Theorem cdj3lem3b 32472

Description: Lemma for cdj3i 32473. 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 31395	. . . . 5 ⊢ (𝐵 +ℋ 𝐴) = (𝐴 +ℋ 𝐵)
5	1	sheli 31246	. . . . . . . . 9 ⊢ (𝑤 ∈ 𝐵 → 𝑤 ∈ ℋ)
6	2	sheli 31246	. . . . . . . . 9 ⊢ (𝑧 ∈ 𝐴 → 𝑧 ∈ ℋ)
7		ax-hvcom 31033	. . . . . . . . 9 ⊢ ((𝑤 ∈ ℋ ∧ 𝑧 ∈ ℋ) → (𝑤 +ℎ 𝑧) = (𝑧 +ℎ 𝑤))
8	5, 6, 7	syl2an 595	. . . . . . . 8 ⊢ ((𝑤 ∈ 𝐵 ∧ 𝑧 ∈ 𝐴) → (𝑤 +ℎ 𝑧) = (𝑧 +ℎ 𝑤))
9	8	eqeq2d 2751	. . . . . . 7 ⊢ ((𝑤 ∈ 𝐵 ∧ 𝑧 ∈ 𝐴) → (𝑥 = (𝑤 +ℎ 𝑧) ↔ 𝑥 = (𝑧 +ℎ 𝑤)))
10	9	rexbidva 3183	. . . . . 6 ⊢ (𝑤 ∈ 𝐵 → (∃𝑧 ∈ 𝐴 𝑥 = (𝑤 +ℎ 𝑧) ↔ ∃𝑧 ∈ 𝐴 𝑥 = (𝑧 +ℎ 𝑤)))
11	10	riotabiia 7425	. . . . 5 ⊢ (℩𝑤 ∈ 𝐵 ∃𝑧 ∈ 𝐴 𝑥 = (𝑤 +ℎ 𝑧)) = (℩𝑤 ∈ 𝐵 ∃𝑧 ∈ 𝐴 𝑥 = (𝑧 +ℎ 𝑤))
12	4, 11	mpteq12i 5272	. . . 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 2771	. . 3 ⊢ 𝑇 = (𝑥 ∈ (𝐵 +ℋ 𝐴) ↦ (℩𝑤 ∈ 𝐵 ∃𝑧 ∈ 𝐴 𝑥 = (𝑤 +ℎ 𝑧)))
14	1, 2, 13	cdj3lem2b 32469	. 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 6920	. . . . . . . 8 ⊢ (𝑥 = 𝑡 → (normℎ‘𝑥) = (normℎ‘𝑡))
16	15	oveq1d 7463	. . . . . . 7 ⊢ (𝑥 = 𝑡 → ((normℎ‘𝑥) + (normℎ‘𝑦)) = ((normℎ‘𝑡) + (normℎ‘𝑦)))
17		fvoveq1 7471	. . . . . . . 8 ⊢ (𝑥 = 𝑡 → (normℎ‘(𝑥 +ℎ 𝑦)) = (normℎ‘(𝑡 +ℎ 𝑦)))
18	17	oveq2d 7464	. . . . . . 7 ⊢ (𝑥 = 𝑡 → (𝑣 · (normℎ‘(𝑥 +ℎ 𝑦))) = (𝑣 · (normℎ‘(𝑡 +ℎ 𝑦))))
19	16, 18	breq12d 5179	. . . . . 6 ⊢ (𝑥 = 𝑡 → (((normℎ‘𝑥) + (normℎ‘𝑦)) ≤ (𝑣 · (normℎ‘(𝑥 +ℎ 𝑦))) ↔ ((normℎ‘𝑡) + (normℎ‘𝑦)) ≤ (𝑣 · (normℎ‘(𝑡 +ℎ 𝑦)))))
20		fveq2 6920	. . . . . . . 8 ⊢ (𝑦 = ℎ → (normℎ‘𝑦) = (normℎ‘ℎ))
21	20	oveq2d 7464	. . . . . . 7 ⊢ (𝑦 = ℎ → ((normℎ‘𝑡) + (normℎ‘𝑦)) = ((normℎ‘𝑡) + (normℎ‘ℎ)))
22		oveq2 7456	. . . . . . . . 9 ⊢ (𝑦 = ℎ → (𝑡 +ℎ 𝑦) = (𝑡 +ℎ ℎ))
23	22	fveq2d 6924	. . . . . . . 8 ⊢ (𝑦 = ℎ → (normℎ‘(𝑡 +ℎ 𝑦)) = (normℎ‘(𝑡 +ℎ ℎ)))
24	23	oveq2d 7464	. . . . . . 7 ⊢ (𝑦 = ℎ → (𝑣 · (normℎ‘(𝑡 +ℎ 𝑦))) = (𝑣 · (normℎ‘(𝑡 +ℎ ℎ))))
25	21, 24	breq12d 5179	. . . . . 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 3247	. . . . 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 31246	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ 𝐵 → 𝑥 ∈ ℋ)
29		normcl 31157	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ ℋ → (normℎ‘𝑥) ∈ ℝ)
30	28, 29	syl 17	. . . . . . . . . . 11 ⊢ (𝑥 ∈ 𝐵 → (normℎ‘𝑥) ∈ ℝ)
31	30	recnd 11318	. . . . . . . . . 10 ⊢ (𝑥 ∈ 𝐵 → (normℎ‘𝑥) ∈ ℂ)
32	2	sheli 31246	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ 𝐴 → 𝑦 ∈ ℋ)
33		normcl 31157	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ ℋ → (normℎ‘𝑦) ∈ ℝ)
34	32, 33	syl 17	. . . . . . . . . . 11 ⊢ (𝑦 ∈ 𝐴 → (normℎ‘𝑦) ∈ ℝ)
35	34	recnd 11318	. . . . . . . . . 10 ⊢ (𝑦 ∈ 𝐴 → (normℎ‘𝑦) ∈ ℂ)
