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Theorem volun 23533
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 1228 . . . . . 6 (((𝐴 ∈ dom vol ∧ 𝐵 ∈ dom vol ∧ (𝐴𝐵) = ∅) ∧ ((vol*‘𝐴) ∈ ℝ ∧ (vol*‘𝐵) ∈ ℝ)) → 𝐴 ∈ dom vol)
2 mblss 23519 . . . . . . . 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 1230 . . . . . . . 8 (((𝐴 ∈ dom vol ∧ 𝐵 ∈ dom vol ∧ (𝐴𝐵) = ∅) ∧ ((vol*‘𝐴) ∈ ℝ ∧ (vol*‘𝐵) ∈ ℝ)) → 𝐵 ∈ dom vol)
5 mblss 23519 . . . . . . . 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 3932 . . . . . 6 (((𝐴 ∈ dom vol ∧ 𝐵 ∈ dom vol ∧ (𝐴𝐵) = ∅) ∧ ((vol*‘𝐴) ∈ ℝ ∧ (vol*‘𝐵) ∈ ℝ)) → (𝐴𝐵) ⊆ ℝ)
8 readdcl 10231 . . . . . . . 8 (((vol*‘𝐴) ∈ ℝ ∧ (vol*‘𝐵) ∈ ℝ) → ((vol*‘𝐴) + (vol*‘𝐵)) ∈ ℝ)
98adantl 473 . . . . . . 7 (((𝐴 ∈ dom vol ∧ 𝐵 ∈ dom vol ∧ (𝐴𝐵) = ∅) ∧ ((vol*‘𝐴) ∈ ℝ ∧ (vol*‘𝐵) ∈ ℝ)) → ((vol*‘𝐴) + (vol*‘𝐵)) ∈ ℝ)
10 simprl 811 . . . . . . . 8 (((𝐴 ∈ dom vol ∧ 𝐵 ∈ dom vol ∧ (𝐴𝐵) = ∅) ∧ ((vol*‘𝐴) ∈ ℝ ∧ (vol*‘𝐵) ∈ ℝ)) → (vol*‘𝐴) ∈ ℝ)
11 simprr 813 . . . . . . . 8 (((𝐴 ∈ dom vol ∧ 𝐵 ∈ dom vol ∧ (𝐴𝐵) = ∅) ∧ ((vol*‘𝐴) ∈ ℝ ∧ (vol*‘𝐵) ∈ ℝ)) → (vol*‘𝐵) ∈ ℝ)
12 ovolun 23487 . . . . . . . 8 (((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ)) → (vol*‘(𝐴𝐵)) ≤ ((vol*‘𝐴) + (vol*‘𝐵)))
133, 10, 6, 11, 12syl22anc 1478 . . . . . . 7 (((𝐴 ∈ dom vol ∧ 𝐵 ∈ dom vol ∧ (𝐴𝐵) = ∅) ∧ ((vol*‘𝐴) ∈ ℝ ∧ (vol*‘𝐵) ∈ ℝ)) → (vol*‘(𝐴𝐵)) ≤ ((vol*‘𝐴) + (vol*‘𝐵)))
14 ovollecl 23471 . . . . . . 7 (((𝐴𝐵) ⊆ ℝ ∧ ((vol*‘𝐴) + (vol*‘𝐵)) ∈ ℝ ∧ (vol*‘(𝐴𝐵)) ≤ ((vol*‘𝐴) + (vol*‘𝐵))) → (vol*‘(𝐴𝐵)) ∈ ℝ)
157, 9, 13, 14syl3anc 1477 . . . . . 6 (((𝐴 ∈ dom vol ∧ 𝐵 ∈ dom vol ∧ (𝐴𝐵) = ∅) ∧ ((vol*‘𝐴) ∈ ℝ ∧ (vol*‘𝐵) ∈ ℝ)) → (vol*‘(𝐴𝐵)) ∈ ℝ)
16 mblsplit 23520 . . . . . 6 ((𝐴 ∈ dom vol ∧ (𝐴𝐵) ⊆ ℝ ∧ (vol*‘(𝐴𝐵)) ∈ ℝ) → (vol*‘(𝐴𝐵)) = ((vol*‘((𝐴𝐵) ∩ 𝐴)) + (vol*‘((𝐴𝐵) ∖ 𝐴))))
171, 7, 15, 16syl3anc 1477 . . . . 5 (((𝐴 ∈ dom vol ∧ 𝐵 ∈ dom vol ∧ (𝐴𝐵) = ∅) ∧ ((vol*‘𝐴) ∈ ℝ ∧ (vol*‘𝐵) ∈ ℝ)) → (vol*‘(𝐴𝐵)) = ((vol*‘((𝐴𝐵) ∩ 𝐴)) + (vol*‘((𝐴𝐵) ∖ 𝐴))))
18 simpl3 1232 . . . . . 6 (((𝐴 ∈ dom vol ∧ 𝐵 ∈ dom vol ∧ (𝐴𝐵) = ∅) ∧ ((vol*‘𝐴) ∈ ℝ ∧ (vol*‘𝐵) ∈ ℝ)) → (𝐴𝐵) = ∅)
19 indir 4018 . . . . . . . . . 10 ((𝐴𝐵) ∩ 𝐴) = ((𝐴𝐴) ∪ (𝐵𝐴))
20 inidm 3965 . . . . . . . . . . . 12 (𝐴𝐴) = 𝐴
21 incom 3948 . . . . . . . . . . . 12 (𝐵𝐴) = (𝐴𝐵)
2220, 21uneq12i 3908 . . . . . . . . . . 11 ((𝐴𝐴) ∪ (𝐵𝐴)) = (𝐴 ∪ (𝐴𝐵))
23 unabs 3997 . . . . . . . . . . 11 (𝐴 ∪ (𝐴𝐵)) = 𝐴
2422, 23eqtri 2782 . . . . . . . . . 10 ((𝐴𝐴) ∪ (𝐵𝐴)) = 𝐴
