Theorem cdj3lem3b 32863

Description: Lemma for cdj3i 32864. 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 31786	. . . . 5 ⊢ (𝐵 +ℋ 𝐴) = (𝐴 +ℋ 𝐵)
5	1	sheli 31637	. . . . . . . . 9 ⊢ (𝑤 ∈ 𝐵 → 𝑤 ∈ ℋ)
6	2	sheli 31637	. . . . . . . . 9 ⊢ (𝑧 ∈ 𝐴 → 𝑧 ∈ ℋ)
7		ax-hvcom 31424	. . . . . . . . 9 ⊢ ((𝑤 ∈ ℋ ∧ 𝑧 ∈ ℋ) → (𝑤 +ℎ 𝑧) = (𝑧 +ℎ 𝑤))
8	5, 6, 7	syl2an 608	. . . . . . . 8 ⊢ ((𝑤 ∈ 𝐵 ∧ 𝑧 ∈ 𝐴) → (𝑤 +ℎ 𝑧) = (𝑧 +ℎ 𝑤))
9	8	eqeq2d 2776	. . . . . . 7 ⊢ ((𝑤 ∈ 𝐵 ∧ 𝑧 ∈ 𝐴) → (𝑥 = (𝑤 +ℎ 𝑧) ↔ 𝑥 = (𝑧 +ℎ 𝑤)))
10	9	rexbidva 3189	. . . . . 6 ⊢ (𝑤 ∈ 𝐵 → (∃𝑧 ∈ 𝐴 𝑥 = (𝑤 +ℎ 𝑧) ↔ ∃𝑧 ∈ 𝐴 𝑥 = (𝑧 +ℎ 𝑤)))
11	10	riotabiia 7396	. . . . 5 ⊢ (℩𝑤 ∈ 𝐵 ∃𝑧 ∈ 𝐴 𝑥 = (𝑤 +ℎ 𝑧)) = (℩𝑤 ∈ 𝐵 ∃𝑧 ∈ 𝐴 𝑥 = (𝑧 +ℎ 𝑤))
12	4, 11	mpteq12i 5210	. . . 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 2791	. . 3 ⊢ 𝑇 = (𝑥 ∈ (𝐵 +ℋ 𝐴) ↦ (℩𝑤 ∈ 𝐵 ∃𝑧 ∈ 𝐴 𝑥 = (𝑤 +ℎ 𝑧)))
14	1, 2, 13	cdj3lem2b 32860	. 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 6885	. . . . . . . 8 ⊢ (𝑥 = 𝑡 → (normℎ‘𝑥) = (normℎ‘𝑡))
16	15	oveq1d 7434	. . . . . . 7 ⊢ (𝑥 = 𝑡 → ((normℎ‘𝑥) + (normℎ‘𝑦)) = ((normℎ‘𝑡) + (normℎ‘𝑦)))
17		fvoveq1 7442	. . . . . . . 8 ⊢ (𝑥 = 𝑡 → (normℎ‘(𝑥 +ℎ 𝑦)) = (normℎ‘(𝑡 +ℎ 𝑦)))
18	17	oveq2d 7435	. . . . . . 7 ⊢ (𝑥 = 𝑡 → (𝑣 · (normℎ‘(𝑥 +ℎ 𝑦))) = (𝑣 · (normℎ‘(𝑡 +ℎ 𝑦))))
19	16, 18	breq12d 5124	. . . . . 6 ⊢ (𝑥 = 𝑡 → (((normℎ‘𝑥) + (normℎ‘𝑦)) ≤ (𝑣 · (normℎ‘(𝑥 +ℎ 𝑦))) ↔ ((normℎ‘𝑡) + (normℎ‘𝑦)) ≤ (𝑣 · (normℎ‘(𝑡 +ℎ 𝑦)))))
20		fveq2 6885	. . . . . . . 8 ⊢ (𝑦 = ℎ → (normℎ‘𝑦) = (normℎ‘ℎ))
21	20	oveq2d 7435	. . . . . . 7 ⊢ (𝑦 = ℎ → ((normℎ‘𝑡) + (normℎ‘𝑦)) = ((normℎ‘𝑡) + (normℎ‘ℎ)))
22		oveq2 7427	. . . . . . . . 9 ⊢ (𝑦 = ℎ → (𝑡 +ℎ 𝑦) = (𝑡 +ℎ ℎ))
23	22	fveq2d 6889	. . . . . . . 8 ⊢ (𝑦 = ℎ → (normℎ‘(𝑡 +ℎ 𝑦)) = (normℎ‘(𝑡 +ℎ ℎ)))
24	23	oveq2d 7435	. . . . . . 7 ⊢ (𝑦 = ℎ → (𝑣 · (normℎ‘(𝑡 +ℎ 𝑦))) = (𝑣 · (normℎ‘(𝑡 +ℎ ℎ))))
25	21, 24	breq12d 5124	. . . . . 6 ⊢ (𝑦 = ℎ → (((normℎ‘𝑡) + (normℎ‘𝑦)) ≤ (𝑣 · (normℎ‘(𝑡 +ℎ 𝑦))) ↔ ((normℎ‘𝑡) + (normℎ‘ℎ)) ≤ (𝑣 · (normℎ‘(𝑡 +ℎ ℎ)))))
26	19, 25	cbvral2vw 3249	. . . . 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))))
27		ralcom 3295	. . . . 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))))
28	1	sheli 31637	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ 𝐵 → 𝑥 ∈ ℋ)
29		normcl 31548	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ ℋ → (normℎ‘𝑥) ∈ ℝ)
30	28, 29	syl 18	. . . . . . . . . . 11 ⊢ (𝑥 ∈ 𝐵 → (normℎ‘𝑥) ∈ ℝ)
31	30	recnd 11252	. . . . . . . . . 10 ⊢ (𝑥 ∈ 𝐵 → (normℎ‘𝑥) ∈ ℂ)
32	2	sheli 31637	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ 𝐴 → 𝑦 ∈ ℋ)
33		normcl 31548	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ ℋ → (normℎ‘𝑦) ∈ ℝ)
34	32, 33	syl 18	. . . . . . . . . . 11 ⊢ (𝑦 ∈ 𝐴 → (normℎ‘𝑦) ∈ ℝ)
35	34	recnd 11252	. . . . . . . . . 10 ⊢ (𝑦 ∈ 𝐴 → (normℎ‘𝑦) ∈ ℂ)
