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Mathbox for Glauco Siliprandi |
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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 11248 | . . 3 ⊢ 0 ∈ ℝ* | |
2 | 1 | a1i 11 | . 2 ⊢ (𝜑 → 0 ∈ ℝ*) |
3 | pnfxr 11255 | . . 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 45092 | . 2 ⊢ (𝜑 → (𝑂‘𝐴) ∈ (0[,]+∞)) |
9 | iccgelb 13367 | . 2 ⊢ ((0 ∈ ℝ* ∧ +∞ ∈ ℝ* ∧ (𝑂‘𝐴) ∈ (0[,]+∞)) → 0 ≤ (𝑂‘𝐴)) | |
10 | 2, 4, 8, 9 | syl3anc 1372 | 1 ⊢ (𝜑 → 0 ≤ (𝑂‘𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1542 ∈ wcel 2107 ⊆ wss 3946 ∪ cuni 4904 class class class wbr 5144 dom cdm 5672 ‘cfv 6535 (class class class)co 7396 0cc0 11097 +∞cpnf 11232 ℝ*cxr 11234 ≤ cle 11236 [,]cicc 13314 OutMeascome 45078 |
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-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5295 ax-nul 5302 ax-pow 5359 ax-pr 5423 ax-un 7712 ax-cnex 11153 ax-resscn 11154 ax-1cn 11155 ax-addrcl 11158 ax-rnegex 11168 ax-cnre 11170 |
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-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-ral 3063 df-rex 3072 df-rab 3434 df-v 3477 df-sbc 3776 df-dif 3949 df-un 3951 df-in 3953 df-ss 3963 df-nul 4321 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4905 df-br 5145 df-opab 5207 df-id 5570 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-iota 6487 df-fun 6537 df-fn 6538 df-f 6539 df-fv 6543 df-ov 7399 df-oprab 7400 df-mpo 7401 df-pnf 11237 df-xr 11239 df-icc 13318 df-ome 45079 |
This theorem is referenced by: omess0 45123 |
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