Theorem cdj3lem3b 32384

Description: Lemma for cdj3i 32385. 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 31307	. . . . 5 ⊢ (𝐵 +ℋ 𝐴) = (𝐴 +ℋ 𝐵)
5	1	sheli 31158	. . . . . . . . 9 ⊢ (𝑤 ∈ 𝐵 → 𝑤 ∈ ℋ)
6	2	sheli 31158	. . . . . . . . 9 ⊢ (𝑧 ∈ 𝐴 → 𝑧 ∈ ℋ)
7		ax-hvcom 30945	. . . . . . . . 9 ⊢ ((𝑤 ∈ ℋ ∧ 𝑧 ∈ ℋ) → (𝑤 +ℎ 𝑧) = (𝑧 +ℎ 𝑤))
8	5, 6, 7	syl2an 596	. . . . . . . 8 ⊢ ((𝑤 ∈ 𝐵 ∧ 𝑧 ∈ 𝐴) → (𝑤 +ℎ 𝑧) = (𝑧 +ℎ 𝑤))
9	8	eqeq2d 2740	. . . . . . 7 ⊢ ((𝑤 ∈ 𝐵 ∧ 𝑧 ∈ 𝐴) → (𝑥 = (𝑤 +ℎ 𝑧) ↔ 𝑥 = (𝑧 +ℎ 𝑤)))
10	9	rexbidva 3151	. . . . . 6 ⊢ (𝑤 ∈ 𝐵 → (∃𝑧 ∈ 𝐴 𝑥 = (𝑤 +ℎ 𝑧) ↔ ∃𝑧 ∈ 𝐴 𝑥 = (𝑧 +ℎ 𝑤)))
11	10	riotabiia 7326	. . . . 5 ⊢ (℩𝑤 ∈ 𝐵 ∃𝑧 ∈ 𝐴 𝑥 = (𝑤 +ℎ 𝑧)) = (℩𝑤 ∈ 𝐵 ∃𝑧 ∈ 𝐴 𝑥 = (𝑧 +ℎ 𝑤))
12	4, 11	mpteq12i 5189	. . . 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 2755	. . 3 ⊢ 𝑇 = (𝑥 ∈ (𝐵 +ℋ 𝐴) ↦ (℩𝑤 ∈ 𝐵 ∃𝑧 ∈ 𝐴 𝑥 = (𝑤 +ℎ 𝑧)))
14	1, 2, 13	cdj3lem2b 32381	. 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 6822	. . . . . . . 8 ⊢ (𝑥 = 𝑡 → (normℎ‘𝑥) = (normℎ‘𝑡))
16	15	oveq1d 7364	. . . . . . 7 ⊢ (𝑥 = 𝑡 → ((normℎ‘𝑥) + (normℎ‘𝑦)) = ((normℎ‘𝑡) + (normℎ‘𝑦)))
17		fvoveq1 7372	. . . . . . . 8 ⊢ (𝑥 = 𝑡 → (normℎ‘(𝑥 +ℎ 𝑦)) = (normℎ‘(𝑡 +ℎ 𝑦)))
18	17	oveq2d 7365	. . . . . . 7 ⊢ (𝑥 = 𝑡 → (𝑣 · (normℎ‘(𝑥 +ℎ 𝑦))) = (𝑣 · (normℎ‘(𝑡 +ℎ 𝑦))))
19	16, 18	breq12d 5105	. . . . . 6 ⊢ (𝑥 = 𝑡 → (((normℎ‘𝑥) + (normℎ‘𝑦)) ≤ (𝑣 · (normℎ‘(𝑥 +ℎ 𝑦))) ↔ ((normℎ‘𝑡) + (normℎ‘𝑦)) ≤ (𝑣 · (normℎ‘(𝑡 +ℎ 𝑦)))))
20		fveq2 6822	. . . . . . . 8 ⊢ (𝑦 = ℎ → (normℎ‘𝑦) = (normℎ‘ℎ))
21	20	oveq2d 7365	. . . . . . 7 ⊢ (𝑦 = ℎ → ((normℎ‘𝑡) + (normℎ‘𝑦)) = ((normℎ‘𝑡) + (normℎ‘ℎ)))
22		oveq2 7357	. . . . . . . . 9 ⊢ (𝑦 = ℎ → (𝑡 +ℎ 𝑦) = (𝑡 +ℎ ℎ))
23	22	fveq2d 6826	. . . . . . . 8 ⊢ (𝑦 = ℎ → (normℎ‘(𝑡 +ℎ 𝑦)) = (normℎ‘(𝑡 +ℎ ℎ)))
24	23	oveq2d 7365	. . . . . . 7 ⊢ (𝑦 = ℎ → (𝑣 · (normℎ‘(𝑡 +ℎ 𝑦))) = (𝑣 · (normℎ‘(𝑡 +ℎ ℎ))))
25	21, 24	breq12d 5105	. . . . . 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 3211	. . . . 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 3257	. . . . 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 31158	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ 𝐵 → 𝑥 ∈ ℋ)
29		normcl 31069	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ ℋ → (normℎ‘𝑥) ∈ ℝ)
30	28, 29	syl 17	. . . . . . . . . . 11 ⊢ (𝑥 ∈ 𝐵 → (normℎ‘𝑥) ∈ ℝ)
31	30	recnd 11143	. . . . . . . . . 10 ⊢ (𝑥 ∈ 𝐵 → (normℎ‘𝑥) ∈ ℂ)
32	2	sheli 31158	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ 𝐴 → 𝑦 ∈ ℋ)
33		normcl 31069	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ ℋ → (normℎ‘𝑦) ∈ ℝ)