36		addcom 11411	. . . . . . . . . 10 ⊢ (((normℎ‘𝑥) ∈ ℂ ∧ (normℎ‘𝑦) ∈ ℂ) → ((normℎ‘𝑥) + (normℎ‘𝑦)) = ((normℎ‘𝑦) + (normℎ‘𝑥)))
37	31, 35, 36	syl2an 608	. . . . . . . . 9 ⊢ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐴) → ((normℎ‘𝑥) + (normℎ‘𝑦)) = ((normℎ‘𝑦) + (normℎ‘𝑥)))
38		ax-hvcom 31424	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ) → (𝑥 +ℎ 𝑦) = (𝑦 +ℎ 𝑥))
39	28, 32, 38	syl2an 608	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐴) → (𝑥 +ℎ 𝑦) = (𝑦 +ℎ 𝑥))
40	39	fveq2d 6889	. . . . . . . . . 10 ⊢ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐴) → (normℎ‘(𝑥 +ℎ 𝑦)) = (normℎ‘(𝑦 +ℎ 𝑥)))
41	40	oveq2d 7435	. . . . . . . . 9 ⊢ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐴) → (𝑣 · (normℎ‘(𝑥 +ℎ 𝑦))) = (𝑣 · (normℎ‘(𝑦 +ℎ 𝑥))))
42	37, 41	breq12d 5124	. . . . . . . 8 ⊢ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐴) → (((normℎ‘𝑥) + (normℎ‘𝑦)) ≤ (𝑣 · (normℎ‘(𝑥 +ℎ 𝑦))) ↔ ((normℎ‘𝑦) + (normℎ‘𝑥)) ≤ (𝑣 · (normℎ‘(𝑦 +ℎ 𝑥)))))
43	42	ralbidva 3188	. . . . . . 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)))))
44	43	ralbiia 3111	. . . . . 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))))
45		fveq2 6885	. . . . . . . . 9 ⊢ (𝑥 = ℎ → (normℎ‘𝑥) = (normℎ‘ℎ))
46	45	oveq2d 7435	. . . . . . . 8 ⊢ (𝑥 = ℎ → ((normℎ‘𝑦) + (normℎ‘𝑥)) = ((normℎ‘𝑦) + (normℎ‘ℎ)))
47		oveq2 7427	. . . . . . . . . 10 ⊢ (𝑥 = ℎ → (𝑦 +ℎ 𝑥) = (𝑦 +ℎ ℎ))
48	47	fveq2d 6889	. . . . . . . . 9 ⊢ (𝑥 = ℎ → (normℎ‘(𝑦 +ℎ 𝑥)) = (normℎ‘(𝑦 +ℎ ℎ)))
49	48	oveq2d 7435	. . . . . . . 8 ⊢ (𝑥 = ℎ → (𝑣 · (normℎ‘(𝑦 +ℎ 𝑥))) = (𝑣 · (normℎ‘(𝑦 +ℎ ℎ))))
50	46, 49	breq12d 5124	. . . . . . 7 ⊢ (𝑥 = ℎ → (((normℎ‘𝑦) + (normℎ‘𝑥)) ≤ (𝑣 · (normℎ‘(𝑦 +ℎ 𝑥))) ↔ ((normℎ‘𝑦) + (normℎ‘ℎ)) ≤ (𝑣 · (normℎ‘(𝑦 +ℎ ℎ)))))
51		fveq2 6885	. . . . . . . . 9 ⊢ (𝑦 = 𝑡 → (normℎ‘𝑦) = (normℎ‘𝑡))
52	51	oveq1d 7434	. . . . . . . 8 ⊢ (𝑦 = 𝑡 → ((normℎ‘𝑦) + (normℎ‘ℎ)) = ((normℎ‘𝑡) + (normℎ‘ℎ)))
53		fvoveq1 7442	. . . . . . . . 9 ⊢ (𝑦 = 𝑡 → (normℎ‘(𝑦 +ℎ ℎ)) = (normℎ‘(𝑡 +ℎ ℎ)))
54	53	oveq2d 7435	. . . . . . . 8 ⊢ (𝑦 = 𝑡 → (𝑣 · (normℎ‘(𝑦 +ℎ ℎ))) = (𝑣 · (normℎ‘(𝑡 +ℎ ℎ))))
55	52, 54	breq12d 5124	. . . . . . 7 ⊢ (𝑦 = 𝑡 → (((normℎ‘𝑦) + (normℎ‘ℎ)) ≤ (𝑣 · (normℎ‘(𝑦 +ℎ ℎ))) ↔ ((normℎ‘𝑡) + (normℎ‘ℎ)) ≤ (𝑣 · (normℎ‘(𝑡 +ℎ ℎ)))))
56	50, 55	cbvral2vw 3249	. . . . . 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))))
57	44, 56	bitr2i 279	. . . . 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))))
58	26, 27, 57	3bitri 300	. . . 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))))
59	58	anbi2i 635	. . 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)))))
60	59	rexbii 3114	. 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)))))
61	2, 1	shscomi 31786	. . . . 5 ⊢ (𝐴 +ℋ 𝐵) = (𝐵 +ℋ 𝐴)
62	61	raleqi 3323	. . . 4 ⊢ (∀𝑢 ∈ (𝐴 +ℋ 𝐵)(normℎ‘(𝑇‘𝑢)) ≤ (𝑣 · (normℎ‘𝑢)) ↔ ∀𝑢 ∈ (𝐵 +ℋ 𝐴)(normℎ‘(𝑇‘𝑢)) ≤ (𝑣 · (normℎ‘𝑢)))
