Theorem cdj3lem3b 31382

Description: Lemma for cdj3i 31383. 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 30305	. . . . 5 ⊢ (𝐵 +ℋ 𝐴) = (𝐴 +ℋ 𝐵)
5	1	sheli 30156	. . . . . . . . 9 ⊢ (𝑤 ∈ 𝐵 → 𝑤 ∈ ℋ)
6	2	sheli 30156	. . . . . . . . 9 ⊢ (𝑧 ∈ 𝐴 → 𝑧 ∈ ℋ)
7		ax-hvcom 29943	. . . . . . . . 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 2747	. . . . . . 7 ⊢ ((𝑤 ∈ 𝐵 ∧ 𝑧 ∈ 𝐴) → (𝑥 = (𝑤 +ℎ 𝑧) ↔ 𝑥 = (𝑧 +ℎ 𝑤)))
10	9	rexbidva 3173	. . . . . 6 ⊢ (𝑤 ∈ 𝐵 → (∃𝑧 ∈ 𝐴 𝑥 = (𝑤 +ℎ 𝑧) ↔ ∃𝑧 ∈ 𝐴 𝑥 = (𝑧 +ℎ 𝑤)))
11	10	riotabiia 7334	. . . . 5 ⊢ (℩𝑤 ∈ 𝐵 ∃𝑧 ∈ 𝐴 𝑥 = (𝑤 +ℎ 𝑧)) = (℩𝑤 ∈ 𝐵 ∃𝑧 ∈ 𝐴 𝑥 = (𝑧 +ℎ 𝑤))
12	4, 11	mpteq12i 5211	. . . 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 2767	. . 3 ⊢ 𝑇 = (𝑥 ∈ (𝐵 +ℋ 𝐴) ↦ (℩𝑤 ∈ 𝐵 ∃𝑧 ∈ 𝐴 𝑥 = (𝑤 +ℎ 𝑧)))
14	1, 2, 13	cdj3lem2b 31379	. 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 6842	. . . . . . . 8 ⊢ (𝑥 = 𝑡 → (normℎ‘𝑥) = (normℎ‘𝑡))
16	15	oveq1d 7372	. . . . . . 7 ⊢ (𝑥 = 𝑡 → ((normℎ‘𝑥) + (normℎ‘𝑦)) = ((normℎ‘𝑡) + (normℎ‘𝑦)))
17		fvoveq1 7380	. . . . . . . 8 ⊢ (𝑥 = 𝑡 → (normℎ‘(𝑥 +ℎ 𝑦)) = (normℎ‘(𝑡 +ℎ 𝑦)))
18	17	oveq2d 7373	. . . . . . 7 ⊢ (𝑥 = 𝑡 → (𝑣 · (normℎ‘(𝑥 +ℎ 𝑦))) = (𝑣 · (normℎ‘(𝑡 +ℎ 𝑦))))
19	16, 18	breq12d 5118	. . . . . 6 ⊢ (𝑥 = 𝑡 → (((normℎ‘𝑥) + (normℎ‘𝑦)) ≤ (𝑣 · (normℎ‘(𝑥 +ℎ 𝑦))) ↔ ((normℎ‘𝑡) + (normℎ‘𝑦)) ≤ (𝑣 · (normℎ‘(𝑡 +ℎ 𝑦)))))
20		fveq2 6842	. . . . . . . 8 ⊢ (𝑦 = ℎ → (normℎ‘𝑦) = (normℎ‘ℎ))
21	20	oveq2d 7373	. . . . . . 7 ⊢ (𝑦 = ℎ → ((normℎ‘𝑡) + (normℎ‘𝑦)) = ((normℎ‘𝑡) + (normℎ‘ℎ)))
22		oveq2 7365	. . . . . . . . 9 ⊢ (𝑦 = ℎ → (𝑡 +ℎ 𝑦) = (𝑡 +ℎ ℎ))
23	22	fveq2d 6846	. . . . . . . 8 ⊢ (𝑦 = ℎ → (normℎ‘(𝑡 +ℎ 𝑦)) = (normℎ‘(𝑡 +ℎ ℎ)))
24	23	oveq2d 7373	. . . . . . 7 ⊢ (𝑦 = ℎ → (𝑣 · (normℎ‘(𝑡 +ℎ 𝑦))) = (𝑣 · (normℎ‘(𝑡 +ℎ ℎ))))
25	21, 24	breq12d 5118	. . . . . 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 3227	. . . . 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 3272	. . . . 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 30156	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ 𝐵 → 𝑥 ∈ ℋ)
29		normcl 30067	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ ℋ → (normℎ‘𝑥) ∈ ℝ)
30	28, 29	syl 17	. . . . . . . . . . 11 ⊢ (𝑥 ∈ 𝐵 → (normℎ‘𝑥) ∈ ℝ)
31	30	recnd 11183	. . . . . . . . . 10 ⊢ (𝑥 ∈ 𝐵 → (normℎ‘𝑥) ∈ ℂ)
32	2	sheli 30156	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ 𝐴 → 𝑦 ∈ ℋ)
