Theorem unmbl 25466

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 25460	. . . 4 ⊢ (𝐴 ∈ dom vol → 𝐴 ⊆ ℝ)
2		mblss 25460	. . . 4 ⊢ (𝐵 ∈ dom vol → 𝐵 ⊆ ℝ)
3	1, 2	anim12i 613	. . 3 ⊢ ((𝐴 ∈ dom vol ∧ 𝐵 ∈ dom vol) → (𝐴 ⊆ ℝ ∧ 𝐵 ⊆ ℝ))
4		unss 4139	. . 3 ⊢ ((𝐴 ⊆ ℝ ∧ 𝐵 ⊆ ℝ) ↔ (𝐴 ∪ 𝐵) ⊆ ℝ)
5	3, 4	sylib 218	. 2 ⊢ ((𝐴 ∈ dom vol ∧ 𝐵 ∈ dom vol) → (𝐴 ∪ 𝐵) ⊆ ℝ)
6		elpwi 4556	. . . 4 ⊢ (𝑥 ∈ 𝒫 ℝ → 𝑥 ⊆ ℝ)
7		inss1 4186	. . . . . . . . 9 ⊢ (𝑥 ∩ (𝐴 ∪ 𝐵)) ⊆ 𝑥
8		ovolsscl 25415	. . . . . . . . 9 ⊢ (((𝑥 ∩ (𝐴 ∪ 𝐵)) ⊆ 𝑥 ∧ 𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) → (vol*‘(𝑥 ∩ (𝐴 ∪ 𝐵))) ∈ ℝ)
9	7, 8	mp3an1 1450	. . . . . . . 8 ⊢ ((𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) → (vol*‘(𝑥 ∩ (𝐴 ∪ 𝐵))) ∈ ℝ)
10	9	adantl 481	. . . . . . 7 ⊢ (((𝐴 ∈ dom vol ∧ 𝐵 ∈ dom vol) ∧ (𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ)) → (vol*‘(𝑥 ∩ (𝐴 ∪ 𝐵))) ∈ ℝ)
11		inss1 4186	. . . . . . . . . 10 ⊢ (𝑥 ∩ 𝐴) ⊆ 𝑥
12		ovolsscl 25415	. . . . . . . . . 10 ⊢ (((𝑥 ∩ 𝐴) ⊆ 𝑥 ∧ 𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) → (vol*‘(𝑥 ∩ 𝐴)) ∈ ℝ)
13	11, 12	mp3an1 1450	. . . . . . . . 9 ⊢ ((𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) → (vol*‘(𝑥 ∩ 𝐴)) ∈ ℝ)
14	13	adantl 481	. . . . . . . 8 ⊢ (((𝐴 ∈ dom vol ∧ 𝐵 ∈ dom vol) ∧ (𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ)) → (vol*‘(𝑥 ∩ 𝐴)) ∈ ℝ)
15		inss1 4186	. . . . . . . . 9 ⊢ ((𝑥 ∖ 𝐴) ∩ 𝐵) ⊆ (𝑥 ∖ 𝐴)
16		difss 4085	. . . . . . . . . 10 ⊢ (𝑥 ∖ 𝐴) ⊆ 𝑥
17		simprl 770	. . . . . . . . . 10 ⊢ (((𝐴 ∈ dom vol ∧ 𝐵 ∈ dom vol) ∧ (𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ)) → 𝑥 ⊆ ℝ)
18	16, 17	sstrid 3942	. . . . . . . . 9 ⊢ (((𝐴 ∈ dom vol ∧ 𝐵 ∈ dom vol) ∧ (𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ)) → (𝑥 ∖ 𝐴) ⊆ ℝ)
19		ovolsscl 25415	. . . . . . . . . . 11 ⊢ (((𝑥 ∖ 𝐴) ⊆ 𝑥 ∧ 𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) → (vol*‘(𝑥 ∖ 𝐴)) ∈ ℝ)
20	16, 19	mp3an1 1450	. . . . . . . . . 10 ⊢ ((𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) → (vol*‘(𝑥 ∖ 𝐴)) ∈ ℝ)
21	20	adantl 481	. . . . . . . . 9 ⊢ (((𝐴 ∈ dom vol ∧ 𝐵 ∈ dom vol) ∧ (𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ)) → (vol*‘(𝑥 ∖ 𝐴)) ∈ ℝ)
22		ovolsscl 25415	. . . . . . . . 9 ⊢ ((((𝑥 ∖ 𝐴) ∩ 𝐵) ⊆ (𝑥 ∖ 𝐴) ∧ (𝑥 ∖ 𝐴) ⊆ ℝ ∧ (vol*‘(𝑥 ∖ 𝐴)) ∈ ℝ) → (vol*‘((𝑥 ∖ 𝐴) ∩ 𝐵)) ∈ ℝ)