36		addcom 11476	. . . . . . . . . 10 ⊢ (((normℎ‘𝑥) ∈ ℂ ∧ (normℎ‘𝑦) ∈ ℂ) → ((normℎ‘𝑥) + (normℎ‘𝑦)) = ((normℎ‘𝑦) + (normℎ‘𝑥)))
37	31, 35, 36	syl2an 595	. . . . . . . . 9 ⊢ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐴) → ((normℎ‘𝑥) + (normℎ‘𝑦)) = ((normℎ‘𝑦) + (normℎ‘𝑥)))
38		ax-hvcom 31033	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ) → (𝑥 +ℎ 𝑦) = (𝑦 +ℎ 𝑥))
39	28, 32, 38	syl2an 595	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐴) → (𝑥 +ℎ 𝑦) = (𝑦 +ℎ 𝑥))
40	39	fveq2d 6924	. . . . . . . . . 10 ⊢ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐴) → (normℎ‘(𝑥 +ℎ 𝑦)) = (normℎ‘(𝑦 +ℎ 𝑥)))
41	40	oveq2d 7464	. . . . . . . . 9 ⊢ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐴) → (𝑣 · (normℎ‘(𝑥 +ℎ 𝑦))) = (𝑣 · (normℎ‘(𝑦 +ℎ 𝑥))))
42	37, 41	breq12d 5179	. . . . . . . 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 3182	. . . . . . 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 3097	. . . . . 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 6920	. . . . . . . . 9 ⊢ (𝑥 = ℎ → (normℎ‘𝑥) = (normℎ‘ℎ))
46	45	oveq2d 7464	. . . . . . . 8 ⊢ (𝑥 = ℎ → ((normℎ‘𝑦) + (normℎ‘𝑥)) = ((normℎ‘𝑦) + (normℎ‘ℎ)))
47		oveq2 7456	. . . . . . . . . 10 ⊢ (𝑥 = ℎ → (𝑦 +ℎ 𝑥) = (𝑦 +ℎ ℎ))
48	47	fveq2d 6924	. . . . . . . . 9 ⊢ (𝑥 = ℎ → (normℎ‘(𝑦 +ℎ 𝑥)) = (normℎ‘(𝑦 +ℎ ℎ)))
49	48	oveq2d 7464	. . . . . . . 8 ⊢ (𝑥 = ℎ → (𝑣 · (normℎ‘(𝑦 +ℎ 𝑥))) = (𝑣 · (normℎ‘(𝑦 +ℎ ℎ))))
50	46, 49	breq12d 5179	. . . . . . 7 ⊢ (𝑥 = ℎ → (((normℎ‘𝑦) + (normℎ‘𝑥)) ≤ (𝑣 · (normℎ‘(𝑦 +ℎ 𝑥))) ↔ ((normℎ‘𝑦) + (normℎ‘ℎ)) ≤ (𝑣 · (normℎ‘(𝑦 +ℎ ℎ)))))
51		fveq2 6920	. . . . . . . . 9 ⊢ (𝑦 = 𝑡 → (normℎ‘𝑦) = (normℎ‘𝑡))
52	51	oveq1d 7463	. . . . . . . 8 ⊢ (𝑦 = 𝑡 → ((normℎ‘𝑦) + (normℎ‘ℎ)) = ((normℎ‘𝑡) + (normℎ‘ℎ)))
53		fvoveq1 7471	. . . . . . . . 9 ⊢ (𝑦 = 𝑡 → (normℎ‘(𝑦 +ℎ ℎ)) = (normℎ‘(𝑡 +ℎ ℎ)))
54	53	oveq2d 7464	. . . . . . . 8 ⊢ (𝑦 = 𝑡 → (𝑣 · (normℎ‘(𝑦 +ℎ ℎ))) = (𝑣 · (normℎ‘(𝑡 +ℎ ℎ))))
55	52, 54	breq12d 5179	. . . . . . 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 3247	. . . . . 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 276	. . . . 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 297	. . . 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 622	. . 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 3100	. 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 31395	. . . . 5 ⊢ (𝐴 +ℋ 𝐵) = (𝐵 +ℋ 𝐴)
62	61	raleqi 3332	. . . 4 ⊢ (∀𝑢 ∈ (𝐴 +ℋ 𝐵)(normℎ‘(𝑇‘𝑢)) ≤ (𝑣 · (normℎ‘𝑢)) ↔ ∀𝑢 ∈ (𝐵 +ℋ 𝐴)(normℎ‘(𝑇‘𝑢)) ≤ (𝑣 · (normℎ‘𝑢)))
