Theorem cdj3lem3b 29299

Description: Lemma for cdj3i 29300. The second-component function 𝑇 is bounded if the subspaces are completely disjoint. (Contributed by NM, 31-May-2005.) (New usage is discouraged.)

Hypotheses

Ref	Expression
cdj3lem2.1	⊢ 𝐴 ∈ Sℋ
cdj3lem2.2	⊢ 𝐵 ∈ Sℋ
cdj3lem3.3	⊢ 𝑇 = (𝑥 ∈ (𝐴 +ℋ 𝐵) ↦ (℩𝑤 ∈ 𝐵 ∃𝑧 ∈ 𝐴 𝑥 = (𝑧 +ℎ 𝑤)))

Assertion

Ref	Expression
cdj3lem3b	⊢ (∃𝑣 ∈ ℝ (0 < 𝑣 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 ((normℎ‘𝑥) + (normℎ‘𝑦)) ≤ (𝑣 · (normℎ‘(𝑥 +ℎ 𝑦)))) → ∃𝑣 ∈ ℝ (0 < 𝑣 ∧ ∀𝑢 ∈ (𝐴 +ℋ 𝐵)(normℎ‘(𝑇‘𝑢)) ≤ (𝑣 · (normℎ‘𝑢))))

Distinct variable groups:   𝑥,𝑦,𝑧,𝑤,𝑣,𝑢,𝐴   𝑥,𝐵,𝑦,𝑧,𝑤,𝑣,𝑢   𝑣,𝑇,𝑢

Allowed substitution hints:   𝑇(𝑥,𝑦,𝑧,𝑤)

