Theorem cdj3lem1 32413

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 3927	. . . . . . . . . . . 12 ⊢ (𝑤 ∈ (𝐴 ∩ 𝐵) ↔ (𝑤 ∈ 𝐴 ∧ 𝑤 ∈ 𝐵))
2		cdj1.2	. . . . . . . . . . . . . 14 ⊢ 𝐵 ∈ Sℋ
3		neg1cn 12147	. . . . . . . . . . . . . 14 ⊢ -1 ∈ ℂ
4		shmulcl 31197	. . . . . . . . . . . . . 14 ⊢ ((𝐵 ∈ Sℋ ∧ -1 ∈ ℂ ∧ 𝑤 ∈ 𝐵) → (-1 ·ℎ 𝑤) ∈ 𝐵)
5	2, 3, 4	mp3an12 1453	. . . . . . . . . . . . 13 ⊢ (𝑤 ∈ 𝐵 → (-1 ·ℎ 𝑤) ∈ 𝐵)
6	5	anim2i 617	. . . . . . . . . . . 12 ⊢ ((𝑤 ∈ 𝐴 ∧ 𝑤 ∈ 𝐵) → (𝑤 ∈ 𝐴 ∧ (-1 ·ℎ 𝑤) ∈ 𝐵))
7	1, 6	sylbi 217	. . . . . . . . . . 11 ⊢ (𝑤 ∈ (𝐴 ∩ 𝐵) → (𝑤 ∈ 𝐴 ∧ (-1 ·ℎ 𝑤) ∈ 𝐵))
8		fveq2 6840	. . . . . . . . . . . . . 14 ⊢ (𝑦 = 𝑤 → (normℎ‘𝑦) = (normℎ‘𝑤))
9	8	oveq1d 7384	. . . . . . . . . . . . 13 ⊢ (𝑦 = 𝑤 → ((normℎ‘𝑦) + (normℎ‘𝑧)) = ((normℎ‘𝑤) + (normℎ‘𝑧)))
10		fvoveq1 7392	. . . . . . . . . . . . . 14 ⊢ (𝑦 = 𝑤 → (normℎ‘(𝑦 +ℎ 𝑧)) = (normℎ‘(𝑤 +ℎ 𝑧)))
11	10	oveq2d 7385	. . . . . . . . . . . . 13 ⊢ (𝑦 = 𝑤 → (𝑥 · (normℎ‘(𝑦 +ℎ 𝑧))) = (𝑥 · (normℎ‘(𝑤 +ℎ 𝑧))))
12	9, 11	breq12d 5115	. . . . . . . . . . . 12 ⊢ (𝑦 = 𝑤 → (((normℎ‘𝑦) + (normℎ‘𝑧)) ≤ (𝑥 · (normℎ‘(𝑦 +ℎ 𝑧))) ↔ ((normℎ‘𝑤) + (normℎ‘𝑧)) ≤ (𝑥 · (normℎ‘(𝑤 +ℎ 𝑧)))))
13		fveq2 6840	. . . . . . . . . . . . . 14 ⊢ (𝑧 = (-1 ·ℎ 𝑤) → (normℎ‘𝑧) = (normℎ‘(-1 ·ℎ 𝑤)))
14	13	oveq2d 7385	. . . . . . . . . . . . 13 ⊢ (𝑧 = (-1 ·ℎ 𝑤) → ((normℎ‘𝑤) + (normℎ‘𝑧)) = ((normℎ‘𝑤) + (normℎ‘(-1 ·ℎ 𝑤))))
15		oveq2 7377	. . . . . . . . . . . . . . 15 ⊢ (𝑧 = (-1 ·ℎ 𝑤) → (𝑤 +ℎ 𝑧) = (𝑤 +ℎ (-1 ·ℎ 𝑤)))
16	15	fveq2d 6844	. . . . . . . . . . . . . 14 ⊢ (𝑧 = (-1 ·ℎ 𝑤) → (normℎ‘(𝑤 +ℎ 𝑧)) = (normℎ‘(𝑤 +ℎ (-1 ·ℎ 𝑤))))
17	16	oveq2d 7385	. . . . . . . . . . . . 13 ⊢ (𝑧 = (-1 ·ℎ 𝑤) → (𝑥 · (normℎ‘(𝑤 +ℎ 𝑧))) = (𝑥 · (normℎ‘(𝑤 +ℎ (-1 ·ℎ 𝑤)))))
18	14, 17	breq12d 5115	. . . . . . . . . . . 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))))))
19	12, 18	rspc2v 3596	. . . . . . . . . . 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))))))
