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Theorem unmbl 23525
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 23519 . . . 4 (𝐴 ∈ dom vol → 𝐴 ⊆ ℝ)
2 mblss 23519 . . . 4 (𝐵 ∈ dom vol → 𝐵 ⊆ ℝ)
31, 2anim12i 600 . . 3 ((𝐴 ∈ dom vol ∧ 𝐵 ∈ dom vol) → (𝐴 ⊆ ℝ ∧ 𝐵 ⊆ ℝ))
4 unss 3938 . . 3 ((𝐴 ⊆ ℝ ∧ 𝐵 ⊆ ℝ) ↔ (𝐴𝐵) ⊆ ℝ)
53, 4sylib 208 . 2 ((𝐴 ∈ dom vol ∧ 𝐵 ∈ dom vol) → (𝐴𝐵) ⊆ ℝ)
6 elpwi 4307 . . . 4 (𝑥 ∈ 𝒫 ℝ → 𝑥 ⊆ ℝ)
7 inss1 3981 . . . . . . . . 9 (𝑥 ∩ (𝐴𝐵)) ⊆ 𝑥
8 ovolsscl 23474 . . . . . . . . 9 (((𝑥 ∩ (𝐴𝐵)) ⊆ 𝑥𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) → (vol*‘(𝑥 ∩ (𝐴𝐵))) ∈ ℝ)
97, 8mp3an1 1559 . . . . . . . 8 ((𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) → (vol*‘(𝑥 ∩ (𝐴𝐵))) ∈ ℝ)
109adantl 467 . . . . . . 7 (((𝐴 ∈ dom vol ∧ 𝐵 ∈ dom vol) ∧ (𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ)) → (vol*‘(𝑥 ∩ (𝐴𝐵))) ∈ ℝ)
11 inss1 3981 . . . . . . . . . 10 (𝑥𝐴) ⊆ 𝑥
12 ovolsscl 23474 . . . . . . . . . 10 (((𝑥𝐴) ⊆ 𝑥𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) → (vol*‘(𝑥𝐴)) ∈ ℝ)
1311, 12mp3an1 1559 . . . . . . . . 9 ((𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) → (vol*‘(𝑥𝐴)) ∈ ℝ)
1413adantl 467 . . . . . . . 8 (((𝐴 ∈ dom vol ∧ 𝐵 ∈ dom vol) ∧ (𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ)) → (vol*‘(𝑥𝐴)) ∈ ℝ)
15 difss 3888 . . . . . . . . . 10 (𝑥𝐴) ⊆ 𝑥
16 simprl 754 . . . . . . . . . 10 (((𝐴 ∈ dom vol ∧ 𝐵 ∈ dom vol) ∧ (𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ)) → 𝑥 ⊆ ℝ)
1715, 16syl5ss 3763 . . . . . . . . 9 (((𝐴 ∈ dom vol ∧ 𝐵 ∈ dom vol) ∧ (𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ)) → (𝑥𝐴) ⊆ ℝ)
18 ovolsscl 23474 . . . . . . . . . . 11 (((𝑥𝐴) ⊆ 𝑥𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) → (vol*‘(𝑥𝐴)) ∈ ℝ)
1915, 18mp3an1 1559 . . . . . . . . . 10 ((𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) → (vol*‘(𝑥𝐴)) ∈ ℝ)
2019adantl 467 . . . . . . . . 9 (((𝐴 ∈ dom vol ∧ 𝐵 ∈ dom vol) ∧ (𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ)) → (vol*‘(𝑥𝐴)) ∈ ℝ)
21 inss1 3981 . . . . . . . . . 10 ((𝑥𝐴) ∩ 𝐵) ⊆ (𝑥𝐴)
22 ovolsscl 23474 . . . . . . . . . 10 ((((𝑥𝐴) ∩ 𝐵) ⊆ (𝑥𝐴) ∧ (𝑥𝐴) ⊆ ℝ ∧ (vol*‘(𝑥𝐴)) ∈ ℝ) → (vol*‘((𝑥𝐴) ∩ 𝐵)) ∈ ℝ)
2321, 22mp3an1 1559 . . . . . . . . 9 (((𝑥𝐴) ⊆ ℝ ∧ (vol*‘(𝑥𝐴)) ∈ ℝ) → (vol*‘((𝑥𝐴) ∩ 𝐵)) ∈ ℝ)
2417, 20, 23syl2anc 573 . . . . . . . 8 (((𝐴 ∈ dom vol ∧ 𝐵 ∈ dom vol) ∧ (𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ)) → (vol*‘((𝑥𝐴) ∩ 𝐵)) ∈ ℝ)
2514, 24readdcld 10271 . . . . . . 7 (((𝐴 ∈ dom vol ∧ 𝐵 ∈ dom vol) ∧ (𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ)) → ((vol*‘(𝑥𝐴)) + (vol*‘((𝑥𝐴) ∩ 𝐵))) ∈ ℝ)