2519, 24eqtri 2782 . . . . . . . . 9 ((𝐴𝐵) ∩ 𝐴) = 𝐴
2625a1i 11 . . . . . . . 8 ((𝐴𝐵) = ∅ → ((𝐴𝐵) ∩ 𝐴) = 𝐴)
2726fveq2d 6357 . . . . . . 7 ((𝐴𝐵) = ∅ → (vol*‘((𝐴𝐵) ∩ 𝐴)) = (vol*‘𝐴))
2821eqeq1i 2765 . . . . . . . . . . 11 ((𝐵𝐴) = ∅ ↔ (𝐴𝐵) = ∅)
29 disj3 4164 . . . . . . . . . . 11 ((𝐵𝐴) = ∅ ↔ 𝐵 = (𝐵𝐴))
3028, 29bitr3i 266 . . . . . . . . . 10 ((𝐴𝐵) = ∅ ↔ 𝐵 = (𝐵𝐴))
3130biimpi 206 . . . . . . . . 9 ((𝐴𝐵) = ∅ → 𝐵 = (𝐵𝐴))
32 uncom 3900 . . . . . . . . . . 11 (𝐴𝐵) = (𝐵𝐴)
3332difeq1i 3867 . . . . . . . . . 10 ((𝐴𝐵) ∖ 𝐴) = ((𝐵𝐴) ∖ 𝐴)
34 difun2 4192 . . . . . . . . . 10 ((𝐵𝐴) ∖ 𝐴) = (𝐵𝐴)
3533, 34eqtri 2782 . . . . . . . . 9 ((𝐴𝐵) ∖ 𝐴) = (𝐵𝐴)
3631, 35syl6reqr 2813 . . . . . . . 8 ((𝐴𝐵) = ∅ → ((𝐴𝐵) ∖ 𝐴) = 𝐵)
3736fveq2d 6357 . . . . . . 7 ((𝐴𝐵) = ∅ → (vol*‘((𝐴𝐵) ∖ 𝐴)) = (vol*‘𝐵))
3827, 37oveq12d 6832 . . . . . 6 ((𝐴𝐵) = ∅ → ((vol*‘((𝐴𝐵) ∩ 𝐴)) + (vol*‘((𝐴𝐵) ∖ 𝐴))) = ((vol*‘𝐴) + (vol*‘𝐵)))
3918, 38syl 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)))
4017, 39eqtrd 2794 . . . 4 (((𝐴 ∈ dom vol ∧ 𝐵 ∈ dom vol ∧ (𝐴𝐵) = ∅) ∧ ((vol*‘𝐴) ∈ ℝ ∧ (vol*‘𝐵) ∈ ℝ)) → (vol*‘(𝐴𝐵)) = ((vol*‘𝐴) + (vol*‘𝐵)))
4140ex 449 . . 3 ((𝐴 ∈ dom vol ∧ 𝐵 ∈ dom vol ∧ (𝐴𝐵) = ∅) → (((vol*‘𝐴) ∈ ℝ ∧ (vol*‘𝐵) ∈ ℝ) → (vol*‘(𝐴𝐵)) = ((vol*‘𝐴) + (vol*‘𝐵))))
42 mblvol 23518 . . . . . 6 (𝐴 ∈ dom vol → (vol‘𝐴) = (vol*‘𝐴))
4342eleq1d 2824 . . . . 5 (𝐴 ∈ dom vol → ((vol‘𝐴) ∈ ℝ ↔ (vol*‘𝐴) ∈ ℝ))
44 mblvol 23518 . . . . . 6 (𝐵 ∈ dom vol → (vol‘𝐵) = (vol*‘𝐵))
4544eleq1d 2824 . . . . 5 (𝐵 ∈ dom vol → ((vol‘𝐵) ∈ ℝ ↔ (vol*‘𝐵) ∈ ℝ))
4643, 45bi2anan9 953 . . . 4 ((𝐴 ∈ dom vol ∧ 𝐵 ∈ dom vol) → (((vol‘𝐴) ∈ ℝ ∧ (vol‘𝐵) ∈ ℝ) ↔ ((vol*‘𝐴) ∈ ℝ ∧ (vol*‘𝐵) ∈ ℝ)))
47463adant3 1127 . . 3 ((𝐴 ∈ dom vol ∧ 𝐵 ∈ dom vol ∧ (𝐴𝐵) = ∅) → (((vol‘𝐴) ∈ ℝ ∧ (vol‘𝐵) ∈ ℝ) ↔ ((vol*‘𝐴) ∈ ℝ ∧ (vol*‘𝐵) ∈ ℝ)))
48 unmbl 23525 . . . . . 6 ((𝐴 ∈ dom vol ∧ 𝐵 ∈ dom vol) → (𝐴𝐵) ∈ dom vol)
49 mblvol 23518 . . . . . 6 ((𝐴𝐵) ∈ dom vol → (vol‘(𝐴𝐵)) = (vol*‘(𝐴𝐵)))
5048, 49syl 17 . . . . 5 ((𝐴 ∈ dom vol ∧ 𝐵 ∈ dom vol) → (vol‘(𝐴𝐵)) = (vol*‘(𝐴𝐵)))
5142, 44oveqan12d 6833 . . . . 5 ((𝐴 ∈ dom vol ∧ 𝐵 ∈ dom vol) → ((vol‘𝐴) + (vol‘𝐵)) = ((vol*‘𝐴) + (vol*‘𝐵)))
5250, 51eqeq12d 2775 . . . 4 ((𝐴 ∈ dom vol ∧ 𝐵 ∈ dom vol) → ((vol‘(𝐴𝐵)) = ((vol‘𝐴) + (vol‘𝐵)) ↔ (vol*‘(𝐴𝐵)) = ((vol*‘𝐴) + (vol*‘𝐵))))
53523adant3 1127 . . 3 ((𝐴 ∈ dom vol ∧ 𝐵 ∈ dom vol ∧ (𝐴𝐵) = ∅) → ((vol‘(𝐴𝐵)) = ((vol‘𝐴) + (vol‘𝐵)) ↔ (vol*‘(𝐴𝐵)) = ((vol*‘𝐴) + (vol*‘𝐵))))
