![]() |
Mathbox for Thierry Arnoux |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > Mathboxes > measle0 | Structured version Visualization version GIF version |
Description: If the measure of a given set is bounded by zero, it is zero. (Contributed by Thierry Arnoux, 20-Oct-2017.) |
Ref | Expression |
---|---|
measle0 | ⊢ ((𝑀 ∈ (measures‘𝑆) ∧ 𝐴 ∈ 𝑆 ∧ (𝑀‘𝐴) ≤ 0) → (𝑀‘𝐴) = 0) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simp3 1129 | . 2 ⊢ ((𝑀 ∈ (measures‘𝑆) ∧ 𝐴 ∈ 𝑆 ∧ (𝑀‘𝐴) ≤ 0) → (𝑀‘𝐴) ≤ 0) | |
2 | measvxrge0 30874 | . . . . 5 ⊢ ((𝑀 ∈ (measures‘𝑆) ∧ 𝐴 ∈ 𝑆) → (𝑀‘𝐴) ∈ (0[,]+∞)) | |
3 | elxrge0 12600 | . . . . 5 ⊢ ((𝑀‘𝐴) ∈ (0[,]+∞) ↔ ((𝑀‘𝐴) ∈ ℝ* ∧ 0 ≤ (𝑀‘𝐴))) | |
4 | 2, 3 | sylib 210 | . . . 4 ⊢ ((𝑀 ∈ (measures‘𝑆) ∧ 𝐴 ∈ 𝑆) → ((𝑀‘𝐴) ∈ ℝ* ∧ 0 ≤ (𝑀‘𝐴))) |
5 | 4 | 3adant3 1123 | . . 3 ⊢ ((𝑀 ∈ (measures‘𝑆) ∧ 𝐴 ∈ 𝑆 ∧ (𝑀‘𝐴) ≤ 0) → ((𝑀‘𝐴) ∈ ℝ* ∧ 0 ≤ (𝑀‘𝐴))) |
6 | 5 | simprd 491 | . 2 ⊢ ((𝑀 ∈ (measures‘𝑆) ∧ 𝐴 ∈ 𝑆 ∧ (𝑀‘𝐴) ≤ 0) → 0 ≤ (𝑀‘𝐴)) |
7 | 5 | simpld 490 | . . 3 ⊢ ((𝑀 ∈ (measures‘𝑆) ∧ 𝐴 ∈ 𝑆 ∧ (𝑀‘𝐴) ≤ 0) → (𝑀‘𝐴) ∈ ℝ*) |
8 | 0xr 10425 | . . 3 ⊢ 0 ∈ ℝ* | |
9 | xrletri3 12302 | . . 3 ⊢ (((𝑀‘𝐴) ∈ ℝ* ∧ 0 ∈ ℝ*) → ((𝑀‘𝐴) = 0 ↔ ((𝑀‘𝐴) ≤ 0 ∧ 0 ≤ (𝑀‘𝐴)))) | |
10 | 7, 8, 9 | sylancl 580 | . 2 ⊢ ((𝑀 ∈ (measures‘𝑆) ∧ 𝐴 ∈ 𝑆 ∧ (𝑀‘𝐴) ≤ 0) → ((𝑀‘𝐴) = 0 ↔ ((𝑀‘𝐴) ≤ 0 ∧ 0 ≤ (𝑀‘𝐴)))) |
11 | 1, 6, 10 | mpbir2and 703 | 1 ⊢ ((𝑀 ∈ (measures‘𝑆) ∧ 𝐴 ∈ 𝑆 ∧ (𝑀‘𝐴) ≤ 0) → (𝑀‘𝐴) = 0) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 198 ∧ wa 386 ∧ w3a 1071 = wceq 1601 ∈ wcel 2107 class class class wbr 4888 ‘cfv 6137 (class class class)co 6924 0cc0 10274 +∞cpnf 10410 ℝ*cxr 10412 ≤ cle 10414 [,]cicc 12495 measurescmeas 30864 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2055 ax-8 2109 ax-9 2116 ax-10 2135 ax-11 2150 ax-12 2163 ax-13 2334 ax-ext 2754 ax-sep 5019 ax-nul 5027 ax-pow 5079 ax-pr 5140 ax-un 7228 ax-cnex 10330 ax-resscn 10331 ax-1cn 10332 ax-addrcl 10335 ax-rnegex 10345 ax-cnre 10347 ax-pre-lttri 10348 ax-pre-lttrn 10349 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3or 1072 df-3an 1073 df-tru 1605 df-fal 1615 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2551 df-eu 2587 df-clab 2764 df-cleq 2770 df-clel 2774 df-nfc 2921 df-ne 2970 df-nel 3076 df-ral 3095 df-rex 3096 df-rab 3099 df-v 3400 df-sbc 3653 df-csb 3752 df-dif 3795 df-un 3797 df-in 3799 df-ss 3806 df-nul 4142 df-if 4308 df-pw 4381 df-sn 4399 df-pr 4401 df-op 4405 df-uni 4674 df-br 4889 df-opab 4951 df-mpt 4968 df-id 5263 df-po 5276 df-so 5277 df-xp 5363 df-rel 5364 df-cnv 5365 df-co 5366 df-dm 5367 df-rn 5368 df-res 5369 df-ima 5370 df-iota 6101 df-fun 6139 df-fn 6140 df-f 6141 df-f1 6142 df-fo 6143 df-f1o 6144 df-fv 6145 df-ov 6927 df-oprab 6928 df-mpt2 6929 df-er 8028 df-en 8244 df-dom 8245 df-sdom 8246 df-pnf 10415 df-mnf 10416 df-xr 10417 df-ltxr 10418 df-le 10419 df-icc 12499 df-esum 30696 df-meas 30865 |
This theorem is referenced by: aean 30913 |
Copyright terms: Public domain | W3C validator |