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Theorem unmbl 23351
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) → (𝐴𝐵) ∈ dom vol)

Proof of Theorem unmbl
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 mblss 23345 . . . 4 (𝐴 ∈ dom vol → 𝐴 ⊆ ℝ)
2 mblss 23345 . . . 4 (𝐵 ∈ dom vol → 𝐵 ⊆ ℝ)
31, 2anim12i 589 . . 3 ((𝐴 ∈ dom vol ∧ 𝐵 ∈ dom vol) → (𝐴 ⊆ ℝ ∧ 𝐵 ⊆ ℝ))
4 unss 3820 . . 3 ((𝐴 ⊆ ℝ ∧ 𝐵 ⊆ ℝ) ↔ (𝐴𝐵) ⊆ ℝ)
53, 4sylib 208 . 2 ((𝐴 ∈ dom vol ∧ 𝐵 ∈ dom vol) → (𝐴𝐵) ⊆ ℝ)
6 elpwi 4201 . . . 4 (𝑥 ∈ 𝒫 ℝ → 𝑥 ⊆ ℝ)
7 inss1 3866 . . . . . . . . 9 (𝑥 ∩ (𝐴𝐵)) ⊆ 𝑥
8 ovolsscl 23300 . . . . . . . . 9 (((𝑥 ∩ (𝐴𝐵)) ⊆ 𝑥𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) → (vol*‘(𝑥 ∩ (𝐴𝐵))) ∈ ℝ)
97, 8mp3an1 1451 . . . . . . . 8 ((𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) → (vol*‘(𝑥 ∩ (𝐴𝐵))) ∈ ℝ)
109adantl 481 . . . . . . 7 (((𝐴 ∈ dom vol ∧ 𝐵 ∈ dom vol) ∧ (𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ)) → (vol*‘(𝑥 ∩ (𝐴𝐵))) ∈ ℝ)
11 inss1 3866 . . . . . . . . . 10 (𝑥𝐴) ⊆ 𝑥
12 ovolsscl 23300 . . . . . . . . . 10 (((𝑥𝐴) ⊆ 𝑥𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) → (vol*‘(𝑥𝐴)) ∈ ℝ)
1311, 12mp3an1 1451 . . . . . . . . 9 ((𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) → (vol*‘(𝑥𝐴)) ∈ ℝ)
1413adantl 481 . . . . . . . 8 (((𝐴 ∈ dom vol ∧ 𝐵 ∈ dom vol) ∧ (𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ)) → (vol*‘(𝑥𝐴)) ∈ ℝ)
15 difss 3770 . . . . . . . . . 10 (𝑥𝐴) ⊆ 𝑥
16 simprl 809 . . . . . . . . . 10 (((𝐴 ∈ dom vol ∧ 𝐵 ∈ dom vol) ∧ (𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ)) → 𝑥 ⊆ ℝ)
1715, 16syl5ss 3647 . . . . . . . . 9 (((𝐴 ∈ dom vol ∧ 𝐵 ∈ dom vol) ∧ (𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ)) → (𝑥𝐴) ⊆ ℝ)
18 ovolsscl 23300 . . . . . . . . . . 11 (((𝑥𝐴) ⊆ 𝑥𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) → (vol*‘(𝑥𝐴)) ∈ ℝ)
1915, 18mp3an1 1451 . . . . . . . . . 10 ((𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) → (vol*‘(𝑥𝐴)) ∈ ℝ)
2019adantl 481 . . . . . . . . 9 (((𝐴 ∈ dom vol ∧ 𝐵 ∈ dom vol) ∧ (𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ)) → (vol*‘(𝑥𝐴)) ∈ ℝ)
