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Theorem unmbl 24144
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 24138 . . . 4 (𝐴 ∈ dom vol → 𝐴 ⊆ ℝ)
2 mblss 24138 . . . 4 (𝐵 ∈ dom vol → 𝐵 ⊆ ℝ)
31, 2anim12i 615 . . 3 ((𝐴 ∈ dom vol ∧ 𝐵 ∈ dom vol) → (𝐴 ⊆ ℝ ∧ 𝐵 ⊆ ℝ))
4 unss 4114 . . 3 ((𝐴 ⊆ ℝ ∧ 𝐵 ⊆ ℝ) ↔ (𝐴𝐵) ⊆ ℝ)
53, 4sylib 221 . 2 ((𝐴 ∈ dom vol ∧ 𝐵 ∈ dom vol) → (𝐴𝐵) ⊆ ℝ)
6 elpwi 4509 . . . 4 (𝑥 ∈ 𝒫 ℝ → 𝑥 ⊆ ℝ)
7 inss1 4158 . . . . . . . . 9 (𝑥 ∩ (𝐴𝐵)) ⊆ 𝑥
8 ovolsscl 24093 . . . . . . . . 9 (((𝑥 ∩ (𝐴𝐵)) ⊆ 𝑥𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) → (vol*‘(𝑥 ∩ (𝐴𝐵))) ∈ ℝ)
97, 8mp3an1 1445 . . . . . . . 8 ((𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) → (vol*‘(𝑥 ∩ (𝐴𝐵))) ∈ ℝ)
109adantl 485 . . . . . . 7 (((𝐴 ∈ dom vol ∧ 𝐵 ∈ dom vol) ∧ (𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ)) → (vol*‘(𝑥 ∩ (𝐴𝐵))) ∈ ℝ)
11 inss1 4158 . . . . . . . . . 10 (𝑥𝐴) ⊆ 𝑥
12 ovolsscl 24093 . . . . . . . . . 10 (((𝑥𝐴) ⊆ 𝑥𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) → (vol*‘(𝑥𝐴)) ∈ ℝ)
1311, 12mp3an1 1445 . . . . . . . . 9 ((𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) → (vol*‘(𝑥𝐴)) ∈ ℝ)
1413adantl 485 . . . . . . . 8 (((𝐴 ∈ dom vol ∧ 𝐵 ∈ dom vol) ∧ (𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ)) → (vol*‘(𝑥𝐴)) ∈ ℝ)
15 inss1 4158 . . . . . . . . 9 ((𝑥𝐴) ∩ 𝐵) ⊆ (𝑥𝐴)
16 difss 4062 . . . . . . . . . 10 (𝑥𝐴) ⊆ 𝑥
17 simprl 770 . . . . . . . . . 10 (((𝐴 ∈ dom vol ∧ 𝐵 ∈ dom vol) ∧ (𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ)) → 𝑥 ⊆ ℝ)
1816, 17sstrid 3929 . . . . . . . . 9 (((𝐴 ∈ dom vol ∧ 𝐵 ∈ dom vol) ∧ (𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ)) → (𝑥𝐴) ⊆ ℝ)
19 ovolsscl 24093 . . . . . . . . . . 11 (((𝑥𝐴) ⊆ 𝑥𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) → (vol*‘(𝑥𝐴)) ∈ ℝ)
2016, 19mp3an1 1445 . . . . . . . . . 10 ((𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) → (vol*‘(𝑥𝐴)) ∈ ℝ)
2120adantl 485 . . . . . . . . 9 (((𝐴 ∈ dom vol ∧ 𝐵 ∈ dom vol) ∧ (𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ)) → (vol*‘(𝑥𝐴)) ∈ ℝ)
22 ovolsscl 24093 . . . . . . . . 9 ((((𝑥𝐴) ∩ 𝐵) ⊆ (𝑥𝐴) ∧ (𝑥𝐴) ⊆ ℝ ∧ (vol*‘(𝑥𝐴)) ∈ ℝ) → (vol*‘((𝑥𝐴) ∩ 𝐵)) ∈ ℝ)
2315, 18, 21, 22mp3an2i 1463 . . . . . . . 8 (((𝐴 ∈ dom vol ∧ 𝐵 ∈ dom vol) ∧ (𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ)) → (vol*‘((𝑥𝐴) ∩ 𝐵)) ∈ ℝ)
2414, 23readdcld 10663 . . . . . . 7 (((𝐴 ∈ dom vol ∧ 𝐵 ∈ dom vol) ∧ (𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ)) → ((vol*‘(𝑥𝐴)) + (vol*‘((𝑥𝐴) ∩ 𝐵))) ∈ ℝ)
