Theorem unmbl 25445

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 25439	. . . 4 ⊢ (𝐴 ∈ dom vol → 𝐴 ⊆ ℝ)
2		mblss 25439	. . . 4 ⊢ (𝐵 ∈ dom vol → 𝐵 ⊆ ℝ)
3	1, 2	anim12i 613	. . 3 ⊢ ((𝐴 ∈ dom vol ∧ 𝐵 ∈ dom vol) → (𝐴 ⊆ ℝ ∧ 𝐵 ⊆ ℝ))
4		unss 4156	. . 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 4573	. . . 4 ⊢ (𝑥 ∈ 𝒫 ℝ → 𝑥 ⊆ ℝ)
7		inss1 4203	. . . . . . . . 9 ⊢ (𝑥 ∩ (𝐴 ∪ 𝐵)) ⊆ 𝑥
8		ovolsscl 25394	. . . . . . . . 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 4203	. . . . . . . . . 10 ⊢ (𝑥 ∩ 𝐴) ⊆ 𝑥
12		ovolsscl 25394	. . . . . . . . . 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 4203	. . . . . . . . 9 ⊢ ((𝑥 ∖ 𝐴) ∩ 𝐵) ⊆ (𝑥 ∖ 𝐴)
16		difss 4102	. . . . . . . . . 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 3961	. . . . . . . . 9 ⊢ (((𝐴 ∈ dom vol ∧ 𝐵 ∈ dom vol) ∧ (𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ)) → (𝑥 ∖ 𝐴) ⊆ ℝ)
19		ovolsscl 25394	. . . . . . . . . . 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 25394	. . . . . . . . 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 11210	. . . . . . 7 ⊢ (((𝐴 ∈ dom vol ∧ 𝐵 ∈ dom vol) ∧ (𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ)) → ((vol*‘(𝑥 ∩ 𝐴)) + (vol*‘((𝑥 ∖ 𝐴) ∩ 𝐵))) ∈ ℝ)
25		difss 4102	. . . . . . . . 9 ⊢ (𝑥 ∖ (𝐴 ∪ 𝐵)) ⊆ 𝑥
26		ovolsscl 25394	. . . . . . . . 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 4175	. . . . . . . . . . . 12 ⊢ ((𝑥 ∖ 𝐴) ∩ 𝐵) = (𝐵 ∩ (𝑥 ∖ 𝐴))
30		indifcom 4249	. . . . . . . . . . . 12 ⊢ (𝐵 ∩ (𝑥 ∖ 𝐴)) = (𝑥 ∩ (𝐵 ∖ 𝐴))
31	29, 30	eqtri 2753	. . . . . . . . . . 11 ⊢ ((𝑥 ∖ 𝐴) ∩ 𝐵) = (𝑥 ∩ (𝐵 ∖ 𝐴))
32	31	uneq2i 4131	. . . . . . . . . 10 ⊢ ((𝑥 ∩ 𝐴) ∪ ((𝑥 ∖ 𝐴) ∩ 𝐵)) = ((𝑥 ∩ 𝐴) ∪ (𝑥 ∩ (𝐵 ∖ 𝐴)))
33		indi 4250	. . . . . . . . . 10 ⊢ (𝑥 ∩ (𝐴 ∪ (𝐵 ∖ 𝐴))) = ((𝑥 ∩ 𝐴) ∪ (𝑥 ∩ (𝐵 ∖ 𝐴)))
34		undif2 4443	. . . . . . . . . . 11 ⊢ (𝐴 ∪ (𝐵 ∖ 𝐴)) = (𝐴 ∪ 𝐵)
35	34	ineq2i 4183	. . . . . . . . . 10 ⊢ (𝑥 ∩ (𝐴 ∪ (𝐵 ∖ 𝐴))) = (𝑥 ∩ (𝐴 ∪ 𝐵))
36	32, 33, 35	3eqtr2ri 2760	. . . . . . . . 9 ⊢ (𝑥 ∩ (𝐴 ∪ 𝐵)) = ((𝑥 ∩ 𝐴) ∪ ((𝑥 ∖ 𝐴) ∩ 𝐵))
37	36	fveq2i 6864	. . . . . . . 8 ⊢ (vol*‘(𝑥 ∩ (𝐴 ∪ 𝐵))) = (vol*‘((𝑥 ∩ 𝐴) ∪ ((𝑥 ∖ 𝐴) ∩ 𝐵)))
38	11, 17	sstrid 3961	. . . . . . . . 9 ⊢ (((𝐴 ∈ dom vol ∧ 𝐵 ∈ dom vol) ∧ (𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ)) → (𝑥 ∩ 𝐴) ⊆ ℝ)
39	15, 18	sstrid 3961	. . . . . . . . 9 ⊢ (((𝐴 ∈ dom vol ∧ 𝐵 ∈ dom vol) ∧ (𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ)) → ((𝑥 ∖ 𝐴) ∩ 𝐵) ⊆ ℝ)
40		ovolun 25407	. . . . . . . . 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 5145	. . . . . . 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 11799	. . . . . 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 25440	. . . . . . . . . 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 4265	. . . . . . . . . . 11 ⊢ (𝑥 ∖ (𝐴 ∪ 𝐵)) = ((𝑥 ∖ 𝐴) ∖ 𝐵)
48	47	fveq2i 6864	. . . . . . . . . 10 ⊢ (vol*‘(𝑥 ∖ (𝐴 ∪ 𝐵))) = (vol*‘((𝑥 ∖ 𝐴) ∖ 𝐵))
49	48	oveq2i 7401	. . . . . . . . 9 ⊢ ((vol*‘((𝑥 ∖ 𝐴) ∩ 𝐵)) + (vol*‘(𝑥 ∖ (𝐴 ∪ 𝐵)))) = ((vol*‘((𝑥 ∖ 𝐴) ∩ 𝐵)) + (vol*‘((𝑥 ∖ 𝐴) ∖ 𝐵)))
50	46, 49	eqtr4di 2783	. . . . . . . 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 7406	. . . . . . 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 25440	. . . . . . . 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 11209	. . . . . . . 8 ⊢ (((𝐴 ∈ dom vol ∧ 𝐵 ∈ dom vol) ∧ (𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ)) → (vol*‘(𝑥 ∩ 𝐴)) ∈ ℂ)
57	23	recnd 11209	. . . . . . . 8 ⊢ (((𝐴 ∈ dom vol ∧ 𝐵 ∈ dom vol) ∧ (𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ)) → (vol*‘((𝑥 ∖ 𝐴) ∩ 𝐵)) ∈ ℂ)
58	28	recnd 11209	. . . . . . . 8 ⊢ (((𝐴 ∈ dom vol ∧ 𝐵 ∈ dom vol) ∧ (𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ)) → (vol*‘(𝑥 ∖ (𝐴 ∪ 𝐵))) ∈ ℂ)
59	56, 57, 58	addassd 11203	. . . . . . 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 2775	. . . . . 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 5138	. . . . 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 3126	. 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 25435	. 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 1540   ∈ wcel 2109  ∀wral 3045   ∖ cdif 3914   ∪ cun 3915   ∩ cin 3916   ⊆ wss 3917  𝒫 cpw 4566   class class class wbr 5110  dom cdm 5641  ‘cfv 6514  (class class class)co 7390  ℝcr 11074   + caddc 11078   ≤ cle 11216  vol*covol 25370  volcvol 25371