Proof of Theorem cdj3lem3b

Dummy variables 𝑡 ℎ are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		cdj3lem2.2	. . 3 ⊢ 𝐵 ∈ Sℋ
2		cdj3lem2.1	. . 3 ⊢ 𝐴 ∈ Sℋ
3		cdj3lem3.3	. . . 4 ⊢ 𝑇 = (𝑥 ∈ (𝐴 +ℋ 𝐵) ↦ (℩𝑤 ∈ 𝐵 ∃𝑧 ∈ 𝐴 𝑥 = (𝑧 +ℎ 𝑤)))
4	1, 2	shscomi 28222	. . . . 5 ⊢ (𝐵 +ℋ 𝐴) = (𝐴 +ℋ 𝐵)
5	1	sheli 28071	. . . . . . . . 9 ⊢ (𝑤 ∈ 𝐵 → 𝑤 ∈ ℋ)
6	2	sheli 28071	. . . . . . . . 9 ⊢ (𝑧 ∈ 𝐴 → 𝑧 ∈ ℋ)
7		ax-hvcom 27858	. . . . . . . . 9 ⊢ ((𝑤 ∈ ℋ ∧ 𝑧 ∈ ℋ) → (𝑤 +ℎ 𝑧) = (𝑧 +ℎ 𝑤))
8	5, 6, 7	syl2an 494	. . . . . . . 8 ⊢ ((𝑤 ∈ 𝐵 ∧ 𝑧 ∈ 𝐴) → (𝑤 +ℎ 𝑧) = (𝑧 +ℎ 𝑤))
9	8	eqeq2d 2632	. . . . . . 7 ⊢ ((𝑤 ∈ 𝐵 ∧ 𝑧 ∈ 𝐴) → (𝑥 = (𝑤 +ℎ 𝑧) ↔ 𝑥 = (𝑧 +ℎ 𝑤)))
10	9	rexbidva 3049	. . . . . 6 ⊢ (𝑤 ∈ 𝐵 → (∃𝑧 ∈ 𝐴 𝑥 = (𝑤 +ℎ 𝑧) ↔ ∃𝑧 ∈ 𝐴 𝑥 = (𝑧 +ℎ 𝑤)))
11	10	riotabiia 6628	. . . . 5 ⊢ (℩𝑤 ∈ 𝐵 ∃𝑧 ∈ 𝐴 𝑥 = (𝑤 +ℎ 𝑧)) = (℩𝑤 ∈ 𝐵 ∃𝑧 ∈ 𝐴 𝑥 = (𝑧 +ℎ 𝑤))
12	4, 11	mpteq12i 4742	. . . 4 ⊢ (𝑥 ∈ (𝐵 +ℋ 𝐴) ↦ (℩𝑤 ∈ 𝐵 ∃𝑧 ∈ 𝐴 𝑥 = (𝑤 +ℎ 𝑧))) = (𝑥 ∈ (𝐴 +ℋ 𝐵) ↦ (℩𝑤 ∈ 𝐵 ∃𝑧 ∈ 𝐴 𝑥 = (𝑧 +ℎ 𝑤)))
13	3, 12	eqtr4i 2647	. . 3 ⊢ 𝑇 = (𝑥 ∈ (𝐵 +ℋ 𝐴) ↦ (℩𝑤 ∈ 𝐵 ∃𝑧 ∈ 𝐴 𝑥 = (𝑤 +ℎ 𝑧)))
14	1, 2, 13	cdj3lem2b 29296	. 2 ⊢ (∃𝑣 ∈ ℝ (0 < 𝑣 ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐴 ((normℎ‘𝑥) + (normℎ‘𝑦)) ≤ (𝑣 · (normℎ‘(𝑥 +ℎ 𝑦)))) → ∃𝑣 ∈ ℝ (0 < 𝑣 ∧ ∀𝑢 ∈ (𝐵 +ℋ 𝐴)(normℎ‘(𝑇‘𝑢)) ≤ (𝑣 · (normℎ‘𝑢))))
15		fveq2 6191	. . . . . . . 8 ⊢ (𝑥 = 𝑡 → (normℎ‘𝑥) = (normℎ‘𝑡))
16	15	oveq1d 6665	. . . . . . 7 ⊢ (𝑥 = 𝑡 → ((normℎ‘𝑥) + (normℎ‘𝑦)) = ((normℎ‘𝑡) + (normℎ‘𝑦)))
17		oveq1 6657	. . . . . . . . 9 ⊢ (𝑥 = 𝑡 → (𝑥 +ℎ 𝑦) = (𝑡 +ℎ 𝑦))
18	17	fveq2d 6195	. . . . . . . 8 ⊢ (𝑥 = 𝑡 → (normℎ‘(𝑥 +ℎ 𝑦)) = (normℎ‘(𝑡 +ℎ 𝑦)))
19	18	oveq2d 6666	. . . . . . 7 ⊢ (𝑥 = 𝑡 → (𝑣 · (normℎ‘(𝑥 +ℎ 𝑦))) = (𝑣 · (normℎ‘(𝑡 +ℎ 𝑦))))
20	16, 19	breq12d 4666	. . . . . 6 ⊢ (𝑥 = 𝑡 → (((normℎ‘𝑥) + (normℎ‘𝑦)) ≤ (𝑣 · (normℎ‘(𝑥 +ℎ 𝑦))) ↔ ((normℎ‘𝑡) + (normℎ‘𝑦)) ≤ (𝑣 · (normℎ‘(𝑡 +ℎ 𝑦)))))
21		fveq2 6191	. . . . . . . 8 ⊢ (𝑦 = ℎ → (normℎ‘𝑦) = (normℎ‘ℎ))
22	21	oveq2d 6666	. . . . . . 7 ⊢ (𝑦 = ℎ → ((normℎ‘𝑡) + (normℎ‘𝑦)) = ((normℎ‘𝑡) + (normℎ‘ℎ)))
23		oveq2 6658	. . . . . . . . 9 ⊢ (𝑦 = ℎ → (𝑡 +ℎ 𝑦) = (𝑡 +ℎ ℎ))
24	23	fveq2d 6195	. . . . . . . 8 ⊢ (𝑦 = ℎ → (normℎ‘(𝑡 +ℎ 𝑦)) = (normℎ‘(𝑡 +ℎ ℎ)))
