Theorem unmbl 25838

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 25832	. . . 4 ⊢ (𝐴 ∈ dom vol → 𝐴 ⊆ ℝ)
2		mblss 25832	. . . 4 ⊢ (𝐵 ∈ dom vol → 𝐵 ⊆ ℝ)
3	1, 2	anim12i 625	. . 3 ⊢ ((𝐴 ∈ dom vol ∧ 𝐵 ∈ dom vol) → (𝐴 ⊆ ℝ ∧ 𝐵 ⊆ ℝ))
4		unss 4136	. . 3 ⊢ ((𝐴 ⊆ ℝ ∧ 𝐵 ⊆ ℝ) ↔ (𝐴 ∪ 𝐵) ⊆ ℝ)
5	3, 4	sylib 221	. 2 ⊢ ((𝐴 ∈ dom vol ∧ 𝐵 ∈ dom vol) → (𝐴 ∪ 𝐵) ⊆ ℝ)
6		elpwi 4564	. . . 4 ⊢ (𝑥 ∈ 𝒫 ℝ → 𝑥 ⊆ ℝ)
7		inss1 4182	. . . . . . . . 9 ⊢ (𝑥 ∩ (𝐴 ∪ 𝐵)) ⊆ 𝑥
8		ovolsscl 25787	. . . . . . . . 9 ⊢ (((𝑥 ∩ (𝐴 ∪ 𝐵)) ⊆ 𝑥 ∧ 𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) → (vol*‘(𝑥 ∩ (𝐴 ∪ 𝐵))) ∈ ℝ)
9	7, 8	mp3an1 1477	. . . . . . . 8 ⊢ ((𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) → (vol*‘(𝑥 ∩ (𝐴 ∪ 𝐵))) ∈ ℝ)
10	9	adantl 487	. . . . . . 7 ⊢ (((𝐴 ∈ dom vol ∧ 𝐵 ∈ dom vol) ∧ (𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ)) → (vol*‘(𝑥 ∩ (𝐴 ∪ 𝐵))) ∈ ℝ)
11		inss1 4182	. . . . . . . . . 10 ⊢ (𝑥 ∩ 𝐴) ⊆ 𝑥
12		ovolsscl 25787	. . . . . . . . . 10 ⊢ (((𝑥 ∩ 𝐴) ⊆ 𝑥 ∧ 𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) → (vol*‘(𝑥 ∩ 𝐴)) ∈ ℝ)
13	11, 12	mp3an1 1477	. . . . . . . . 9 ⊢ ((𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) → (vol*‘(𝑥 ∩ 𝐴)) ∈ ℝ)
14	13	adantl 487	. . . . . . . 8 ⊢ (((𝐴 ∈ dom vol ∧ 𝐵 ∈ dom vol) ∧ (𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ)) → (vol*‘(𝑥 ∩ 𝐴)) ∈ ℝ)
15		inss1 4182	. . . . . . . . 9 ⊢ ((𝑥 ∖ 𝐴) ∩ 𝐵) ⊆ (𝑥 ∖ 𝐴)
16		difss 4083	. . . . . . . . . 10 ⊢ (𝑥 ∖ 𝐴) ⊆ 𝑥
17		simprl 783	. . . . . . . . . 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 25787	. . . . . . . . . . 11 ⊢ (((𝑥 ∖ 𝐴) ⊆ 𝑥 ∧ 𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) → (vol*‘(𝑥 ∖ 𝐴)) ∈ ℝ)
20	16, 19	mp3an1 1477	. . . . . . . . . 10 ⊢ ((𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) → (vol*‘(𝑥 ∖ 𝐴)) ∈ ℝ)
21	20	adantl 487	. . . . . . . . 9 ⊢ (((𝐴 ∈ dom vol ∧ 𝐵 ∈ dom vol) ∧ (𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ)) → (vol*‘(𝑥 ∖ 𝐴)) ∈ ℝ)
22		ovolsscl 25787	. . . . . . . . 9 ⊢ ((((𝑥 ∖ 𝐴) ∩ 𝐵) ⊆ (𝑥 ∖ 𝐴) ∧ (𝑥 ∖ 𝐴) ⊆ ℝ ∧ (vol*‘(𝑥 ∖ 𝐴)) ∈ ℝ) → (vol*‘((𝑥 ∖ 𝐴) ∩ 𝐵)) ∈ ℝ)
23	15, 18, 21, 22	mp3an2i 1495	. . . . . . . 8 ⊢ (((𝐴 ∈ dom vol ∧ 𝐵 ∈ dom vol) ∧ (𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ)) → (vol*‘((𝑥 ∖ 𝐴) ∩ 𝐵)) ∈ ℝ)
24	14, 23	readdcld 11319	. . . . . . 7 ⊢ (((𝐴 ∈ dom vol ∧ 𝐵 ∈ dom vol) ∧ (𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ)) → ((vol*‘(𝑥 ∩ 𝐴)) + (vol*‘((𝑥 ∖ 𝐴) ∩ 𝐵))) ∈ ℝ)
25		difss 4083	. . . . . . . . 9 ⊢ (𝑥 ∖ (𝐴 ∪ 𝐵)) ⊆ 𝑥
26		ovolsscl 25787	. . . . . . . . 9 ⊢ (((𝑥 ∖ (𝐴 ∪ 𝐵)) ⊆ 𝑥 ∧ 𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) → (vol*‘(𝑥 ∖ (𝐴 ∪ 𝐵))) ∈ ℝ)
27	25, 26	mp3an1 1477	. . . . . . . 8 ⊢ ((𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) → (vol*‘(𝑥 ∖ (𝐴 ∪ 𝐵))) ∈ ℝ)
28	27	adantl 487	. . . . . . 7 ⊢ (((𝐴 ∈ dom vol ∧ 𝐵 ∈ dom vol) ∧ (𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ)) → (vol*‘(𝑥 ∖ (𝐴 ∪ 𝐵))) ∈ ℝ)
29		incom 4155	. . . . . . . . . . . 12 ⊢ ((𝑥 ∖ 𝐴) ∩ 𝐵) = (𝐵 ∩ (𝑥 ∖ 𝐴))
30		indifcom 4229	. . . . . . . . . . . 12 ⊢ (𝐵 ∩ (𝑥 ∖ 𝐴)) = (𝑥 ∩ (𝐵 ∖ 𝐴))
31	29, 30	eqtri 2784	. . . . . . . . . . 11 ⊢ ((𝑥 ∖ 𝐴) ∩ 𝐵) = (𝑥 ∩ (𝐵 ∖ 𝐴))