23	15, 18, 21, 22	mp3an2i 1468	. . . . . . . 8 ⊢ (((𝐴 ∈ dom vol ∧ 𝐵 ∈ dom vol) ∧ (𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ)) → (vol*‘((𝑥 ∖ 𝐴) ∩ 𝐵)) ∈ ℝ)
24	14, 23	readdcld 11148	. . . . . . 7 ⊢ (((𝐴 ∈ dom vol ∧ 𝐵 ∈ dom vol) ∧ (𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ)) → ((vol*‘(𝑥 ∩ 𝐴)) + (vol*‘((𝑥 ∖ 𝐴) ∩ 𝐵))) ∈ ℝ)
25		difss 4085	. . . . . . . . 9 ⊢ (𝑥 ∖ (𝐴 ∪ 𝐵)) ⊆ 𝑥
26		ovolsscl 25415	. . . . . . . . 9 ⊢ (((𝑥 ∖ (𝐴 ∪ 𝐵)) ⊆ 𝑥 ∧ 𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) → (vol*‘(𝑥 ∖ (𝐴 ∪ 𝐵))) ∈ ℝ)
27	25, 26	mp3an1 1450	. . . . . . . 8 ⊢ ((𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) → (vol*‘(𝑥 ∖ (𝐴 ∪ 𝐵))) ∈ ℝ)
28	27	adantl 481	. . . . . . 7 ⊢ (((𝐴 ∈ dom vol ∧ 𝐵 ∈ dom vol) ∧ (𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ)) → (vol*‘(𝑥 ∖ (𝐴 ∪ 𝐵))) ∈ ℝ)
29		incom 4158	. . . . . . . . . . . 12 ⊢ ((𝑥 ∖ 𝐴) ∩ 𝐵) = (𝐵 ∩ (𝑥 ∖ 𝐴))
30		indifcom 4232	. . . . . . . . . . . 12 ⊢ (𝐵 ∩ (𝑥 ∖ 𝐴)) = (𝑥 ∩ (𝐵 ∖ 𝐴))
31	29, 30	eqtri 2756	. . . . . . . . . . 11 ⊢ ((𝑥 ∖ 𝐴) ∩ 𝐵) = (𝑥 ∩ (𝐵 ∖ 𝐴))
32	31	uneq2i 4114	. . . . . . . . . 10 ⊢ ((𝑥 ∩ 𝐴) ∪ ((𝑥 ∖ 𝐴) ∩ 𝐵)) = ((𝑥 ∩ 𝐴) ∪ (𝑥 ∩ (𝐵 ∖ 𝐴)))
33		indi 4233	. . . . . . . . . 10 ⊢ (𝑥 ∩ (𝐴 ∪ (𝐵 ∖ 𝐴))) = ((𝑥 ∩ 𝐴) ∪ (𝑥 ∩ (𝐵 ∖ 𝐴)))
34		undif2 4426	. . . . . . . . . . 11 ⊢ (𝐴 ∪ (𝐵 ∖ 𝐴)) = (𝐴 ∪ 𝐵)
35	34	ineq2i 4166	. . . . . . . . . 10 ⊢ (𝑥 ∩ (𝐴 ∪ (𝐵 ∖ 𝐴))) = (𝑥 ∩ (𝐴 ∪ 𝐵))
36	32, 33, 35	3eqtr2ri 2763	. . . . . . . . 9 ⊢ (𝑥 ∩ (𝐴 ∪ 𝐵)) = ((𝑥 ∩ 𝐴) ∪ ((𝑥 ∖ 𝐴) ∩ 𝐵))
37	36	fveq2i 6831	. . . . . . . 8 ⊢ (vol*‘(𝑥 ∩ (𝐴 ∪ 𝐵))) = (vol*‘((𝑥 ∩ 𝐴) ∪ ((𝑥 ∖ 𝐴) ∩ 𝐵)))
38	11, 17	sstrid 3942	. . . . . . . . 9 ⊢ (((𝐴 ∈ dom vol ∧ 𝐵 ∈ dom vol) ∧ (𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ)) → (𝑥 ∩ 𝐴) ⊆ ℝ)
39	15, 18	sstrid 3942	. . . . . . . . 9 ⊢ (((𝐴 ∈ dom vol ∧ 𝐵 ∈ dom vol) ∧ (𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ)) → ((𝑥 ∖ 𝐴) ∩ 𝐵) ⊆ ℝ)
40		ovolun 25428	. . . . . . . . 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 838	. . . . . . . 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 5128	. . . . . . 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 11738	. . . . . 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 768	. . . . . . . . . 10 ⊢ (((𝐴 ∈ dom vol ∧ 𝐵 ∈ dom vol) ∧ (𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ)) → 𝐵 ∈ dom vol)
