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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 44809 | . . . . . 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 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 4581 | . . . 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 2796 | . . 3 ⊢ (𝜑 → 𝒫 𝑋 = dom 𝑂) |
12 | 11 | feq2d 6659 | . 2 ⊢ (𝜑 → (𝑂:𝒫 𝑋⟶(0[,]+∞) ↔ 𝑂:dom 𝑂⟶(0[,]+∞))) |
13 | 6, 12 | mpbird 257 | 1 ⊢ (𝜑 → 𝑂:𝒫 𝑋⟶(0[,]+∞)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 397 = wceq 1542 ∈ wcel 2107 ∀wral 3065 ∅c0 4287 𝒫 cpw 4565 ∪ cuni 4870 class class class wbr 5110 dom cdm 5638 ↾ cres 5640 ⟶wf 6497 ‘cfv 6501 (class class class)co 7362 ωcom 7807 ≼ cdom 8888 0cc0 11058 +∞cpnf 11193 ≤ cle 11197 [,]cicc 13274 Σ^csumge0 44677 OutMeascome 44804 |
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 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-ext 2708 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-sb 2069 df-clab 2715 df-cleq 2729 df-clel 2815 df-ral 3066 df-rab 3411 df-v 3450 df-dif 3918 df-un 3920 df-in 3922 df-ss 3932 df-nul 4288 df-if 4492 df-pw 4567 df-sn 4592 df-pr 4594 df-op 4598 df-uni 4871 df-br 5111 df-opab 5173 df-rel 5645 df-cnv 5646 df-co 5647 df-dm 5648 df-rn 5649 df-res 5650 df-iota 6453 df-fun 6503 df-fn 6504 df-f 6505 df-fv 6509 df-ome 44805 |
This theorem is referenced by: omecl 44818 omeunle 44831 omeiunle 44832 caratheodory 44843 |
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