Theorem cdj3lem1 29293

Description: A property of "𝐴 and 𝐵 are completely disjoint subspaces." Part of Lemma 5 of [Holland] p. 1520. (Contributed by NM, 23-May-2005.) (New usage is discouraged.)

Hypotheses

Ref	Expression
cdj1.1	⊢ 𝐴 ∈ Sℋ
cdj1.2	⊢ 𝐵 ∈ Sℋ

Assertion

Ref	Expression
cdj3lem1	⊢ (∃𝑥 ∈ ℝ (0 < 𝑥 ∧ ∀𝑦 ∈ 𝐴 ∀𝑧 ∈ 𝐵 ((normℎ‘𝑦) + (normℎ‘𝑧)) ≤ (𝑥 · (normℎ‘(𝑦 +ℎ 𝑧)))) → (𝐴 ∩ 𝐵) = 0ℋ)

Distinct variable groups:   𝑥,𝑦,𝑧,𝐴   𝑥,𝐵,𝑦,𝑧

Proof of Theorem cdj3lem1

Dummy variable 𝑤 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		elin 3796	. . . . . . . . . . . 12 ⊢ (𝑤 ∈ (𝐴 ∩ 𝐵) ↔ (𝑤 ∈ 𝐴 ∧ 𝑤 ∈ 𝐵))
2		cdj1.2	. . . . . . . . . . . . . 14 ⊢ 𝐵 ∈ Sℋ
3		neg1cn 11124	. . . . . . . . . . . . . 14 ⊢ -1 ∈ ℂ
4		shmulcl 28075	. . . . . . . . . . . . . 14 ⊢ ((𝐵 ∈ Sℋ ∧ -1 ∈ ℂ ∧ 𝑤 ∈ 𝐵) → (-1 ·ℎ 𝑤) ∈ 𝐵)
5	2, 3, 4	mp3an12 1414	. . . . . . . . . . . . 13 ⊢ (𝑤 ∈ 𝐵 → (-1 ·ℎ 𝑤) ∈ 𝐵)
6	5	anim2i 593	. . . . . . . . . . . 12 ⊢ ((𝑤 ∈ 𝐴 ∧ 𝑤 ∈ 𝐵) → (𝑤 ∈ 𝐴 ∧ (-1 ·ℎ 𝑤) ∈ 𝐵))
7	1, 6	sylbi 207	. . . . . . . . . . 11 ⊢ (𝑤 ∈ (𝐴 ∩ 𝐵) → (𝑤 ∈ 𝐴 ∧ (-1 ·ℎ 𝑤) ∈ 𝐵))
8		fveq2 6191	. . . . . . . . . . . . . 14 ⊢ (𝑦 = 𝑤 → (normℎ‘𝑦) = (normℎ‘𝑤))
9	8	oveq1d 6665	. . . . . . . . . . . . 13 ⊢ (𝑦 = 𝑤 → ((normℎ‘𝑦) + (normℎ‘𝑧)) = ((normℎ‘𝑤) + (normℎ‘𝑧)))
10		oveq1 6657	. . . . . . . . . . . . . . 15 ⊢ (𝑦 = 𝑤 → (𝑦 +ℎ 𝑧) = (𝑤 +ℎ 𝑧))
11	10	fveq2d 6195	. . . . . . . . . . . . . 14 ⊢ (𝑦 = 𝑤 → (normℎ‘(𝑦 +ℎ 𝑧)) = (normℎ‘(𝑤 +ℎ 𝑧)))
12	11	oveq2d 6666	. . . . . . . . . . . . 13 ⊢ (𝑦 = 𝑤 → (𝑥 · (normℎ‘(𝑦 +ℎ 𝑧))) = (𝑥 · (normℎ‘(𝑤 +ℎ 𝑧))))
13	9, 12	breq12d 4666	. . . . . . . . . . . 12 ⊢ (𝑦 = 𝑤 → (((normℎ‘𝑦) + (normℎ‘𝑧)) ≤ (𝑥 · (normℎ‘(𝑦 +ℎ 𝑧))) ↔ ((normℎ‘𝑤) + (normℎ‘𝑧)) ≤ (𝑥 · (normℎ‘(𝑤 +ℎ 𝑧)))))
14		fveq2 6191	. . . . . . . . . . . . . 14 ⊢ (𝑧 = (-1 ·ℎ 𝑤) → (normℎ‘𝑧) = (normℎ‘(-1 ·ℎ 𝑤)))
