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Theorem unmbl 24904
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 24898 . . . 4 (𝐴 ∈ dom vol → 𝐴 ⊆ ℝ)
2 mblss 24898 . . . 4 (𝐵 ∈ dom vol → 𝐵 ⊆ ℝ)
31, 2anim12i 614 . . 3 ((𝐴 ∈ dom vol ∧ 𝐵 ∈ dom vol) → (𝐴 ⊆ ℝ ∧ 𝐵 ⊆ ℝ))
4 unss 4145 . . 3 ((𝐴 ⊆ ℝ ∧ 𝐵 ⊆ ℝ) ↔ (𝐴𝐵) ⊆ ℝ)
53, 4sylib 217 . 2 ((𝐴 ∈ dom vol ∧ 𝐵 ∈ dom vol) → (𝐴𝐵) ⊆ ℝ)
6 elpwi 4568 . . . 4 (𝑥 ∈ 𝒫 ℝ → 𝑥 ⊆ ℝ)
7 inss1 4189 . . . . . . . . 9 (𝑥 ∩ (𝐴𝐵)) ⊆ 𝑥
8 ovolsscl 24853 . . . . . . . . 9 (((𝑥 ∩ (𝐴𝐵)) ⊆ 𝑥𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) → (vol*‘(𝑥 ∩ (𝐴𝐵))) ∈ ℝ)
97, 8mp3an1 1449 . . . . . . . 8 ((𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) → (vol*‘(𝑥 ∩ (𝐴𝐵))) ∈ ℝ)
109adantl 483 . . . . . . 7 (((𝐴 ∈ dom vol ∧ 𝐵 ∈ dom vol) ∧ (𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ)) → (vol*‘(𝑥 ∩ (𝐴𝐵))) ∈ ℝ)
11 inss1 4189 . . . . . . . . . 10 (𝑥𝐴) ⊆ 𝑥
12 ovolsscl 24853 . . . . . . . . . 10 (((𝑥𝐴) ⊆ 𝑥𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) → (vol*‘(𝑥𝐴)) ∈ ℝ)
1311, 12mp3an1 1449 . . . . . . . . 9 ((𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) → (vol*‘(𝑥𝐴)) ∈ ℝ)
1413adantl 483 . . . . . . . 8 (((𝐴 ∈ dom vol ∧ 𝐵 ∈ dom vol) ∧ (𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ)) → (vol*‘(𝑥𝐴)) ∈ ℝ)
15 inss1 4189 . . . . . . . . 9 ((𝑥𝐴) ∩ 𝐵) ⊆ (𝑥𝐴)
16 difss 4092 . . . . . . . . . 10 (𝑥𝐴) ⊆ 𝑥
17 simprl 770 . . . . . . . . . 10 (((𝐴 ∈ dom vol ∧ 𝐵 ∈ dom vol) ∧ (𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ)) → 𝑥 ⊆ ℝ)
1816, 17sstrid 3956 . . . . . . . . 9 (((𝐴 ∈ dom vol ∧ 𝐵 ∈ dom vol) ∧ (𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ)) → (𝑥𝐴) ⊆ ℝ)
19 ovolsscl 24853 . . . . . . . . . . 11 (((𝑥𝐴) ⊆ 𝑥𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) → (vol*‘(𝑥𝐴)) ∈ ℝ)
2016, 19mp3an1 1449 . . . . . . . . . 10 ((𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) → (vol*‘(𝑥𝐴)) ∈ ℝ)
2120adantl 483 . . . . . . . . 9 (((𝐴 ∈ dom vol ∧ 𝐵 ∈ dom vol) ∧ (𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ)) → (vol*‘(𝑥𝐴)) ∈ ℝ)
22 ovolsscl 24853 . . . . . . . . 9 ((((𝑥𝐴) ∩ 𝐵) ⊆ (𝑥𝐴) ∧ (𝑥𝐴) ⊆ ℝ ∧ (vol*‘(𝑥𝐴)) ∈ ℝ) → (vol*‘((𝑥𝐴) ∩ 𝐵)) ∈ ℝ)
2315, 18, 21, 22mp3an2i 1467 . . . . . . . 8 (((𝐴 ∈ dom vol ∧ 𝐵 ∈ dom vol) ∧ (𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ)) → (vol*‘((𝑥𝐴) ∩ 𝐵)) ∈ ℝ)
2414, 23readdcld 11185 . . . . . . 7 (((𝐴 ∈ dom vol ∧ 𝐵 ∈ dom vol) ∧ (𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ)) → ((vol*‘(𝑥𝐴)) + (vol*‘((𝑥𝐴) ∩ 𝐵))) ∈ ℝ)
25 difss 4092 . . . . . . . . 9 (𝑥 ∖ (𝐴𝐵)) ⊆ 𝑥
