Theorem unmbl 24701

Description: A union of measurable sets is measurable. (Contributed by Mario Carneiro, 18-Mar-2014.)

Assertion

Ref	Expression
unmbl	⊢ ((𝐴 ∈ dom vol ∧ 𝐵 ∈ dom vol) → (𝐴 ∪ 𝐵) ∈ dom vol)

Proof of Theorem unmbl

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		mblss 24695	. . . 4 ⊢ (𝐴 ∈ dom vol → 𝐴 ⊆ ℝ)
2		mblss 24695	. . . 4 ⊢ (𝐵 ∈ dom vol → 𝐵 ⊆ ℝ)
3	1, 2	anim12i 613	. . 3 ⊢ ((𝐴 ∈ dom vol ∧ 𝐵 ∈ dom vol) → (𝐴 ⊆ ℝ ∧ 𝐵 ⊆ ℝ))
4		unss 4118	. . 3 ⊢ ((𝐴 ⊆ ℝ ∧ 𝐵 ⊆ ℝ) ↔ (𝐴 ∪ 𝐵) ⊆ ℝ)
5	3, 4	sylib 217	. 2 ⊢ ((𝐴 ∈ dom vol ∧ 𝐵 ∈ dom vol) → (𝐴 ∪ 𝐵) ⊆ ℝ)
6		elpwi 4542	. . . 4 ⊢ (𝑥 ∈ 𝒫 ℝ → 𝑥 ⊆ ℝ)
7		inss1 4162	. . . . . . . . 9 ⊢ (𝑥 ∩ (𝐴 ∪ 𝐵)) ⊆ 𝑥
8		ovolsscl 24650	. . . . . . . . 9 ⊢ (((𝑥 ∩ (𝐴 ∪ 𝐵)) ⊆ 𝑥 ∧ 𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) → (vol*‘(𝑥 ∩ (𝐴 ∪ 𝐵))) ∈ ℝ)
9	7, 8	mp3an1 1447	. . . . . . . 8 ⊢ ((𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) → (vol*‘(𝑥 ∩ (𝐴 ∪ 𝐵))) ∈ ℝ)
10	9	adantl 482	. . . . . . 7 ⊢ (((𝐴 ∈ dom vol ∧ 𝐵 ∈ dom vol) ∧ (𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ)) → (vol*‘(𝑥 ∩ (𝐴 ∪ 𝐵))) ∈ ℝ)
11		inss1 4162	. . . . . . . . . 10 ⊢ (𝑥 ∩ 𝐴) ⊆ 𝑥
12		ovolsscl 24650	. . . . . . . . . 10 ⊢ (((𝑥 ∩ 𝐴) ⊆ 𝑥 ∧ 𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) → (vol*‘(𝑥 ∩ 𝐴)) ∈ ℝ)
13	11, 12	mp3an1 1447	. . . . . . . . 9 ⊢ ((𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) → (vol*‘(𝑥 ∩ 𝐴)) ∈ ℝ)
14	13	adantl 482	. . . . . . . 8 ⊢ (((𝐴 ∈ dom vol ∧ 𝐵 ∈ dom vol) ∧ (𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ)) → (vol*‘(𝑥 ∩ 𝐴)) ∈ ℝ)
15		inss1 4162	. . . . . . . . 9 ⊢ ((𝑥 ∖ 𝐴) ∩ 𝐵) ⊆ (𝑥 ∖ 𝐴)
16		difss 4066	. . . . . . . . . 10 ⊢ (𝑥 ∖ 𝐴) ⊆ 𝑥
17		simprl 768	. . . . . . . . . 10 ⊢ (((𝐴 ∈ dom vol ∧ 𝐵 ∈ dom vol) ∧ (𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ)) → 𝑥 ⊆ ℝ)
18	16, 17	sstrid 3932	. . . . . . . . 9 ⊢ (((𝐴 ∈ dom vol ∧ 𝐵 ∈ dom vol) ∧ (𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ)) → (𝑥 ∖ 𝐴) ⊆ ℝ)
