Metamath Proof Explorer


Theorem ovnf

Description: The Lebesgue outer measure is a function that maps sets to nonnegative extended reals. This is step (a)(i) of the proof of Proposition 115D (a) of Fremlin1 p. 30. (Contributed by Glauco Siliprandi, 11-Oct-2020)

Ref Expression
Hypothesis ovnf.1 φXFin
Assertion ovnf φvoln*X:𝒫X0+∞

Proof

Step Hyp Ref Expression
1 ovnf.1 φXFin
2 0e0iccpnf 00+∞
3 2 a1i φy𝒫X00+∞
4 0xr 0*
5 4 a1i φy𝒫X0*
6 pnfxr +∞*
7 6 a1i φy𝒫X+∞*
8 1 adantr φy𝒫XXFin
9 elpwi y𝒫XyX
10 9 adantl φy𝒫XyX
11 eqid z*|i2XyjkX.ijkz=sum^jkXvol.ijk=z*|i2XyjkX.ijkz=sum^jkXvol.ijk
12 8 10 11 ovnsupge0 φy𝒫Xz*|i2XyjkX.ijkz=sum^jkXvol.ijk0+∞
13 8 10 11 ovnpnfelsup φy𝒫X+∞z*|i2XyjkX.ijkz=sum^jkXvol.ijk
14 13 ne0d φy𝒫Xz*|i2XyjkX.ijkz=sum^jkXvol.ijk
15 5 7 12 14 inficc φy𝒫Xsupz*|i2XyjkX.ijkz=sum^jkXvol.ijk*<0+∞
16 3 15 ifcld φy𝒫XifX=0supz*|i2XyjkX.ijkz=sum^jkXvol.ijk*<0+∞
17 eqid y𝒫XifX=0supz*|i2XyjkX.ijkz=sum^jkXvol.ijk*<=y𝒫XifX=0supz*|i2XyjkX.ijkz=sum^jkXvol.ijk*<
18 16 17 fmptd φy𝒫XifX=0supz*|i2XyjkX.ijkz=sum^jkXvol.ijk*<:𝒫X0+∞
19 1 ovnval φvoln*X=y𝒫XifX=0supz*|i2XyjkX.ijkz=sum^jkXvol.ijk*<
20 19 feq1d φvoln*X:𝒫X0+∞y𝒫XifX=0supz*|i2XyjkX.ijkz=sum^jkXvol.ijk*<:𝒫X0+∞
21 18 20 mpbird φvoln*X:𝒫X0+∞