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Theorem norm-ii-i 31796

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 11347	. . . . . . . . . . 11 ⊢ 1 ∈ ℝ
2		ax-1cn 11297	. . . . . . . . . . . 12 ⊢ 1 ∈ ℂ
3	2	cjrebi 15384	. . . . . . . . . . 11 ⊢ (1 ∈ ℝ ↔ (∗‘1) = 1)
4	1, 3	mpbi 233	. . . . . . . . . 10 ⊢ (∗‘1) = 1
5	4	oveq1i 7461	. . . . . . . . 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 31740	. . . . . . . . . 10 ⊢ (𝐵 ·_ih 𝐴) ∈ ℂ
9	8	mullidi 11353	. . . . . . . . 9 ⊢ (1 · (𝐵 ·_ih 𝐴)) = (𝐵 ·_ih 𝐴)
10	5, 9	eqtri 2786	. . . . . . . 8 ⊢ ((∗‘1) · (𝐵 ·_ih 𝐴)) = (𝐵 ·_ih 𝐴)
11	7, 6	hicli 31740	. . . . . . . . 9 ⊢ (𝐴 ·_ih 𝐵) ∈ ℂ
12	11	mullidi 11353	. . . . . . . 8 ⊢ (1 · (𝐴 ·_ih 𝐵)) = (𝐴 ·_ih 𝐵)
13	10, 12	oveq12i 7463	. . . . . . 7 ⊢ (((∗‘1) · (𝐵 ·_ih 𝐴)) + (1 · (𝐴 ·_ih 𝐵))) = ((𝐵 ·_ih 𝐴) + (𝐴 ·_ih 𝐵))
14		abs1 15507	. . . . . . . 8 ⊢ (abs‘1) = 1
15	2, 6, 7, 14	normlem7 31775	. . . . . . 7 ⊢ (((∗‘1) · (𝐵 ·_ih 𝐴)) + (1 · (𝐴 ·_ih 𝐵))) ≤ (2 · ((√‘(𝐴 ·_ih 𝐴)) · (√‘(𝐵 ·_ih 𝐵))))
16	13, 15	eqbrtrri 5134	. . . . . 6 ⊢ ((𝐵 ·_ih 𝐴) + (𝐴 ·_ih 𝐵)) ≤ (2 · ((√‘(𝐴 ·_ih 𝐴)) · (√‘(𝐵 ·_ih 𝐵))))
17		eqid 2763	. . . . . . . . . 10 ⊢ -(((∗‘1) · (𝐵 ·_ih 𝐴)) + (1 · (𝐴 ·_ih 𝐵))) = -(((∗‘1) · (𝐵 ·_ih 𝐴)) + (1 · (𝐴 ·_ih 𝐵)))
18	2, 6, 7, 17	normlem2 31770	. . . . . . . . 9 ⊢ -(((∗‘1) · (𝐵 ·_ih 𝐴)) + (1 · (𝐴 ·_ih 𝐵))) ∈ ℝ
19	2	cjcli 15379	. . . . . . . . . . . 12 ⊢ (∗‘1) ∈ ℂ
20	19, 8	mulcli 11355	. . . . . . . . . . 11 ⊢ ((∗‘1) · (𝐵 ·_ih 𝐴)) ∈ ℂ
21	2, 11	mulcli 11355	. . . . . . . . . . 11 ⊢ (1 · (𝐴 ·_ih 𝐵)) ∈ ℂ
22	20, 21	addcli 11354	. . . . . . . . . 10 ⊢ (((∗‘1) · (𝐵 ·_ih 𝐴)) + (1 · (𝐴 ·_ih 𝐵))) ∈ ℂ
23	22	negrebi 11671	. . . . . . . . 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 2861	. . . . . . 7 ⊢ ((𝐵 ·_ih 𝐴) + (𝐴 ·_ih 𝐵)) ∈ ℝ
26		2re 12457	. . . . . . . 8 ⊢ 2 ∈ ℝ
27		hiidge0 31757	. . . . . . . . . . 11 ⊢ (𝐴 ∈ ℋ → 0 ≤ (𝐴 ·_ih 𝐴))
28	7, 27	ax-mp 5	. . . . . . . . . 10 ⊢ 0 ≤ (𝐴 ·_ih 𝐴)
29		hiidrcl 31754	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ ℋ → (𝐴 ·_ih 𝐴) ∈ ℝ)