19		ovolsscl 24650	. . . . . . . . . . 11 ⊢ (((𝑥 ∖ 𝐴) ⊆ 𝑥 ∧ 𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) → (vol*‘(𝑥 ∖ 𝐴)) ∈ ℝ)
20	16, 19	mp3an1 1447	. . . . . . . . . 10 ⊢ ((𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) → (vol*‘(𝑥 ∖ 𝐴)) ∈ ℝ)
21	20	adantl 482	. . . . . . . . 9 ⊢ (((𝐴 ∈ dom vol ∧ 𝐵 ∈ dom vol) ∧ (𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ)) → (vol*‘(𝑥 ∖ 𝐴)) ∈ ℝ)
22		ovolsscl 24650	. . . . . . . . 9 ⊢ ((((𝑥 ∖ 𝐴) ∩ 𝐵) ⊆ (𝑥 ∖ 𝐴) ∧ (𝑥 ∖ 𝐴) ⊆ ℝ ∧ (vol*‘(𝑥 ∖ 𝐴)) ∈ ℝ) → (vol*‘((𝑥 ∖ 𝐴) ∩ 𝐵)) ∈ ℝ)
23	15, 18, 21, 22	mp3an2i 1465	. . . . . . . 8 ⊢ (((𝐴 ∈ dom vol ∧ 𝐵 ∈ dom vol) ∧ (𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ)) → (vol*‘((𝑥 ∖ 𝐴) ∩ 𝐵)) ∈ ℝ)
24	14, 23	readdcld 11004	. . . . . . 7 ⊢ (((𝐴 ∈ dom vol ∧ 𝐵 ∈ dom vol) ∧ (𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ)) → ((vol*‘(𝑥 ∩ 𝐴)) + (vol*‘((𝑥 ∖ 𝐴) ∩ 𝐵))) ∈ ℝ)
25		difss 4066	. . . . . . . . 9 ⊢ (𝑥 ∖ (𝐴 ∪ 𝐵)) ⊆ 𝑥
26		ovolsscl 24650	. . . . . . . . 9 ⊢ (((𝑥 ∖ (𝐴 ∪ 𝐵)) ⊆ 𝑥 ∧ 𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) → (vol*‘(𝑥 ∖ (𝐴 ∪ 𝐵))) ∈ ℝ)
27	25, 26	mp3an1 1447	. . . . . . . 8 ⊢ ((𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) → (vol*‘(𝑥 ∖ (𝐴 ∪ 𝐵))) ∈ ℝ)
28	27	adantl 482	. . . . . . 7 ⊢ (((𝐴 ∈ dom vol ∧ 𝐵 ∈ dom vol) ∧ (𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ)) → (vol*‘(𝑥 ∖ (𝐴 ∪ 𝐵))) ∈ ℝ)
29		incom 4135	. . . . . . . . . . . 12 ⊢ ((𝑥 ∖ 𝐴) ∩ 𝐵) = (𝐵 ∩ (𝑥 ∖ 𝐴))
30		indifcom 4206	. . . . . . . . . . . 12 ⊢ (𝐵 ∩ (𝑥 ∖ 𝐴)) = (𝑥 ∩ (𝐵 ∖ 𝐴))
31	29, 30	eqtri 2766	. . . . . . . . . . 11 ⊢ ((𝑥 ∖ 𝐴) ∩ 𝐵) = (𝑥 ∩ (𝐵 ∖ 𝐴))
32	31	uneq2i 4094	. . . . . . . . . 10 ⊢ ((𝑥 ∩ 𝐴) ∪ ((𝑥 ∖ 𝐴) ∩ 𝐵)) = ((𝑥 ∩ 𝐴) ∪ (𝑥 ∩ (𝐵 ∖ 𝐴)))
33		indi 4207	. . . . . . . . . 10 ⊢ (𝑥 ∩ (𝐴 ∪ (𝐵 ∖ 𝐴))) = ((𝑥 ∩ 𝐴) ∪ (𝑥 ∩ (𝐵 ∖ 𝐴)))
34		undif2 4410	. . . . . . . . . . 11 ⊢ (𝐴 ∪ (𝐵 ∖ 𝐴)) = (𝐴 ∪ 𝐵)