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 2702  ax-sep 5254  ax-nul 5264  ax-pow 5323  ax-pr 5390  ax-un 7714  ax-cnex 11131  ax-resscn 11132  ax-1cn 11133  ax-icn 11134  ax-addcl 11135  ax-addrcl 11136  ax-mulcl 11137  ax-mulrcl 11138  ax-mulcom 11139  ax-addass 11140  ax-mulass 11141  ax-distr 11142  ax-i2m1 11143  ax-1ne0 11144  ax-1rid 11145  ax-rnegex 11146  ax-rrecex 11147  ax-cnre 11148  ax-pre-lttri 11149  ax-pre-lttrn 11150  ax-pre-ltadd 11151  ax-pre-mulgt0 11152  ax-pre-sup 11153

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 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-nel 3031  df-ral 3046  df-rex 3055  df-rmo 3356  df-reu 3357  df-rab 3409  df-v 3452  df-sbc 3757  df-csb 3866  df-dif 3920  df-un 3922  df-in 3924  df-ss 3934  df-pss 3937  df-nul 4300  df-if 4492  df-pw 4568  df-sn 4593  df-pr 4595  df-op 4599  df-uni 4875  df-iun 4960  df-br 5111  df-opab 5173  df-mpt 5192  df-tr 5218  df-id 5536  df-eprel 5541  df-po 5549  df-so 5550  df-fr 5594  df-we 5596  df-xp 5647  df-rel 5648  df-cnv 5649  df-co 5650  df-dm 5651  df-rn 5652  df-res 5653  df-ima 5654  df-pred 6277  df-ord 6338  df-on 6339  df-lim 6340  df-suc 6341  df-iota 6467  df-fun 6516  df-fn 6517  df-f 6518  df-f1 6519  df-fo 6520  df-f1o 6521  df-fv 6522  df-riota 7347  df-ov 7393  df-oprab 7394  df-mpo 7395  df-om 7846  df-1st 7971  df-2nd 7972  df-frecs 8263  df-wrecs 8294  df-recs 8343  df-rdg 8381  df-er 8674  df-map 8804  df-en 8922  df-dom 8923  df-sdom 8924  df-sup 9400  df-inf 9401  df-pnf 11217  df-mnf 11218  df-xr 11219  df-ltxr 11220  df-le 11221  df-sub 11414  df-neg 11415  df-div 11843  df-nn 12194  df-2 12256  df-3 12257  df-n0 12450  df-z 12537  df-uz 12801  df-q 12915  df-rp 12959  df-ioo 13317  df-ico 13319  df-icc 13320  df-fz 13476  df-fl 13761  df-seq 13974  df-exp 14034  df-cj 15072  df-re 15073  df-im 15074  df-sqrt 15208  df-abs 15209  df-ovol 25372  df-vol 25373

This theorem is referenced by:  inmbl  25450  finiunmbl  25452  volun  25453  voliunlem1  25458  icombl1  25471  iccmbl  25474  uniiccmbl  25498  mbfimaicc  25539  mbfeqalem2  25550  mbfres2  25553  mbfmax  25557  itgss3  25723  ismblfin  37662  mbfposadd  37668  cnambfre  37669  itg2addnclem2  37673  iblabsnclem  37684  ftc1anclem1  37694  ftc1anclem5  37698  iocmbl  43209