15	14	oveq2d 6666	. . . . . . . . . . . . 13 ⊢ (𝑧 = (-1 ·ℎ 𝑤) → ((normℎ‘𝑤) + (normℎ‘𝑧)) = ((normℎ‘𝑤) + (normℎ‘(-1 ·ℎ 𝑤))))
16		oveq2 6658	. . . . . . . . . . . . . . 15 ⊢ (𝑧 = (-1 ·ℎ 𝑤) → (𝑤 +ℎ 𝑧) = (𝑤 +ℎ (-1 ·ℎ 𝑤)))
17	16	fveq2d 6195	. . . . . . . . . . . . . 14 ⊢ (𝑧 = (-1 ·ℎ 𝑤) → (normℎ‘(𝑤 +ℎ 𝑧)) = (normℎ‘(𝑤 +ℎ (-1 ·ℎ 𝑤))))
18	17	oveq2d 6666	. . . . . . . . . . . . 13 ⊢ (𝑧 = (-1 ·ℎ 𝑤) → (𝑥 · (normℎ‘(𝑤 +ℎ 𝑧))) = (𝑥 · (normℎ‘(𝑤 +ℎ (-1 ·ℎ 𝑤)))))
19	15, 18	breq12d 4666	. . . . . . . . . . . 12 ⊢ (𝑧 = (-1 ·ℎ 𝑤) → (((normℎ‘𝑤) + (normℎ‘𝑧)) ≤ (𝑥 · (normℎ‘(𝑤 +ℎ 𝑧))) ↔ ((normℎ‘𝑤) + (normℎ‘(-1 ·ℎ 𝑤))) ≤ (𝑥 · (normℎ‘(𝑤 +ℎ (-1 ·ℎ 𝑤))))))
20	13, 19	rspc2v 3322	. . . . . . . . . . 11 ⊢ ((𝑤 ∈ 𝐴 ∧ (-1 ·ℎ 𝑤) ∈ 𝐵) → (∀𝑦 ∈ 𝐴 ∀𝑧 ∈ 𝐵 ((normℎ‘𝑦) + (normℎ‘𝑧)) ≤ (𝑥 · (normℎ‘(𝑦 +ℎ 𝑧))) → ((normℎ‘𝑤) + (normℎ‘(-1 ·ℎ 𝑤))) ≤ (𝑥 · (normℎ‘(𝑤 +ℎ (-1 ·ℎ 𝑤))))))
21	7, 20	syl 17	. . . . . . . . . 10 ⊢ (𝑤 ∈ (𝐴 ∩ 𝐵) → (∀𝑦 ∈ 𝐴 ∀𝑧 ∈ 𝐵 ((normℎ‘𝑦) + (normℎ‘𝑧)) ≤ (𝑥 · (normℎ‘(𝑦 +ℎ 𝑧))) → ((normℎ‘𝑤) + (normℎ‘(-1 ·ℎ 𝑤))) ≤ (𝑥 · (normℎ‘(𝑤 +ℎ (-1 ·ℎ 𝑤))))))
22	21	adantl 482	. . . . . . . . 9 ⊢ ((𝑥 ∈ ℝ ∧ 𝑤 ∈ (𝐴 ∩ 𝐵)) → (∀𝑦 ∈ 𝐴 ∀𝑧 ∈ 𝐵 ((normℎ‘𝑦) + (normℎ‘𝑧)) ≤ (𝑥 · (normℎ‘(𝑦 +ℎ 𝑧))) → ((normℎ‘𝑤) + (normℎ‘(-1 ·ℎ 𝑤))) ≤ (𝑥 · (normℎ‘(𝑤 +ℎ (-1 ·ℎ 𝑤))))))
23		cdj1.1	. . . . . . . . . . . 12 ⊢ 𝐴 ∈ Sℋ
24	23, 2	shincli 28221	. . . . . . . . . . 11 ⊢ (𝐴 ∩ 𝐵) ∈ Sℋ
25	24	sheli 28071	. . . . . . . . . 10 ⊢ (𝑤 ∈ (𝐴 ∩ 𝐵) → 𝑤 ∈ ℋ)
26		normneg 28001	. . . . . . . . . . . . . . 15 ⊢ (𝑤 ∈ ℋ → (normℎ‘(-1 ·ℎ 𝑤)) = (normℎ‘𝑤))
27	26	oveq2d 6666	. . . . . . . . . . . . . 14 ⊢ (𝑤 ∈ ℋ → ((normℎ‘𝑤) + (normℎ‘(-1 ·ℎ 𝑤))) = ((normℎ‘𝑤) + (normℎ‘𝑤)))
28		normcl 27982	. . . . . . . . . . . . . . . 16 ⊢ (𝑤 ∈ ℋ → (normℎ‘𝑤) ∈ ℝ)
29	28	recnd 10068	. . . . . . . . . . . . . . 15 ⊢ (𝑤 ∈ ℋ → (normℎ‘𝑤) ∈ ℂ)
