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Theorem caragensplit 40018
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 (𝜑𝑂 ∈ OutMeas)
caragensplit.s 𝑆 = (CaraGen‘𝑂)
caragensplit.x 𝑋 = dom 𝑂
caragensplit.e (𝜑𝐸𝑆)
caragensplit.a (𝜑𝐴𝑋)
Assertion
Ref Expression
caragensplit (𝜑 → ((𝑂‘(𝐴𝐸)) +𝑒 (𝑂‘(𝐴𝐸))) = (𝑂𝐴))

Proof of Theorem caragensplit
Dummy variable 𝑎 is distinct from all other variables.
StepHypRef Expression
1 caragensplit.a . . . 4 (𝜑𝐴𝑋)
2 caragensplit.o . . . . . . 7 (𝜑𝑂 ∈ OutMeas)
3 caragensplit.x . . . . . . 7 𝑋 = dom 𝑂
42, 3unidmex 38699 . . . . . 6 (𝜑𝑋 ∈ V)
5 ssexg 4764 . . . . . 6 ((𝐴𝑋𝑋 ∈ V) → 𝐴 ∈ V)
61, 4, 5syl2anc 692 . . . . 5 (𝜑𝐴 ∈ V)
7 elpwg 4138 . . . . 5 (𝐴 ∈ V → (𝐴 ∈ 𝒫 𝑋𝐴𝑋))
86, 7syl 17 . . . 4 (𝜑 → (𝐴 ∈ 𝒫 𝑋𝐴𝑋))
91, 8mpbird 247 . . 3 (𝜑𝐴 ∈ 𝒫 𝑋)
103pweqi 4134 . . 3 𝒫 𝑋 = 𝒫 dom 𝑂
119, 10syl6eleq 2708 . 2 (𝜑𝐴 ∈ 𝒫 dom 𝑂)
12 caragensplit.e . . . 4 (𝜑𝐸𝑆)
13 caragensplit.s . . . . 5 𝑆 = (CaraGen‘𝑂)
142, 13caragenel 40013 . . . 4 (𝜑 → (𝐸𝑆 ↔ (𝐸 ∈ 𝒫 dom 𝑂 ∧ ∀𝑎 ∈ 𝒫 dom 𝑂((𝑂‘(𝑎𝐸)) +𝑒 (𝑂‘(𝑎𝐸))) = (𝑂𝑎))))
1512, 14mpbid 222 . . 3 (𝜑 → (𝐸 ∈ 𝒫 dom 𝑂 ∧ ∀𝑎 ∈ 𝒫 dom 𝑂((𝑂‘(𝑎𝐸)) +𝑒 (𝑂‘(𝑎𝐸))) = (𝑂𝑎)))
1615simprd 479 . 2 (𝜑 → ∀𝑎 ∈ 𝒫 dom 𝑂((𝑂‘(𝑎𝐸)) +𝑒 (𝑂‘(𝑎𝐸))) = (𝑂𝑎))
17 ineq1 3785 . . . . . 6 (𝑎 = 𝐴 → (𝑎𝐸) = (𝐴𝐸))
1817fveq2d 6152 . . . . 5 (𝑎 = 𝐴 → (𝑂‘(𝑎𝐸)) = (𝑂‘(𝐴𝐸)))
19 difeq1 3699 . . . . . 6 (𝑎 = 𝐴 → (𝑎𝐸) = (𝐴𝐸))
2019fveq2d 6152 . . . . 5 (𝑎 = 𝐴 → (𝑂‘(𝑎𝐸)) = (𝑂‘(𝐴𝐸)))
2118, 20oveq12d 6622 . . . 4 (𝑎 = 𝐴 → ((𝑂‘(𝑎𝐸)) +𝑒 (𝑂‘(𝑎𝐸))) = ((𝑂‘(𝐴𝐸)) +𝑒 (𝑂‘(𝐴𝐸))))
22 fveq2 6148 . . . 4 (𝑎 = 𝐴 → (𝑂𝑎) = (𝑂𝐴))
2321, 22eqeq12d 2636 . . 3 (𝑎 = 𝐴 → (((𝑂‘(𝑎𝐸)) +𝑒 (𝑂‘(𝑎𝐸))) = (𝑂𝑎) ↔ ((𝑂‘(𝐴𝐸)) +𝑒 (𝑂‘(𝐴𝐸))) = (𝑂𝐴)))
2423rspcva 3293 . 2 ((𝐴 ∈ 𝒫 dom 𝑂 ∧ ∀𝑎 ∈ 𝒫 dom 𝑂((𝑂‘(𝑎𝐸)) +𝑒 (𝑂‘(𝑎𝐸))) = (𝑂𝑎)) → ((𝑂‘(𝐴𝐸)) +𝑒 (𝑂‘(𝐴𝐸))) = (𝑂𝐴))
2511, 16, 24syl2anc 692 1 (𝜑 → ((𝑂‘(𝐴𝐸)) +𝑒 (𝑂‘(𝐴𝐸))) = (𝑂𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384   = wceq 1480  wcel 1987  wral 2907  Vcvv 3186  cdif 3552  cin 3554  wss 3555  𝒫 cpw 4130   cuni 4402  dom cdm 5074  cfv 5847  (class class class)co 6604   +𝑒 cxad 11888  OutMeascome 40007  CaraGenccaragen 40009
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3188  df-sbc 3418  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-op 4155  df-uni 4403  df-br 4614  df-opab 4674  df-mpt 4675  df-id 4989  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-iota 5810  df-fun 5849  df-fv 5855  df-ov 6607  df-caragen 40010
This theorem is referenced by:  caragenuncllem  40030  carageniuncllem1  40039  carageniuncllem2  40040  caratheodorylem1  40044
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