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Theorem volun 24932
Description: The Lebesgue measure function is finitely additive. (Contributed by Mario Carneiro, 18-Mar-2014.)
Assertion
Ref Expression
volun (((𝐴 ∈ dom vol ∧ 𝐵 ∈ dom vol ∧ (𝐴𝐵) = ∅) ∧ ((vol‘𝐴) ∈ ℝ ∧ (vol‘𝐵) ∈ ℝ)) → (vol‘(𝐴𝐵)) = ((vol‘𝐴) + (vol‘𝐵)))

Proof of Theorem volun
StepHypRef Expression
1 simpl1 1192 . . . . . 6 (((𝐴 ∈ dom vol ∧ 𝐵 ∈ dom vol ∧ (𝐴𝐵) = ∅) ∧ ((vol*‘𝐴) ∈ ℝ ∧ (vol*‘𝐵) ∈ ℝ)) → 𝐴 ∈ dom vol)
2 mblss 24918 . . . . . . . 8 (𝐴 ∈ dom vol → 𝐴 ⊆ ℝ)
31, 2syl 17 . . . . . . 7 (((𝐴 ∈ dom vol ∧ 𝐵 ∈ dom vol ∧ (𝐴𝐵) = ∅) ∧ ((vol*‘𝐴) ∈ ℝ ∧ (vol*‘𝐵) ∈ ℝ)) → 𝐴 ⊆ ℝ)
4 simpl2 1193 . . . . . . . 8 (((𝐴 ∈ dom vol ∧ 𝐵 ∈ dom vol ∧ (𝐴𝐵) = ∅) ∧ ((vol*‘𝐴) ∈ ℝ ∧ (vol*‘𝐵) ∈ ℝ)) → 𝐵 ∈ dom vol)
5 mblss 24918 . . . . . . . 8 (𝐵 ∈ dom vol → 𝐵 ⊆ ℝ)
64, 5syl 17 . . . . . . 7 (((𝐴 ∈ dom vol ∧ 𝐵 ∈ dom vol ∧ (𝐴𝐵) = ∅) ∧ ((vol*‘𝐴) ∈ ℝ ∧ (vol*‘𝐵) ∈ ℝ)) → 𝐵 ⊆ ℝ)
73, 6unssd 4150 . . . . . 6 (((𝐴 ∈ dom vol ∧ 𝐵 ∈ dom vol ∧ (𝐴𝐵) = ∅) ∧ ((vol*‘𝐴) ∈ ℝ ∧ (vol*‘𝐵) ∈ ℝ)) → (𝐴𝐵) ⊆ ℝ)
8 readdcl 11142 . . . . . . . 8 (((vol*‘𝐴) ∈ ℝ ∧ (vol*‘𝐵) ∈ ℝ) → ((vol*‘𝐴) + (vol*‘𝐵)) ∈ ℝ)
98adantl 483 . . . . . . 7 (((𝐴 ∈ dom vol ∧ 𝐵 ∈ dom vol ∧ (𝐴𝐵) = ∅) ∧ ((vol*‘𝐴) ∈ ℝ ∧ (vol*‘𝐵) ∈ ℝ)) → ((vol*‘𝐴) + (vol*‘𝐵)) ∈ ℝ)
10 simprl 770 . . . . . . . 8 (((𝐴 ∈ dom vol ∧ 𝐵 ∈ dom vol ∧ (𝐴𝐵) = ∅) ∧ ((vol*‘𝐴) ∈ ℝ ∧ (vol*‘𝐵) ∈ ℝ)) → (vol*‘𝐴) ∈ ℝ)
11 simprr 772 . . . . . . . 8 (((𝐴 ∈ dom vol ∧ 𝐵 ∈ dom vol ∧ (𝐴𝐵) = ∅) ∧ ((vol*‘𝐴) ∈ ℝ ∧ (vol*‘𝐵) ∈ ℝ)) → (vol*‘𝐵) ∈ ℝ)
12 ovolun 24886 . . . . . . . 8 (((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ)) → (vol*‘(𝐴𝐵)) ≤ ((vol*‘𝐴) + (vol*‘𝐵)))
133, 10, 6, 11, 12syl22anc 838 . . . . . . 7 (((𝐴 ∈ dom vol ∧ 𝐵 ∈ dom vol ∧ (𝐴𝐵) = ∅) ∧ ((vol*‘𝐴) ∈ ℝ ∧ (vol*‘𝐵) ∈ ℝ)) → (vol*‘(𝐴𝐵)) ≤ ((vol*‘𝐴) + (vol*‘𝐵)))
14 ovollecl 24870 . . . . . . 7 (((𝐴𝐵) ⊆ ℝ ∧ ((vol*‘𝐴) + (vol*‘𝐵)) ∈ ℝ ∧ (vol*‘(𝐴𝐵)) ≤ ((vol*‘𝐴) + (vol*‘𝐵))) → (vol*‘(𝐴𝐵)) ∈ ℝ)
