-z 12698 df-uz 12970 df-rp 13125 df-seq 14149 df-exp 14209 df-cj 15272 df-re 15273 df-im 15274 df-sqrt 15408 df-abs 15409 df-hnorm 31576 df-hvsub 31579

This theorem is used by: norm-ii 31746 norm3difi 31755

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