34	32, 33	syl 17	. . . . . . . . . . 11 ⊢ (𝑦 ∈ 𝐴 → (normℎ‘𝑦) ∈ ℝ)
35	34	recnd 11143	. . . . . . . . . 10 ⊢ (𝑦 ∈ 𝐴 → (normℎ‘𝑦) ∈ ℂ)
36		addcom 11302	. . . . . . . . . 10 ⊢ (((normℎ‘𝑥) ∈ ℂ ∧ (normℎ‘𝑦) ∈ ℂ) → ((normℎ‘𝑥) + (normℎ‘𝑦)) = ((normℎ‘𝑦) + (normℎ‘𝑥)))
37	31, 35, 36	syl2an 596	. . . . . . . . 9 ⊢ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐴) → ((normℎ‘𝑥) + (normℎ‘𝑦)) = ((normℎ‘𝑦) + (normℎ‘𝑥)))
38		ax-hvcom 30945	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ) → (𝑥 +ℎ 𝑦) = (𝑦 +ℎ 𝑥))
39	28, 32, 38	syl2an 596	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐴) → (𝑥 +ℎ 𝑦) = (𝑦 +ℎ 𝑥))
40	39	fveq2d 6826	. . . . . . . . . 10 ⊢ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐴) → (normℎ‘(𝑥 +ℎ 𝑦)) = (normℎ‘(𝑦 +ℎ 𝑥)))
41	40	oveq2d 7365	. . . . . . . . 9 ⊢ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐴) → (𝑣 · (normℎ‘(𝑥 +ℎ 𝑦))) = (𝑣 · (normℎ‘(𝑦 +ℎ 𝑥))))
42	37, 41	breq12d 5105	. . . . . . . 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 3150	. . . . . . 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 3073	. . . . . 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 6822	. . . . . . . . 9 ⊢ (𝑥 = ℎ → (normℎ‘𝑥) = (normℎ‘ℎ))
46	45	oveq2d 7365	. . . . . . . 8 ⊢ (𝑥 = ℎ → ((normℎ‘𝑦) + (normℎ‘𝑥)) = ((normℎ‘𝑦) + (normℎ‘ℎ)))
47		oveq2 7357	. . . . . . . . . 10 ⊢ (𝑥 = ℎ → (𝑦 +ℎ 𝑥) = (𝑦 +ℎ ℎ))
48	47	fveq2d 6826	. . . . . . . . 9 ⊢ (𝑥 = ℎ → (normℎ‘(𝑦 +ℎ 𝑥)) = (normℎ‘(𝑦 +ℎ ℎ)))
49	48	oveq2d 7365	. . . . . . . 8 ⊢ (𝑥 = ℎ → (𝑣 · (normℎ‘(𝑦 +ℎ 𝑥))) = (𝑣 · (normℎ‘(𝑦 +ℎ ℎ))))
50	46, 49	breq12d 5105	. . . . . . 7 ⊢ (𝑥 = ℎ → (((normℎ‘𝑦) + (normℎ‘𝑥)) ≤ (𝑣 · (normℎ‘(𝑦 +ℎ 𝑥))) ↔ ((normℎ‘𝑦) + (normℎ‘ℎ)) ≤ (𝑣 · (normℎ‘(𝑦 +ℎ ℎ)))))
51		fveq2 6822	. . . . . . . . 9 ⊢ (𝑦 = 𝑡 → (normℎ‘𝑦) = (normℎ‘𝑡))
52	51	oveq1d 7364	. . . . . . . 8 ⊢ (𝑦 = 𝑡 → ((normℎ‘𝑦) + (normℎ‘ℎ)) = ((normℎ‘𝑡) + (normℎ‘ℎ)))
53		fvoveq1 7372	. . . . . . . . 9 ⊢ (𝑦 = 𝑡 → (normℎ‘(𝑦 +ℎ ℎ)) = (normℎ‘(𝑡 +ℎ ℎ)))
54	53	oveq2d 7365	. . . . . . . 8 ⊢ (𝑦 = 𝑡 → (𝑣 · (normℎ‘(𝑦 +ℎ ℎ))) = (𝑣 · (normℎ‘(𝑡 +ℎ ℎ))))
55	52, 54	breq12d 5105	. . . . . . 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 3211	. . . . . 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 623	. . 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 3076	. 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 31307	. . . . 5 ⊢ (𝐴 +ℋ 𝐵) = (𝐵 +ℋ 𝐴)
62	61	raleqi 3287	. . . 4 ⊢ (∀𝑢 ∈ (𝐴 +ℋ 𝐵)(normℎ‘(𝑇‘𝑢)) ≤ (𝑣 · (normℎ‘𝑢)) ↔ ∀𝑢 ∈ (𝐵 +ℋ 𝐴)(normℎ‘(𝑇‘𝑢)) ≤ (𝑣 · (normℎ‘𝑢)))
