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Theorem caragensplit 43154
 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 41699 . . . . . 6 (𝜑𝑋 ∈ V)
5 ssexg 5191 . . . . . 6 ((𝐴𝑋𝑋 ∈ V) → 𝐴 ∈ V)
61, 4, 5syl2anc 587 . . . . 5 (𝜑𝐴 ∈ V)
7 elpwg 4500 . . . . 5 (𝐴 ∈ V → (𝐴 ∈ 𝒫 𝑋𝐴𝑋))
86, 7syl 17 . . . 4 (𝜑 → (𝐴 ∈ 𝒫 𝑋𝐴𝑋))
91, 8mpbird 260 . . 3 (𝜑𝐴 ∈ 𝒫 𝑋)
103pweqi 4515 . . 3 𝒫 𝑋 = 𝒫 dom 𝑂
119, 10eleqtrdi 2900 . 2 (𝜑𝐴 ∈ 𝒫 dom 𝑂)
12 caragensplit.e . . . 4 (𝜑𝐸𝑆)
13 caragensplit.s . . . . 5 𝑆 = (CaraGen‘𝑂)
142, 13caragenel 43149 . . . 4 (𝜑 → (𝐸𝑆 ↔ (𝐸 ∈ 𝒫 dom 𝑂 ∧ ∀𝑎 ∈ 𝒫 dom 𝑂((𝑂‘(𝑎𝐸)) +𝑒 (𝑂‘(𝑎𝐸))) = (𝑂𝑎))))
1512, 14mpbid 235 . . 3 (𝜑 → (𝐸 ∈ 𝒫 dom 𝑂 ∧ ∀𝑎 ∈ 𝒫 dom 𝑂((𝑂‘(𝑎𝐸)) +𝑒 (𝑂‘(𝑎𝐸))) = (𝑂𝑎)))
1615simprd 499 . 2 (𝜑 → ∀𝑎 ∈ 𝒫 dom 𝑂((𝑂‘(𝑎𝐸)) +𝑒 (𝑂‘(𝑎𝐸))) = (𝑂𝑎))
17 ineq1 4131 . . . . . 6 (𝑎 = 𝐴 → (𝑎𝐸) = (𝐴𝐸))
1817fveq2d 6649 . . . . 5 (𝑎 = 𝐴 → (𝑂‘(𝑎𝐸)) = (𝑂‘(𝐴𝐸)))
19 difeq1 4043 . . . . . 6 (𝑎 = 𝐴 → (𝑎𝐸) = (𝐴𝐸))
2019fveq2d 6649 . . . . 5 (𝑎 = 𝐴 → (𝑂‘(𝑎𝐸)) = (𝑂‘(𝐴𝐸)))
2118, 20oveq12d 7153 . . . 4 (𝑎 = 𝐴 → ((𝑂‘(𝑎𝐸)) +𝑒 (𝑂‘(𝑎𝐸))) = ((𝑂‘(𝐴𝐸)) +𝑒 (𝑂‘(𝐴𝐸))))
22 fveq2 6645 . . . 4 (𝑎 = 𝐴 → (𝑂𝑎) = (𝑂𝐴))
2321, 22eqeq12d 2814 . . 3 (𝑎 = 𝐴 → (((𝑂‘(𝑎𝐸)) +𝑒 (𝑂‘(𝑎𝐸))) = (𝑂𝑎) ↔ ((𝑂‘(𝐴𝐸)) +𝑒 (𝑂‘(𝐴𝐸))) = (𝑂𝐴)))
2423rspcva 3569 . 2 ((𝐴 ∈ 𝒫 dom 𝑂 ∧ ∀𝑎 ∈ 𝒫 dom 𝑂((𝑂‘(𝑎𝐸)) +𝑒 (𝑂‘(𝑎𝐸))) = (𝑂𝑎)) → ((𝑂‘(𝐴𝐸)) +𝑒 (𝑂‘(𝐴𝐸))) = (𝑂𝐴))
2511, 16, 24syl2anc 587 1 (𝜑 → ((𝑂‘(𝐴𝐸)) +𝑒 (𝑂‘(𝐴𝐸))) = (𝑂𝐴))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   = wceq 1538   ∈ wcel 2111  ∀wral 3106  Vcvv 3441   ∖ cdif 3878   ∩ cin 3880   ⊆ wss 3881  𝒫 cpw 4497  ∪ cuni 4800  dom cdm 5519  ‘cfv 6324  (class class class)co 7135   +𝑒 cxad 12495  OutMeascome 43143  CaraGenccaragen 43145 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 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5167  ax-nul 5174  ax-pow 5231  ax-pr 5295  ax-un 7443 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ral 3111  df-rex 3112  df-rab 3115  df-v 3443  df-sbc 3721  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4801  df-br 5031  df-opab 5093  df-mpt 5111  df-id 5425  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-iota 6283  df-fun 6326  df-fv 6332  df-ov 7138  df-caragen 43146 This theorem is referenced by:  caragenuncllem  43166  carageniuncllem1  43175  carageniuncllem2  43176  caratheodorylem1  43180
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