20	7, 19	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))))))
21	20	adantl 481	. . . . . . . . 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))))))
22		cdj1.1	. . . . . . . . . . . 12 ⊢ 𝐴 ∈ Sℋ
23	22, 2	shincli 31341	. . . . . . . . . . 11 ⊢ (𝐴 ∩ 𝐵) ∈ Sℋ
24	23	sheli 31193	. . . . . . . . . 10 ⊢ (𝑤 ∈ (𝐴 ∩ 𝐵) → 𝑤 ∈ ℋ)
25		normneg 31123	. . . . . . . . . . . . . . 15 ⊢ (𝑤 ∈ ℋ → (normℎ‘(-1 ·ℎ 𝑤)) = (normℎ‘𝑤))
26	25	oveq2d 7385	. . . . . . . . . . . . . 14 ⊢ (𝑤 ∈ ℋ → ((normℎ‘𝑤) + (normℎ‘(-1 ·ℎ 𝑤))) = ((normℎ‘𝑤) + (normℎ‘𝑤)))
27		normcl 31104	. . . . . . . . . . . . . . . 16 ⊢ (𝑤 ∈ ℋ → (normℎ‘𝑤) ∈ ℝ)
28	27	recnd 11178	. . . . . . . . . . . . . . 15 ⊢ (𝑤 ∈ ℋ → (normℎ‘𝑤) ∈ ℂ)
29	28	2timesd 12401	. . . . . . . . . . . . . 14 ⊢ (𝑤 ∈ ℋ → (2 · (normℎ‘𝑤)) = ((normℎ‘𝑤) + (normℎ‘𝑤)))
30	26, 29	eqtr4d 2767	. . . . . . . . . . . . 13 ⊢ (𝑤 ∈ ℋ → ((normℎ‘𝑤) + (normℎ‘(-1 ·ℎ 𝑤))) = (2 · (normℎ‘𝑤)))
31	30	adantl 481	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ ℝ ∧ 𝑤 ∈ ℋ) → ((normℎ‘𝑤) + (normℎ‘(-1 ·ℎ 𝑤))) = (2 · (normℎ‘𝑤)))
32		hvnegid 31006	. . . . . . . . . . . . . . . . 17 ⊢ (𝑤 ∈ ℋ → (𝑤 +ℎ (-1 ·ℎ 𝑤)) = 0ℎ)
33	32	fveq2d 6844	. . . . . . . . . . . . . . . 16 ⊢ (𝑤 ∈ ℋ → (normℎ‘(𝑤 +ℎ (-1 ·ℎ 𝑤))) = (normℎ‘0ℎ))
34		norm0 31107	. . . . . . . . . . . . . . . 16 ⊢ (normℎ‘0ℎ) = 0
35	33, 34	eqtrdi 2780	. . . . . . . . . . . . . . 15 ⊢ (𝑤 ∈ ℋ → (normℎ‘(𝑤 +ℎ (-1 ·ℎ 𝑤))) = 0)
36	35	oveq2d 7385	. . . . . . . . . . . . . 14 ⊢ (𝑤 ∈ ℋ → (𝑥 · (normℎ‘(𝑤 +ℎ (-1 ·ℎ 𝑤)))) = (𝑥 · 0))
37		recn 11134	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ∈ ℝ → 𝑥 ∈ ℂ)
38	37	mul01d 11349	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ ℝ → (𝑥 · 0) = 0)
39	36, 38	sylan9eqr 2786	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ ℝ ∧ 𝑤 ∈ ℋ) → (𝑥 · (normℎ‘(𝑤 +ℎ (-1 ·ℎ 𝑤)))) = 0)
40		2t0e0 12326	. . . . . . . . . . . . 13 ⊢ (2 · 0) = 0