25	24	oveq2d 6666	. . . . . . 7 ⊢ (𝑦 = ℎ → (𝑣 · (normℎ‘(𝑡 +ℎ 𝑦))) = (𝑣 · (normℎ‘(𝑡 +ℎ ℎ))))
26	22, 25	breq12d 4666	. . . . . 6 ⊢ (𝑦 = ℎ → (((normℎ‘𝑡) + (normℎ‘𝑦)) ≤ (𝑣 · (normℎ‘(𝑡 +ℎ 𝑦))) ↔ ((normℎ‘𝑡) + (normℎ‘ℎ)) ≤ (𝑣 · (normℎ‘(𝑡 +ℎ ℎ)))))
27	20, 26	cbvral2v 3179	. . . . 5 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 ((normℎ‘𝑥) + (normℎ‘𝑦)) ≤ (𝑣 · (normℎ‘(𝑥 +ℎ 𝑦))) ↔ ∀𝑡 ∈ 𝐴 ∀ℎ ∈ 𝐵 ((normℎ‘𝑡) + (normℎ‘ℎ)) ≤ (𝑣 · (normℎ‘(𝑡 +ℎ ℎ))))
28		ralcom 3098	. . . . 5 ⊢ (∀𝑡 ∈ 𝐴 ∀ℎ ∈ 𝐵 ((normℎ‘𝑡) + (normℎ‘ℎ)) ≤ (𝑣 · (normℎ‘(𝑡 +ℎ ℎ))) ↔ ∀ℎ ∈ 𝐵 ∀𝑡 ∈ 𝐴 ((normℎ‘𝑡) + (normℎ‘ℎ)) ≤ (𝑣 · (normℎ‘(𝑡 +ℎ ℎ))))
29	1	sheli 28071	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ 𝐵 → 𝑥 ∈ ℋ)
30		normcl 27982	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ ℋ → (normℎ‘𝑥) ∈ ℝ)
31	29, 30	syl 17	. . . . . . . . . . 11 ⊢ (𝑥 ∈ 𝐵 → (normℎ‘𝑥) ∈ ℝ)
32	31	recnd 10068	. . . . . . . . . 10 ⊢ (𝑥 ∈ 𝐵 → (normℎ‘𝑥) ∈ ℂ)
33	2	sheli 28071	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ 𝐴 → 𝑦 ∈ ℋ)
34		normcl 27982	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ ℋ → (normℎ‘𝑦) ∈ ℝ)
35	33, 34	syl 17	. . . . . . . . . . 11 ⊢ (𝑦 ∈ 𝐴 → (normℎ‘𝑦) ∈ ℝ)
36	35	recnd 10068	. . . . . . . . . 10 ⊢ (𝑦 ∈ 𝐴 → (normℎ‘𝑦) ∈ ℂ)
37		addcom 10222	. . . . . . . . . 10 ⊢ (((normℎ‘𝑥) ∈ ℂ ∧ (normℎ‘𝑦) ∈ ℂ) → ((normℎ‘𝑥) + (normℎ‘𝑦)) = ((normℎ‘𝑦) + (normℎ‘𝑥)))
38	32, 36, 37	syl2an 494	. . . . . . . . 9 ⊢ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐴) → ((normℎ‘𝑥) + (normℎ‘𝑦)) = ((normℎ‘𝑦) + (normℎ‘𝑥)))
39		ax-hvcom 27858	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ) → (𝑥 +ℎ 𝑦) = (𝑦 +ℎ 𝑥))
40	29, 33, 39	syl2an 494	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐴) → (𝑥 +ℎ 𝑦) = (𝑦 +ℎ 𝑥))
41	40	fveq2d 6195	. . . . . . . . . 10 ⊢ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐴) → (normℎ‘(𝑥 +ℎ 𝑦)) = (normℎ‘(𝑦 +ℎ 𝑥)))
42	41	oveq2d 6666	. . . . . . . . 9 ⊢ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐴) → (𝑣 · (normℎ‘(𝑥 +ℎ 𝑦))) = (𝑣 · (normℎ‘(𝑦 +ℎ 𝑥))))
43	38, 42	breq12d 4666	. . . . . . . 8 ⊢ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐴) → (((normℎ‘𝑥) + (normℎ‘𝑦)) ≤ (𝑣 · (normℎ‘(𝑥 +ℎ 𝑦))) ↔ ((normℎ‘𝑦) + (normℎ‘𝑥)) ≤ (𝑣 · (normℎ‘(𝑦 +ℎ 𝑥)))))