35	34	ineq2i 4143	. . . . . . . . . 10 ⊢ (𝑥 ∩ (𝐴 ∪ (𝐵 ∖ 𝐴))) = (𝑥 ∩ (𝐴 ∪ 𝐵))
36	32, 33, 35	3eqtr2ri 2773	. . . . . . . . 9 ⊢ (𝑥 ∩ (𝐴 ∪ 𝐵)) = ((𝑥 ∩ 𝐴) ∪ ((𝑥 ∖ 𝐴) ∩ 𝐵))
37	36	fveq2i 6777	. . . . . . . 8 ⊢ (vol*‘(𝑥 ∩ (𝐴 ∪ 𝐵))) = (vol*‘((𝑥 ∩ 𝐴) ∪ ((𝑥 ∖ 𝐴) ∩ 𝐵)))
38	11, 17	sstrid 3932	. . . . . . . . 9 ⊢ (((𝐴 ∈ dom vol ∧ 𝐵 ∈ dom vol) ∧ (𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ)) → (𝑥 ∩ 𝐴) ⊆ ℝ)
39	15, 18	sstrid 3932	. . . . . . . . 9 ⊢ (((𝐴 ∈ dom vol ∧ 𝐵 ∈ dom vol) ∧ (𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ)) → ((𝑥 ∖ 𝐴) ∩ 𝐵) ⊆ ℝ)
40		ovolun 24663	. . . . . . . . 9 ⊢ ((((𝑥 ∩ 𝐴) ⊆ ℝ ∧ (vol*‘(𝑥 ∩ 𝐴)) ∈ ℝ) ∧ (((𝑥 ∖ 𝐴) ∩ 𝐵) ⊆ ℝ ∧ (vol*‘((𝑥 ∖ 𝐴) ∩ 𝐵)) ∈ ℝ)) → (vol*‘((𝑥 ∩ 𝐴) ∪ ((𝑥 ∖ 𝐴) ∩ 𝐵))) ≤ ((vol*‘(𝑥 ∩ 𝐴)) + (vol*‘((𝑥 ∖ 𝐴) ∩ 𝐵))))
41	38, 14, 39, 23, 40	syl22anc 836	. . . . . . . 8 ⊢ (((𝐴 ∈ dom vol ∧ 𝐵 ∈ dom vol) ∧ (𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ)) → (vol*‘((𝑥 ∩ 𝐴) ∪ ((𝑥 ∖ 𝐴) ∩ 𝐵))) ≤ ((vol*‘(𝑥 ∩ 𝐴)) + (vol*‘((𝑥 ∖ 𝐴) ∩ 𝐵))))
42	37, 41	eqbrtrid 5109	. . . . . . 7 ⊢ (((𝐴 ∈ dom vol ∧ 𝐵 ∈ dom vol) ∧ (𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ)) → (vol*‘(𝑥 ∩ (𝐴 ∪ 𝐵))) ≤ ((vol*‘(𝑥 ∩ 𝐴)) + (vol*‘((𝑥 ∖ 𝐴) ∩ 𝐵))))
43	10, 24, 28, 42	leadd1dd 11589	. . . . . 6 ⊢ (((𝐴 ∈ dom vol ∧ 𝐵 ∈ dom vol) ∧ (𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ)) → ((vol*‘(𝑥 ∩ (𝐴 ∪ 𝐵))) + (vol*‘(𝑥 ∖ (𝐴 ∪ 𝐵)))) ≤ (((vol*‘(𝑥 ∩ 𝐴)) + (vol*‘((𝑥 ∖ 𝐴) ∩ 𝐵))) + (vol*‘(𝑥 ∖ (𝐴 ∪ 𝐵)))))
44		simplr 766	. . . . . . . . . 10 ⊢ (((𝐴 ∈ dom vol ∧ 𝐵 ∈ dom vol) ∧ (𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ)) → 𝐵 ∈ dom vol)
45		mblsplit 24696	. . . . . . . . . 10 ⊢ ((𝐵 ∈ dom vol ∧ (𝑥 ∖ 𝐴) ⊆ ℝ ∧ (vol*‘(𝑥 ∖ 𝐴)) ∈ ℝ) → (vol*‘(𝑥 ∖ 𝐴)) = ((vol*‘((𝑥 ∖ 𝐴) ∩ 𝐵)) + (vol*‘((𝑥 ∖ 𝐴) ∖ 𝐵))))
46	44, 18, 21, 45	syl3anc 1370	. . . . . . . . 9 ⊢ (((𝐴 ∈ dom vol ∧ 𝐵 ∈ dom vol) ∧ (𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ)) → (vol*‘(𝑥 ∖ 𝐴)) = ((vol*‘((𝑥 ∖ 𝐴) ∩ 𝐵)) + (vol*‘((𝑥 ∖ 𝐴) ∖ 𝐵))))