33		normcl 30067	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ ℋ → (normℎ‘𝑦) ∈ ℝ)
34	32, 33	syl 17	. . . . . . . . . . 11 ⊢ (𝑦 ∈ 𝐴 → (normℎ‘𝑦) ∈ ℝ)
35	34	recnd 11183	. . . . . . . . . 10 ⊢ (𝑦 ∈ 𝐴 → (normℎ‘𝑦) ∈ ℂ)
36		addcom 11341	. . . . . . . . . 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 29943	. . . . . . . . . . . 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 6846	. . . . . . . . . 10 ⊢ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐴) → (normℎ‘(𝑥 +ℎ 𝑦)) = (normℎ‘(𝑦 +ℎ 𝑥)))
41	40	oveq2d 7373	. . . . . . . . 9 ⊢ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐴) → (𝑣 · (normℎ‘(𝑥 +ℎ 𝑦))) = (𝑣 · (normℎ‘(𝑦 +ℎ 𝑥))))
42	37, 41	breq12d 5118	. . . . . . . 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 3172	. . . . . . 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 3094	. . . . . 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 6842	. . . . . . . . 9 ⊢ (𝑥 = ℎ → (normℎ‘𝑥) = (normℎ‘ℎ))
46	45	oveq2d 7373	. . . . . . . 8 ⊢ (𝑥 = ℎ → ((normℎ‘𝑦) + (normℎ‘𝑥)) = ((normℎ‘𝑦) + (normℎ‘ℎ)))
47		oveq2 7365	. . . . . . . . . 10 ⊢ (𝑥 = ℎ → (𝑦 +ℎ 𝑥) = (𝑦 +ℎ ℎ))
48	47	fveq2d 6846	. . . . . . . . 9 ⊢ (𝑥 = ℎ → (normℎ‘(𝑦 +ℎ 𝑥)) = (normℎ‘(𝑦 +ℎ ℎ)))
49	48	oveq2d 7373	. . . . . . . 8 ⊢ (𝑥 = ℎ → (𝑣 · (normℎ‘(𝑦 +ℎ 𝑥))) = (𝑣 · (normℎ‘(𝑦 +ℎ ℎ))))
50	46, 49	breq12d 5118	. . . . . . 7 ⊢ (𝑥 = ℎ → (((normℎ‘𝑦) + (normℎ‘𝑥)) ≤ (𝑣 · (normℎ‘(𝑦 +ℎ 𝑥))) ↔ ((normℎ‘𝑦) + (normℎ‘ℎ)) ≤ (𝑣 · (normℎ‘(𝑦 +ℎ ℎ)))))
51		fveq2 6842	. . . . . . . . 9 ⊢ (𝑦 = 𝑡 → (normℎ‘𝑦) = (normℎ‘𝑡))
52	51	oveq1d 7372	. . . . . . . 8 ⊢ (𝑦 = 𝑡 → ((normℎ‘𝑦) + (normℎ‘ℎ)) = ((normℎ‘𝑡) + (normℎ‘ℎ)))
53		fvoveq1 7380	. . . . . . . . 9 ⊢ (𝑦 = 𝑡 → (normℎ‘(𝑦 +ℎ ℎ)) = (normℎ‘(𝑡 +ℎ ℎ)))
54	53	oveq2d 7373	. . . . . . . 8 ⊢ (𝑦 = 𝑡 → (𝑣 · (normℎ‘(𝑦 +ℎ ℎ))) = (𝑣 · (normℎ‘(𝑡 +ℎ ℎ))))
55	52, 54	breq12d 5118	. . . . . . 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 3227	. . . . . 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 275	. . . . 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 296	. . . 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 3097	. 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 30305	. . . . 5 ⊢ (𝐴 +ℋ 𝐵) = (𝐵 +ℋ 𝐴)
