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 44032 | . . . . . 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 231 | . . . 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 765 | . . 3 ⊢ (𝜑 → ((𝑂:dom 𝑂⟶(0[,]+∞) ∧ dom 𝑂 = 𝒫 ∪ dom 𝑂) ∧ (𝑂‘∅) = 0)) |
6 | 5 | simplld 765 | . 2 ⊢ (𝜑 → 𝑂:dom 𝑂⟶(0[,]+∞)) |
7 | omef.x | . . . . 5 ⊢ 𝑋 = ∪ dom 𝑂 | |
8 | 7 | pweqi 4551 | . . . 4 ⊢ 𝒫 𝑋 = 𝒫 ∪ dom 𝑂 |
9 | simp-4r 781 | . . . . 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 2797 | . . 3 ⊢ (𝜑 → 𝒫 𝑋 = dom 𝑂) |
12 | 11 | feq2d 6586 | . 2 ⊢ (𝜑 → (𝑂:𝒫 𝑋⟶(0[,]+∞) ↔ 𝑂:dom 𝑂⟶(0[,]+∞))) |
13 | 6, 12 | mpbird 256 | 1 ⊢ (𝜑 → 𝑂:𝒫 𝑋⟶(0[,]+∞)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 396 = wceq 1539 ∈ wcel 2106 ∀wral 3064 ∅c0 4256 𝒫 cpw 4533 ∪ cuni 4839 class class class wbr 5074 dom cdm 5589 ↾ cres 5591 ⟶wf 6429 ‘cfv 6433 (class class class)co 7275 ωcom 7712 ≼ cdom 8731 0cc0 10871 +∞cpnf 11006 ≤ cle 11010 [,]cicc 13082 Σ^csumge0 43900 OutMeascome 44027 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-ext 2709 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-sb 2068 df-clab 2716 df-cleq 2730 df-clel 2816 df-ral 3069 df-rab 3073 df-v 3434 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-nul 4257 df-if 4460 df-pw 4535 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4840 df-br 5075 df-opab 5137 df-rel 5596 df-cnv 5597 df-co 5598 df-dm 5599 df-rn 5600 df-res 5601 df-iota 6391 df-fun 6435 df-fn 6436 df-f 6437 df-fv 6441 df-ome 44028 |
This theorem is referenced by: omecl 44041 omeunle 44054 omeiunle 44055 caratheodory 44066 |
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