47		difun1 4223	. . . . . . . . . . 11 ⊢ (𝑥 ∖ (𝐴 ∪ 𝐵)) = ((𝑥 ∖ 𝐴) ∖ 𝐵)
48	47	fveq2i 6777	. . . . . . . . . 10 ⊢ (vol*‘(𝑥 ∖ (𝐴 ∪ 𝐵))) = (vol*‘((𝑥 ∖ 𝐴) ∖ 𝐵))
49	48	oveq2i 7286	. . . . . . . . 9 ⊢ ((vol*‘((𝑥 ∖ 𝐴) ∩ 𝐵)) + (vol*‘(𝑥 ∖ (𝐴 ∪ 𝐵)))) = ((vol*‘((𝑥 ∖ 𝐴) ∩ 𝐵)) + (vol*‘((𝑥 ∖ 𝐴) ∖ 𝐵)))
50	46, 49	eqtr4di 2796	. . . . . . . 8 ⊢ (((𝐴 ∈ dom vol ∧ 𝐵 ∈ dom vol) ∧ (𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ)) → (vol*‘(𝑥 ∖ 𝐴)) = ((vol*‘((𝑥 ∖ 𝐴) ∩ 𝐵)) + (vol*‘(𝑥 ∖ (𝐴 ∪ 𝐵)))))
51	50	oveq2d 7291	. . . . . . 7 ⊢ (((𝐴 ∈ dom vol ∧ 𝐵 ∈ dom vol) ∧ (𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ)) → ((vol*‘(𝑥 ∩ 𝐴)) + (vol*‘(𝑥 ∖ 𝐴))) = ((vol*‘(𝑥 ∩ 𝐴)) + ((vol*‘((𝑥 ∖ 𝐴) ∩ 𝐵)) + (vol*‘(𝑥 ∖ (𝐴 ∪ 𝐵))))))
52		simpll 764	. . . . . . . 8 ⊢ (((𝐴 ∈ dom vol ∧ 𝐵 ∈ dom vol) ∧ (𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ)) → 𝐴 ∈ dom vol)
53		simprr 770	. . . . . . . 8 ⊢ (((𝐴 ∈ dom vol ∧ 𝐵 ∈ dom vol) ∧ (𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ)) → (vol*‘𝑥) ∈ ℝ)
54		mblsplit 24696	. . . . . . . 8 ⊢ ((𝐴 ∈ dom vol ∧ 𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) → (vol*‘𝑥) = ((vol*‘(𝑥 ∩ 𝐴)) + (vol*‘(𝑥 ∖ 𝐴))))
55	52, 17, 53, 54	syl3anc 1370	. . . . . . 7 ⊢ (((𝐴 ∈ dom vol ∧ 𝐵 ∈ dom vol) ∧ (𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ)) → (vol*‘𝑥) = ((vol*‘(𝑥 ∩ 𝐴)) + (vol*‘(𝑥 ∖ 𝐴))))
56	14	recnd 11003	. . . . . . . 8 ⊢ (((𝐴 ∈ dom vol ∧ 𝐵 ∈ dom vol) ∧ (𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ)) → (vol*‘(𝑥 ∩ 𝐴)) ∈ ℂ)
57	23	recnd 11003	. . . . . . . 8 ⊢ (((𝐴 ∈ dom vol ∧ 𝐵 ∈ dom vol) ∧ (𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ)) → (vol*‘((𝑥 ∖ 𝐴) ∩ 𝐵)) ∈ ℂ)
58	28	recnd 11003	. . . . . . . 8 ⊢ (((𝐴 ∈ dom vol ∧ 𝐵 ∈ dom vol) ∧ (𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ)) → (vol*‘(𝑥 ∖ (𝐴 ∪ 𝐵))) ∈ ℂ)