62	61	raleqi 3311	. . . 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 3097	. 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 291	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 396   = wceq 1541   ∈ wcel 2106  ∀wral 3064  ∃wrex 3073   class class class wbr 5105   ↦ cmpt 5188  ‘cfv 6496  ℩crio 7312  (class class class)co 7357  ℂcc 11049  ℝcr 11050  0cc0 11051   + caddc 11054   · cmul 11056   < clt 11189   ≤ cle 11190   ℋchba 29861   +_ℎ cva 29862  norm_ℎcno 29865   S_ℋ csh 29870   +_ℋ cph 29873

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-rep 5242  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672  ax-cnex 11107  ax-resscn 11108  ax-1cn 11109  ax-icn 11110  ax-addcl 11111  ax-addrcl 11112  ax-mulcl 11113  ax-mulrcl 11114  ax-mulcom 11115  ax-addass 11116  ax-mulass 11117  ax-distr 11118  ax-i2m1 11119  ax-1ne0 11120  ax-1rid 11121  ax-rnegex 11122  ax-rrecex 11123  ax-cnre 11124  ax-pre-lttri 11125  ax-pre-lttrn 11126  ax-pre-ltadd 11127  ax-pre-mulgt0 11128  ax-pre-sup 11129  ax-hilex 29941  ax-hfvadd 29942  ax-hvcom 29943  ax-hvass 29944  ax-hv0cl 29945  ax-hvaddid 29946  ax-hfvmul 29947  ax-hvmulid 29948  ax-hvmulass 29949  ax-hvdistr1 29950  ax-hvdistr2 29951  ax-hvmul0 29952  ax-hfi 30021  ax-his1 30024  ax-his3 30026  ax-his4 30027

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3065  df-rex 3074  df-rmo 3353  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-int 4908  df-iun 4956  df-br 5106  df-opab 5168  df-mpt 5189  df-tr 5223  df-id 5531  df-eprel 5537  df-po 5545  df-so 5546  df-fr 5588  df-we 5590  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-pred 6253  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-riota 7313  df-ov 7360  df-oprab 7361  df-mpo 7362  df-om 7803  df-2nd 7922  df-frecs 8212  df-wrecs 8243  df-recs 8317  df-rdg 8356  df-er 8648  df-en 8884  df-dom 8885  df-sdom 8886  df-sup 9378  df-pnf 11191  df-mnf 11192  df-xr 11193  df-ltxr 11194  df-le 11195  df-sub 11387  df-neg 11388  df-div 11813  df-nn 12154  df-2 12216  df-3 12217  df-n0 12414  df-z 12500  df-uz 12764  df-rp 12916  df-seq 13907  df-exp 13968  df-cj 14984  df-re 14985  df-im 14986  df-sqrt 15120  df-abs 15121  df-grpo 29435  df-ablo 29487  df-hnorm 29910  df-hvsub 29913  df-sh 30149  df-ch0 30195  df-shs 30250

This theorem is referenced by:  cdj3i  31383