21 inss1 3866 . . . . . . . . . 10 ((𝑥𝐴) ∩ 𝐵) ⊆ (𝑥𝐴)
22 ovolsscl 23300 . . . . . . . . . 10 ((((𝑥𝐴) ∩ 𝐵) ⊆ (𝑥𝐴) ∧ (𝑥𝐴) ⊆ ℝ ∧ (vol*‘(𝑥𝐴)) ∈ ℝ) → (vol*‘((𝑥𝐴) ∩ 𝐵)) ∈ ℝ)
2321, 22mp3an1 1451 . . . . . . . . 9 (((𝑥𝐴) ⊆ ℝ ∧ (vol*‘(𝑥𝐴)) ∈ ℝ) → (vol*‘((𝑥𝐴) ∩ 𝐵)) ∈ ℝ)
2417, 20, 23syl2anc 694 . . . . . . . 8 (((𝐴 ∈ dom vol ∧ 𝐵 ∈ dom vol) ∧ (𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ)) → (vol*‘((𝑥𝐴) ∩ 𝐵)) ∈ ℝ)
2514, 24readdcld 10107 . . . . . . 7 (((𝐴 ∈ dom vol ∧ 𝐵 ∈ dom vol) ∧ (𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ)) → ((vol*‘(𝑥𝐴)) + (vol*‘((𝑥𝐴) ∩ 𝐵))) ∈ ℝ)
26 difss 3770 . . . . . . . . 9 (𝑥 ∖ (𝐴𝐵)) ⊆ 𝑥
27 ovolsscl 23300 . . . . . . . . 9 (((𝑥 ∖ (𝐴𝐵)) ⊆ 𝑥𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) → (vol*‘(𝑥 ∖ (𝐴𝐵))) ∈ ℝ)
2826, 27mp3an1 1451 . . . . . . . 8 ((𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) → (vol*‘(𝑥 ∖ (𝐴𝐵))) ∈ ℝ)
2928adantl 481 . . . . . . 7 (((𝐴 ∈ dom vol ∧ 𝐵 ∈ dom vol) ∧ (𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ)) → (vol*‘(𝑥 ∖ (𝐴𝐵))) ∈ ℝ)
30 incom 3838 . . . . . . . . . . . 12 ((𝑥𝐴) ∩ 𝐵) = (𝐵 ∩ (𝑥𝐴))
31 indifcom 3905 . . . . . . . . . . . 12 (𝐵 ∩ (𝑥𝐴)) = (𝑥 ∩ (𝐵𝐴))
3230, 31eqtri 2673 . . . . . . . . . . 11 ((𝑥𝐴) ∩ 𝐵) = (𝑥 ∩ (𝐵𝐴))
3332uneq2i 3797 . . . . . . . . . 10 ((𝑥𝐴) ∪ ((𝑥𝐴) ∩ 𝐵)) = ((𝑥𝐴) ∪ (𝑥 ∩ (𝐵𝐴)))
34 indi 3906 . . . . . . . . . 10 (𝑥 ∩ (𝐴 ∪ (𝐵𝐴))) = ((𝑥𝐴) ∪ (𝑥 ∩ (𝐵𝐴)))
35 undif2 4077 . . . . . . . . . . 11 (𝐴 ∪ (𝐵𝐴)) = (𝐴𝐵)
3635ineq2i 3844 . . . . . . . . . 10 (𝑥 ∩ (𝐴 ∪ (𝐵𝐴))) = (𝑥 ∩ (𝐴𝐵))
3733, 34, 363eqtr2ri 2680 . . . . . . . . 9 (𝑥 ∩ (𝐴𝐵)) = ((𝑥𝐴) ∪ ((𝑥𝐴) ∩ 𝐵))
3837fveq2i 6232 . . . . . . . 8 (vol*‘(𝑥 ∩ (𝐴𝐵))) = (vol*‘((𝑥𝐴) ∪ ((𝑥𝐴) ∩ 𝐵)))
3911, 16syl5ss 3647 . . . . . . . . 9 (((𝐴 ∈ dom vol ∧ 𝐵 ∈ dom vol) ∧ (𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ)) → (𝑥𝐴) ⊆ ℝ)
4021, 17syl5ss 3647 . . . . . . . . 9 (((𝐴 ∈ dom vol ∧ 𝐵 ∈ dom vol) ∧ (𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ)) → ((𝑥𝐴) ∩ 𝐵) ⊆ ℝ)