26 ovolsscl 24853 . . . . . . . . 9 (((𝑥 ∖ (𝐴𝐵)) ⊆ 𝑥𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) → (vol*‘(𝑥 ∖ (𝐴𝐵))) ∈ ℝ)
2725, 26mp3an1 1449 . . . . . . . 8 ((𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) → (vol*‘(𝑥 ∖ (𝐴𝐵))) ∈ ℝ)
2827adantl 483 . . . . . . 7 (((𝐴 ∈ dom vol ∧ 𝐵 ∈ dom vol) ∧ (𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ)) → (vol*‘(𝑥 ∖ (𝐴𝐵))) ∈ ℝ)
29 incom 4162 . . . . . . . . . . . 12 ((𝑥𝐴) ∩ 𝐵) = (𝐵 ∩ (𝑥𝐴))
30 indifcom 4233 . . . . . . . . . . . 12 (𝐵 ∩ (𝑥𝐴)) = (𝑥 ∩ (𝐵𝐴))
3129, 30eqtri 2765 . . . . . . . . . . 11 ((𝑥𝐴) ∩ 𝐵) = (𝑥 ∩ (𝐵𝐴))
3231uneq2i 4121 . . . . . . . . . 10 ((𝑥𝐴) ∪ ((𝑥𝐴) ∩ 𝐵)) = ((𝑥𝐴) ∪ (𝑥 ∩ (𝐵𝐴)))
33 indi 4234 . . . . . . . . . 10 (𝑥 ∩ (𝐴 ∪ (𝐵𝐴))) = ((𝑥𝐴) ∪ (𝑥 ∩ (𝐵𝐴)))
34 undif2 4437 . . . . . . . . . . 11 (𝐴 ∪ (𝐵𝐴)) = (𝐴𝐵)
3534ineq2i 4170 . . . . . . . . . 10 (𝑥 ∩ (𝐴 ∪ (𝐵𝐴))) = (𝑥 ∩ (𝐴𝐵))
3632, 33, 353eqtr2ri 2772 . . . . . . . . 9 (𝑥 ∩ (𝐴𝐵)) = ((𝑥𝐴) ∪ ((𝑥𝐴) ∩ 𝐵))
3736fveq2i 6846 . . . . . . . 8 (vol*‘(𝑥 ∩ (𝐴𝐵))) = (vol*‘((𝑥𝐴) ∪ ((𝑥𝐴) ∩ 𝐵)))
3811, 17sstrid 3956 . . . . . . . . 9 (((𝐴 ∈ dom vol ∧ 𝐵 ∈ dom vol) ∧ (𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ)) → (𝑥𝐴) ⊆ ℝ)
3915, 18sstrid 3956 . . . . . . . . 9 (((𝐴 ∈ dom vol ∧ 𝐵 ∈ dom vol) ∧ (𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ)) → ((𝑥𝐴) ∩ 𝐵) ⊆ ℝ)
40 ovolun 24866 . . . . . . . . 9 ((((𝑥𝐴) ⊆ ℝ ∧ (vol*‘(𝑥𝐴)) ∈ ℝ) ∧ (((𝑥𝐴) ∩ 𝐵) ⊆ ℝ ∧ (vol*‘((𝑥𝐴) ∩ 𝐵)) ∈ ℝ)) → (vol*‘((𝑥𝐴) ∪ ((𝑥𝐴) ∩ 𝐵))) ≤ ((vol*‘(𝑥𝐴)) + (vol*‘((𝑥𝐴) ∩ 𝐵))))
4138, 14, 39, 23, 40syl22anc 838 . . . . . . . 8 (((𝐴 ∈ dom vol ∧ 𝐵 ∈ dom vol) ∧ (𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ)) → (vol*‘((𝑥𝐴) ∪ ((𝑥𝐴) ∩ 𝐵))) ≤ ((vol*‘(𝑥𝐴)) + (vol*‘((𝑥𝐴) ∩ 𝐵))))
4237, 41eqbrtrid 5141 . . . . . . 7 (((𝐴 ∈ dom vol ∧ 𝐵 ∈ dom vol) ∧ (𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ)) → (vol*‘(𝑥 ∩ (𝐴𝐵))) ≤ ((vol*‘(𝑥𝐴)) + (vol*‘((𝑥𝐴) ∩ 𝐵))))
4310, 24, 28, 42leadd1dd 11770 . . . . . 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)))))
44 simplr 768 . . . . . . . . . 10 (((𝐴 ∈ dom vol ∧ 𝐵 ∈ dom vol) ∧ (𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ)) → 𝐵 ∈ dom vol)
45 mblsplit 24899 . . . . . . . . . 10 ((𝐵 ∈ dom vol ∧ (𝑥𝐴) ⊆ ℝ ∧ (vol*‘(𝑥𝐴)) ∈ ℝ) → (vol*‘(𝑥𝐴)) = ((vol*‘((𝑥𝐴) ∩ 𝐵)) + (vol*‘((𝑥𝐴) ∖ 𝐵))))
4644, 18, 21, 45syl3anc 1372 . . . . . . . . 9 (((𝐴 ∈ dom vol ∧ 𝐵 ∈ dom vol) ∧ (𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ)) → (vol*‘(𝑥𝐴)) = ((vol*‘((𝑥𝐴) ∩ 𝐵)) + (vol*‘((𝑥𝐴) ∖ 𝐵))))
47 difun1 4250 . . . . . . . . . . 11 (𝑥 ∖ (𝐴𝐵)) = ((𝑥𝐴) ∖ 𝐵)