25 difss 4062 . . . . . . . . 9 (𝑥 ∖ (𝐴𝐵)) ⊆ 𝑥
26 ovolsscl 24093 . . . . . . . . 9 (((𝑥 ∖ (𝐴𝐵)) ⊆ 𝑥𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) → (vol*‘(𝑥 ∖ (𝐴𝐵))) ∈ ℝ)
2725, 26mp3an1 1445 . . . . . . . 8 ((𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) → (vol*‘(𝑥 ∖ (𝐴𝐵))) ∈ ℝ)
2827adantl 485 . . . . . . 7 (((𝐴 ∈ dom vol ∧ 𝐵 ∈ dom vol) ∧ (𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ)) → (vol*‘(𝑥 ∖ (𝐴𝐵))) ∈ ℝ)
29 incom 4131 . . . . . . . . . . . 12 ((𝑥𝐴) ∩ 𝐵) = (𝐵 ∩ (𝑥𝐴))
30 indifcom 4202 . . . . . . . . . . . 12 (𝐵 ∩ (𝑥𝐴)) = (𝑥 ∩ (𝐵𝐴))
3129, 30eqtri 2824 . . . . . . . . . . 11 ((𝑥𝐴) ∩ 𝐵) = (𝑥 ∩ (𝐵𝐴))
3231uneq2i 4090 . . . . . . . . . 10 ((𝑥𝐴) ∪ ((𝑥𝐴) ∩ 𝐵)) = ((𝑥𝐴) ∪ (𝑥 ∩ (𝐵𝐴)))
33 indi 4203 . . . . . . . . . 10 (𝑥 ∩ (𝐴 ∪ (𝐵𝐴))) = ((𝑥𝐴) ∪ (𝑥 ∩ (𝐵𝐴)))
34 undif2 4386 . . . . . . . . . . 11 (𝐴 ∪ (𝐵𝐴)) = (𝐴𝐵)
3534ineq2i 4139 . . . . . . . . . 10 (𝑥 ∩ (𝐴 ∪ (𝐵𝐴))) = (𝑥 ∩ (𝐴𝐵))
3632, 33, 353eqtr2ri 2831 . . . . . . . . 9 (𝑥 ∩ (𝐴𝐵)) = ((𝑥𝐴) ∪ ((𝑥𝐴) ∩ 𝐵))
3736fveq2i 6652 . . . . . . . 8 (vol*‘(𝑥 ∩ (𝐴𝐵))) = (vol*‘((𝑥𝐴) ∪ ((𝑥𝐴) ∩ 𝐵)))
3811, 17sstrid 3929 . . . . . . . . 9 (((𝐴 ∈ dom vol ∧ 𝐵 ∈ dom vol) ∧ (𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ)) → (𝑥𝐴) ⊆ ℝ)
3915, 18sstrid 3929 . . . . . . . . 9 (((𝐴 ∈ dom vol ∧ 𝐵 ∈ dom vol) ∧ (𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ)) → ((𝑥𝐴) ∩ 𝐵) ⊆ ℝ)
40 ovolun 24106 . . . . . . . . 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 837 . . . . . . . 8 (((𝐴 ∈ dom vol ∧ 𝐵 ∈ dom vol) ∧ (𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ)) → (vol*‘((𝑥𝐴) ∪ ((𝑥𝐴) ∩ 𝐵))) ≤ ((vol*‘(𝑥𝐴)) + (vol*‘((𝑥𝐴) ∩ 𝐵))))
4237, 41eqbrtrid 5068 . . . . . . 7 (((𝐴 ∈ dom vol ∧ 𝐵 ∈ dom vol) ∧ (𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ)) → (vol*‘(𝑥 ∩ (𝐴𝐵))) ≤ ((vol*‘(𝑥𝐴)) + (vol*‘((𝑥𝐴) ∩ 𝐵))))
4310, 24, 28, 42leadd1dd 11247 . . . . . 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 24139 . . . . . . . . . 10 ((𝐵 ∈ dom vol ∧ (𝑥𝐴) ⊆ ℝ ∧ (vol*‘(𝑥𝐴)) ∈ ℝ) → (vol*‘(𝑥𝐴)) = ((vol*‘((𝑥𝐴) ∩ 𝐵)) + (vol*‘((𝑥𝐴) ∖ 𝐵))))
4644, 18, 21, 45syl3anc 1368 . . . . . . . . 9 (((𝐴 ∈ dom vol ∧ 𝐵 ∈ dom vol) ∧ (𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ)) → (vol*‘(𝑥𝐴)) = ((vol*‘((𝑥𝐴) ∩ 𝐵)) + (vol*‘((𝑥𝐴) ∖ 𝐵))))
47 difun1 4217 . . . . . . . . . . 11 (𝑥 ∖ (𝐴𝐵)) = ((𝑥𝐴) ∖ 𝐵)