45		mblsplit 25461	. . . . . . . . . 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 1373	. . . . . . . . 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 4248	. . . . . . . . . . 11 ⊢ (𝑥 ∖ (𝐴 ∪ 𝐵)) = ((𝑥 ∖ 𝐴) ∖ 𝐵)
48	47	fveq2i 6831	. . . . . . . . . 10 ⊢ (vol*‘(𝑥 ∖ (𝐴 ∪ 𝐵))) = (vol*‘((𝑥 ∖ 𝐴) ∖ 𝐵))
49	48	oveq2i 7363	. . . . . . . . 9 ⊢ ((vol*‘((𝑥 ∖ 𝐴) ∩ 𝐵)) + (vol*‘(𝑥 ∖ (𝐴 ∪ 𝐵)))) = ((vol*‘((𝑥 ∖ 𝐴) ∩ 𝐵)) + (vol*‘((𝑥 ∖ 𝐴) ∖ 𝐵)))
50	46, 49	eqtr4di 2786	. . . . . . . 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 7368	. . . . . . 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 766	. . . . . . . 8 ⊢ (((𝐴 ∈ dom vol ∧ 𝐵 ∈ dom vol) ∧ (𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ)) → 𝐴 ∈ dom vol)
53		simprr 772	. . . . . . . 8 ⊢ (((𝐴 ∈ dom vol ∧ 𝐵 ∈ dom vol) ∧ (𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ)) → (vol*‘𝑥) ∈ ℝ)
54		mblsplit 25461	. . . . . . . 8 ⊢ ((𝐴 ∈ dom vol ∧ 𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) → (vol*‘𝑥) = ((vol*‘(𝑥 ∩ 𝐴)) + (vol*‘(𝑥 ∖ 𝐴))))
55	52, 17, 53, 54	syl3anc 1373	. . . . . . 7 ⊢ (((𝐴 ∈ dom vol ∧ 𝐵 ∈ dom vol) ∧ (𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ)) → (vol*‘𝑥) = ((vol*‘(𝑥 ∩ 𝐴)) + (vol*‘(𝑥 ∖ 𝐴))))
56	14	recnd 11147	. . . . . . . 8 ⊢ (((𝐴 ∈ dom vol ∧ 𝐵 ∈ dom vol) ∧ (𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ)) → (vol*‘(𝑥 ∩ 𝐴)) ∈ ℂ)
57	23	recnd 11147	. . . . . . . 8 ⊢ (((𝐴 ∈ dom vol ∧ 𝐵 ∈ dom vol) ∧ (𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ)) → (vol*‘((𝑥 ∖ 𝐴) ∩ 𝐵)) ∈ ℂ)
58	28	recnd 11147	. . . . . . . 8 ⊢ (((𝐴 ∈ dom vol ∧ 𝐵 ∈ dom vol) ∧ (𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ)) → (vol*‘(𝑥 ∖ (𝐴 ∪ 𝐵))) ∈ ℂ)
59	56, 57, 58	addassd 11141	. . . . . . 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 2778	. . . . . 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 5121	. . . . 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 456	. . . 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 3125	. 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 25456	. 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 395   = wceq 1541   ∈ wcel 2113  ∀wral 3048   ∖ cdif 3895   ∪ cun 3896   ∩ cin 3897   ⊆ wss 3898  𝒫 cpw 4549   class class class wbr 5093  dom cdm 5619  ‘cfv 6486  (class class class)co 7352  ℝcr 11012   + caddc 11016   ≤ cle 11154  vol*covol 25391  volcvol 25392