32	31	uneq2i 4112	. . . . . . . . . 10 ⊢ ((𝑥 ∩ 𝐴) ∪ ((𝑥 ∖ 𝐴) ∩ 𝐵)) = ((𝑥 ∩ 𝐴) ∪ (𝑥 ∩ (𝐵 ∖ 𝐴)))
33		indi 4230	. . . . . . . . . 10 ⊢ (𝑥 ∩ (𝐴 ∪ (𝐵 ∖ 𝐴))) = ((𝑥 ∩ 𝐴) ∪ (𝑥 ∩ (𝐵 ∖ 𝐴)))
34		undif2 4431	. . . . . . . . . . 11 ⊢ (𝐴 ∪ (𝐵 ∖ 𝐴)) = (𝐴 ∪ 𝐵)
35	34	ineq2i 4163	. . . . . . . . . 10 ⊢ (𝑥 ∩ (𝐴 ∪ (𝐵 ∖ 𝐴))) = (𝑥 ∩ (𝐴 ∪ 𝐵))
36	32, 33, 35	3eqtr2ri 2791	. . . . . . . . 9 ⊢ (𝑥 ∩ (𝐴 ∪ 𝐵)) = ((𝑥 ∩ 𝐴) ∪ ((𝑥 ∖ 𝐴) ∩ 𝐵))
37	36	fveq2i 6880	. . . . . . . 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 25800	. . . . . . . . 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 852	. . . . . . . 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 5140	. . . . . . 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 11911	. . . . . 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 781	. . . . . . . . . 10 ⊢ (((𝐴 ∈ dom vol ∧ 𝐵 ∈ dom vol) ∧ (𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ)) → 𝐵 ∈ dom vol)
45		mblsplit 25833	. . . . . . . . . 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 1398	. . . . . . . . 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 4245	. . . . . . . . . . 11 ⊢ (𝑥 ∖ (𝐴 ∪ 𝐵)) = ((𝑥 ∖ 𝐴) ∖ 𝐵)
48	47	fveq2i 6880	. . . . . . . . . 10 ⊢ (vol*‘(𝑥 ∖ (𝐴 ∪ 𝐵))) = (vol*‘((𝑥 ∖ 𝐴) ∖ 𝐵))
49	48	oveq2i 7423	. . . . . . . . 9 ⊢ ((vol*‘((𝑥 ∖ 𝐴) ∩ 𝐵)) + (vol*‘(𝑥 ∖ (𝐴 ∪ 𝐵)))) = ((vol*‘((𝑥 ∖ 𝐴) ∩ 𝐵)) + (vol*‘((𝑥 ∖ 𝐴) ∖ 𝐵)))
50	46, 49	eqtr4di 2814	. . . . . . . 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 7428	. . . . . . 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 779	. . . . . . . 8 ⊢ (((𝐴 ∈ dom vol ∧ 𝐵 ∈ dom vol) ∧ (𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ)) → 𝐴 ∈ dom vol)
53		simprr 785	. . . . . . . 8 ⊢ (((𝐴 ∈ dom vol ∧ 𝐵 ∈ dom vol) ∧ (𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ)) → (vol*‘𝑥) ∈ ℝ)
54		mblsplit 25833	. . . . . . . 8 ⊢ ((𝐴 ∈ dom vol ∧ 𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) → (vol*‘𝑥) = ((vol*‘(𝑥 ∩ 𝐴)) + (vol*‘(𝑥 ∖ 𝐴))))
55	52, 17, 53, 54	syl3anc 1398	. . . . . . 7 ⊢ (((𝐴 ∈ dom vol ∧ 𝐵 ∈ dom vol) ∧ (𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ)) → (vol*‘𝑥) = ((vol*‘(𝑥 ∩ 𝐴)) + (vol*‘(𝑥 ∖ 𝐴))))
56	14	recnd 11318	. . . . . . . 8 ⊢ (((𝐴 ∈ dom vol ∧ 𝐵 ∈ dom vol) ∧ (𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ)) → (vol*‘(𝑥 ∩ 𝐴)) ∈ ℂ)
57	23	recnd 11318	. . . . . . . 8 ⊢ (((𝐴 ∈ dom vol ∧ 𝐵 ∈ dom vol) ∧ (𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ)) → (vol*‘((𝑥 ∖ 𝐴) ∩ 𝐵)) ∈ ℂ)
58	28	recnd 11318	. . . . . . . 8 ⊢ (((𝐴 ∈ dom vol ∧ 𝐵 ∈ dom vol) ∧ (𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ)) → (vol*‘(𝑥 ∖ (𝐴 ∪ 𝐵))) ∈ ℂ)
59	56, 57, 58	addassd 11312	. . . . . . 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 2806	. . . . . 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 5133	. . . . 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 462	. . . 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 605	. . 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 3155	. 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 25828	. 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 595	1 ⊢ ((𝐴 ∈ dom vol ∧ 𝐵 ∈ dom vol) → (𝐴 ∪ 𝐵) ∈ dom vol)

