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Theorem unmbl 23029
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 23023 . . . 4 (𝐴 ∈ dom vol → 𝐴 ⊆ ℝ)
2 mblss 23023 . . . 4 (𝐵 ∈ dom vol → 𝐵 ⊆ ℝ)
31, 2anim12i 587 . . 3 ((𝐴 ∈ dom vol ∧ 𝐵 ∈ dom vol) → (𝐴 ⊆ ℝ ∧ 𝐵 ⊆ ℝ))
4 unss 3748 . . 3 ((𝐴 ⊆ ℝ ∧ 𝐵 ⊆ ℝ) ↔ (𝐴𝐵) ⊆ ℝ)
53, 4sylib 206 . 2 ((𝐴 ∈ dom vol ∧ 𝐵 ∈ dom vol) → (𝐴𝐵) ⊆ ℝ)
6 elpwi 4116 . . . 4 (𝑥 ∈ 𝒫 ℝ → 𝑥 ⊆ ℝ)
7 inss1 3794 . . . . . . . . 9 (𝑥 ∩ (𝐴𝐵)) ⊆ 𝑥
8 ovolsscl 22978 . . . . . . . . 9 (((𝑥 ∩ (𝐴𝐵)) ⊆ 𝑥𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) → (vol*‘(𝑥 ∩ (𝐴𝐵))) ∈ ℝ)
97, 8mp3an1 1402 . . . . . . . 8 ((𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) → (vol*‘(𝑥 ∩ (𝐴𝐵))) ∈ ℝ)
109adantl 480 . . . . . . 7 (((𝐴 ∈ dom vol ∧ 𝐵 ∈ dom vol) ∧ (𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ)) → (vol*‘(𝑥 ∩ (𝐴𝐵))) ∈ ℝ)
11 inss1 3794 . . . . . . . . . 10 (𝑥𝐴) ⊆ 𝑥
12 ovolsscl 22978 . . . . . . . . . 10 (((𝑥𝐴) ⊆ 𝑥𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) → (vol*‘(𝑥𝐴)) ∈ ℝ)
1311, 12mp3an1 1402 . . . . . . . . 9 ((𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) → (vol*‘(𝑥𝐴)) ∈ ℝ)
1413adantl 480 . . . . . . . 8 (((𝐴 ∈ dom vol ∧ 𝐵 ∈ dom vol) ∧ (𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ)) → (vol*‘(𝑥𝐴)) ∈ ℝ)
15 difss 3698 . . . . . . . . . 10 (𝑥𝐴) ⊆ 𝑥
16 simprl 789 . . . . . . . . . 10 (((𝐴 ∈ dom vol ∧ 𝐵 ∈ dom vol) ∧ (𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ)) → 𝑥 ⊆ ℝ)
1715, 16syl5ss 3578 . . . . . . . . 9 (((𝐴 ∈ dom vol ∧ 𝐵 ∈ dom vol) ∧ (𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ)) → (𝑥𝐴) ⊆ ℝ)
18 ovolsscl 22978 . . . . . . . . . . 11 (((𝑥𝐴) ⊆ 𝑥𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) → (vol*‘(𝑥𝐴)) ∈ ℝ)
1915, 18mp3an1 1402 . . . . . . . . . 10 ((𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) → (vol*‘(𝑥𝐴)) ∈ ℝ)
2019adantl 480 . . . . . . . . 9 (((𝐴 ∈ dom vol ∧ 𝐵 ∈ dom vol) ∧ (𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ)) → (vol*‘(𝑥𝐴)) ∈ ℝ)
21 inss1 3794 . . . . . . . . . 10 ((𝑥𝐴) ∩ 𝐵) ⊆ (𝑥𝐴)
22 ovolsscl 22978 . . . . . . . . . 10 ((((𝑥𝐴) ∩ 𝐵) ⊆ (𝑥𝐴) ∧ (𝑥𝐴) ⊆ ℝ ∧ (vol*‘(𝑥𝐴)) ∈ ℝ) → (vol*‘((𝑥𝐴) ∩ 𝐵)) ∈ ℝ)
2321, 22mp3an1 1402 . . . . . . . . 9 (((𝑥𝐴) ⊆ ℝ ∧ (vol*‘(𝑥𝐴)) ∈ ℝ) → (vol*‘((𝑥𝐴) ∩ 𝐵)) ∈ ℝ)
2417, 20, 23syl2anc 690 . . . . . . . 8 (((𝐴 ∈ dom vol ∧ 𝐵 ∈ dom vol) ∧ (𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ)) → (vol*‘((𝑥𝐴) ∩ 𝐵)) ∈ ℝ)
2514, 24readdcld 9925 . . . . . . 7 (((𝐴 ∈ dom vol ∧ 𝐵 ∈ dom vol) ∧ (𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ)) → ((vol*‘(𝑥𝐴)) + (vol*‘((𝑥𝐴) ∩ 𝐵))) ∈ ℝ)