41 ovolun 23313 . . . . . . . . 9 ((((𝑥𝐴) ⊆ ℝ ∧ (vol*‘(𝑥𝐴)) ∈ ℝ) ∧ (((𝑥𝐴) ∩ 𝐵) ⊆ ℝ ∧ (vol*‘((𝑥𝐴) ∩ 𝐵)) ∈ ℝ)) → (vol*‘((𝑥𝐴) ∪ ((𝑥𝐴) ∩ 𝐵))) ≤ ((vol*‘(𝑥𝐴)) + (vol*‘((𝑥𝐴) ∩ 𝐵))))
4239, 14, 40, 24, 41syl22anc 1367 . . . . . . . 8 (((𝐴 ∈ dom vol ∧ 𝐵 ∈ dom vol) ∧ (𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ)) → (vol*‘((𝑥𝐴) ∪ ((𝑥𝐴) ∩ 𝐵))) ≤ ((vol*‘(𝑥𝐴)) + (vol*‘((𝑥𝐴) ∩ 𝐵))))
4338, 42syl5eqbr 4720 . . . . . . 7 (((𝐴 ∈ dom vol ∧ 𝐵 ∈ dom vol) ∧ (𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ)) → (vol*‘(𝑥 ∩ (𝐴𝐵))) ≤ ((vol*‘(𝑥𝐴)) + (vol*‘((𝑥𝐴) ∩ 𝐵))))
4410, 25, 29, 43leadd1dd 10679 . . . . . 6 (((𝐴 ∈ dom vol ∧ 𝐵 ∈ dom vol) ∧ (𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ)) → ((vol*‘(𝑥 ∩ (𝐴𝐵))) + (vol*‘(𝑥 ∖ (𝐴𝐵)))) ≤ (((vol*‘(𝑥𝐴)) + (vol*‘((𝑥𝐴) ∩ 𝐵))) + (vol*‘(𝑥 ∖ (𝐴𝐵)))))
45 simplr 807 . . . . . . . . . 10 (((𝐴 ∈ dom vol ∧ 𝐵 ∈ dom vol) ∧ (𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ)) → 𝐵 ∈ dom vol)
46 mblsplit 23346 . . . . . . . . . 10 ((𝐵 ∈ dom vol ∧ (𝑥𝐴) ⊆ ℝ ∧ (vol*‘(𝑥𝐴)) ∈ ℝ) → (vol*‘(𝑥𝐴)) = ((vol*‘((𝑥𝐴) ∩ 𝐵)) + (vol*‘((𝑥𝐴) ∖ 𝐵))))
4745, 17, 20, 46syl3anc 1366 . . . . . . . . 9 (((𝐴 ∈ dom vol ∧ 𝐵 ∈ dom vol) ∧ (𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ)) → (vol*‘(𝑥𝐴)) = ((vol*‘((𝑥𝐴) ∩ 𝐵)) + (vol*‘((𝑥𝐴) ∖ 𝐵))))
48 difun1 3920 . . . . . . . . . . 11 (𝑥 ∖ (𝐴𝐵)) = ((𝑥𝐴) ∖ 𝐵)
4948fveq2i 6232 . . . . . . . . . 10 (vol*‘(𝑥 ∖ (𝐴𝐵))) = (vol*‘((𝑥𝐴) ∖ 𝐵))
5049oveq2i 6701 . . . . . . . . 9 ((vol*‘((𝑥𝐴) ∩ 𝐵)) + (vol*‘(𝑥 ∖ (𝐴𝐵)))) = ((vol*‘((𝑥𝐴) ∩ 𝐵)) + (vol*‘((𝑥𝐴) ∖ 𝐵)))
5147, 50syl6eqr 2703 . . . . . . . 8 (((𝐴 ∈ dom vol ∧ 𝐵 ∈ dom vol) ∧ (𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ)) → (vol*‘(𝑥𝐴)) = ((vol*‘((𝑥𝐴) ∩ 𝐵)) + (vol*‘(𝑥 ∖ (𝐴𝐵)))))
5251oveq2d 6706 . . . . . . 7 (((𝐴 ∈ dom vol ∧ 𝐵 ∈ dom vol) ∧ (𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ)) → ((vol*‘(𝑥𝐴)) + (vol*‘(𝑥𝐴))) = ((vol*‘(𝑥𝐴)) + ((vol*‘((𝑥𝐴) ∩ 𝐵)) + (vol*‘(𝑥 ∖ (𝐴𝐵))))))
53 simpll 805 . . . . . . . 8 (((𝐴 ∈ dom vol ∧ 𝐵 ∈ dom vol) ∧ (𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ)) → 𝐴 ∈ dom vol)