30	7, 29	ax-mp 5	. . . . . . . . . . 11 ⊢ (𝐴 ·_ih 𝐴) ∈ ℝ
31	30	sqrtcli 15583	. . . . . . . . . 10 ⊢ (0 ≤ (𝐴 ·_ih 𝐴) → (√‘(𝐴 ·_ih 𝐴)) ∈ ℝ)
32	28, 31	ax-mp 5	. . . . . . . . 9 ⊢ (√‘(𝐴 ·_ih 𝐴)) ∈ ℝ
33		hiidge0 31757	. . . . . . . . . . 11 ⊢ (𝐵 ∈ ℋ → 0 ≤ (𝐵 ·_ih 𝐵))
34	6, 33	ax-mp 5	. . . . . . . . . 10 ⊢ 0 ≤ (𝐵 ·_ih 𝐵)
35		hiidrcl 31754	. . . . . . . . . . . 12 ⊢ (𝐵 ∈ ℋ → (𝐵 ·_ih 𝐵) ∈ ℝ)
36	6, 35	ax-mp 5	. . . . . . . . . . 11 ⊢ (𝐵 ·_ih 𝐵) ∈ ℝ
37	36	sqrtcli 15583	. . . . . . . . . 10 ⊢ (0 ≤ (𝐵 ·_ih 𝐵) → (√‘(𝐵 ·_ih 𝐵)) ∈ ℝ)
38	34, 37	ax-mp 5	. . . . . . . . 9 ⊢ (√‘(𝐵 ·_ih 𝐵)) ∈ ℝ
39	32, 38	remulcli 11364	. . . . . . . 8 ⊢ ((√‘(𝐴 ·_ih 𝐴)) · (√‘(𝐵 ·_ih 𝐵))) ∈ ℝ
40	26, 39	remulcli 11364	. . . . . . 7 ⊢ (2 · ((√‘(𝐴 ·_ih 𝐴)) · (√‘(𝐵 ·_ih 𝐵)))) ∈ ℝ
41	30, 36	readdcli 11363	. . . . . . 7 ⊢ ((𝐴 ·_ih 𝐴) + (𝐵 ·_ih 𝐵)) ∈ ℝ
42	25, 40, 41	leadd2i 11912	. . . . . 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 31776	. . . . . 6 ⊢ ((𝐴 +_ℎ 𝐵) ·_ih (𝐴 +_ℎ 𝐵)) = (((𝐴 ·_ih 𝐴) + (𝐵 ·_ih 𝐵)) + ((𝐴 ·_ih 𝐵) + (𝐵 ·_ih 𝐴)))
45	11, 8	addcomi 11540	. . . . . . 7 ⊢ ((𝐴 ·_ih 𝐵) + (𝐵 ·_ih 𝐴)) = ((𝐵 ·_ih 𝐴) + (𝐴 ·_ih 𝐵))
46	45	oveq2i 7462	. . . . . 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 2786	. . . . 5 ⊢ ((𝐴 +_ℎ 𝐵) ·_ih (𝐴 +_ℎ 𝐵)) = (((𝐴 ·_ih 𝐴) + (𝐵 ·_ih 𝐵)) + ((𝐵 ·_ih 𝐴) + (𝐴 ·_ih 𝐵)))
48	32	recni 11362	. . . . . . 7 ⊢ (√‘(𝐴 ·_ih 𝐴)) ∈ ℂ
49	38	recni 11362	. . . . . . 7 ⊢ (√‘(𝐵 ·_ih 𝐵)) ∈ ℂ
50	48, 49	binom2i 14397	. . . . . 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 14364	. . . . . . 7 ⊢ ((√‘(𝐴 ·_ih 𝐴))↑2) ∈ ℂ
52		2cn 12458	. . . . . . . 8 ⊢ 2 ∈ ℂ
53	48, 49	mulcli 11355	. . . . . . . 8 ⊢ ((√‘(𝐴 ·_ih 𝐴)) · (√‘(𝐵 ·_ih 𝐵))) ∈ ℂ
54	52, 53	mulcli 11355	. . . . . . 7 ⊢ (2 · ((√‘(𝐴 ·_ih 𝐴)) · (√‘(𝐵 ·_ih 𝐵)))) ∈ ℂ
55	49	sqcli 14364	. . . . . . 7 ⊢ ((√‘(𝐵 ·_ih 𝐵))↑2) ∈ ℂ
56	51, 54, 55	add32i 11573	. . . . . 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 15587	. . . . . . . . 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 15587	. . . . . . . . 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 7463	. . . . . . 7 ⊢ (((√‘(𝐴 ·_ih 𝐴))↑2) + ((√‘(𝐵 ·_ih 𝐵))↑2)) = ((𝐴 ·_ih 𝐴) + (𝐵 ·_ih 𝐵))