157, 9, 13, 14syl3anc 1372 . . . . . 6 (((𝐴 ∈ dom vol ∧ 𝐵 ∈ dom vol ∧ (𝐴𝐵) = ∅) ∧ ((vol*‘𝐴) ∈ ℝ ∧ (vol*‘𝐵) ∈ ℝ)) → (vol*‘(𝐴𝐵)) ∈ ℝ)
16 mblsplit 24919 . . . . . 6 ((𝐴 ∈ dom vol ∧ (𝐴𝐵) ⊆ ℝ ∧ (vol*‘(𝐴𝐵)) ∈ ℝ) → (vol*‘(𝐴𝐵)) = ((vol*‘((𝐴𝐵) ∩ 𝐴)) + (vol*‘((𝐴𝐵) ∖ 𝐴))))
171, 7, 15, 16syl3anc 1372 . . . . 5 (((𝐴 ∈ dom vol ∧ 𝐵 ∈ dom vol ∧ (𝐴𝐵) = ∅) ∧ ((vol*‘𝐴) ∈ ℝ ∧ (vol*‘𝐵) ∈ ℝ)) → (vol*‘(𝐴𝐵)) = ((vol*‘((𝐴𝐵) ∩ 𝐴)) + (vol*‘((𝐴𝐵) ∖ 𝐴))))
18 simpl3 1194 . . . . . 6 (((𝐴 ∈ dom vol ∧ 𝐵 ∈ dom vol ∧ (𝐴𝐵) = ∅) ∧ ((vol*‘𝐴) ∈ ℝ ∧ (vol*‘𝐵) ∈ ℝ)) → (𝐴𝐵) = ∅)
19 indir 4239 . . . . . . . . . 10 ((𝐴𝐵) ∩ 𝐴) = ((𝐴𝐴) ∪ (𝐵𝐴))
20 inidm 4182 . . . . . . . . . . . 12 (𝐴𝐴) = 𝐴
21 incom 4165 . . . . . . . . . . . 12 (𝐵𝐴) = (𝐴𝐵)
2220, 21uneq12i 4125 . . . . . . . . . . 11 ((𝐴𝐴) ∪ (𝐵𝐴)) = (𝐴 ∪ (𝐴𝐵))
23 unabs 4218 . . . . . . . . . . 11 (𝐴 ∪ (𝐴𝐵)) = 𝐴
2422, 23eqtri 2761 . . . . . . . . . 10 ((𝐴𝐴) ∪ (𝐵𝐴)) = 𝐴
2519, 24eqtri 2761 . . . . . . . . 9 ((𝐴𝐵) ∩ 𝐴) = 𝐴
2625a1i 11 . . . . . . . 8 ((𝐴𝐵) = ∅ → ((𝐴𝐵) ∩ 𝐴) = 𝐴)
2726fveq2d 6850 . . . . . . 7 ((𝐴𝐵) = ∅ → (vol*‘((𝐴𝐵) ∩ 𝐴)) = (vol*‘𝐴))
28 uncom 4117 . . . . . . . . . . 11 (𝐴𝐵) = (𝐵𝐴)
2928difeq1i 4082 . . . . . . . . . 10 ((𝐴𝐵) ∖ 𝐴) = ((𝐵𝐴) ∖ 𝐴)
30 difun2 4444 . . . . . . . . . 10 ((𝐵𝐴) ∖ 𝐴) = (𝐵𝐴)
3129, 30eqtri 2761 . . . . . . . . 9 ((𝐴𝐵) ∖ 𝐴) = (𝐵𝐴)
3221eqeq1i 2738 . . . . . . . . . 10 ((𝐵𝐴) = ∅ ↔ (𝐴𝐵) = ∅)
33 disj3 4417 . . . . . . . . . 10 ((𝐵𝐴) = ∅ ↔ 𝐵 = (𝐵𝐴))
3432, 33sylbb1 236 . . . . . . . . 9 ((𝐴𝐵) = ∅ → 𝐵 = (𝐵𝐴))
3531, 34eqtr4id 2792 . . . . . . . 8 ((𝐴𝐵) = ∅ → ((𝐴𝐵) ∖ 𝐴) = 𝐵)
3635fveq2d 6850 . . . . . . 7 ((𝐴𝐵) = ∅ → (vol*‘((𝐴𝐵) ∖ 𝐴)) = (vol*‘𝐵))
3727, 36oveq12d 7379 . . . . . 6 ((𝐴𝐵) = ∅ → ((vol*‘((𝐴𝐵) ∩ 𝐴)) + (vol*‘((𝐴𝐵) ∖ 𝐴))) = ((vol*‘𝐴) + (vol*‘𝐵)))
3818, 37syl 17 . . . . 5 (((𝐴 ∈ dom vol ∧ 𝐵 ∈ dom vol ∧ (𝐴𝐵) = ∅) ∧ ((vol*‘𝐴) ∈ ℝ ∧ (vol*‘𝐵) ∈ ℝ)) → ((vol*‘((𝐴𝐵) ∩ 𝐴)) + (vol*‘((𝐴𝐵) ∖ 𝐴))) = ((vol*‘𝐴) + (vol*‘𝐵)))
3917, 38eqtrd 2773 . . . 4 (((𝐴 ∈ dom vol ∧ 𝐵 ∈ dom vol ∧ (𝐴𝐵) = ∅) ∧ ((vol*‘𝐴) ∈ ℝ ∧ (vol*‘𝐵) ∈ ℝ)) → (vol*‘(𝐴𝐵)) = ((vol*‘𝐴) + (vol*‘𝐵)))