26 difss 3888 . . . . . . . . 9 (𝑥 ∖ (𝐴𝐵)) ⊆ 𝑥
27 ovolsscl 23474 . . . . . . . . 9 (((𝑥 ∖ (𝐴𝐵)) ⊆ 𝑥𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) → (vol*‘(𝑥 ∖ (𝐴𝐵))) ∈ ℝ)
2826, 27mp3an1 1559 . . . . . . . 8 ((𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) → (vol*‘(𝑥 ∖ (𝐴𝐵))) ∈ ℝ)
2928adantl 467 . . . . . . 7 (((𝐴 ∈ dom vol ∧ 𝐵 ∈ dom vol) ∧ (𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ)) → (vol*‘(𝑥 ∖ (𝐴𝐵))) ∈ ℝ)
30 incom 3956 . . . . . . . . . . . 12 ((𝑥𝐴) ∩ 𝐵) = (𝐵 ∩ (𝑥𝐴))
31 indifcom 4021 . . . . . . . . . . . 12 (𝐵 ∩ (𝑥𝐴)) = (𝑥 ∩ (𝐵𝐴))
3230, 31eqtri 2793 . . . . . . . . . . 11 ((𝑥𝐴) ∩ 𝐵) = (𝑥 ∩ (𝐵𝐴))
3332uneq2i 3915 . . . . . . . . . 10 ((𝑥𝐴) ∪ ((𝑥𝐴) ∩ 𝐵)) = ((𝑥𝐴) ∪ (𝑥 ∩ (𝐵𝐴)))
34 indi 4022 . . . . . . . . . 10 (𝑥 ∩ (𝐴 ∪ (𝐵𝐴))) = ((𝑥𝐴) ∪ (𝑥 ∩ (𝐵𝐴)))
35 undif2 4186 . . . . . . . . . . 11 (𝐴 ∪ (𝐵𝐴)) = (𝐴𝐵)
3635ineq2i 3962 . . . . . . . . . 10 (𝑥 ∩ (𝐴 ∪ (𝐵𝐴))) = (𝑥 ∩ (𝐴𝐵))
3733, 34, 363eqtr2ri 2800 . . . . . . . . 9 (𝑥 ∩ (𝐴𝐵)) = ((𝑥𝐴) ∪ ((𝑥𝐴) ∩ 𝐵))
3837fveq2i 6335 . . . . . . . 8 (vol*‘(𝑥 ∩ (𝐴𝐵))) = (vol*‘((𝑥𝐴) ∪ ((𝑥𝐴) ∩ 𝐵)))
3911, 16syl5ss 3763 . . . . . . . . 9 (((𝐴 ∈ dom vol ∧ 𝐵 ∈ dom vol) ∧ (𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ)) → (𝑥𝐴) ⊆ ℝ)
4021, 17syl5ss 3763 . . . . . . . . 9 (((𝐴 ∈ dom vol ∧ 𝐵 ∈ dom vol) ∧ (𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ)) → ((𝑥𝐴) ∩ 𝐵) ⊆ ℝ)
41 ovolun 23487 . . . . . . . . 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 1477 . . . . . . . 8 (((𝐴 ∈ dom vol ∧ 𝐵 ∈ dom vol) ∧ (𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ)) → (vol*‘((𝑥𝐴) ∪ ((𝑥𝐴) ∩ 𝐵))) ≤ ((vol*‘(𝑥𝐴)) + (vol*‘((𝑥𝐴) ∩ 𝐵))))
4338, 42syl5eqbr 4821 . . . . . . 7 (((𝐴 ∈ dom vol ∧ 𝐵 ∈ dom vol) ∧ (𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ)) → (vol*‘(𝑥 ∩ (𝐴𝐵))) ≤ ((vol*‘(𝑥𝐴)) + (vol*‘((𝑥𝐴) ∩ 𝐵))))
4410, 25, 29, 43leadd1dd 10843 . . . . . 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 752 . . . . . . . . . 10 (((𝐴 ∈ dom vol ∧ 𝐵 ∈ dom vol) ∧ (𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ)) → 𝐵 ∈ dom vol)
46 mblsplit 23520 . . . . . . . . . 10 ((𝐵 ∈ dom vol ∧ (𝑥𝐴) ⊆ ℝ ∧ (vol*‘(𝑥𝐴)) ∈ ℝ) → (vol*‘(𝑥𝐴)) = ((vol*‘((𝑥𝐴) ∩ 𝐵)) + (vol*‘((𝑥𝐴) ∖ 𝐵))))
4745, 17, 20, 46syl3anc 1476 . . . . . . . . 9 (((𝐴 ∈ dom vol ∧ 𝐵 ∈ dom vol) ∧ (𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ)) → (vol*‘(𝑥𝐴)) = ((vol*‘((𝑥𝐴) ∩ 𝐵)) + (vol*‘((𝑥𝐴) ∖ 𝐵))))
48 difun1 4036 . . . . . . . . . . 11 (𝑥 ∖ (𝐴𝐵)) = ((𝑥𝐴) ∖ 𝐵)