62	61	oveq1i 7461	. . . . . 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 2790	. . . . 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 5141	. . . 4 ⊢ ((𝐴 +_ℎ 𝐵) ·_ih (𝐴 +_ℎ 𝐵)) ≤ (((√‘(𝐴 ·_ih 𝐴)) + (√‘(𝐵 ·_ih 𝐵)))↑2)
65	7, 6	hvaddcli 31677	. . . . . 6 ⊢ (𝐴 +_ℎ 𝐵) ∈ ℋ
66		hiidge0 31757	. . . . . 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 11363	. . . . . 6 ⊢ ((√‘(𝐴 ·_ih 𝐴)) + (√‘(𝐵 ·_ih 𝐵))) ∈ ℝ
69	68	sqge0i 14371	. . . . 5 ⊢ 0 ≤ (((√‘(𝐴 ·_ih 𝐴)) + (√‘(𝐵 ·_ih 𝐵)))↑2)
70		hiidrcl 31754	. . . . . . 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 14369	. . . . . 6 ⊢ (((√‘(𝐴 ·_ih 𝐴)) + (√‘(𝐵 ·_ih 𝐵)))↑2) ∈ ℝ
73	71, 72	sqrtlei 15600	. . . . 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 15588	. . . . . 6 ⊢ (0 ≤ (𝐴 ·_ih 𝐴) → 0 ≤ (√‘(𝐴 ·_ih 𝐴)))
77	28, 76	ax-mp 5	. . . . 5 ⊢ 0 ≤ (√‘(𝐴 ·_ih 𝐴))
78	36	sqrtge0i 15588	. . . . . 6 ⊢ (0 ≤ (𝐵 ·_ih 𝐵) → 0 ≤ (√‘(𝐵 ·_ih 𝐵)))
79	34, 78	ax-mp 5	. . . . 5 ⊢ 0 ≤ (√‘(𝐵 ·_ih 𝐵))
80	32, 38	addge0i 11896	. . . . 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 15586	. . . 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 5136	. 2 ⊢ (√‘((𝐴 +_ℎ 𝐵) ·_ih (𝐴 +_ℎ 𝐵))) ≤ ((√‘(𝐴 ·_ih 𝐴)) + (√‘(𝐵 ·_ih 𝐵)))
85		normval 31783	. . 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 31783	. . . 4 ⊢ (𝐴 ∈ ℋ → (norm_ℎ‘𝐴) = (√‘(𝐴 ·_ih 𝐴)))
88	7, 87	ax-mp 5	. . 3 ⊢ (norm_ℎ‘𝐴) = (√‘(𝐴 ·_ih 𝐴))
89		normval 31783	. . . 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 7463	. 2 ⊢ ((norm_ℎ‘𝐴) + (norm_ℎ‘𝐵)) = ((√‘(𝐴 ·_ih 𝐴)) + (√‘(𝐵 ·_ih 𝐵)))
92	84, 86, 91	3brtr4i 5141	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 1570 ∈ wcel 2147 class class class wbr 5108 ‘cfv 6566 (class class class)co 7451 ℝcr 11238 0cc0 11239 1c1 11240 + caddc 11242 · cmul 11244 ≤ cle 11383 -cneg 11581 2c2 12437 ↑cexp 14244 ∗ccj 15306 √csqrt 15443 ℋchba 31578 +_ℎ cva 31579 ·_ih csp 31581 norm_ℎcno 31582

