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Theorem caragensplit 41212
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 39708 . . . . . 6 (𝜑𝑋 ∈ V)
5 ssexg 4948 . . . . . 6 ((𝐴𝑋𝑋 ∈ V) → 𝐴 ∈ V)
61, 4, 5syl2anc 696 . . . . 5 (𝜑𝐴 ∈ V)
7 elpwg 4302 . . . . 5 (𝐴 ∈ V → (𝐴 ∈ 𝒫 𝑋𝐴𝑋))
86, 7syl 17 . . . 4 (𝜑 → (𝐴 ∈ 𝒫 𝑋𝐴𝑋))
91, 8mpbird 247 . . 3 (𝜑𝐴 ∈ 𝒫 𝑋)
103pweqi 4298 . . 3 𝒫 𝑋 = 𝒫 dom 𝑂
119, 10syl6eleq 2841 . 2 (𝜑𝐴 ∈ 𝒫 dom 𝑂)
12 caragensplit.e . . . 4 (𝜑𝐸𝑆)
13 caragensplit.s . . . . 5 𝑆 = (CaraGen‘𝑂)
142, 13caragenel 41207 . . . 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 482 . 2 (𝜑 → ∀𝑎 ∈ 𝒫 dom 𝑂((𝑂‘(𝑎𝐸)) +𝑒 (𝑂‘(𝑎𝐸))) = (𝑂𝑎))
17 ineq1 3942 . . . . . 6 (𝑎 = 𝐴 → (𝑎𝐸) = (𝐴𝐸))
1817fveq2d 6348 . . . . 5 (𝑎 = 𝐴 → (𝑂‘(𝑎𝐸)) = (𝑂‘(𝐴𝐸)))
19 difeq1 3856 . . . . . 6 (𝑎 = 𝐴 → (𝑎𝐸) = (𝐴𝐸))
2019fveq2d 6348 . . . . 5 (𝑎 = 𝐴 → (𝑂‘(𝑎𝐸)) = (𝑂‘(𝐴𝐸)))
2118, 20oveq12d 6823 . . . 4 (𝑎 = 𝐴 → ((𝑂‘(𝑎𝐸)) +𝑒 (𝑂‘(𝑎𝐸))) = ((𝑂‘(𝐴𝐸)) +𝑒 (𝑂‘(𝐴𝐸))))
22 fveq2 6344 . . . 4 (𝑎 = 𝐴 → (𝑂𝑎) = (𝑂𝐴))
2321, 22eqeq12d 2767 . . 3 (𝑎 = 𝐴 → (((𝑂‘(𝑎𝐸)) +𝑒 (𝑂‘(𝑎𝐸))) = (𝑂𝑎) ↔ ((𝑂‘(𝐴𝐸)) +𝑒 (𝑂‘(𝐴𝐸))) = (𝑂𝐴)))
2423rspcva 3439 . 2 ((𝐴 ∈ 𝒫 dom 𝑂 ∧ ∀𝑎 ∈ 𝒫 dom 𝑂((𝑂‘(𝑎𝐸)) +𝑒 (𝑂‘(𝑎𝐸))) = (𝑂𝑎)) → ((𝑂‘(𝐴𝐸)) +𝑒 (𝑂‘(𝐴𝐸))) = (𝑂𝐴))
2511, 16, 24syl2anc 696 1 (𝜑 → ((𝑂‘(𝐴𝐸)) +𝑒 (𝑂‘(𝐴𝐸))) = (𝑂𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383   = wceq 1624  wcel 2131  wral 3042  Vcvv 3332  cdif 3704  cin 3706  wss 3707  𝒫 cpw 4294   cuni 4580  dom cdm 5258  cfv 6041  (class class class)co 6805   +𝑒 cxad 12129  OutMeascome 41201  CaraGenccaragen 41203
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1863  ax-4 1878  ax-5 1980  ax-6 2046  ax-7 2082  ax-8 2133  ax-9 2140  ax-10 2160  ax-11 2175  ax-12 2188  ax-13 2383  ax-ext 2732  ax-sep 4925  ax-nul 4933  ax-pow 4984  ax-pr 5047  ax-un 7106
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1627  df-ex 1846  df-nf 1851  df-sb 2039  df-eu 2603  df-mo 2604  df-clab 2739  df-cleq 2745  df-clel 2748  df-nfc 2883  df-ral 3047  df-rex 3048  df-rab 3051  df-v 3334  df-sbc 3569  df-dif 3710  df-un 3712  df-in 3714  df-ss 3721  df-nul 4051  df-if 4223  df-pw 4296  df-sn 4314  df-pr 4316  df-op 4320  df-uni 4581  df-br 4797  df-opab 4857  df-mpt 4874  df-id 5166  df-xp 5264  df-rel 5265  df-cnv 5266  df-co 5267  df-dm 5268  df-rn 5269  df-iota 6004  df-fun 6043  df-fv 6049  df-ov 6808  df-caragen 41204
This theorem is referenced by:  caragenuncllem  41224  carageniuncllem1  41233  carageniuncllem2  41234  caratheodorylem1  41238
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