26 difss 3698 . . . . . . . . 9 (𝑥 ∖ (𝐴𝐵)) ⊆ 𝑥
27 ovolsscl 22978 . . . . . . . . 9 (((𝑥 ∖ (𝐴𝐵)) ⊆ 𝑥𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) → (vol*‘(𝑥 ∖ (𝐴𝐵))) ∈ ℝ)
2826, 27mp3an1 1402 . . . . . . . 8 ((𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) → (vol*‘(𝑥 ∖ (𝐴𝐵))) ∈ ℝ)
2928adantl 480 . . . . . . 7 (((𝐴 ∈ dom vol ∧ 𝐵 ∈ dom vol) ∧ (𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ)) → (vol*‘(𝑥 ∖ (𝐴𝐵))) ∈ ℝ)
30 incom 3766 . . . . . . . . . . . 12 ((𝑥𝐴) ∩ 𝐵) = (𝐵 ∩ (𝑥𝐴))
31 indifcom 3830 . . . . . . . . . . . 12 (𝐵 ∩ (𝑥𝐴)) = (𝑥 ∩ (𝐵𝐴))
3230, 31eqtri 2631 . . . . . . . . . . 11 ((𝑥𝐴) ∩ 𝐵) = (𝑥 ∩ (𝐵𝐴))
3332uneq2i 3725 . . . . . . . . . 10 ((𝑥𝐴) ∪ ((𝑥𝐴) ∩ 𝐵)) = ((𝑥𝐴) ∪ (𝑥 ∩ (𝐵𝐴)))
34 indi 3831 . . . . . . . . . 10 (𝑥 ∩ (𝐴 ∪ (𝐵𝐴))) = ((𝑥𝐴) ∪ (𝑥 ∩ (𝐵𝐴)))
35 undif2 3995 . . . . . . . . . . 11 (𝐴 ∪ (𝐵𝐴)) = (𝐴𝐵)
3635ineq2i 3772 . . . . . . . . . 10 (𝑥 ∩ (𝐴 ∪ (𝐵𝐴))) = (𝑥 ∩ (𝐴𝐵))
3733, 34, 363eqtr2ri 2638 . . . . . . . . 9 (𝑥 ∩ (𝐴𝐵)) = ((𝑥𝐴) ∪ ((𝑥𝐴) ∩ 𝐵))
3837fveq2i 6091 . . . . . . . 8 (vol*‘(𝑥 ∩ (𝐴𝐵))) = (vol*‘((𝑥𝐴) ∪ ((𝑥𝐴) ∩ 𝐵)))
3911, 16syl5ss 3578 . . . . . . . . 9 (((𝐴 ∈ dom vol ∧ 𝐵 ∈ dom vol) ∧ (𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ)) → (𝑥𝐴) ⊆ ℝ)
4021, 17syl5ss 3578 . . . . . . . . 9 (((𝐴 ∈ dom vol ∧ 𝐵 ∈ dom vol) ∧ (𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ)) → ((𝑥𝐴) ∩ 𝐵) ⊆ ℝ)
41 ovolun 22991 . . . . . . . . 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 1318 . . . . . . . 8 (((𝐴 ∈ dom vol ∧ 𝐵 ∈ dom vol) ∧ (𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ)) → (vol*‘((𝑥𝐴) ∪ ((𝑥𝐴) ∩ 𝐵))) ≤ ((vol*‘(𝑥𝐴)) + (vol*‘((𝑥𝐴) ∩ 𝐵))))
4338, 42syl5eqbr 4612 . . . . . . 7 (((𝐴 ∈ dom vol ∧ 𝐵 ∈ dom vol) ∧ (𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ)) → (vol*‘(𝑥 ∩ (𝐴𝐵))) ≤ ((vol*‘(𝑥𝐴)) + (vol*‘((𝑥𝐴) ∩ 𝐵))))
4410, 25, 29, 43leadd1dd 10490 . . . . . 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 787 . . . . . . . . . 10 (((𝐴 ∈ dom vol ∧ 𝐵 ∈ dom vol) ∧ (𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ)) → 𝐵 ∈ dom vol)
46 mblsplit 23024 . . . . . . . . . 10 ((𝐵 ∈ dom vol ∧ (𝑥𝐴) ⊆ ℝ ∧ (vol*‘(𝑥𝐴)) ∈ ℝ) → (vol*‘(𝑥𝐴)) = ((vol*‘((𝑥𝐴) ∩ 𝐵)) + (vol*‘((𝑥𝐴) ∖ 𝐵))))
4745, 17, 20, 46syl3anc 1317 . . . . . . . . 9 (((𝐴 ∈ dom vol ∧ 𝐵 ∈ dom vol) ∧ (𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ)) → (vol*‘(𝑥𝐴)) = ((vol*‘((𝑥𝐴) ∩ 𝐵)) + (vol*‘((𝑥𝐴) ∖ 𝐵))))
48 difun1 3845 . . . . . . . . . . 11 (𝑥 ∖ (𝐴𝐵)) = ((𝑥𝐴) ∖ 𝐵)
4948fveq2i 6091 . . . . . . . . . 10 (vol*‘(𝑥 ∖ (𝐴𝐵))) = (vol*‘((𝑥𝐴) ∖ 𝐵))
5049oveq2i 6538 . . . . . . . . 9 ((vol*‘((𝑥𝐴) ∩ 𝐵)) + (vol*‘(𝑥 ∖ (𝐴𝐵)))) = ((vol*‘((𝑥𝐴) ∩ 𝐵)) + (vol*‘((𝑥𝐴) ∖ 𝐵)))
