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Mirrors > Home > MPE Home > Th. List > Mathboxes > measge0 | Structured version Visualization version GIF version |
Description: A measure is nonnegative. (Contributed by Thierry Arnoux, 9-Mar-2018.) |
Ref | Expression |
---|---|
measge0 | ⊢ ((𝑀 ∈ (measures‘𝑆) ∧ 𝐴 ∈ 𝑆) → 0 ≤ (𝑀‘𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | measvxrge0 31459 | . . 3 ⊢ ((𝑀 ∈ (measures‘𝑆) ∧ 𝐴 ∈ 𝑆) → (𝑀‘𝐴) ∈ (0[,]+∞)) | |
2 | elxrge0 12839 | . . 3 ⊢ ((𝑀‘𝐴) ∈ (0[,]+∞) ↔ ((𝑀‘𝐴) ∈ ℝ* ∧ 0 ≤ (𝑀‘𝐴))) | |
3 | 1, 2 | sylib 220 | . 2 ⊢ ((𝑀 ∈ (measures‘𝑆) ∧ 𝐴 ∈ 𝑆) → ((𝑀‘𝐴) ∈ ℝ* ∧ 0 ≤ (𝑀‘𝐴))) |
4 | 3 | simprd 498 | 1 ⊢ ((𝑀 ∈ (measures‘𝑆) ∧ 𝐴 ∈ 𝑆) → 0 ≤ (𝑀‘𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 ∈ wcel 2110 class class class wbr 5059 ‘cfv 6350 (class class class)co 7150 0cc0 10531 +∞cpnf 10666 ℝ*cxr 10668 ≤ cle 10670 [,]cicc 12735 measurescmeas 31449 |
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 2156 ax-12 2172 ax-ext 2793 ax-sep 5196 ax-nul 5203 ax-pow 5259 ax-pr 5322 ax-un 7455 ax-cnex 10587 ax-resscn 10588 ax-1cn 10589 ax-addrcl 10592 ax-rnegex 10602 ax-cnre 10604 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-fal 1546 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-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3497 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4833 df-br 5060 df-opab 5122 df-mpt 5140 df-id 5455 df-xp 5556 df-rel 5557 df-cnv 5558 df-co 5559 df-dm 5560 df-rn 5561 df-iota 6309 df-fun 6352 df-fn 6353 df-f 6354 df-fv 6358 df-ov 7153 df-oprab 7154 df-mpo 7155 df-pnf 10671 df-mnf 10672 df-xr 10673 df-ltxr 10674 df-le 10675 df-icc 12739 df-esum 31282 df-meas 31450 |
This theorem is referenced by: sibfof 31593 |
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