30	29	2timesd 11275	. . . . . . . . . . . . . 14 ⊢ (𝑤 ∈ ℋ → (2 · (normℎ‘𝑤)) = ((normℎ‘𝑤) + (normℎ‘𝑤)))
31	27, 30	eqtr4d 2659	. . . . . . . . . . . . 13 ⊢ (𝑤 ∈ ℋ → ((normℎ‘𝑤) + (normℎ‘(-1 ·ℎ 𝑤))) = (2 · (normℎ‘𝑤)))
32	31	adantl 482	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ ℝ ∧ 𝑤 ∈ ℋ) → ((normℎ‘𝑤) + (normℎ‘(-1 ·ℎ 𝑤))) = (2 · (normℎ‘𝑤)))
33		hvnegid 27884	. . . . . . . . . . . . . . . . 17 ⊢ (𝑤 ∈ ℋ → (𝑤 +ℎ (-1 ·ℎ 𝑤)) = 0ℎ)
34	33	fveq2d 6195	. . . . . . . . . . . . . . . 16 ⊢ (𝑤 ∈ ℋ → (normℎ‘(𝑤 +ℎ (-1 ·ℎ 𝑤))) = (normℎ‘0ℎ))
35		norm0 27985	. . . . . . . . . . . . . . . 16 ⊢ (normℎ‘0ℎ) = 0
36	34, 35	syl6eq 2672	. . . . . . . . . . . . . . 15 ⊢ (𝑤 ∈ ℋ → (normℎ‘(𝑤 +ℎ (-1 ·ℎ 𝑤))) = 0)
37	36	oveq2d 6666	. . . . . . . . . . . . . 14 ⊢ (𝑤 ∈ ℋ → (𝑥 · (normℎ‘(𝑤 +ℎ (-1 ·ℎ 𝑤)))) = (𝑥 · 0))
38		recn 10026	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ∈ ℝ → 𝑥 ∈ ℂ)
39	38	mul01d 10235	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ ℝ → (𝑥 · 0) = 0)
40	37, 39	sylan9eqr 2678	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ ℝ ∧ 𝑤 ∈ ℋ) → (𝑥 · (normℎ‘(𝑤 +ℎ (-1 ·ℎ 𝑤)))) = 0)
41		2t0e0 11183	. . . . . . . . . . . . 13 ⊢ (2 · 0) = 0
42	40, 41	syl6eqr 2674	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ ℝ ∧ 𝑤 ∈ ℋ) → (𝑥 · (normℎ‘(𝑤 +ℎ (-1 ·ℎ 𝑤)))) = (2 · 0))
43	32, 42	breq12d 4666	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ ℝ ∧ 𝑤 ∈ ℋ) → (((normℎ‘𝑤) + (normℎ‘(-1 ·ℎ 𝑤))) ≤ (𝑥 · (normℎ‘(𝑤 +ℎ (-1 ·ℎ 𝑤)))) ↔ (2 · (normℎ‘𝑤)) ≤ (2 · 0)))
44		0re 10040	. . . . . . . . . . . . . . 15 ⊢ 0 ∈ ℝ
45		letri3 10123	. . . . . . . . . . . . . . 15 ⊢ (((normℎ‘𝑤) ∈ ℝ ∧ 0 ∈ ℝ) → ((normℎ‘𝑤) = 0 ↔ ((normℎ‘𝑤) ≤ 0 ∧ 0 ≤ (normℎ‘𝑤))))
46	28, 44, 45	sylancl 694	. . . . . . . . . . . . . 14 ⊢ (𝑤 ∈ ℋ → ((normℎ‘𝑤) = 0 ↔ ((normℎ‘𝑤) ≤ 0 ∧ 0 ≤ (normℎ‘𝑤))))
47		normge0 27983	. . . . . . . . . . . . . . 15 ⊢ (𝑤 ∈ ℋ → 0 ≤ (normℎ‘𝑤))