5147, 50syl6eqr 2661 . . . . . . . 8 (((𝐴 ∈ dom vol ∧ 𝐵 ∈ dom vol) ∧ (𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ)) → (vol*‘(𝑥𝐴)) = ((vol*‘((𝑥𝐴) ∩ 𝐵)) + (vol*‘(𝑥 ∖ (𝐴𝐵)))))
5251oveq2d 6543 . . . . . . 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 785 . . . . . . . 8 (((𝐴 ∈ dom vol ∧ 𝐵 ∈ dom vol) ∧ (𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ)) → 𝐴 ∈ dom vol)
54 simprr 791 . . . . . . . 8 (((𝐴 ∈ dom vol ∧ 𝐵 ∈ dom vol) ∧ (𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ)) → (vol*‘𝑥) ∈ ℝ)
55 mblsplit 23024 . . . . . . . 8 ((𝐴 ∈ dom vol ∧ 𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) → (vol*‘𝑥) = ((vol*‘(𝑥𝐴)) + (vol*‘(𝑥𝐴))))
5653, 16, 54, 55syl3anc 1317 . . . . . . 7 (((𝐴 ∈ dom vol ∧ 𝐵 ∈ dom vol) ∧ (𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ)) → (vol*‘𝑥) = ((vol*‘(𝑥𝐴)) + (vol*‘(𝑥𝐴))))
5714recnd 9924 . . . . . . . 8 (((𝐴 ∈ dom vol ∧ 𝐵 ∈ dom vol) ∧ (𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ)) → (vol*‘(𝑥𝐴)) ∈ ℂ)
5824recnd 9924 . . . . . . . 8 (((𝐴 ∈ dom vol ∧ 𝐵 ∈ dom vol) ∧ (𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ)) → (vol*‘((𝑥𝐴) ∩ 𝐵)) ∈ ℂ)
5929recnd 9924 . . . . . . . 8 (((𝐴 ∈ dom vol ∧ 𝐵 ∈ dom vol) ∧ (𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ)) → (vol*‘(𝑥 ∖ (𝐴𝐵))) ∈ ℂ)
6057, 58, 59addassd 9918 . . . . . . 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 2653 . . . . . 6 (((𝐴 ∈ dom vol ∧ 𝐵 ∈ dom vol) ∧ (𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ)) → (vol*‘𝑥) = (((vol*‘(𝑥𝐴)) + (vol*‘((𝑥𝐴) ∩ 𝐵))) + (vol*‘(𝑥 ∖ (𝐴𝐵)))))
6244, 61breqtrrd 4605 . . . . 5 (((𝐴 ∈ dom vol ∧ 𝐵 ∈ dom vol) ∧ (𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ)) → ((vol*‘(𝑥 ∩ (𝐴𝐵))) + (vol*‘(𝑥 ∖ (𝐴𝐵)))) ≤ (vol*‘𝑥))
6362expr 640 . . . 4 (((𝐴 ∈ dom vol ∧ 𝐵 ∈ dom vol) ∧ 𝑥 ⊆ ℝ) → ((vol*‘𝑥) ∈ ℝ → ((vol*‘(𝑥 ∩ (𝐴𝐵))) + (vol*‘(𝑥 ∖ (𝐴𝐵)))) ≤ (vol*‘𝑥)))
646, 63sylan2 489 . . 3 (((𝐴 ∈ dom vol ∧ 𝐵 ∈ dom vol) ∧ 𝑥 ∈ 𝒫 ℝ) → ((vol*‘𝑥) ∈ ℝ → ((vol*‘(𝑥 ∩ (𝐴𝐵))) + (vol*‘(𝑥 ∖ (𝐴𝐵)))) ≤ (vol*‘𝑥)))
6564ralrimiva 2948 . 2 ((𝐴 ∈ dom vol ∧ 𝐵 ∈ dom vol) → ∀𝑥 ∈ 𝒫 ℝ((vol*‘𝑥) ∈ ℝ → ((vol*‘(𝑥 ∩ (𝐴𝐵))) + (vol*‘(𝑥 ∖ (𝐴𝐵)))) ≤ (vol*‘𝑥)))
66 ismbl2 23019 . 2 ((𝐴𝐵) ∈ dom vol ↔ ((𝐴𝐵) ⊆ ℝ ∧ ∀𝑥 ∈ 𝒫 ℝ((vol*‘𝑥) ∈ ℝ → ((vol*‘(𝑥 ∩ (𝐴𝐵))) + (vol*‘(𝑥 ∖ (𝐴𝐵)))) ≤ (vol*‘𝑥))))
675, 65, 66sylanbrc 694 1 ((𝐴 ∈ dom vol ∧ 𝐵 ∈ dom vol) → (𝐴𝐵) ∈ dom vol)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382   = wceq 1474  wcel 1976  wral 2895  cdif 3536  cun 3537  cin 3538  wss 3539  𝒫 cpw 4107   class class class wbr 4577  dom cdm 5028  cfv 5790  (class class class)co 6527  cr 9791   + caddc 9795  cle 9931  vol*covol 22955  volcvol 22956