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2145  ∀wral 3077   ∖ cdif 3896   ∪ cun 3897   ∩ cin 3898   ⊆ wss 3899  𝒫 cpw 4557   class class class wbr 5103  dom cdm 5651  ‘cfv 6531  (class class class)co 7412  ℝcr 11180   + caddc 11184   ≤ cle 11325  vol*covol 25763  volcvol 25764

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2733  ax-sep 5249  ax-nul 5260  ax-pow 5327  ax-pr 5391  ax-un 7740  ax-cnex 11237  ax-resscn 11238  ax-1cn 11239  ax-icn 11240  ax-addcl 11241  ax-addrcl 11242  ax-mulcl 11243  ax-mulrcl 11244  ax-mulcom 11245  ax-addass 11246  ax-mulass 11247  ax-distr 11248  ax-i2m1 11249  ax-1ne0 11250  ax-1rid 11251  ax-rnegex 11252  ax-rrecex 11253  ax-cnre 11254  ax-pre-lttri 11255  ax-pre-lttrn 11256  ax-pre-ltadd 11257  ax-pre-mulgt0 11258  ax-pre-sup 11259

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-nel 3063  df-ral 3078  df-rex 3088  df-rmo 3366  df-reu 3367  df-rab 3414  df-v 3453  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-iun 4953  df-br 5104  df-opab 5168  df-mpt 5187  df-tr 5213  df-id 5546  df-eprel 5551  df-po 5559  df-so 5560  df-fr 5604  df-we 5606  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-pred 6297  df-ord 6358  df-on 6359  df-lim 6360  df-suc 6361  df-iota 6487  df-fun 6533  df-fn 6534  df-f 6535  df-f1 6536  df-fo 6537  df-f1o 6538  df-fv 6539  df-riota 7369  df-ov 7415  df-oprab 7416  df-mpo 7417  df-om 7867  df-1st 7990  df-2nd 7991  df-frecs 8283  df-wrecs 8314  df-recs 8363  df-rdg 8402  df-er 8701  df-map 8833  df-en 8958  df-dom 8959  df-sdom 8960  df-sup 9418  df-inf 9419  df-pnf 11326  df-mnf 11327  df-xr 11328  df-ltxr 11329  df-le 11330  df-sub 11524  df-neg 11525  df-div 11955  df-nn 12317  df-2 12386  df-3 12387  df-n0 12588  df-z 12675  df-uz 12947  df-q 13057  df-rp 13102  df-ioo 13461  df-ico 13463  df-icc 13464  df-fz 13621  df-fl 13912  df-seq 14125  df-exp 14185  df-cj 15246  df-re 15247  df-im 15248  df-sqrt 15382  df-abs 15383  df-ovol 25765  df-vol 25766

This theorem is used by:  inmbl  25843  finiunmbl  25845  volun  25846  voliunlem1  25851  icombl1  25864  iccmbl  25867  uniiccmbl  25891  mbfimaicc  25932  mbfeqalem2  25943  mbfres2  25946  mbfmax  25950  itgss3  26115  ismblfin  38547  mbfposadd  38553  cnambfre  38554  itg2addnclem2  38558  iblabsnclem  38569  ftc1anclem1  38579  ftc1anclem5  38583  iocmbl  44173