63	62	anbi2i 623	. . 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 3076	. 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 1540   ∈ wcel 2109  ∀wral 3044  ∃wrex 3053   class class class wbr 5092   ↦ cmpt 5173  ‘cfv 6482  ℩crio 7305  (class class class)co 7349  ℂcc 11007  ℝcr 11008  0cc0 11009   + caddc 11012   · cmul 11014   < clt 11149   ≤ cle 11150   ℋchba 30863   +_ℎ cva 30864  norm_ℎcno 30867   S_ℋ csh 30872   +_ℋ cph 30875

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-rep 5218  ax-sep 5235  ax-nul 5245  ax-pow 5304  ax-pr 5371  ax-un 7671  ax-cnex 11065  ax-resscn 11066  ax-1cn 11067  ax-icn 11068  ax-addcl 11069  ax-addrcl 11070  ax-mulcl 11071  ax-mulrcl 11072  ax-mulcom 11073  ax-addass 11074  ax-mulass 11075  ax-distr 11076  ax-i2m1 11077  ax-1ne0 11078  ax-1rid 11079  ax-rnegex 11080  ax-rrecex 11081  ax-cnre 11082  ax-pre-lttri 11083  ax-pre-lttrn 11084  ax-pre-ltadd 11085  ax-pre-mulgt0 11086  ax-pre-sup 11087  ax-hilex 30943  ax-hfvadd 30944  ax-hvcom 30945  ax-hvass 30946  ax-hv0cl 30947  ax-hvaddid 30948  ax-hfvmul 30949  ax-hvmulid 30950  ax-hvmulass 30951  ax-hvdistr1 30952  ax-hvdistr2 30953  ax-hvmul0 30954  ax-hfi 31023  ax-his1 31026  ax-his3 31028  ax-his4 31029

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 3343  df-reu 3344  df-rab 3395  df-v 3438  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-pss 3923  df-nul 4285  df-if 4477  df-pw 4553  df-sn 4578  df-pr 4580  df-op 4584  df-uni 4859  df-int 4897  df-iun 4943  df-br 5093  df-opab 5155  df-mpt 5174  df-tr 5200  df-id 5514  df-eprel 5519  df-po 5527  df-so 5528  df-fr 5572  df-we 5574  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-res 5631  df-ima 5632  df-pred 6249  df-ord 6310  df-on 6311  df-lim 6312  df-suc 6313  df-iota 6438  df-fun 6484  df-fn 6485  df-f 6486  df-f1 6487  df-fo 6488  df-f1o 6489  df-fv 6490  df-riota 7306  df-ov 7352  df-oprab 7353  df-mpo 7354  df-om 7800  df-2nd 7925  df-frecs 8214  df-wrecs 8245  df-recs 8294  df-rdg 8332  df-er 8625  df-en 8873  df-dom 8874  df-sdom 8875  df-sup 9332  df-pnf 11151  df-mnf 11152  df-xr 11153  df-ltxr 11154  df-le 11155  df-sub 11349  df-neg 11350  df-div 11778  df-nn 12129  df-2 12191  df-3 12192  df-n0 12385  df-z 12472  df-uz 12736  df-rp 12894  df-seq 13909  df-exp 13969  df-cj 15006  df-re 15007  df-im 15008  df-sqrt 15142  df-abs 15143  df-grpo 30437  df-ablo 30489  df-hnorm 30912  df-hvsub 30915  df-sh 31151  df-ch0 31197  df-shs 31252

This theorem is referenced by:  cdj3i  32385