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Theorem caragensplit 45947
Description: If 𝐸 is in the set generated by the Caratheodory's method, then it splits any set 𝐴 in two parts such that the sum of the outer measures of the two parts is equal to the outer measure of the whole set 𝐴. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
Hypotheses
Ref Expression
caragensplit.o (πœ‘ β†’ 𝑂 ∈ OutMeas)
caragensplit.s 𝑆 = (CaraGenβ€˜π‘‚)
caragensplit.x 𝑋 = βˆͺ dom 𝑂
caragensplit.e (πœ‘ β†’ 𝐸 ∈ 𝑆)
caragensplit.a (πœ‘ β†’ 𝐴 βŠ† 𝑋)
Assertion
Ref Expression
caragensplit (πœ‘ β†’ ((π‘‚β€˜(𝐴 ∩ 𝐸)) +𝑒 (π‘‚β€˜(𝐴 βˆ– 𝐸))) = (π‘‚β€˜π΄))

Proof of Theorem caragensplit
Dummy variable π‘Ž is distinct from all other variables.
StepHypRef Expression
1 caragensplit.a . . . 4 (πœ‘ β†’ 𝐴 βŠ† 𝑋)
2 caragensplit.o . . . . . . 7 (πœ‘ β†’ 𝑂 ∈ OutMeas)
3 caragensplit.x . . . . . . 7 𝑋 = βˆͺ dom 𝑂
42, 3unidmex 44475 . . . . . 6 (πœ‘ β†’ 𝑋 ∈ V)
5 ssexg 5319 . . . . . 6 ((𝐴 βŠ† 𝑋 ∧ 𝑋 ∈ V) β†’ 𝐴 ∈ V)
61, 4, 5syl2anc 582 . . . . 5 (πœ‘ β†’ 𝐴 ∈ V)
7 elpwg 4602 . . . . 5 (𝐴 ∈ V β†’ (𝐴 ∈ 𝒫 𝑋 ↔ 𝐴 βŠ† 𝑋))
86, 7syl 17 . . . 4 (πœ‘ β†’ (𝐴 ∈ 𝒫 𝑋 ↔ 𝐴 βŠ† 𝑋))
91, 8mpbird 256 . . 3 (πœ‘ β†’ 𝐴 ∈ 𝒫 𝑋)
103pweqi 4615 . . 3 𝒫 𝑋 = 𝒫 βˆͺ dom 𝑂
119, 10eleqtrdi 2835 . 2 (πœ‘ β†’ 𝐴 ∈ 𝒫 βˆͺ dom 𝑂)
12 caragensplit.e . . . 4 (πœ‘ β†’ 𝐸 ∈ 𝑆)
13 caragensplit.s . . . . 5 𝑆 = (CaraGenβ€˜π‘‚)
142, 13caragenel 45942 . . . 4 (πœ‘ β†’ (𝐸 ∈ 𝑆 ↔ (𝐸 ∈ 𝒫 βˆͺ dom 𝑂 ∧ βˆ€π‘Ž ∈ 𝒫 βˆͺ dom 𝑂((π‘‚β€˜(π‘Ž ∩ 𝐸)) +𝑒 (π‘‚β€˜(π‘Ž βˆ– 𝐸))) = (π‘‚β€˜π‘Ž))))
1512, 14mpbid 231 . . 3 (πœ‘ β†’ (𝐸 ∈ 𝒫 βˆͺ dom 𝑂 ∧ βˆ€π‘Ž ∈ 𝒫 βˆͺ dom 𝑂((π‘‚β€˜(π‘Ž ∩ 𝐸)) +𝑒 (π‘‚β€˜(π‘Ž βˆ– 𝐸))) = (π‘‚β€˜π‘Ž)))
1615simprd 494 . 2 (πœ‘ β†’ βˆ€π‘Ž ∈ 𝒫 βˆͺ dom 𝑂((π‘‚β€˜(π‘Ž ∩ 𝐸)) +𝑒 (π‘‚β€˜(π‘Ž βˆ– 𝐸))) = (π‘‚β€˜π‘Ž))
17 ineq1 4200 . . . . . 6 (π‘Ž = 𝐴 β†’ (π‘Ž ∩ 𝐸) = (𝐴 ∩ 𝐸))
