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Mirrors > Home > MPE Home > Th. List > Mathboxes > measvxrge0 | Structured version Visualization version GIF version |
Description: The values of a measure are positive extended reals. (Contributed by Thierry Arnoux, 26-Dec-2016.) |
Ref | Expression |
---|---|
measvxrge0 | ⊢ ((𝑀 ∈ (measures‘𝑆) ∧ 𝐴 ∈ 𝑆) → (𝑀‘𝐴) ∈ (0[,]+∞)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | measfrge0 33950 | . 2 ⊢ (𝑀 ∈ (measures‘𝑆) → 𝑀:𝑆⟶(0[,]+∞)) | |
2 | 1 | ffvelcdmda 7093 | 1 ⊢ ((𝑀 ∈ (measures‘𝑆) ∧ 𝐴 ∈ 𝑆) → (𝑀‘𝐴) ∈ (0[,]+∞)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 394 ∈ wcel 2098 ‘cfv 6549 (class class class)co 7419 0cc0 11140 +∞cpnf 11277 [,]cicc 13362 measurescmeas 33942 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-sep 5300 ax-nul 5307 ax-pow 5365 ax-pr 5429 ax-un 7741 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2930 df-ral 3051 df-rex 3060 df-rab 3419 df-v 3463 df-sbc 3774 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-nul 4323 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4910 df-br 5150 df-opab 5212 df-mpt 5233 df-id 5576 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-iota 6501 df-fun 6551 df-fn 6552 df-f 6553 df-fv 6557 df-ov 7422 df-esum 33775 df-meas 33943 |
This theorem is referenced by: measge0 33954 measle0 33955 measxun2 33957 measun 33958 measvunilem 33959 measvuni 33961 measssd 33962 measunl 33963 measiun 33965 meascnbl 33966 measinb 33968 measdivcst 33971 measdivcstALTV 33972 sibfinima 34087 prob01 34161 probmeasb 34178 |
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