4039ex 414 . . 3 ((𝐴 ∈ dom vol ∧ 𝐵 ∈ dom vol ∧ (𝐴𝐵) = ∅) → (((vol*‘𝐴) ∈ ℝ ∧ (vol*‘𝐵) ∈ ℝ) → (vol*‘(𝐴𝐵)) = ((vol*‘𝐴) + (vol*‘𝐵))))
41 mblvol 24917 . . . . . 6 (𝐴 ∈ dom vol → (vol‘𝐴) = (vol*‘𝐴))
4241eleq1d 2819 . . . . 5 (𝐴 ∈ dom vol → ((vol‘𝐴) ∈ ℝ ↔ (vol*‘𝐴) ∈ ℝ))
43 mblvol 24917 . . . . . 6 (𝐵 ∈ dom vol → (vol‘𝐵) = (vol*‘𝐵))
4443eleq1d 2819 . . . . 5 (𝐵 ∈ dom vol → ((vol‘𝐵) ∈ ℝ ↔ (vol*‘𝐵) ∈ ℝ))
4542, 44bi2anan9 638 . . . 4 ((𝐴 ∈ dom vol ∧ 𝐵 ∈ dom vol) → (((vol‘𝐴) ∈ ℝ ∧ (vol‘𝐵) ∈ ℝ) ↔ ((vol*‘𝐴) ∈ ℝ ∧ (vol*‘𝐵) ∈ ℝ)))
46453adant3 1133 . . 3 ((𝐴 ∈ dom vol ∧ 𝐵 ∈ dom vol ∧ (𝐴𝐵) = ∅) → (((vol‘𝐴) ∈ ℝ ∧ (vol‘𝐵) ∈ ℝ) ↔ ((vol*‘𝐴) ∈ ℝ ∧ (vol*‘𝐵) ∈ ℝ)))
47 unmbl 24924 . . . . . 6 ((𝐴 ∈ dom vol ∧ 𝐵 ∈ dom vol) → (𝐴𝐵) ∈ dom vol)
48 mblvol 24917 . . . . . 6 ((𝐴𝐵) ∈ dom vol → (vol‘(𝐴𝐵)) = (vol*‘(𝐴𝐵)))
4947, 48syl 17 . . . . 5 ((𝐴 ∈ dom vol ∧ 𝐵 ∈ dom vol) → (vol‘(𝐴𝐵)) = (vol*‘(𝐴𝐵)))
5041, 43oveqan12d 7380 . . . . 5 ((𝐴 ∈ dom vol ∧ 𝐵 ∈ dom vol) → ((vol‘𝐴) + (vol‘𝐵)) = ((vol*‘𝐴) + (vol*‘𝐵)))
5149, 50eqeq12d 2749 . . . 4 ((𝐴 ∈ dom vol ∧ 𝐵 ∈ dom vol) → ((vol‘(𝐴𝐵)) = ((vol‘𝐴) + (vol‘𝐵)) ↔ (vol*‘(𝐴𝐵)) = ((vol*‘𝐴) + (vol*‘𝐵))))
52513adant3 1133 . . 3 ((𝐴 ∈ dom vol ∧ 𝐵 ∈ dom vol ∧ (𝐴𝐵) = ∅) → ((vol‘(𝐴𝐵)) = ((vol‘𝐴) + (vol‘𝐵)) ↔ (vol*‘(𝐴𝐵)) = ((vol*‘𝐴) + (vol*‘𝐵))))
5340, 46, 523imtr4d 294 . 2 ((𝐴 ∈ dom vol ∧ 𝐵 ∈ dom vol ∧ (𝐴𝐵) = ∅) → (((vol‘𝐴) ∈ ℝ ∧ (vol‘𝐵) ∈ ℝ) → (vol‘(𝐴𝐵)) = ((vol‘𝐴) + (vol‘𝐵))))
5453imp 408 1 (((𝐴 ∈ dom vol ∧ 𝐵 ∈ dom vol ∧ (𝐴𝐵) = ∅) ∧ ((vol‘𝐴) ∈ ℝ ∧ (vol‘𝐵) ∈ ℝ)) → (vol‘(𝐴𝐵)) = ((vol‘𝐴) + (vol‘𝐵)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 397  w3a 1088   = wceq 1542  wcel 2107  cdif 3911  cun 3912  cin 3913  wss 3914  c0 4286   class class class wbr 5109  dom cdm 5637  cfv 6500  (class class class)co 7361  cr 11058   + caddc 11062  cle 11198  vol*covol 24849  volcvol 24850