44	43	ralbidva 2985	. . . . . . 7 ⊢ (𝑥 ∈ 𝐵 → (∀𝑦 ∈ 𝐴 ((normℎ‘𝑥) + (normℎ‘𝑦)) ≤ (𝑣 · (normℎ‘(𝑥 +ℎ 𝑦))) ↔ ∀𝑦 ∈ 𝐴 ((normℎ‘𝑦) + (normℎ‘𝑥)) ≤ (𝑣 · (normℎ‘(𝑦 +ℎ 𝑥)))))
45	44	ralbiia 2979	. . . . . 6 ⊢ (∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐴 ((normℎ‘𝑥) + (normℎ‘𝑦)) ≤ (𝑣 · (normℎ‘(𝑥 +ℎ 𝑦))) ↔ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐴 ((normℎ‘𝑦) + (normℎ‘𝑥)) ≤ (𝑣 · (normℎ‘(𝑦 +ℎ 𝑥))))
46		fveq2 6191	. . . . . . . . 9 ⊢ (𝑥 = ℎ → (normℎ‘𝑥) = (normℎ‘ℎ))
47	46	oveq2d 6666	. . . . . . . 8 ⊢ (𝑥 = ℎ → ((normℎ‘𝑦) + (normℎ‘𝑥)) = ((normℎ‘𝑦) + (normℎ‘ℎ)))
48		oveq2 6658	. . . . . . . . . 10 ⊢ (𝑥 = ℎ → (𝑦 +ℎ 𝑥) = (𝑦 +ℎ ℎ))
49	48	fveq2d 6195	. . . . . . . . 9 ⊢ (𝑥 = ℎ → (normℎ‘(𝑦 +ℎ 𝑥)) = (normℎ‘(𝑦 +ℎ ℎ)))
50	49	oveq2d 6666	. . . . . . . 8 ⊢ (𝑥 = ℎ → (𝑣 · (normℎ‘(𝑦 +ℎ 𝑥))) = (𝑣 · (normℎ‘(𝑦 +ℎ ℎ))))
51	47, 50	breq12d 4666	. . . . . . 7 ⊢ (𝑥 = ℎ → (((normℎ‘𝑦) + (normℎ‘𝑥)) ≤ (𝑣 · (normℎ‘(𝑦 +ℎ 𝑥))) ↔ ((normℎ‘𝑦) + (normℎ‘ℎ)) ≤ (𝑣 · (normℎ‘(𝑦 +ℎ ℎ)))))
52		fveq2 6191	. . . . . . . . 9 ⊢ (𝑦 = 𝑡 → (normℎ‘𝑦) = (normℎ‘𝑡))
53	52	oveq1d 6665	. . . . . . . 8 ⊢ (𝑦 = 𝑡 → ((normℎ‘𝑦) + (normℎ‘ℎ)) = ((normℎ‘𝑡) + (normℎ‘ℎ)))
54		oveq1 6657	. . . . . . . . . 10 ⊢ (𝑦 = 𝑡 → (𝑦 +ℎ ℎ) = (𝑡 +ℎ ℎ))
55	54	fveq2d 6195	. . . . . . . . 9 ⊢ (𝑦 = 𝑡 → (normℎ‘(𝑦 +ℎ ℎ)) = (normℎ‘(𝑡 +ℎ ℎ)))
56	55	oveq2d 6666	. . . . . . . 8 ⊢ (𝑦 = 𝑡 → (𝑣 · (normℎ‘(𝑦 +ℎ ℎ))) = (𝑣 · (normℎ‘(𝑡 +ℎ ℎ))))
57	53, 56	breq12d 4666	. . . . . . 7 ⊢ (𝑦 = 𝑡 → (((normℎ‘𝑦) + (normℎ‘ℎ)) ≤ (𝑣 · (normℎ‘(𝑦 +ℎ ℎ))) ↔ ((normℎ‘𝑡) + (normℎ‘ℎ)) ≤ (𝑣 · (normℎ‘(𝑡 +ℎ ℎ)))))
58	51, 57	cbvral2v 3179	. . . . . 6 ⊢ (∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐴 ((normℎ‘𝑦) + (normℎ‘𝑥)) ≤ (𝑣 · (normℎ‘(𝑦 +ℎ 𝑥))) ↔ ∀ℎ ∈ 𝐵 ∀𝑡 ∈ 𝐴 ((normℎ‘𝑡) + (normℎ‘ℎ)) ≤ (𝑣 · (normℎ‘(𝑡 +ℎ ℎ))))
59	45, 58	bitr2i 265	. . . . 5 ⊢ (∀ℎ ∈ 𝐵 ∀𝑡 ∈ 𝐴 ((normℎ‘𝑡) + (normℎ‘ℎ)) ≤ (𝑣 · (normℎ‘(𝑡 +ℎ ℎ))) ↔ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐴 ((normℎ‘𝑥) + (normℎ‘𝑦)) ≤ (𝑣 · (normℎ‘(𝑥 +ℎ 𝑦))))