4948fveq2i 6335 . . . . . . . . . 10 (vol*‘(𝑥 ∖ (𝐴𝐵))) = (vol*‘((𝑥𝐴) ∖ 𝐵))
5049oveq2i 6804 . . . . . . . . 9 ((vol*‘((𝑥𝐴) ∩ 𝐵)) + (vol*‘(𝑥 ∖ (𝐴𝐵)))) = ((vol*‘((𝑥𝐴) ∩ 𝐵)) + (vol*‘((𝑥𝐴) ∖ 𝐵)))
5147, 50syl6eqr 2823 . . . . . . . 8 (((𝐴 ∈ dom vol ∧ 𝐵 ∈ dom vol) ∧ (𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ)) → (vol*‘(𝑥𝐴)) = ((vol*‘((𝑥𝐴) ∩ 𝐵)) + (vol*‘(𝑥 ∖ (𝐴𝐵)))))
5251oveq2d 6809 . . . . . . 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 750 . . . . . . . 8 (((𝐴 ∈ dom vol ∧ 𝐵 ∈ dom vol) ∧ (𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ)) → 𝐴 ∈ dom vol)
54 simprr 756 . . . . . . . 8 (((𝐴 ∈ dom vol ∧ 𝐵 ∈ dom vol) ∧ (𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ)) → (vol*‘𝑥) ∈ ℝ)
55 mblsplit 23520 . . . . . . . 8 ((𝐴 ∈ dom vol ∧ 𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) → (vol*‘𝑥) = ((vol*‘(𝑥𝐴)) + (vol*‘(𝑥𝐴))))
5653, 16, 54, 55syl3anc 1476 . . . . . . 7 (((𝐴 ∈ dom vol ∧ 𝐵 ∈ dom vol) ∧ (𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ)) → (vol*‘𝑥) = ((vol*‘(𝑥𝐴)) + (vol*‘(𝑥𝐴))))
5714recnd 10270 . . . . . . . 8 (((𝐴 ∈ dom vol ∧ 𝐵 ∈ dom vol) ∧ (𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ)) → (vol*‘(𝑥𝐴)) ∈ ℂ)
5824recnd 10270 . . . . . . . 8 (((𝐴 ∈ dom vol ∧ 𝐵 ∈ dom vol) ∧ (𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ)) → (vol*‘((𝑥𝐴) ∩ 𝐵)) ∈ ℂ)
5929recnd 10270 . . . . . . . 8 (((𝐴 ∈ dom vol ∧ 𝐵 ∈ dom vol) ∧ (𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ)) → (vol*‘(𝑥 ∖ (𝐴𝐵))) ∈ ℂ)
6057, 58, 59addassd 10264 . . . . . . 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 2815 . . . . . 6 (((𝐴 ∈ dom vol ∧ 𝐵 ∈ dom vol) ∧ (𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ)) → (vol*‘𝑥) = (((vol*‘(𝑥𝐴)) + (vol*‘((𝑥𝐴) ∩ 𝐵))) + (vol*‘(𝑥 ∖ (𝐴𝐵)))))
6244, 61breqtrrd 4814 . . . . 5 (((𝐴 ∈ dom vol ∧ 𝐵 ∈ dom vol) ∧ (𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ)) → ((vol*‘(𝑥 ∩ (𝐴𝐵))) + (vol*‘(𝑥 ∖ (𝐴𝐵)))) ≤ (vol*‘𝑥))
6362expr 444 . . . 4 (((𝐴 ∈ dom vol ∧ 𝐵 ∈ dom vol) ∧ 𝑥 ⊆ ℝ) → ((vol*‘𝑥) ∈ ℝ → ((vol*‘(𝑥 ∩ (𝐴𝐵))) + (vol*‘(𝑥 ∖ (𝐴𝐵)))) ≤ (vol*‘𝑥)))
646, 63sylan2 580 . . 3 (((𝐴 ∈ dom vol ∧ 𝐵 ∈ dom vol) ∧ 𝑥 ∈ 𝒫 ℝ) → ((vol*‘𝑥) ∈ ℝ → ((vol*‘(𝑥 ∩ (𝐴𝐵))) + (vol*‘(𝑥 ∖ (𝐴𝐵)))) ≤ (vol*‘𝑥)))
6564ralrimiva 3115 . 2 ((𝐴 ∈ dom vol ∧ 𝐵 ∈ dom vol) → ∀𝑥 ∈ 𝒫 ℝ((vol*‘𝑥) ∈ ℝ → ((vol*‘(𝑥 ∩ (𝐴𝐵))) + (vol*‘(𝑥 ∖ (𝐴𝐵)))) ≤ (vol*‘𝑥)))
66 ismbl2 23515 . 2 ((𝐴𝐵) ∈ dom vol ↔ ((𝐴𝐵) ⊆ ℝ ∧ ∀𝑥 ∈ 𝒫 ℝ((vol*‘𝑥) ∈ ℝ → ((vol*‘(𝑥 ∩ (𝐴𝐵))) + (vol*‘(𝑥 ∖ (𝐴𝐵)))) ≤ (vol*‘𝑥))))
