Users' Mathboxes Mathbox for Glauco Siliprandi < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  caragensplit Structured version   Visualization version   GIF version

Theorem caragensplit 42659
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 41189 . . . . . 6 (𝜑𝑋 ∈ V)
5 ssexg 5218 . . . . . 6 ((𝐴𝑋𝑋 ∈ V) → 𝐴 ∈ V)
61, 4, 5syl2anc 584 . . . . 5 (𝜑𝐴 ∈ V)
7 elpwg 4541 . . . . 5 (𝐴 ∈ V → (𝐴 ∈ 𝒫 𝑋𝐴𝑋))
86, 7syl 17 . . . 4 (𝜑 → (𝐴 ∈ 𝒫 𝑋𝐴𝑋))
91, 8mpbird 258 . . 3 (𝜑𝐴 ∈ 𝒫 𝑋)
103pweqi 4539 . . 3 𝒫 𝑋 = 𝒫 dom 𝑂
119, 10eleqtrdi 2920 . 2 (𝜑𝐴 ∈ 𝒫 dom 𝑂)
12 caragensplit.e . . . 4 (𝜑𝐸𝑆)
13 caragensplit.s . . . . 5 𝑆 = (CaraGen‘𝑂)
142, 13caragenel 42654 . . . 4 (𝜑 → (𝐸𝑆 ↔ (𝐸 ∈ 𝒫 dom 𝑂 ∧ ∀𝑎 ∈ 𝒫 dom 𝑂((𝑂‘(𝑎𝐸)) +𝑒 (𝑂‘(𝑎𝐸))) = (𝑂𝑎))))
1512, 14mpbid 233 . . 3 (𝜑 → (𝐸 ∈ 𝒫 dom 𝑂 ∧ ∀𝑎 ∈ 𝒫 dom 𝑂((𝑂‘(𝑎𝐸)) +𝑒 (𝑂‘(𝑎𝐸))) = (𝑂𝑎)))
1615simprd 496 . 2 (𝜑 → ∀𝑎 ∈ 𝒫 dom 𝑂((𝑂‘(𝑎𝐸)) +𝑒 (𝑂‘(𝑎𝐸))) = (𝑂𝑎))
17 ineq1 4178 . . . . . 6 (𝑎 = 𝐴 → (𝑎𝐸) = (𝐴𝐸))
1817fveq2d 6667 . . . . 5 (𝑎 = 𝐴 → (𝑂‘(𝑎𝐸)) = (𝑂‘(𝐴𝐸)))
19 difeq1 4089 . . . . . 6 (𝑎 = 𝐴 → (𝑎𝐸) = (𝐴𝐸))
2019fveq2d 6667 . . . . 5 (𝑎 = 𝐴 → (𝑂‘(𝑎𝐸)) = (𝑂‘(𝐴𝐸)))
2118, 20oveq12d 7163 . . . 4 (𝑎 = 𝐴 → ((𝑂‘(𝑎𝐸)) +𝑒 (𝑂‘(𝑎𝐸))) = ((𝑂‘(𝐴𝐸)) +𝑒 (𝑂‘(𝐴𝐸))))
22 fveq2 6663 . . . 4 (𝑎 = 𝐴 → (𝑂𝑎) = (𝑂𝐴))
2321, 22eqeq12d 2834 . . 3 (𝑎 = 𝐴 → (((𝑂‘(𝑎𝐸)) +𝑒 (𝑂‘(𝑎𝐸))) = (𝑂𝑎) ↔ ((𝑂‘(𝐴𝐸)) +𝑒 (𝑂‘(𝐴𝐸))) = (𝑂𝐴)))
2423rspcva 3618 . 2 ((𝐴 ∈ 𝒫 dom 𝑂 ∧ ∀𝑎 ∈ 𝒫 dom 𝑂((𝑂‘(𝑎𝐸)) +𝑒 (𝑂‘(𝑎𝐸))) = (𝑂𝑎)) → ((𝑂‘(𝐴𝐸)) +𝑒 (𝑂‘(𝐴𝐸))) = (𝑂𝐴))
2511, 16, 24syl2anc 584 1 (𝜑 → ((𝑂‘(𝐴𝐸)) +𝑒 (𝑂‘(𝐴𝐸))) = (𝑂𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  wa 396   = wceq 1528  wcel 2105  wral 3135  Vcvv 3492  cdif 3930  cin 3932  wss 3933  𝒫 cpw 4535   cuni 4830  dom cdm 5548  cfv 6348  (class class class)co 7145   +𝑒 cxad 12493  OutMeascome 42648  CaraGenccaragen 42650
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ral 3140  df-rex 3141  df-rab 3144  df-v 3494  df-sbc 3770  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-br 5058  df-opab 5120  df-mpt 5138  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-iota 6307  df-fun 6350  df-fv 6356  df-ov 7148  df-caragen 42651
This theorem is referenced by:  caragenuncllem  42671  carageniuncllem1  42680  carageniuncllem2  42681  caratheodorylem1  42685
  Copyright terms: Public domain W3C validator