63	62	anbi2i 635	. . 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))))
64	63	rexbii 3114	. 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))))
65	14, 60, 64	3imtr4i 295	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))))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2146  ∀wral 3081  ∃wrex 3091   class class class wbr 5111   ↦ cmpt 5194  ‘cfv 6540  ℩crio 7375  (class class class)co 7419  ℂcc 11113  ℝcr 11114  0cc0 11115   + caddc 11118   · cmul 11120   < clt 11258   ≤ cle 11259   ℋchba 31342   +_ℎ cva 31343  norm_ℎcno 31346   S_ℋ csh 31351   +_ℋ cph 31354

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 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2737  ax-rep 5240  ax-sep 5259  ax-nul 5271  ax-pow 5338  ax-pr 5406  ax-un 7742  ax-cnex 11171  ax-resscn 11172  ax-1cn 11173  ax-icn 11174  ax-addcl 11175  ax-addrcl 11176  ax-mulcl 11177  ax-mulrcl 11178  ax-mulcom 11179  ax-addass 11180  ax-mulass 11181  ax-distr 11182  ax-i2m1 11183  ax-1ne0 11184  ax-1rid 11185  ax-rnegex 11186  ax-rrecex 11187  ax-cnre 11188  ax-pre-lttri 11189  ax-pre-lttrn 11190  ax-pre-ltadd 11191  ax-pre-mulgt0 11192  ax-pre-sup 11193  ax-hilex 31422  ax-hfvadd 31423  ax-hvcom 31424  ax-hvass 31425  ax-hv0cl 31426  ax-hvaddid 31427  ax-hfvmul 31428  ax-hvmulid 31429  ax-hvmulass 31430  ax-hvdistr1 31431  ax-hvdistr2 31432  ax-hvmul0 31433  ax-hfi 31502  ax-his1 31505  ax-his3 31507  ax-his4 31508

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 2100  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-nel 3067  df-ral 3082  df-rex 3092  df-rmo 3371  df-reu 3372  df-rab 3419  df-v 3459  df-sbc 3747  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-pss 3926  df-nul 4287  df-if 4490  df-pw 4566  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4875  df-int 4915  df-iun 4960  df-br 5112  df-opab 5176  df-mpt 5195  df-tr 5221  df-id 5558  df-eprel 5563  df-po 5571  df-so 5572  df-fr 5616  df-we 5618  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-res 5675  df-ima 5676  df-pred 6306  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6496  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-riota 7376  df-ov 7422  df-oprab 7423  df-mpo 7424  df-om 7869  df-2nd 7993  df-frecs 8284  df-wrecs 8315  df-recs 8364  df-rdg 8403  df-er 8700  df-en 8950  df-dom 8951  df-sdom 8952  df-sup 9409  df-pnf 11260  df-mnf 11261  df-xr 11262  df-ltxr 11263  df-le 11264  df-sub 11458  df-neg 11459  df-div 11887  df-nn 12249  df-2 12318  df-3 12319  df-n0 12520  df-z 12607  df-uz 12879  df-rp 13033  df-seq 14056  df-exp 14116  df-cj 15174  df-re 15175  df-im 15176  df-sqrt 15310  df-abs 15311  df-grpo 30916  df-ablo 30968  df-hnorm 31391  df-hvsub 31394  df-sh 31630  df-ch0 31676  df-shs 31731

This theorem is used by:  cdj3i  32864