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-ext 2705  ax-sep 5236  ax-nul 5246  ax-pow 5305  ax-pr 5372  ax-un 7674  ax-cnex 11069  ax-resscn 11070  ax-1cn 11071  ax-icn 11072  ax-addcl 11073  ax-addrcl 11074  ax-mulcl 11075  ax-mulrcl 11076  ax-mulcom 11077  ax-addass 11078  ax-mulass 11079  ax-distr 11080  ax-i2m1 11081  ax-1ne0 11082  ax-1rid 11083  ax-rnegex 11084  ax-rrecex 11085  ax-cnre 11086  ax-pre-lttri 11087  ax-pre-lttrn 11088  ax-pre-ltadd 11089  ax-pre-mulgt0 11090  ax-pre-sup 11091

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2725  df-clel 2808  df-nfc 2882  df-ne 2930  df-nel 3034  df-ral 3049  df-rex 3058  df-rmo 3347  df-reu 3348  df-rab 3397  df-v 3439  df-sbc 3738  df-csb 3847  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-pss 3918  df-nul 4283  df-if 4475  df-pw 4551  df-sn 4576  df-pr 4578  df-op 4582  df-uni 4859  df-iun 4943  df-br 5094  df-opab 5156  df-mpt 5175  df-tr 5201  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 6253  df-ord 6314  df-on 6315  df-lim 6316  df-suc 6317  df-iota 6442  df-fun 6488  df-fn 6489  df-f 6490  df-f1 6491  df-fo 6492  df-f1o 6493  df-fv 6494  df-riota 7309  df-ov 7355  df-oprab 7356  df-mpo 7357  df-om 7803  df-1st 7927  df-2nd 7928  df-frecs 8217  df-wrecs 8248  df-recs 8297  df-rdg 8335  df-er 8628  df-map 8758  df-en 8876  df-dom 8877  df-sdom 8878  df-sup 9333  df-inf 9334  df-pnf 11155  df-mnf 11156  df-xr 11157  df-ltxr 11158  df-le 11159  df-sub 11353  df-neg 11354  df-div 11782  df-nn 12133  df-2 12195  df-3 12196  df-n0 12389  df-z 12476  df-uz 12739  df-q 12849  df-rp 12893  df-ioo 13251  df-ico 13253  df-icc 13254  df-fz 13410  df-fl 13698  df-seq 13911  df-exp 13971  df-cj 15008  df-re 15009  df-im 15010  df-sqrt 15144  df-abs 15145  df-ovol 25393  df-vol 25394

This theorem is referenced by:  inmbl  25471  finiunmbl  25473  volun  25474  voliunlem1  25479  icombl1  25492  iccmbl  25495  uniiccmbl  25519  mbfimaicc  25560  mbfeqalem2  25571  mbfres2  25574  mbfmax  25578  itgss3  25744  ismblfin  37721  mbfposadd  37727  cnambfre  37728  itg2addnclem2  37732  iblabsnclem  37743  ftc1anclem1  37753  ftc1anclem5  37757  iocmbl  43330