4847fveq2i 6652 . . . . . . . . . 10 (vol*‘(𝑥 ∖ (𝐴𝐵))) = (vol*‘((𝑥𝐴) ∖ 𝐵))
4948oveq2i 7150 . . . . . . . . 9 ((vol*‘((𝑥𝐴) ∩ 𝐵)) + (vol*‘(𝑥 ∖ (𝐴𝐵)))) = ((vol*‘((𝑥𝐴) ∩ 𝐵)) + (vol*‘((𝑥𝐴) ∖ 𝐵)))
5046, 49eqtr4di 2854 . . . . . . . 8 (((𝐴 ∈ dom vol ∧ 𝐵 ∈ dom vol) ∧ (𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ)) → (vol*‘(𝑥𝐴)) = ((vol*‘((𝑥𝐴) ∩ 𝐵)) + (vol*‘(𝑥 ∖ (𝐴𝐵)))))
5150oveq2d 7155 . . . . . . 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 24139 . . . . . . . 8 ((𝐴 ∈ dom vol ∧ 𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) → (vol*‘𝑥) = ((vol*‘(𝑥𝐴)) + (vol*‘(𝑥𝐴))))
5552, 17, 53, 54syl3anc 1368 . . . . . . 7 (((𝐴 ∈ dom vol ∧ 𝐵 ∈ dom vol) ∧ (𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ)) → (vol*‘𝑥) = ((vol*‘(𝑥𝐴)) + (vol*‘(𝑥𝐴))))
5614recnd 10662 . . . . . . . 8 (((𝐴 ∈ dom vol ∧ 𝐵 ∈ dom vol) ∧ (𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ)) → (vol*‘(𝑥𝐴)) ∈ ℂ)
5723recnd 10662 . . . . . . . 8 (((𝐴 ∈ dom vol ∧ 𝐵 ∈ dom vol) ∧ (𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ)) → (vol*‘((𝑥𝐴) ∩ 𝐵)) ∈ ℂ)
5828recnd 10662 . . . . . . . 8 (((𝐴 ∈ dom vol ∧ 𝐵 ∈ dom vol) ∧ (𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ)) → (vol*‘(𝑥 ∖ (𝐴𝐵))) ∈ ℂ)
5956, 57, 58addassd 10656 . . . . . . 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 2846 . . . . . 6 (((𝐴 ∈ dom vol ∧ 𝐵 ∈ dom vol) ∧ (𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ)) → (vol*‘𝑥) = (((vol*‘(𝑥𝐴)) + (vol*‘((𝑥𝐴) ∩ 𝐵))) + (vol*‘(𝑥 ∖ (𝐴𝐵)))))
6143, 60breqtrrd 5061 . . . . 5 (((𝐴 ∈ dom vol ∧ 𝐵 ∈ dom vol) ∧ (𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ)) → ((vol*‘(𝑥 ∩ (𝐴𝐵))) + (vol*‘(𝑥 ∖ (𝐴𝐵)))) ≤ (vol*‘𝑥))
6261expr 460 . . . 4 (((𝐴 ∈ dom vol ∧ 𝐵 ∈ dom vol) ∧ 𝑥 ⊆ ℝ) → ((vol*‘𝑥) ∈ ℝ → ((vol*‘(𝑥 ∩ (𝐴𝐵))) + (vol*‘(𝑥 ∖ (𝐴𝐵)))) ≤ (vol*‘𝑥)))
636, 62sylan2 595 . . 3 (((𝐴 ∈ dom vol ∧ 𝐵 ∈ dom vol) ∧ 𝑥 ∈ 𝒫 ℝ) → ((vol*‘𝑥) ∈ ℝ → ((vol*‘(𝑥 ∩ (𝐴𝐵))) + (vol*‘(𝑥 ∖ (𝐴𝐵)))) ≤ (vol*‘𝑥)))
6463ralrimiva 3152 . 2 ((𝐴 ∈ dom vol ∧ 𝐵 ∈ dom vol) → ∀𝑥 ∈ 𝒫 ℝ((vol*‘𝑥) ∈ ℝ → ((vol*‘(𝑥 ∩ (𝐴𝐵))) + (vol*‘(𝑥 ∖ (𝐴𝐵)))) ≤ (vol*‘𝑥)))
65 ismbl2 24134 . 2 ((𝐴𝐵) ∈ dom vol ↔ ((𝐴𝐵) ⊆ ℝ ∧ ∀𝑥 ∈ 𝒫 ℝ((vol*‘𝑥) ∈ ℝ → ((vol*‘(𝑥 ∩ (𝐴𝐵))) + (vol*‘(𝑥 ∖ (𝐴𝐵)))) ≤ (vol*‘𝑥))))
665, 64, 65sylanbrc 586 1 ((𝐴 ∈ dom vol ∧ 𝐵 ∈ dom vol) → (𝐴𝐵) ∈ dom vol)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399   = wceq 1538  wcel 2112  wral 3109  cdif 3881  cun 3882  cin 3883  wss 3884  𝒫 cpw 4500   class class class wbr 5033  dom cdm 5523  cfv 6328  (class class class)co 7139  cr 10529   + caddc 10533  cle 10669  vol*covol 24069  volcvol 24070