59	56, 57, 58	addassd 10997	. . . . . . 7 ⊢ (((𝐴 ∈ dom vol ∧ 𝐵 ∈ dom vol) ∧ (𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ)) → (((vol*‘(𝑥 ∩ 𝐴)) + (vol*‘((𝑥 ∖ 𝐴) ∩ 𝐵))) + (vol*‘(𝑥 ∖ (𝐴 ∪ 𝐵)))) = ((vol*‘(𝑥 ∩ 𝐴)) + ((vol*‘((𝑥 ∖ 𝐴) ∩ 𝐵)) + (vol*‘(𝑥 ∖ (𝐴 ∪ 𝐵))))))
60	51, 55, 59	3eqtr4d 2788	. . . . . 6 ⊢ (((𝐴 ∈ dom vol ∧ 𝐵 ∈ dom vol) ∧ (𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ)) → (vol*‘𝑥) = (((vol*‘(𝑥 ∩ 𝐴)) + (vol*‘((𝑥 ∖ 𝐴) ∩ 𝐵))) + (vol*‘(𝑥 ∖ (𝐴 ∪ 𝐵)))))
61	43, 60	breqtrrd 5102	. . . . 5 ⊢ (((𝐴 ∈ dom vol ∧ 𝐵 ∈ dom vol) ∧ (𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ)) → ((vol*‘(𝑥 ∩ (𝐴 ∪ 𝐵))) + (vol*‘(𝑥 ∖ (𝐴 ∪ 𝐵)))) ≤ (vol*‘𝑥))
62	61	expr 457	. . . 4 ⊢ (((𝐴 ∈ dom vol ∧ 𝐵 ∈ dom vol) ∧ 𝑥 ⊆ ℝ) → ((vol*‘𝑥) ∈ ℝ → ((vol*‘(𝑥 ∩ (𝐴 ∪ 𝐵))) + (vol*‘(𝑥 ∖ (𝐴 ∪ 𝐵)))) ≤ (vol*‘𝑥)))
63	6, 62	sylan2 593	. . 3 ⊢ (((𝐴 ∈ dom vol ∧ 𝐵 ∈ dom vol) ∧ 𝑥 ∈ 𝒫 ℝ) → ((vol*‘𝑥) ∈ ℝ → ((vol*‘(𝑥 ∩ (𝐴 ∪ 𝐵))) + (vol*‘(𝑥 ∖ (𝐴 ∪ 𝐵)))) ≤ (vol*‘𝑥)))
64	63	ralrimiva 3103	. 2 ⊢ ((𝐴 ∈ dom vol ∧ 𝐵 ∈ dom vol) → ∀𝑥 ∈ 𝒫 ℝ((vol*‘𝑥) ∈ ℝ → ((vol*‘(𝑥 ∩ (𝐴 ∪ 𝐵))) + (vol*‘(𝑥 ∖ (𝐴 ∪ 𝐵)))) ≤ (vol*‘𝑥)))
65		ismbl2 24691	. 2 ⊢ ((𝐴 ∪ 𝐵) ∈ dom vol ↔ ((𝐴 ∪ 𝐵) ⊆ ℝ ∧ ∀𝑥 ∈ 𝒫 ℝ((vol*‘𝑥) ∈ ℝ → ((vol*‘(𝑥 ∩ (𝐴 ∪ 𝐵))) + (vol*‘(𝑥 ∖ (𝐴 ∪ 𝐵)))) ≤ (vol*‘𝑥))))
66	5, 64, 65	sylanbrc 583	1 ⊢ ((𝐴 ∈ dom vol ∧ 𝐵 ∈ dom vol) → (𝐴 ∪ 𝐵) ∈ dom vol)

Syntax hints:   → wi 4   ∧ wa 396   = wceq 1539   ∈ wcel 2106  ∀wral 3064   ∖ cdif 3884   ∪ cun 3885   ∩ cin 3886   ⊆ wss 3887  𝒫 cpw 4533   class class class wbr 5074  dom cdm 5589  ‘cfv 6433  (class class class)co 7275  ℝcr 10870   + caddc 10874   ≤ cle 11010  vol*covol 24626  volcvol 24627