41	39, 40	eqtr4di 2782	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ ℝ ∧ 𝑤 ∈ ℋ) → (𝑥 · (normℎ‘(𝑤 +ℎ (-1 ·ℎ 𝑤)))) = (2 · 0))
42	31, 41	breq12d 5115	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ ℝ ∧ 𝑤 ∈ ℋ) → (((normℎ‘𝑤) + (normℎ‘(-1 ·ℎ 𝑤))) ≤ (𝑥 · (normℎ‘(𝑤 +ℎ (-1 ·ℎ 𝑤)))) ↔ (2 · (normℎ‘𝑤)) ≤ (2 · 0)))
43		0re 11152	. . . . . . . . . . . . . . 15 ⊢ 0 ∈ ℝ
44		letri3 11235	. . . . . . . . . . . . . . 15 ⊢ (((normℎ‘𝑤) ∈ ℝ ∧ 0 ∈ ℝ) → ((normℎ‘𝑤) = 0 ↔ ((normℎ‘𝑤) ≤ 0 ∧ 0 ≤ (normℎ‘𝑤))))
45	27, 43, 44	sylancl 586	. . . . . . . . . . . . . 14 ⊢ (𝑤 ∈ ℋ → ((normℎ‘𝑤) = 0 ↔ ((normℎ‘𝑤) ≤ 0 ∧ 0 ≤ (normℎ‘𝑤))))
46		normge0 31105	. . . . . . . . . . . . . . 15 ⊢ (𝑤 ∈ ℋ → 0 ≤ (normℎ‘𝑤))
47	46	biantrud 531	. . . . . . . . . . . . . 14 ⊢ (𝑤 ∈ ℋ → ((normℎ‘𝑤) ≤ 0 ↔ ((normℎ‘𝑤) ≤ 0 ∧ 0 ≤ (normℎ‘𝑤))))
48		2re 12236	. . . . . . . . . . . . . . . . 17 ⊢ 2 ∈ ℝ
49		2pos 12265	. . . . . . . . . . . . . . . . 17 ⊢ 0 < 2
50	48, 49	pm3.2i 470	. . . . . . . . . . . . . . . 16 ⊢ (2 ∈ ℝ ∧ 0 < 2)
51		lemul2 12011	. . . . . . . . . . . . . . . 16 ⊢ (((normℎ‘𝑤) ∈ ℝ ∧ 0 ∈ ℝ ∧ (2 ∈ ℝ ∧ 0 < 2)) → ((normℎ‘𝑤) ≤ 0 ↔ (2 · (normℎ‘𝑤)) ≤ (2 · 0)))
52	43, 50, 51	mp3an23 1455	. . . . . . . . . . . . . . 15 ⊢ ((normℎ‘𝑤) ∈ ℝ → ((normℎ‘𝑤) ≤ 0 ↔ (2 · (normℎ‘𝑤)) ≤ (2 · 0)))
53	27, 52	syl 17	. . . . . . . . . . . . . 14 ⊢ (𝑤 ∈ ℋ → ((normℎ‘𝑤) ≤ 0 ↔ (2 · (normℎ‘𝑤)) ≤ (2 · 0)))
54	45, 47, 53	3bitr2rd 308	. . . . . . . . . . . . 13 ⊢ (𝑤 ∈ ℋ → ((2 · (normℎ‘𝑤)) ≤ (2 · 0) ↔ (normℎ‘𝑤) = 0))
55		norm-i 31108	. . . . . . . . . . . . 13 ⊢ (𝑤 ∈ ℋ → ((normℎ‘𝑤) = 0 ↔ 𝑤 = 0ℎ))
56	54, 55	bitrd 279	. . . . . . . . . . . 12 ⊢ (𝑤 ∈ ℋ → ((2 · (normℎ‘𝑤)) ≤ (2 · 0) ↔ 𝑤 = 0ℎ))
57	56	adantl 481	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ ℝ ∧ 𝑤 ∈ ℋ) → ((2 · (normℎ‘𝑤)) ≤ (2 · 0) ↔ 𝑤 = 0ℎ))
58	42, 57	bitrd 279	. . . . . . . . . 10 ⊢ ((𝑥 ∈ ℝ ∧ 𝑤 ∈ ℋ) → (((normℎ‘𝑤) + (normℎ‘(-1 ·ℎ 𝑤))) ≤ (𝑥 · (normℎ‘(𝑤 +ℎ (-1 ·ℎ 𝑤)))) ↔ 𝑤 = 0ℎ))