63	62	anbi2i 622	. . 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 3100	. 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 292	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 395   = wceq 1537   ∈ wcel 2108  ∀wral 3067  ∃wrex 3076   class class class wbr 5166   ↦ cmpt 5249  ‘cfv 6573  ℩crio 7403  (class class class)co 7448  ℂcc 11182  ℝcr 11183  0cc0 11184   + caddc 11187   · cmul 11189   < clt 11324   ≤ cle 11325   ℋchba 30951   +_ℎ cva 30952  norm_ℎcno 30955   S_ℋ csh 30960   +_ℋ cph 30963

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-rep 5303  ax-sep 5317  ax-nul 5324  ax-pow 5383  ax-pr 5447  ax-un 7770  ax-cnex 11240  ax-resscn 11241  ax-1cn 11242  ax-icn 11243  ax-addcl 11244  ax-addrcl 11245  ax-mulcl 11246  ax-mulrcl 11247  ax-mulcom 11248  ax-addass 11249  ax-mulass 11250  ax-distr 11251  ax-i2m1 11252  ax-1ne0 11253  ax-1rid 11254  ax-rnegex 11255  ax-rrecex 11256  ax-cnre 11257  ax-pre-lttri 11258  ax-pre-lttrn 11259  ax-pre-ltadd 11260  ax-pre-mulgt0 11261  ax-pre-sup 11262  ax-hilex 31031  ax-hfvadd 31032  ax-hvcom 31033  ax-hvass 31034  ax-hv0cl 31035  ax-hvaddid 31036  ax-hfvmul 31037  ax-hvmulid 31038  ax-hvmulass 31039  ax-hvdistr1 31040  ax-hvdistr2 31041  ax-hvmul0 31042  ax-hfi 31111  ax-his1 31114  ax-his3 31116  ax-his4 31117

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3or 1088  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-nel 3053  df-ral 3068  df-rex 3077  df-rmo 3388  df-reu 3389  df-rab 3444  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-pss 3996  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-int 4971  df-iun 5017  df-br 5167  df-opab 5229  df-mpt 5250  df-tr 5284  df-id 5593  df-eprel 5599  df-po 5607  df-so 5608  df-fr 5652  df-we 5654  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-pred 6332  df-ord 6398  df-on 6399  df-lim 6400  df-suc 6401  df-iota 6525  df-fun 6575  df-fn 6576  df-f 6577  df-f1 6578  df-fo 6579  df-f1o 6580  df-fv 6581  df-riota 7404  df-ov 7451  df-oprab 7452  df-mpo 7453  df-om 7904  df-2nd 8031  df-frecs 8322  df-wrecs 8353  df-recs 8427  df-rdg 8466  df-er 8763  df-en 9004  df-dom 9005  df-sdom 9006  df-sup 9511  df-pnf 11326  df-mnf 11327  df-xr 11328  df-ltxr 11329  df-le 11330  df-sub 11522  df-neg 11523  df-div 11948  df-nn 12294  df-2 12356  df-3 12357  df-n0 12554  df-z 12640  df-uz 12904  df-rp 13058  df-seq 14053  df-exp 14113  df-cj 15148  df-re 15149  df-im 15150  df-sqrt 15284  df-abs 15285  df-grpo 30525  df-ablo 30577  df-hnorm 31000  df-hvsub 31003  df-sh 31239  df-ch0 31285  df-shs 31340

This theorem is referenced by:  cdj3i  32473