1817fveq2d 6894 . . . . 5 (π‘Ž = 𝐴 β†’ (π‘‚β€˜(π‘Ž ∩ 𝐸)) = (π‘‚β€˜(𝐴 ∩ 𝐸)))
19 difeq1 4108 . . . . . 6 (π‘Ž = 𝐴 β†’ (π‘Ž βˆ– 𝐸) = (𝐴 βˆ– 𝐸))
2019fveq2d 6894 . . . . 5 (π‘Ž = 𝐴 β†’ (π‘‚β€˜(π‘Ž βˆ– 𝐸)) = (π‘‚β€˜(𝐴 βˆ– 𝐸)))
2118, 20oveq12d 7431 . . . 4 (π‘Ž = 𝐴 β†’ ((π‘‚β€˜(π‘Ž ∩ 𝐸)) +𝑒 (π‘‚β€˜(π‘Ž βˆ– 𝐸))) = ((π‘‚β€˜(𝐴 ∩ 𝐸)) +𝑒 (π‘‚β€˜(𝐴 βˆ– 𝐸))))
22 fveq2 6890 . . . 4 (π‘Ž = 𝐴 β†’ (π‘‚β€˜π‘Ž) = (π‘‚β€˜π΄))
2321, 22eqeq12d 2741 . . 3 (π‘Ž = 𝐴 β†’ (((π‘‚β€˜(π‘Ž ∩ 𝐸)) +𝑒 (π‘‚β€˜(π‘Ž βˆ– 𝐸))) = (π‘‚β€˜π‘Ž) ↔ ((π‘‚β€˜(𝐴 ∩ 𝐸)) +𝑒 (π‘‚β€˜(𝐴 βˆ– 𝐸))) = (π‘‚β€˜π΄)))
2423rspcva 3601 . 2 ((𝐴 ∈ 𝒫 βˆͺ dom 𝑂 ∧ βˆ€π‘Ž ∈ 𝒫 βˆͺ dom 𝑂((π‘‚β€˜(π‘Ž ∩ 𝐸)) +𝑒 (π‘‚β€˜(π‘Ž βˆ– 𝐸))) = (π‘‚β€˜π‘Ž)) β†’ ((π‘‚β€˜(𝐴 ∩ 𝐸)) +𝑒 (π‘‚β€˜(𝐴 βˆ– 𝐸))) = (π‘‚β€˜π΄))
2511, 16, 24syl2anc 582 1 (πœ‘ β†’ ((π‘‚β€˜(𝐴 ∩ 𝐸)) +𝑒 (π‘‚β€˜(𝐴 βˆ– 𝐸))) = (π‘‚β€˜π΄))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 394   = wceq 1533   ∈ wcel 2098  βˆ€wral 3051  Vcvv 3463   βˆ– cdif 3938   ∩ cin 3940   βŠ† wss 3941  π’« cpw 4599  βˆͺ cuni 4904  dom cdm 5673  β€˜cfv 6543  (class class class)co 7413   +𝑒 cxad 13117  OutMeascome 45936  CaraGenccaragen 45938
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-sep 5295  ax-nul 5302  ax-pow 5360  ax-pr 5424  ax-un 7735
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ral 3052  df-rex 3061  df-rab 3420  df-v 3465  df-dif 3944  df-un 3946  df-in 3948  df-ss 3958  df-nul 4320  df-if 4526  df-pw 4601  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4905  df-br 5145  df-opab 5207  df-mpt 5228  df-id 5571  df-xp 5679  df-rel 5680  df-cnv 5681  df-co 5682  df-dm 5683  df-rn 5684  df-iota 6495  df-fun 6545  df-fv 6551  df-ov 7416  df-caragen 45939
This theorem is referenced by:  caragenuncllem  45959  carageniuncllem1  45968  carageniuncllem2  45969  caratheodorylem1  45973
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