4847fveq2i 6846 . . . . . . . . . 10 (vol*‘(𝑥 ∖ (𝐴𝐵))) = (vol*‘((𝑥𝐴) ∖ 𝐵))
4948oveq2i 7369 . . . . . . . . 9 ((vol*‘((𝑥𝐴) ∩ 𝐵)) + (vol*‘(𝑥 ∖ (𝐴𝐵)))) = ((vol*‘((𝑥𝐴) ∩ 𝐵)) + (vol*‘((𝑥𝐴) ∖ 𝐵)))
5046, 49eqtr4di 2795 . . . . . . . 8 (((𝐴 ∈ dom vol ∧ 𝐵 ∈ dom vol) ∧ (𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ)) → (vol*‘(𝑥𝐴)) = ((vol*‘((𝑥𝐴) ∩ 𝐵)) + (vol*‘(𝑥 ∖ (𝐴𝐵)))))
5150oveq2d 7374 . . . . . . 7 (((𝐴 ∈ dom vol ∧ 𝐵 ∈ dom vol) ∧ (𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ)) → ((vol*‘(𝑥𝐴)) + (vol*‘(𝑥𝐴))) = ((vol*‘(𝑥𝐴)) + ((vol*‘((𝑥𝐴) ∩ 𝐵)) + (vol*‘(𝑥 ∖ (𝐴𝐵))))))
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 24899 . . . . . . . 8 ((𝐴 ∈ dom vol ∧ 𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) → (vol*‘𝑥) = ((vol*‘(𝑥𝐴)) + (vol*‘(𝑥𝐴))))
5552, 17, 53, 54syl3anc 1372 . . . . . . 7 (((𝐴 ∈ dom vol ∧ 𝐵 ∈ dom vol) ∧ (𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ)) → (vol*‘𝑥) = ((vol*‘(𝑥𝐴)) + (vol*‘(𝑥𝐴))))
5614recnd 11184 . . . . . . . 8 (((𝐴 ∈ dom vol ∧ 𝐵 ∈ dom vol) ∧ (𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ)) → (vol*‘(𝑥𝐴)) ∈ ℂ)
5723recnd 11184 . . . . . . . 8 (((𝐴 ∈ dom vol ∧ 𝐵 ∈ dom vol) ∧ (𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ)) → (vol*‘((𝑥𝐴) ∩ 𝐵)) ∈ ℂ)
5828recnd 11184 . . . . . . . 8 (((𝐴 ∈ dom vol ∧ 𝐵 ∈ dom vol) ∧ (𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ)) → (vol*‘(𝑥 ∖ (𝐴𝐵))) ∈ ℂ)
5956, 57, 58addassd 11178 . . . . . . 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))))))
6051, 55, 593eqtr4d 2787 . . . . . 6 (((𝐴 ∈ dom vol ∧ 𝐵 ∈ dom vol) ∧ (𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ)) → (vol*‘𝑥) = (((vol*‘(𝑥𝐴)) + (vol*‘((𝑥𝐴) ∩ 𝐵))) + (vol*‘(𝑥 ∖ (𝐴𝐵)))))
6143, 60breqtrrd 5134 . . . . 5 (((𝐴 ∈ dom vol ∧ 𝐵 ∈ dom vol) ∧ (𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ)) → ((vol*‘(𝑥 ∩ (𝐴𝐵))) + (vol*‘(𝑥 ∖ (𝐴𝐵)))) ≤ (vol*‘𝑥))
6261expr 458 . . . 4 (((𝐴 ∈ dom vol ∧ 𝐵 ∈ dom vol) ∧ 𝑥 ⊆ ℝ) → ((vol*‘𝑥) ∈ ℝ → ((vol*‘(𝑥 ∩ (𝐴𝐵))) + (vol*‘(𝑥 ∖ (𝐴𝐵)))) ≤ (vol*‘𝑥)))
636, 62sylan2 594 . . 3 (((𝐴 ∈ dom vol ∧ 𝐵 ∈ dom vol) ∧ 𝑥 ∈ 𝒫 ℝ) → ((vol*‘𝑥) ∈ ℝ → ((vol*‘(𝑥 ∩ (𝐴𝐵))) + (vol*‘(𝑥 ∖ (𝐴𝐵)))) ≤ (vol*‘𝑥)))
6463ralrimiva 3144 . 2 ((𝐴 ∈ dom vol ∧ 𝐵 ∈ dom vol) → ∀𝑥 ∈ 𝒫 ℝ((vol*‘𝑥) ∈ ℝ → ((vol*‘(𝑥 ∩ (𝐴𝐵))) + (vol*‘(𝑥 ∖ (𝐴𝐵)))) ≤ (vol*‘𝑥)))
65 ismbl2 24894 . 2 ((𝐴𝐵) ∈ dom vol ↔ ((𝐴𝐵) ⊆ ℝ ∧ ∀𝑥 ∈ 𝒫 ℝ((vol*‘𝑥) ∈ ℝ → ((vol*‘(𝑥 ∩ (𝐴𝐵))) + (vol*‘(𝑥 ∖ (𝐴𝐵)))) ≤ (vol*‘𝑥))))