48	47	biantrud 528	. . . . . . . . . . . . . 14 ⊢ (𝑤 ∈ ℋ → ((normℎ‘𝑤) ≤ 0 ↔ ((normℎ‘𝑤) ≤ 0 ∧ 0 ≤ (normℎ‘𝑤))))
49		2re 11090	. . . . . . . . . . . . . . . . 17 ⊢ 2 ∈ ℝ
50		2pos 11112	. . . . . . . . . . . . . . . . 17 ⊢ 0 < 2
51	49, 50	pm3.2i 471	. . . . . . . . . . . . . . . 16 ⊢ (2 ∈ ℝ ∧ 0 < 2)
52		lemul2 10876	. . . . . . . . . . . . . . . 16 ⊢ (((normℎ‘𝑤) ∈ ℝ ∧ 0 ∈ ℝ ∧ (2 ∈ ℝ ∧ 0 < 2)) → ((normℎ‘𝑤) ≤ 0 ↔ (2 · (normℎ‘𝑤)) ≤ (2 · 0)))
53	44, 51, 52	mp3an23 1416	. . . . . . . . . . . . . . 15 ⊢ ((normℎ‘𝑤) ∈ ℝ → ((normℎ‘𝑤) ≤ 0 ↔ (2 · (normℎ‘𝑤)) ≤ (2 · 0)))
54	28, 53	syl 17	. . . . . . . . . . . . . 14 ⊢ (𝑤 ∈ ℋ → ((normℎ‘𝑤) ≤ 0 ↔ (2 · (normℎ‘𝑤)) ≤ (2 · 0)))
55	46, 48, 54	3bitr2rd 297	. . . . . . . . . . . . 13 ⊢ (𝑤 ∈ ℋ → ((2 · (normℎ‘𝑤)) ≤ (2 · 0) ↔ (normℎ‘𝑤) = 0))
56		norm-i 27986	. . . . . . . . . . . . 13 ⊢ (𝑤 ∈ ℋ → ((normℎ‘𝑤) = 0 ↔ 𝑤 = 0ℎ))
57	55, 56	bitrd 268	. . . . . . . . . . . 12 ⊢ (𝑤 ∈ ℋ → ((2 · (normℎ‘𝑤)) ≤ (2 · 0) ↔ 𝑤 = 0ℎ))
58	57	adantl 482	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ ℝ ∧ 𝑤 ∈ ℋ) → ((2 · (normℎ‘𝑤)) ≤ (2 · 0) ↔ 𝑤 = 0ℎ))
59	43, 58	bitrd 268	. . . . . . . . . 10 ⊢ ((𝑥 ∈ ℝ ∧ 𝑤 ∈ ℋ) → (((normℎ‘𝑤) + (normℎ‘(-1 ·ℎ 𝑤))) ≤ (𝑥 · (normℎ‘(𝑤 +ℎ (-1 ·ℎ 𝑤)))) ↔ 𝑤 = 0ℎ))
60	25, 59	sylan2 491	. . . . . . . . 9 ⊢ ((𝑥 ∈ ℝ ∧ 𝑤 ∈ (𝐴 ∩ 𝐵)) → (((normℎ‘𝑤) + (normℎ‘(-1 ·ℎ 𝑤))) ≤ (𝑥 · (normℎ‘(𝑤 +ℎ (-1 ·ℎ 𝑤)))) ↔ 𝑤 = 0ℎ))
61	22, 60	sylibd 229	. . . . . . . 8 ⊢ ((𝑥 ∈ ℝ ∧ 𝑤 ∈ (𝐴 ∩ 𝐵)) → (∀𝑦 ∈ 𝐴 ∀𝑧 ∈ 𝐵 ((normℎ‘𝑦) + (normℎ‘𝑧)) ≤ (𝑥 · (normℎ‘(𝑦 +ℎ 𝑧))) → 𝑤 = 0ℎ))
62	61	impancom 456	. . . . . . 7 ⊢ ((𝑥 ∈ ℝ ∧ ∀𝑦 ∈ 𝐴 ∀𝑧 ∈ 𝐵 ((normℎ‘𝑦) + (normℎ‘𝑧)) ≤ (𝑥 · (normℎ‘(𝑦 +ℎ 𝑧)))) → (𝑤 ∈ (𝐴 ∩ 𝐵) → 𝑤 = 0ℎ))
63		elch0 28111	. . . . . . 7 ⊢ (𝑤 ∈ 0ℋ ↔ 𝑤 = 0ℎ)