59	24, 58	sylan2 593	. . . . . . . . 9 ⊢ ((𝑥 ∈ ℝ ∧ 𝑤 ∈ (𝐴 ∩ 𝐵)) → (((normℎ‘𝑤) + (normℎ‘(-1 ·ℎ 𝑤))) ≤ (𝑥 · (normℎ‘(𝑤 +ℎ (-1 ·ℎ 𝑤)))) ↔ 𝑤 = 0ℎ))
60	21, 59	sylibd 239	. . . . . . . 8 ⊢ ((𝑥 ∈ ℝ ∧ 𝑤 ∈ (𝐴 ∩ 𝐵)) → (∀𝑦 ∈ 𝐴 ∀𝑧 ∈ 𝐵 ((normℎ‘𝑦) + (normℎ‘𝑧)) ≤ (𝑥 · (normℎ‘(𝑦 +ℎ 𝑧))) → 𝑤 = 0ℎ))
61	60	impancom 451	. . . . . . 7 ⊢ ((𝑥 ∈ ℝ ∧ ∀𝑦 ∈ 𝐴 ∀𝑧 ∈ 𝐵 ((normℎ‘𝑦) + (normℎ‘𝑧)) ≤ (𝑥 · (normℎ‘(𝑦 +ℎ 𝑧)))) → (𝑤 ∈ (𝐴 ∩ 𝐵) → 𝑤 = 0ℎ))
62		elch0 31233	. . . . . . 7 ⊢ (𝑤 ∈ 0ℋ ↔ 𝑤 = 0ℎ)
63	61, 62	imbitrrdi 252	. . . . . 6 ⊢ ((𝑥 ∈ ℝ ∧ ∀𝑦 ∈ 𝐴 ∀𝑧 ∈ 𝐵 ((normℎ‘𝑦) + (normℎ‘𝑧)) ≤ (𝑥 · (normℎ‘(𝑦 +ℎ 𝑧)))) → (𝑤 ∈ (𝐴 ∩ 𝐵) → 𝑤 ∈ 0ℋ))
64	63	ssrdv 3949	. . . . 5 ⊢ ((𝑥 ∈ ℝ ∧ ∀𝑦 ∈ 𝐴 ∀𝑧 ∈ 𝐵 ((normℎ‘𝑦) + (normℎ‘𝑧)) ≤ (𝑥 · (normℎ‘(𝑦 +ℎ 𝑧)))) → (𝐴 ∩ 𝐵) ⊆ 0ℋ)
65	64	ex 412	. . . 4 ⊢ (𝑥 ∈ ℝ → (∀𝑦 ∈ 𝐴 ∀𝑧 ∈ 𝐵 ((normℎ‘𝑦) + (normℎ‘𝑧)) ≤ (𝑥 · (normℎ‘(𝑦 +ℎ 𝑧))) → (𝐴 ∩ 𝐵) ⊆ 0ℋ))
66		shle0 31421	. . . . 5 ⊢ ((𝐴 ∩ 𝐵) ∈ Sℋ → ((𝐴 ∩ 𝐵) ⊆ 0ℋ ↔ (𝐴 ∩ 𝐵) = 0ℋ))
67	23, 66	ax-mp 5	. . . 4 ⊢ ((𝐴 ∩ 𝐵) ⊆ 0ℋ ↔ (𝐴 ∩ 𝐵) = 0ℋ)
68	65, 67	imbitrdi 251	. . 3 ⊢ (𝑥 ∈ ℝ → (∀𝑦 ∈ 𝐴 ∀𝑧 ∈ 𝐵 ((normℎ‘𝑦) + (normℎ‘𝑧)) ≤ (𝑥 · (normℎ‘(𝑦 +ℎ 𝑧))) → (𝐴 ∩ 𝐵) = 0ℋ))
69	68	adantld 490	. 2 ⊢ (𝑥 ∈ ℝ → ((0 < 𝑥 ∧ ∀𝑦 ∈ 𝐴 ∀𝑧 ∈ 𝐵 ((normℎ‘𝑦) + (normℎ‘𝑧)) ≤ (𝑥 · (normℎ‘(𝑦 +ℎ 𝑧)))) → (𝐴 ∩ 𝐵) = 0ℋ))
70	69	rexlimiv 3127	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 206   ∧ wa 395   = wceq 1540   ∈ wcel 2109  ∀wral 3044  ∃wrex 3053   ∩ cin 3910   ⊆ wss 3911   class class class wbr 5102  ‘cfv 6499  (class class class)co 7369  ℂcc 11042  ℝcr 11043  0cc0 11044  1c1 11045   + caddc 11047   · cmul 11049   < clt 11184   ≤ cle 11185  -cneg 11382  2c2 12217   ℋchba 30898   +_ℎ cva 30899   ·_ℎ csm 30900  norm_ℎcno 30902  0_ℎc0v 30903   S_ℋ csh 30907  0_ℋc0h 30914