54 simprr 811 . . . . . . . 8 (((𝐴 ∈ dom vol ∧ 𝐵 ∈ dom vol) ∧ (𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ)) → (vol*‘𝑥) ∈ ℝ)
55 mblsplit 23346 . . . . . . . 8 ((𝐴 ∈ dom vol ∧ 𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) → (vol*‘𝑥) = ((vol*‘(𝑥𝐴)) + (vol*‘(𝑥𝐴))))
5653, 16, 54, 55syl3anc 1366 . . . . . . 7 (((𝐴 ∈ dom vol ∧ 𝐵 ∈ dom vol) ∧ (𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ)) → (vol*‘𝑥) = ((vol*‘(𝑥𝐴)) + (vol*‘(𝑥𝐴))))
5714recnd 10106 . . . . . . . 8 (((𝐴 ∈ dom vol ∧ 𝐵 ∈ dom vol) ∧ (𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ)) → (vol*‘(𝑥𝐴)) ∈ ℂ)
5824recnd 10106 . . . . . . . 8 (((𝐴 ∈ dom vol ∧ 𝐵 ∈ dom vol) ∧ (𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ)) → (vol*‘((𝑥𝐴) ∩ 𝐵)) ∈ ℂ)
5929recnd 10106 . . . . . . . 8 (((𝐴 ∈ dom vol ∧ 𝐵 ∈ dom vol) ∧ (𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ)) → (vol*‘(𝑥 ∖ (𝐴𝐵))) ∈ ℂ)
6057, 58, 59addassd 10100 . . . . . . 7 (((𝐴 ∈ dom vol ∧ 𝐵 ∈ dom vol) ∧ (𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ)) → (((vol*‘(𝑥𝐴)) + (vol*‘((𝑥𝐴) ∩ 𝐵))) + (vol*‘(𝑥 ∖ (𝐴𝐵)))) = ((vol*‘(𝑥𝐴)) + ((vol*‘((𝑥𝐴) ∩ 𝐵)) + (vol*‘(𝑥 ∖ (𝐴𝐵))))))
6152, 56, 603eqtr4d 2695 . . . . . 6 (((𝐴 ∈ dom vol ∧ 𝐵 ∈ dom vol) ∧ (𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ)) → (vol*‘𝑥) = (((vol*‘(𝑥𝐴)) + (vol*‘((𝑥𝐴) ∩ 𝐵))) + (vol*‘(𝑥 ∖ (𝐴𝐵)))))
6244, 61breqtrrd 4713 . . . . 5 (((𝐴 ∈ dom vol ∧ 𝐵 ∈ dom vol) ∧ (𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ)) → ((vol*‘(𝑥 ∩ (𝐴𝐵))) + (vol*‘(𝑥 ∖ (𝐴𝐵)))) ≤ (vol*‘𝑥))
6362expr 642 . . . 4 (((𝐴 ∈ dom vol ∧ 𝐵 ∈ dom vol) ∧ 𝑥 ⊆ ℝ) → ((vol*‘𝑥) ∈ ℝ → ((vol*‘(𝑥 ∩ (𝐴𝐵))) + (vol*‘(𝑥 ∖ (𝐴𝐵)))) ≤ (vol*‘𝑥)))
646, 63sylan2 490 . . 3 (((𝐴 ∈ dom vol ∧ 𝐵 ∈ dom vol) ∧ 𝑥 ∈ 𝒫 ℝ) → ((vol*‘𝑥) ∈ ℝ → ((vol*‘(𝑥 ∩ (𝐴𝐵))) + (vol*‘(𝑥 ∖ (𝐴𝐵)))) ≤ (vol*‘𝑥)))
6564ralrimiva 2995 . 2 ((𝐴 ∈ dom vol ∧ 𝐵 ∈ dom vol) → ∀𝑥 ∈ 𝒫 ℝ((vol*‘𝑥) ∈ ℝ → ((vol*‘(𝑥 ∩ (𝐴𝐵))) + (vol*‘(𝑥 ∖ (𝐴𝐵)))) ≤ (vol*‘𝑥)))
66 ismbl2 23341 . 2 ((𝐴𝐵) ∈ dom vol ↔ ((𝐴𝐵) ⊆ ℝ ∧ ∀𝑥 ∈ 𝒫 ℝ((vol*‘𝑥) ∈ ℝ → ((vol*‘(𝑥 ∩ (𝐴𝐵))) + (vol*‘(𝑥 ∖ (𝐴𝐵)))) ≤ (vol*‘𝑥))))