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-8 1978  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2033  ax-13 2233  ax-ext 2589  ax-sep 4703  ax-nul 4712  ax-pow 4764  ax-pr 4828  ax-un 6824  ax-cnex 9848  ax-resscn 9849  ax-1cn 9850  ax-icn 9851  ax-addcl 9852  ax-addrcl 9853  ax-mulcl 9854  ax-mulrcl 9855  ax-mulcom 9856  ax-addass 9857  ax-mulass 9858  ax-distr 9859  ax-i2m1 9860  ax-1ne0 9861  ax-1rid 9862  ax-rnegex 9863  ax-rrecex 9864  ax-cnre 9865  ax-pre-lttri 9866  ax-pre-lttrn 9867  ax-pre-ltadd 9868  ax-pre-mulgt0 9869  ax-pre-sup 9870
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3or 1031  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-eu 2461  df-mo 2462  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-ne 2781  df-nel 2782  df-ral 2900  df-rex 2901  df-reu 2902  df-rmo 2903  df-rab 2904  df-v 3174  df-sbc 3402  df-csb 3499  df-dif 3542  df-un 3544  df-in 3546  df-ss 3553  df-pss 3555  df-nul 3874  df-if 4036  df-pw 4109  df-sn 4125  df-pr 4127  df-tp 4129  df-op 4131  df-uni 4367  df-iun 4451  df-br 4578  df-opab 4638  df-mpt 4639  df-tr 4675  df-eprel 4939  df-id 4943  df-po 4949  df-so 4950  df-fr 4987  df-we 4989  df-xp 5034  df-rel 5035  df-cnv 5036  df-co 5037  df-dm 5038  df-rn 5039  df-res 5040  df-ima 5041  df-pred 5583  df-ord 5629  df-on 5630  df-lim 5631  df-suc 5632  df-iota 5754  df-fun 5792  df-fn 5793  df-f 5794  df-f1 5795  df-fo 5796  df-f1o 5797  df-fv 5798  df-riota 6489  df-ov 6530  df-oprab 6531  df-mpt2 6532  df-om 6935  df-1st 7036  df-2nd 7037  df-wrecs 7271  df-recs 7332  df-rdg 7370  df-er 7606  df-map 7723  df-en 7819  df-dom 7820  df-sdom 7821  df-sup 8208  df-inf 8209  df-pnf 9932  df-mnf 9933  df-xr 9934  df-ltxr 9935  df-le 9936  df-sub 10119  df-neg 10120  df-div 10534  df-nn 10868  df-2 10926  df-3 10927  df-n0 11140  df-z 11211  df-uz 11520  df-q 11621  df-rp 11665  df-ioo 12006  df-ico 12008  df-icc 12009  df-fz 12153  df-fl 12410  df-seq 12619  df-exp 12678  df-cj 13633  df-re 13634  df-im 13635  df-sqrt 13769  df-abs 13770  df-ovol 22957  df-vol 22958
This theorem is referenced by:  inmbl  23034  finiunmbl  23036  volun  23037  voliunlem1  23042  icombl1  23055  iccmbl  23058  uniiccmbl  23081  mbfimaicc  23123  mbfeqalem  23132  mbfres2  23135  mbfmax  23139  itgss3  23304  ismblfin  32416  mbfposadd  32423  cnambfre  32424  itg2addnclem2  32428  iblabsnclem  32439  ftc1anclem1  32451  ftc1anclem5  32455  iocmbl  36613
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