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Mathbox for Thierry Arnoux |
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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 1137 | . 2 ⊢ ((𝑀 ∈ (measures‘𝑆) ∧ 𝐴 ∈ 𝑆 ∧ (𝑀‘𝐴) ≤ 0) → (𝑀‘𝐴) ≤ 0) | |
2 | measvxrge0 34186 | . . . . 5 ⊢ ((𝑀 ∈ (measures‘𝑆) ∧ 𝐴 ∈ 𝑆) → (𝑀‘𝐴) ∈ (0[,]+∞)) | |
3 | elxrge0 13494 | . . . . 5 ⊢ ((𝑀‘𝐴) ∈ (0[,]+∞) ↔ ((𝑀‘𝐴) ∈ ℝ* ∧ 0 ≤ (𝑀‘𝐴))) | |
4 | 2, 3 | sylib 218 | . . . 4 ⊢ ((𝑀 ∈ (measures‘𝑆) ∧ 𝐴 ∈ 𝑆) → ((𝑀‘𝐴) ∈ ℝ* ∧ 0 ≤ (𝑀‘𝐴))) |
5 | 4 | 3adant3 1131 | . . 3 ⊢ ((𝑀 ∈ (measures‘𝑆) ∧ 𝐴 ∈ 𝑆 ∧ (𝑀‘𝐴) ≤ 0) → ((𝑀‘𝐴) ∈ ℝ* ∧ 0 ≤ (𝑀‘𝐴))) |
6 | 5 | simprd 495 | . 2 ⊢ ((𝑀 ∈ (measures‘𝑆) ∧ 𝐴 ∈ 𝑆 ∧ (𝑀‘𝐴) ≤ 0) → 0 ≤ (𝑀‘𝐴)) |
7 | 5 | simpld 494 | . . 3 ⊢ ((𝑀 ∈ (measures‘𝑆) ∧ 𝐴 ∈ 𝑆 ∧ (𝑀‘𝐴) ≤ 0) → (𝑀‘𝐴) ∈ ℝ*) |
8 | 0xr 11306 | . . 3 ⊢ 0 ∈ ℝ* | |
9 | xrletri3 13193 | . . 3 ⊢ (((𝑀‘𝐴) ∈ ℝ* ∧ 0 ∈ ℝ*) → ((𝑀‘𝐴) = 0 ↔ ((𝑀‘𝐴) ≤ 0 ∧ 0 ≤ (𝑀‘𝐴)))) | |
10 | 7, 8, 9 | sylancl 586 | . 2 ⊢ ((𝑀 ∈ (measures‘𝑆) ∧ 𝐴 ∈ 𝑆 ∧ (𝑀‘𝐴) ≤ 0) → ((𝑀‘𝐴) = 0 ↔ ((𝑀‘𝐴) ≤ 0 ∧ 0 ≤ (𝑀‘𝐴)))) |
11 | 1, 6, 10 | mpbir2and 713 | 1 ⊢ ((𝑀 ∈ (measures‘𝑆) ∧ 𝐴 ∈ 𝑆 ∧ (𝑀‘𝐴) ≤ 0) → (𝑀‘𝐴) = 0) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∧ w3a 1086 = wceq 1537 ∈ wcel 2106 class class class wbr 5148 ‘cfv 6563 (class class class)co 7431 0cc0 11153 +∞cpnf 11290 ℝ*cxr 11292 ≤ cle 11294 [,]cicc 13387 measurescmeas 34176 |
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 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-cnex 11209 ax-resscn 11210 ax-1cn 11211 ax-addrcl 11214 ax-rnegex 11224 ax-cnre 11226 ax-pre-lttri 11227 ax-pre-lttrn 11228 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5583 df-po 5597 df-so 5598 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-ov 7434 df-oprab 7435 df-mpo 7436 df-er 8744 df-en 8985 df-dom 8986 df-sdom 8987 df-pnf 11295 df-mnf 11296 df-xr 11297 df-ltxr 11298 df-le 11299 df-icc 13391 df-esum 34009 df-meas 34177 |
This theorem is referenced by: aean 34225 |
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