675, 65, 66sylanbrc 699 1 ((𝐴 ∈ dom vol ∧ 𝐵 ∈ dom vol) → (𝐴𝐵) ∈ dom vol)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383   = wceq 1523  wcel 2030  wral 2941  cdif 3604  cun 3605  cin 3606  wss 3607  𝒫 cpw 4191   class class class wbr 4685  dom cdm 5143  cfv 5926  (class class class)co 6690  cr 9973   + caddc 9977  cle 10113  vol*covol 23277  volcvol 23278
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991  ax-cnex 10030  ax-resscn 10031  ax-1cn 10032  ax-icn 10033  ax-addcl 10034  ax-addrcl 10035  ax-mulcl 10036  ax-mulrcl 10037  ax-mulcom 10038  ax-addass 10039  ax-mulass 10040  ax-distr 10041  ax-i2m1 10042  ax-1ne0 10043  ax-1rid 10044  ax-rnegex 10045  ax-rrecex 10046  ax-cnre 10047  ax-pre-lttri 10048  ax-pre-lttrn 10049  ax-pre-ltadd 10050  ax-pre-mulgt0 10051  ax-pre-sup 10052
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-nel 2927  df-ral 2946  df-rex 2947  df-reu 2948  df-rmo 2949  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-pred 5718  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-riota 6651  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-om 7108  df-1st 7210  df-2nd 7211  df-wrecs 7452  df-recs 7513  df-rdg 7551  df-er 7787  df-map 7901  df-en 7998  df-dom 7999  df-sdom 8000  df-sup 8389  df-inf 8390  df-pnf 10114  df-mnf 10115  df-xr 10116  df-ltxr 10117  df-le 10118  df-sub 10306  df-neg 10307  df-div 10723  df-nn 11059  df-2 11117  df-3 11118  df-n0 11331  df-z 11416  df-uz 11726  df-q 11827  df-rp 11871  df-ioo 12217  df-ico 12219  df-icc 12220  df-fz 12365  df-fl 12633  df-seq 12842  df-exp 12901  df-cj 13883  df-re 13884  df-im 13885  df-sqrt 14019  df-abs 14020  df-ovol 23279  df-vol 23280
This theorem is referenced by:  inmbl  23356  finiunmbl  23358  volun  23359  voliunlem1  23364  icombl1  23377  iccmbl  23380  uniiccmbl  23404  mbfimaicc  23445  mbfeqalem  23454  mbfres2  23457  mbfmax  23461  itgss3  23626  ismblfin  33580  mbfposadd  33587  cnambfre  33588  itg2addnclem2  33592  iblabsnclem  33603  ftc1anclem1  33615  ftc1anclem5  33619  iocmbl  38115
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