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  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 2709  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352  ax-un 7588  ax-cnex 10927  ax-resscn 10928  ax-1cn 10929  ax-icn 10930  ax-addcl 10931  ax-addrcl 10932  ax-mulcl 10933  ax-mulrcl 10934  ax-mulcom 10935  ax-addass 10936  ax-mulass 10937  ax-distr 10938  ax-i2m1 10939  ax-1ne0 10940  ax-1rid 10941  ax-rnegex 10942  ax-rrecex 10943  ax-cnre 10944  ax-pre-lttri 10945  ax-pre-lttrn 10946  ax-pre-ltadd 10947  ax-pre-mulgt0 10948  ax-pre-sup 10949

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3069  df-rex 3070  df-rmo 3071  df-reu 3072  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-pss 3906  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-iun 4926  df-br 5075  df-opab 5137  df-mpt 5158  df-tr 5192  df-id 5489  df-eprel 5495  df-po 5503  df-so 5504  df-fr 5544  df-we 5546  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-pred 6202  df-ord 6269  df-on 6270  df-lim 6271  df-suc 6272  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438  df-fo 6439  df-f1o 6440  df-fv 6441  df-riota 7232  df-ov 7278  df-oprab 7279  df-mpo 7280  df-om 7713  df-1st 7831  df-2nd 7832  df-frecs 8097  df-wrecs 8128  df-recs 8202  df-rdg 8241  df-er 8498  df-map 8617  df-en 8734  df-dom 8735  df-sdom 8736  df-sup 9201  df-inf 9202  df-pnf 11011  df-mnf 11012  df-xr 11013  df-ltxr 11014  df-le 11015  df-sub 11207  df-neg 11208  df-div 11633  df-nn 11974  df-2 12036  df-3 12037  df-n0 12234  df-z 12320  df-uz 12583  df-q 12689  df-rp 12731  df-ioo 13083  df-ico 13085  df-icc 13086  df-fz 13240  df-fl 13512  df-seq 13722  df-exp 13783  df-cj 14810  df-re 14811  df-im 14812  df-sqrt 14946  df-abs 14947  df-ovol 24628  df-vol 24629

This theorem is referenced by:  inmbl  24706  finiunmbl  24708  volun  24709  voliunlem1  24714  icombl1  24727  iccmbl  24730  uniiccmbl  24754  mbfimaicc  24795  mbfeqalem2  24806  mbfres2  24809  mbfmax  24813  itgss3  24979  ismblfin  35818  mbfposadd  35824  cnambfre  35825  itg2addnclem2  35829  iblabsnclem  35840  ftc1anclem1  35850  ftc1anclem5  35854  iocmbl  41044