64	62, 63	syl6ibr 242	. . . . . 6 ⊢ ((𝑥 ∈ ℝ ∧ ∀𝑦 ∈ 𝐴 ∀𝑧 ∈ 𝐵 ((normℎ‘𝑦) + (normℎ‘𝑧)) ≤ (𝑥 · (normℎ‘(𝑦 +ℎ 𝑧)))) → (𝑤 ∈ (𝐴 ∩ 𝐵) → 𝑤 ∈ 0ℋ))
65	64	ssrdv 3609	. . . . 5 ⊢ ((𝑥 ∈ ℝ ∧ ∀𝑦 ∈ 𝐴 ∀𝑧 ∈ 𝐵 ((normℎ‘𝑦) + (normℎ‘𝑧)) ≤ (𝑥 · (normℎ‘(𝑦 +ℎ 𝑧)))) → (𝐴 ∩ 𝐵) ⊆ 0ℋ)
66	65	ex 450	. . . 4 ⊢ (𝑥 ∈ ℝ → (∀𝑦 ∈ 𝐴 ∀𝑧 ∈ 𝐵 ((normℎ‘𝑦) + (normℎ‘𝑧)) ≤ (𝑥 · (normℎ‘(𝑦 +ℎ 𝑧))) → (𝐴 ∩ 𝐵) ⊆ 0ℋ))
67		shle0 28301	. . . . 5 ⊢ ((𝐴 ∩ 𝐵) ∈ Sℋ → ((𝐴 ∩ 𝐵) ⊆ 0ℋ ↔ (𝐴 ∩ 𝐵) = 0ℋ))
68	24, 67	ax-mp 5	. . . 4 ⊢ ((𝐴 ∩ 𝐵) ⊆ 0ℋ ↔ (𝐴 ∩ 𝐵) = 0ℋ)
69	66, 68	syl6ib 241	. . 3 ⊢ (𝑥 ∈ ℝ → (∀𝑦 ∈ 𝐴 ∀𝑧 ∈ 𝐵 ((normℎ‘𝑦) + (normℎ‘𝑧)) ≤ (𝑥 · (normℎ‘(𝑦 +ℎ 𝑧))) → (𝐴 ∩ 𝐵) = 0ℋ))
70	69	adantld 483	. 2 ⊢ (𝑥 ∈ ℝ → ((0 < 𝑥 ∧ ∀𝑦 ∈ 𝐴 ∀𝑧 ∈ 𝐵 ((normℎ‘𝑦) + (normℎ‘𝑧)) ≤ (𝑥 · (normℎ‘(𝑦 +ℎ 𝑧)))) → (𝐴 ∩ 𝐵) = 0ℋ))
71	70	rexlimiv 3027	1 ⊢ (∃𝑥 ∈ ℝ (0 < 𝑥 ∧ ∀𝑦 ∈ 𝐴 ∀𝑧 ∈ 𝐵 ((normℎ‘𝑦) + (normℎ‘𝑧)) ≤ (𝑥 · (normℎ‘(𝑦 +ℎ 𝑧)))) → (𝐴 ∩ 𝐵) = 0ℋ)

Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 384   = wceq 1483   ∈ wcel 1990  ∀wral 2912  ∃wrex 2913   ∩ cin 3573   ⊆ wss 3574   class class class wbr 4653  ‘cfv 5888  (class class class)co 6650  ℂcc 9934  ℝcr 9935  0cc0 9936  1c1 9937   + caddc 9939   · cmul 9941   < clt 10074   ≤ cle 10075  -cneg 10267  2c2 11070   ℋchil 27776   +_ℎ cva 27777   ·_ℎ csm 27778  norm_ℎcno 27780  0_ℎc0v 27781   S_ℋ csh 27785  0_ℋc0h 27792

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-hilex 27856  ax-hfvadd 27857  ax-hvcom 27858  ax-hv0cl 27860  ax-hvaddid 27861  ax-hfvmul 27862  ax-hvmulid 27863  ax-hvmulass 27864  ax-hvdistr1 27865  ax-hvdistr2 27866  ax-hvmul0 27867  ax-hfi 27936  ax-his1 27939  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-int 4476  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-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  df-sh 28064  df-ch0 28110

This theorem is referenced by:  cdj3lem2b  29296  cdj3i  29300