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Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > omef | Structured version Visualization version GIF version |
Description: An outer measure is a function that maps to nonnegative extended reals. (Contributed by Glauco Siliprandi, 17-Aug-2020.) |
Ref | Expression |
---|---|
omef.o | ⊢ (𝜑 → 𝑂 ∈ OutMeas) |
omef.x | ⊢ 𝑋 = ∪ dom 𝑂 |
Ref | Expression |
---|---|
omef | ⊢ (𝜑 → 𝑂:𝒫 𝑋⟶(0[,]+∞)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | omef.o | . . . . 5 ⊢ (𝜑 → 𝑂 ∈ OutMeas) | |
2 | isome 43133 | . . . . . 6 ⊢ (𝑂 ∈ OutMeas → (𝑂 ∈ OutMeas ↔ ((((𝑂:dom 𝑂⟶(0[,]+∞) ∧ dom 𝑂 = 𝒫 ∪ dom 𝑂) ∧ (𝑂‘∅) = 0) ∧ ∀𝑦 ∈ 𝒫 ∪ dom 𝑂∀𝑧 ∈ 𝒫 𝑦(𝑂‘𝑧) ≤ (𝑂‘𝑦)) ∧ ∀𝑦 ∈ 𝒫 dom 𝑂(𝑦 ≼ ω → (𝑂‘∪ 𝑦) ≤ (Σ^‘(𝑂 ↾ 𝑦)))))) | |
3 | 1, 2 | syl 17 | . . . . 5 ⊢ (𝜑 → (𝑂 ∈ OutMeas ↔ ((((𝑂:dom 𝑂⟶(0[,]+∞) ∧ dom 𝑂 = 𝒫 ∪ dom 𝑂) ∧ (𝑂‘∅) = 0) ∧ ∀𝑦 ∈ 𝒫 ∪ dom 𝑂∀𝑧 ∈ 𝒫 𝑦(𝑂‘𝑧) ≤ (𝑂‘𝑦)) ∧ ∀𝑦 ∈ 𝒫 dom 𝑂(𝑦 ≼ ω → (𝑂‘∪ 𝑦) ≤ (Σ^‘(𝑂 ↾ 𝑦)))))) |
4 | 1, 3 | mpbid 235 | . . . 4 ⊢ (𝜑 → ((((𝑂:dom 𝑂⟶(0[,]+∞) ∧ dom 𝑂 = 𝒫 ∪ dom 𝑂) ∧ (𝑂‘∅) = 0) ∧ ∀𝑦 ∈ 𝒫 ∪ dom 𝑂∀𝑧 ∈ 𝒫 𝑦(𝑂‘𝑧) ≤ (𝑂‘𝑦)) ∧ ∀𝑦 ∈ 𝒫 dom 𝑂(𝑦 ≼ ω → (𝑂‘∪ 𝑦) ≤ (Σ^‘(𝑂 ↾ 𝑦))))) |
5 | 4 | simplld 767 | . . 3 ⊢ (𝜑 → ((𝑂:dom 𝑂⟶(0[,]+∞) ∧ dom 𝑂 = 𝒫 ∪ dom 𝑂) ∧ (𝑂‘∅) = 0)) |
6 | 5 | simplld 767 | . 2 ⊢ (𝜑 → 𝑂:dom 𝑂⟶(0[,]+∞)) |
7 | omef.x | . . . . 5 ⊢ 𝑋 = ∪ dom 𝑂 | |
8 | 7 | pweqi 4515 | . . . 4 ⊢ 𝒫 𝑋 = 𝒫 ∪ dom 𝑂 |
9 | simp-4r 783 | . . . . 5 ⊢ (((((𝑂:dom 𝑂⟶(0[,]+∞) ∧ dom 𝑂 = 𝒫 ∪ dom 𝑂) ∧ (𝑂‘∅) = 0) ∧ ∀𝑦 ∈ 𝒫 ∪ dom 𝑂∀𝑧 ∈ 𝒫 𝑦(𝑂‘𝑧) ≤ (𝑂‘𝑦)) ∧ ∀𝑦 ∈ 𝒫 dom 𝑂(𝑦 ≼ ω → (𝑂‘∪ 𝑦) ≤ (Σ^‘(𝑂 ↾ 𝑦)))) → dom 𝑂 = 𝒫 ∪ dom 𝑂) | |
10 | 4, 9 | syl 17 | . . . 4 ⊢ (𝜑 → dom 𝑂 = 𝒫 ∪ dom 𝑂) |
11 | 8, 10 | eqtr4id 2852 | . . 3 ⊢ (𝜑 → 𝒫 𝑋 = dom 𝑂) |
12 | 11 | feq2d 6473 | . 2 ⊢ (𝜑 → (𝑂:𝒫 𝑋⟶(0[,]+∞) ↔ 𝑂:dom 𝑂⟶(0[,]+∞))) |
13 | 6, 12 | mpbird 260 | 1 ⊢ (𝜑 → 𝑂:𝒫 𝑋⟶(0[,]+∞)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 = wceq 1538 ∈ wcel 2111 ∀wral 3106 ∅c0 4243 𝒫 cpw 4497 ∪ cuni 4800 class class class wbr 5030 dom cdm 5519 ↾ cres 5521 ⟶wf 6320 ‘cfv 6324 (class class class)co 7135 ωcom 7560 ≼ cdom 8490 0cc0 10526 +∞cpnf 10661 ≤ cle 10665 [,]cicc 12729 Σ^csumge0 43001 OutMeascome 43128 |
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 |
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-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ral 3111 df-rab 3115 df-v 3443 df-un 3886 df-in 3888 df-ss 3898 df-pw 4499 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-br 5031 df-opab 5093 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-fv 6332 df-ome 43129 |
This theorem is referenced by: omecl 43142 omeunle 43155 omeiunle 43156 caratheodory 43167 |
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