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Mirrors > Home > MPE Home > Th. List > Mathboxes > omege0 | Structured version Visualization version GIF version |
Description: If the outer measure of a set is greater than or equal to 0. (Contributed by Glauco Siliprandi, 24-Dec-2020.) |
Ref | Expression |
---|---|
omege0.o | ⊢ (𝜑 → 𝑂 ∈ OutMeas) |
omege0.x | ⊢ 𝑋 = ∪ dom 𝑂 |
omege0.a | ⊢ (𝜑 → 𝐴 ⊆ 𝑋) |
Ref | Expression |
---|---|
omege0 | ⊢ (𝜑 → 0 ≤ (𝑂‘𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 0xr 10687 | . . 3 ⊢ 0 ∈ ℝ* | |
2 | 1 | a1i 11 | . 2 ⊢ (𝜑 → 0 ∈ ℝ*) |
3 | pnfxr 10694 | . . 3 ⊢ +∞ ∈ ℝ* | |
4 | 3 | a1i 11 | . 2 ⊢ (𝜑 → +∞ ∈ ℝ*) |
5 | omege0.o | . . 3 ⊢ (𝜑 → 𝑂 ∈ OutMeas) | |
6 | omege0.x | . . 3 ⊢ 𝑋 = ∪ dom 𝑂 | |
7 | omege0.a | . . 3 ⊢ (𝜑 → 𝐴 ⊆ 𝑋) | |
8 | 5, 6, 7 | omecl 42784 | . 2 ⊢ (𝜑 → (𝑂‘𝐴) ∈ (0[,]+∞)) |
9 | iccgelb 12792 | . 2 ⊢ ((0 ∈ ℝ* ∧ +∞ ∈ ℝ* ∧ (𝑂‘𝐴) ∈ (0[,]+∞)) → 0 ≤ (𝑂‘𝐴)) | |
10 | 2, 4, 8, 9 | syl3anc 1367 | 1 ⊢ (𝜑 → 0 ≤ (𝑂‘𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1533 ∈ wcel 2110 ⊆ wss 3935 ∪ cuni 4837 class class class wbr 5065 dom cdm 5554 ‘cfv 6354 (class class class)co 7155 0cc0 10536 +∞cpnf 10671 ℝ*cxr 10673 ≤ cle 10675 [,]cicc 12740 OutMeascome 42770 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-sep 5202 ax-nul 5209 ax-pow 5265 ax-pr 5329 ax-un 7460 ax-cnex 10592 ax-resscn 10593 ax-1cn 10594 ax-addrcl 10597 ax-rnegex 10607 ax-cnre 10609 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-sbc 3772 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4567 df-pr 4569 df-op 4573 df-uni 4838 df-br 5066 df-opab 5128 df-id 5459 df-xp 5560 df-rel 5561 df-cnv 5562 df-co 5563 df-dm 5564 df-rn 5565 df-res 5566 df-iota 6313 df-fun 6356 df-fn 6357 df-f 6358 df-fv 6362 df-ov 7158 df-oprab 7159 df-mpo 7160 df-pnf 10676 df-xr 10678 df-icc 12744 df-ome 42771 |
This theorem is referenced by: omess0 42815 |
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