5441, 47, 533imtr4d 283 . 2 ((𝐴 ∈ dom vol ∧ 𝐵 ∈ dom vol ∧ (𝐴𝐵) = ∅) → (((vol‘𝐴) ∈ ℝ ∧ (vol‘𝐵) ∈ ℝ) → (vol‘(𝐴𝐵)) = ((vol‘𝐴) + (vol‘𝐵))))
5554imp 444 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 196  wa 383  w3a 1072   = wceq 1632  wcel 2139  cdif 3712  cun 3713  cin 3714  wss 3715  c0 4058   class class class wbr 4804  dom cdm 5266  cfv 6049  (class class class)co 6814  cr 10147   + caddc 10151  cle 10287  vol*covol 23451  volcvol 23452
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7115  ax-cnex 10204  ax-resscn 10205  ax-1cn 10206  ax-icn 10207  ax-addcl 10208  ax-addrcl 10209  ax-mulcl 10210  ax-mulrcl 10211  ax-mulcom 10212  ax-addass 10213  ax-mulass 10214  ax-distr 10215  ax-i2m1 10216  ax-1ne0 10217  ax-1rid 10218  ax-rnegex 10219  ax-rrecex 10220  ax-cnre 10221  ax-pre-lttri 10222  ax-pre-lttrn 10223  ax-pre-ltadd 10224  ax-pre-mulgt0 10225  ax-pre-sup 10226
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-nel 3036  df-ral 3055  df-rex 3056  df-reu 3057  df-rmo 3058  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-pss 3731  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-tp 4326  df-op 4328  df-uni 4589  df-iun 4674  df-br 4805  df-opab 4865  df-mpt 4882  df-tr 4905  df-id 5174  df-eprel 5179  df-po 5187  df-so 5188  df-fr 5225  df-we 5227  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-pred 5841  df-ord 5887  df-on 5888  df-lim 5889  df-suc 5890  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-f1 6054  df-fo 6055  df-f1o 6056  df-fv 6057  df-riota 6775  df-ov 6817  df-oprab 6818  df-mpt2 6819  df-om 7232  df-1st 7334  df-2nd 7335  df-wrecs 7577  df-recs 7638  df-rdg 7676  df-er 7913  df-map 8027  df-en 8124  df-dom 8125  df-sdom 8126  df-sup 8515  df-inf 8516  df-pnf 10288  df-mnf 10289  df-xr 10290  df-ltxr 10291  df-le 10292  df-sub 10480  df-neg 10481  df-div 10897  df-nn 11233  df-2 11291  df-3 11292  df-n0 11505  df-z 11590  df-uz 11900  df-q 12002  df-rp 12046  df-ioo 12392  df-ico 12394  df-icc 12395  df-fz 12540  df-fl 12807  df-seq 13016  df-exp 13075  df-cj 14058  df-re 14059  df-im 14060  df-sqrt 14194  df-abs 14195  df-ovol 23453  df-vol 23454
This theorem is referenced by:  volinun  23534  volfiniun  23535  volsup  23544  ovolioo  23556  ismblfin  33781  volioc  40709  volico  40721
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