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-sep 5246  ax-nul 5256  ax-pow 5315  ax-pr 5382  ax-un 7691  ax-cnex 11100  ax-resscn 11101  ax-1cn 11102  ax-icn 11103  ax-addcl 11104  ax-addrcl 11105  ax-mulcl 11106  ax-mulrcl 11107  ax-mulcom 11108  ax-addass 11109  ax-mulass 11110  ax-distr 11111  ax-i2m1 11112  ax-1ne0 11113  ax-1rid 11114  ax-rnegex 11115  ax-rrecex 11116  ax-cnre 11117  ax-pre-lttri 11118  ax-pre-lttrn 11119  ax-pre-ltadd 11120  ax-pre-mulgt0 11121  ax-pre-sup 11122  ax-hilex 30978  ax-hfvadd 30979  ax-hvcom 30980  ax-hv0cl 30982  ax-hvaddid 30983  ax-hfvmul 30984  ax-hvmulid 30985  ax-hvmulass 30986  ax-hvdistr1 30987  ax-hvdistr2 30988  ax-hvmul0 30989  ax-hfi 31058  ax-his1 31061  ax-his3 31063  ax-his4 31064

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-nel 3030  df-ral 3045  df-rex 3054  df-rmo 3351  df-reu 3352  df-rab 3403  df-v 3446  df-sbc 3751  df-csb 3860  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-pss 3931  df-nul 4293  df-if 4485  df-pw 4561  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4868  df-int 4907  df-iun 4953  df-br 5103  df-opab 5165  df-mpt 5184  df-tr 5210  df-id 5526  df-eprel 5531  df-po 5539  df-so 5540  df-fr 5584  df-we 5586  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-pred 6262  df-ord 6323  df-on 6324  df-lim 6325  df-suc 6326  df-iota 6452  df-fun 6501  df-fn 6502  df-f 6503  df-f1 6504  df-fo 6505  df-f1o 6506  df-fv 6507  df-riota 7326  df-ov 7372  df-oprab 7373  df-mpo 7374  df-om 7823  df-2nd 7948  df-frecs 8237  df-wrecs 8268  df-recs 8317  df-rdg 8355  df-er 8648  df-en 8896  df-dom 8897  df-sdom 8898  df-sup 9369  df-pnf 11186  df-mnf 11187  df-xr 11188  df-ltxr 11189  df-le 11190  df-sub 11383  df-neg 11384  df-div 11812  df-nn 12163  df-2 12225  df-3 12226  df-n0 12419  df-z 12506  df-uz 12770  df-rp 12928  df-seq 13943  df-exp 14003  df-cj 15041  df-re 15042  df-im 15043  df-sqrt 15177  df-abs 15178  df-hnorm 30947  df-hvsub 30950  df-sh 31186  df-ch0 31232

This theorem is referenced by:  cdj3lem2b  32416  cdj3i  32420