665, 64, 65sylanbrc 584 1 ((𝐴 ∈ dom vol ∧ 𝐵 ∈ dom vol) → (𝐴𝐵) ∈ dom vol)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 397   = wceq 1542  wcel 2107  wral 3065  cdif 3908  cun 3909  cin 3910  wss 3911  𝒫 cpw 4561   class class class wbr 5106  dom cdm 5634  cfv 6497  (class class class)co 7358  cr 11051   + caddc 11055  cle 11191  vol*covol 24829  volcvol 24830
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2708  ax-sep 5257  ax-nul 5264  ax-pow 5321  ax-pr 5385  ax-un 7673  ax-cnex 11108  ax-resscn 11109  ax-1cn 11110  ax-icn 11111  ax-addcl 11112  ax-addrcl 11113  ax-mulcl 11114  ax-mulrcl 11115  ax-mulcom 11116  ax-addass 11117  ax-mulass 11118  ax-distr 11119  ax-i2m1 11120  ax-1ne0 11121  ax-1rid 11122  ax-rnegex 11123  ax-rrecex 11124  ax-cnre 11125  ax-pre-lttri 11126  ax-pre-lttrn 11127  ax-pre-ltadd 11128  ax-pre-mulgt0 11129  ax-pre-sup 11130
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2890  df-ne 2945  df-nel 3051  df-ral 3066  df-rex 3075  df-rmo 3354  df-reu 3355  df-rab 3409  df-v 3448  df-sbc 3741  df-csb 3857  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-pss 3930  df-nul 4284  df-if 4488  df-pw 4563  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-iun 4957  df-br 5107  df-opab 5169  df-mpt 5190  df-tr 5224  df-id 5532  df-eprel 5538  df-po 5546  df-so 5547  df-fr 5589  df-we 5591  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-res 5646  df-ima 5647  df-pred 6254  df-ord 6321  df-on 6322  df-lim 6323  df-suc 6324  df-iota 6449  df-fun 6499  df-fn 6500  df-f 6501  df-f1 6502  df-fo 6503  df-f1o 6504  df-fv 6505  df-riota 7314  df-ov 7361  df-oprab 7362  df-mpo 7363  df-om 7804  df-1st 7922  df-2nd 7923  df-frecs 8213  df-wrecs 8244  df-recs 8318  df-rdg 8357  df-er 8649  df-map 8768  df-en 8885  df-dom 8886  df-sdom 8887  df-sup 9379  df-inf 9380  df-pnf 11192  df-mnf 11193  df-xr 11194  df-ltxr 11195  df-le 11196  df-sub 11388  df-neg 11389  df-div 11814  df-nn 12155  df-2 12217  df-3 12218  df-n0 12415  df-z 12501  df-uz 12765  df-q 12875  df-rp 12917  df-ioo 13269  df-ico 13271  df-icc 13272  df-fz 13426  df-fl 13698  df-seq 13908  df-exp 13969  df-cj 14985  df-re 14986  df-im 14987  df-sqrt 15121  df-abs 15122  df-ovol 24831  df-vol 24832
This theorem is referenced by:  inmbl  24909  finiunmbl  24911  volun  24912  voliunlem1  24917  icombl1  24930  iccmbl  24933  uniiccmbl  24957  mbfimaicc  24998  mbfeqalem2  25009  mbfres2  25012  mbfmax  25016  itgss3  25182  ismblfin  36122  mbfposadd  36128  cnambfre  36129  itg2addnclem2  36133  iblabsnclem  36144  ftc1anclem1  36154  ftc1anclem5  36158  iocmbl  41550
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