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 1945 ax-6 2002 ax-7 2043 ax-8 2149 ax-9 2157 ax-10 2180 ax-11 2197 ax-12 2216 ax-ext 2735 ax-sep 5255 ax-nul 5266 ax-pow 5334 ax-pr 5398 ax-un 7783 ax-cnex 11295 ax-resscn 11296 ax-1cn 11297 ax-icn 11298 ax-addcl 11299 ax-addrcl 11300 ax-mulcl 11301 ax-mulrcl 11302 ax-mulcom 11303 ax-addass 11304 ax-mulass 11305 ax-distr 11306 ax-i2m1 11307 ax-1ne0 11308 ax-1rid 11309 ax-rnegex 11310 ax-rrecex 11311 ax-cnre 11312 ax-pre-lttri 11313 ax-pre-lttrn 11314 ax-pre-ltadd 11315 ax-pre-mulgt0 11316 ax-pre-sup 11317 ax-hfvadd 31659 ax-hv0cl 31662 ax-hfvmul 31664 ax-hvmulass 31666 ax-hvmul0 31669 ax-hfi 31738 ax-his1 31741 ax-his2 31742 ax-his3 31743 ax-his4 31744

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 2102 df-mo 2567 df-eu 2597 df-clab 2742 df-cleq 2755 df-clel 2839 df-nfc 2913 df-ne 2960 df-nel 3066 df-ral 3081 df-rex 3091 df-rmo 3369 df-reu 3370 df-rab 3417 df-v 3456 df-sbc 3743 df-csb 3852 df-ss 3901 df-pss 4017 df-dif 4050 df-in 4094 df-un 4154 df-nul 4282 df-if 4486 df-pw 4562 df-sn 4589 df-pr 4591 df-op 4841 df-uni 4873 df-iun 4958 df-br 5109 df-opab 5174 df-mpt 5193 df-tr 5219 df-id 5553 df-eprel 5558 df-po 5566 df-so 5567 df-fr 5611 df-we 5613 df-xp 5664 df-rel 5665 df-cnv 5666 df-co 5667 df-dm 5668 df-rn 5669 df-res 5670 df-ima 5671 df-pred 6330 df-ord 6391 df-on 6392 df-lim 6393 df-suc 6394 df-iota 6520 df-fun 6568 df-fn 6569 df-f 6570 df-f1 6571 df-fo 6572 df-f1o 6573 df-fv 6574 df-riota 7408 df-ov 7454 df-oprab 7455 df-mpo 7456 df-om 7912 df-2nd 8036 df-frecs 8337 df-wrecs 8368 df-recs 8417 df-rdg 8456 df-er 8755 df-en 9012 df-dom 9013 df-sdom 9014 df-sup 9471 df-pnf 11384 df-mnf 11385 df-xr 11386 df-ltxr 11387 df-le 11388 df-sub 11582 df-neg 11583 df-div 12014 df-nn 12376 df-2 12445 df-3 12446 df-4 12447 df-n0 12647 df-z 12734 df-uz 13006 df-rp 13161 df-seq 14185 df-exp 14245 df-cj 15309 df-re 15310 df-im 15311 df-sqrt 15445 df-abs 15446 df-hnorm 31627 df-hvsub 31630

This theorem is used by: norm-ii 31797 norm3difi 31806

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