60	27, 28, 59	3bitri 286	. . . 4 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 ((normℎ‘𝑥) + (normℎ‘𝑦)) ≤ (𝑣 · (normℎ‘(𝑥 +ℎ 𝑦))) ↔ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐴 ((normℎ‘𝑥) + (normℎ‘𝑦)) ≤ (𝑣 · (normℎ‘(𝑥 +ℎ 𝑦))))
61	60	anbi2i 730	. . 3 ⊢ ((0 < 𝑣 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 ((normℎ‘𝑥) + (normℎ‘𝑦)) ≤ (𝑣 · (normℎ‘(𝑥 +ℎ 𝑦)))) ↔ (0 < 𝑣 ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐴 ((normℎ‘𝑥) + (normℎ‘𝑦)) ≤ (𝑣 · (normℎ‘(𝑥 +ℎ 𝑦)))))
62	61	rexbii 3041	. 2 ⊢ (∃𝑣 ∈ ℝ (0 < 𝑣 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 ((normℎ‘𝑥) + (normℎ‘𝑦)) ≤ (𝑣 · (normℎ‘(𝑥 +ℎ 𝑦)))) ↔ ∃𝑣 ∈ ℝ (0 < 𝑣 ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐴 ((normℎ‘𝑥) + (normℎ‘𝑦)) ≤ (𝑣 · (normℎ‘(𝑥 +ℎ 𝑦)))))
63	2, 1	shscomi 28222	. . . . 5 ⊢ (𝐴 +ℋ 𝐵) = (𝐵 +ℋ 𝐴)
64	63	raleqi 3142	. . . 4 ⊢ (∀𝑢 ∈ (𝐴 +ℋ 𝐵)(normℎ‘(𝑇‘𝑢)) ≤ (𝑣 · (normℎ‘𝑢)) ↔ ∀𝑢 ∈ (𝐵 +ℋ 𝐴)(normℎ‘(𝑇‘𝑢)) ≤ (𝑣 · (normℎ‘𝑢)))
65	64	anbi2i 730	. . 3 ⊢ ((0 < 𝑣 ∧ ∀𝑢 ∈ (𝐴 +ℋ 𝐵)(normℎ‘(𝑇‘𝑢)) ≤ (𝑣 · (normℎ‘𝑢))) ↔ (0 < 𝑣 ∧ ∀𝑢 ∈ (𝐵 +ℋ 𝐴)(normℎ‘(𝑇‘𝑢)) ≤ (𝑣 · (normℎ‘𝑢))))
66	65	rexbii 3041	. 2 ⊢ (∃𝑣 ∈ ℝ (0 < 𝑣 ∧ ∀𝑢 ∈ (𝐴 +ℋ 𝐵)(normℎ‘(𝑇‘𝑢)) ≤ (𝑣 · (normℎ‘𝑢))) ↔ ∃𝑣 ∈ ℝ (0 < 𝑣 ∧ ∀𝑢 ∈ (𝐵 +ℋ 𝐴)(normℎ‘(𝑇‘𝑢)) ≤ (𝑣 · (normℎ‘𝑢))))
67	14, 62, 66	3imtr4i 281	1 ⊢ (∃𝑣 ∈ ℝ (0 < 𝑣 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 ((normℎ‘𝑥) + (normℎ‘𝑦)) ≤ (𝑣 · (normℎ‘(𝑥 +ℎ 𝑦)))) → ∃𝑣 ∈ ℝ (0 < 𝑣 ∧ ∀𝑢 ∈ (𝐴 +ℋ 𝐵)(normℎ‘(𝑇‘𝑢)) ≤ (𝑣 · (normℎ‘𝑢))))

Syntax hints:   → wi 4   ∧ wa 384   = wceq 1483   ∈ wcel 1990  ∀wral 2912  ∃wrex 2913   class class class wbr 4653   ↦ cmpt 4729  ‘cfv 5888  ℩crio 6610  (class class class)co 6650  ℂcc 9934  ℝcr 9935  0cc0 9936   + caddc 9939   · cmul 9941   < clt 10074   ≤ cle 10075   ℋchil 27776   +_ℎ cva 27777  norm_ℎcno 27780   S_ℋ csh 27785   +_ℋ cph 27788

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-rep 4771  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-hvass 27859  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-grpo 27347  df-ablo 27399  df-hnorm 27825  df-hvsub 27828  df-sh 28064  df-ch0 28110  df-shs 28167

This theorem is referenced by:  cdj3i  29300