675, 65, 66sylanbrc 572 1 ((𝐴 ∈ dom vol ∧ 𝐵 ∈ dom vol) → (𝐴𝐵) ∈ dom vol)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382   = wceq 1631  wcel 2145  wral 3061  cdif 3720  cun 3721  cin 3722  wss 3723  𝒫 cpw 4297   class class class wbr 4786  dom cdm 5249  cfv 6031  (class class class)co 6793  cr 10137   + caddc 10141  cle 10277  vol*covol 23450  volcvol 23451
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4915  ax-nul 4923  ax-pow 4974  ax-pr 5034  ax-un 7096  ax-cnex 10194  ax-resscn 10195  ax-1cn 10196  ax-icn 10197  ax-addcl 10198  ax-addrcl 10199  ax-mulcl 10200  ax-mulrcl 10201  ax-mulcom 10202  ax-addass 10203  ax-mulass 10204  ax-distr 10205  ax-i2m1 10206  ax-1ne0 10207  ax-1rid 10208  ax-rnegex 10209  ax-rrecex 10210  ax-cnre 10211  ax-pre-lttri 10212  ax-pre-lttrn 10213  ax-pre-ltadd 10214  ax-pre-mulgt0 10215  ax-pre-sup 10216
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3or 1072  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-nel 3047  df-ral 3066  df-rex 3067  df-reu 3068  df-rmo 3069  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-pss 3739  df-nul 4064  df-if 4226  df-pw 4299  df-sn 4317  df-pr 4319  df-tp 4321  df-op 4323  df-uni 4575  df-iun 4656  df-br 4787  df-opab 4847  df-mpt 4864  df-tr 4887  df-id 5157  df-eprel 5162  df-po 5170  df-so 5171  df-fr 5208  df-we 5210  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-pred 5823  df-ord 5869  df-on 5870  df-lim 5871  df-suc 5872  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-f1 6036  df-fo 6037  df-f1o 6038  df-fv 6039  df-riota 6754  df-ov 6796  df-oprab 6797  df-mpt2 6798  df-om 7213  df-1st 7315  df-2nd 7316  df-wrecs 7559  df-recs 7621  df-rdg 7659  df-er 7896  df-map 8011  df-en 8110  df-dom 8111  df-sdom 8112  df-sup 8504  df-inf 8505  df-pnf 10278  df-mnf 10279  df-xr 10280  df-ltxr 10281  df-le 10282  df-sub 10470  df-neg 10471  df-div 10887  df-nn 11223  df-2 11281  df-3 11282  df-n0 11495  df-z 11580  df-uz 11889  df-q 11992  df-rp 12036  df-ioo 12384  df-ico 12386  df-icc 12387  df-fz 12534  df-fl 12801  df-seq 13009  df-exp 13068  df-cj 14047  df-re 14048  df-im 14049  df-sqrt 14183  df-abs 14184  df-ovol 23452  df-vol 23453
This theorem is referenced by:  inmbl  23530  finiunmbl  23532  volun  23533  voliunlem1  23538  icombl1  23551  iccmbl  23554  uniiccmbl  23578  mbfimaicc  23619  mbfeqalem  23629  mbfres2  23632  mbfmax  23636  itgss3  23801  ismblfin  33783  mbfposadd  33789  cnambfre  33790  itg2addnclem2  33794  iblabsnclem  33805  ftc1anclem1  33817  ftc1anclem5  33821  iocmbl  38324
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