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 2704  ax-sep 5260  ax-nul 5267  ax-pow 5324  ax-pr 5388  ax-un 7676  ax-cnex 11115  ax-resscn 11116  ax-1cn 11117  ax-icn 11118  ax-addcl 11119  ax-addrcl 11120  ax-mulcl 11121  ax-mulrcl 11122  ax-mulcom 11123  ax-addass 11124  ax-mulass 11125  ax-distr 11126  ax-i2m1 11127  ax-1ne0 11128  ax-1rid 11129  ax-rnegex 11130  ax-rrecex 11131  ax-cnre 11132  ax-pre-lttri 11133  ax-pre-lttrn 11134  ax-pre-ltadd 11135  ax-pre-mulgt0 11136  ax-pre-sup 11137
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 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3352  df-reu 3353  df-rab 3407  df-v 3449  df-sbc 3744  df-csb 3860  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-pss 3933  df-nul 4287  df-if 4491  df-pw 4566  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4870  df-iun 4960  df-br 5110  df-opab 5172  df-mpt 5193  df-tr 5227  df-id 5535  df-eprel 5541  df-po 5549  df-so 5550  df-fr 5592  df-we 5594  df-xp 5643  df-rel 5644  df-cnv 5645  df-co 5646  df-dm 5647  df-rn 5648  df-res 5649  df-ima 5650  df-pred 6257  df-ord 6324  df-on 6325  df-lim 6326  df-suc 6327  df-iota 6452  df-fun 6502  df-fn 6503  df-f 6504  df-f1 6505  df-fo 6506  df-f1o 6507  df-fv 6508  df-riota 7317  df-ov 7364  df-oprab 7365  df-mpo 7366  df-om 7807  df-1st 7925  df-2nd 7926  df-frecs 8216  df-wrecs 8247  df-recs 8321  df-rdg 8360  df-er 8654  df-map 8773  df-en 8890  df-dom 8891  df-sdom 8892  df-sup 9386  df-inf 9387  df-pnf 11199  df-mnf 11200  df-xr 11201  df-ltxr 11202  df-le 11203  df-sub 11395  df-neg 11396  df-div 11821  df-nn 12162  df-2 12224  df-3 12225  df-n0 12422  df-z 12508  df-uz 12772  df-q 12882  df-rp 12924  df-ioo 13277  df-ico 13279  df-icc 13280  df-fz 13434  df-fl 13706  df-seq 13916  df-exp 13977  df-cj 14993  df-re 14994  df-im 14995  df-sqrt 15129  df-abs 15130  df-ovol 24851  df-vol 24852
This theorem is referenced by:  volinun  24933  volfiniun  24934  volsup  24943  ovolioo  24955  ismblfin  36169  volioc  44303  volico  44314
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