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-sep 5170  ax-nul 5177  ax-pow 5234  ax-pr 5298  ax-un 7445  ax-cnex 10586  ax-resscn 10587  ax-1cn 10588  ax-icn 10589  ax-addcl 10590  ax-addrcl 10591  ax-mulcl 10592  ax-mulrcl 10593  ax-mulcom 10594  ax-addass 10595  ax-mulass 10596  ax-distr 10597  ax-i2m1 10598  ax-1ne0 10599  ax-1rid 10600  ax-rnegex 10601  ax-rrecex 10602  ax-cnre 10603  ax-pre-lttri 10604  ax-pre-lttrn 10605  ax-pre-ltadd 10606  ax-pre-mulgt0 10607  ax-pre-sup 10608
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ne 2991  df-nel 3095  df-ral 3114  df-rex 3115  df-reu 3116  df-rmo 3117  df-rab 3118  df-v 3446  df-sbc 3724  df-csb 3832  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-pss 3903  df-nul 4247  df-if 4429  df-pw 4502  df-sn 4529  df-pr 4531  df-tp 4533  df-op 4535  df-uni 4804  df-iun 4886  df-br 5034  df-opab 5096  df-mpt 5114  df-tr 5140  df-id 5428  df-eprel 5433  df-po 5442  df-so 5443  df-fr 5482  df-we 5484  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-rn 5534  df-res 5535  df-ima 5536  df-pred 6120  df-ord 6166  df-on 6167  df-lim 6168  df-suc 6169  df-iota 6287  df-fun 6330  df-fn 6331  df-f 6332  df-f1 6333  df-fo 6334  df-f1o 6335  df-fv 6336  df-riota 7097  df-ov 7142  df-oprab 7143  df-mpo 7144  df-om 7565  df-1st 7675  df-2nd 7676  df-wrecs 7934  df-recs 7995  df-rdg 8033  df-er 8276  df-map 8395  df-en 8497  df-dom 8498  df-sdom 8499  df-sup 8894  df-inf 8895  df-pnf 10670  df-mnf 10671  df-xr 10672  df-ltxr 10673  df-le 10674  df-sub 10865  df-neg 10866  df-div 11291  df-nn 11630  df-2 11692  df-3 11693  df-n0 11890  df-z 11974  df-uz 12236  df-q 12341  df-rp 12382  df-ioo 12734  df-ico 12736  df-icc 12737  df-fz 12890  df-fl 13161  df-seq 13369  df-exp 13430  df-cj 14453  df-re 14454  df-im 14455  df-sqrt 14589  df-abs 14590  df-ovol 24071  df-vol 24072
This theorem is referenced by:  inmbl  24149  finiunmbl  24151  volun  24152  voliunlem1  24157  icombl1  24170  iccmbl  24173  uniiccmbl  24197  mbfimaicc  24238  mbfeqalem2  24249  mbfres2  24252  mbfmax  24256  itgss3  24421  ismblfin  35091  mbfposadd  35097  cnambfre  35098  itg2addnclem2  35102  iblabsnclem  35113  ftc1anclem1  35123  ftc1anclem5  35127  iocmbl  40150
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