![]() |
Mathbox for Thierry Arnoux |
< Previous
Next >
Nearby theorems |
|
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 30606 | . 2 ⊢ (𝑀 ∈ (measures‘𝑆) → 𝑀:𝑆⟶(0[,]+∞)) | |
2 | 1 | ffvelrnda 6502 | 1 ⊢ ((𝑀 ∈ (measures‘𝑆) ∧ 𝐴 ∈ 𝑆) → (𝑀‘𝐴) ∈ (0[,]+∞)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 382 ∈ wcel 2145 ‘cfv 6031 (class class class)co 6793 0cc0 10138 +∞cpnf 10273 [,]cicc 12383 measurescmeas 30598 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1870 ax-4 1885 ax-5 1991 ax-6 2057 ax-7 2093 ax-8 2147 ax-9 2154 ax-10 2174 ax-11 2190 ax-12 2203 ax-13 2408 ax-ext 2751 ax-sep 4915 ax-nul 4923 ax-pow 4974 ax-pr 5034 ax-un 7096 |
This theorem depends on definitions: df-bi 197 df-an 383 df-or 837 df-3an 1073 df-tru 1634 df-fal 1637 df-ex 1853 df-nf 1858 df-sb 2050 df-eu 2622 df-mo 2623 df-clab 2758 df-cleq 2764 df-clel 2767 df-nfc 2902 df-ne 2944 df-ral 3066 df-rex 3067 df-rab 3070 df-v 3353 df-sbc 3588 df-csb 3683 df-dif 3726 df-un 3728 df-in 3730 df-ss 3737 df-nul 4064 df-if 4226 df-pw 4299 df-sn 4317 df-pr 4319 df-op 4323 df-uni 4575 df-br 4787 df-opab 4847 df-mpt 4864 df-id 5157 df-xp 5255 df-rel 5256 df-cnv 5257 df-co 5258 df-dm 5259 df-rn 5260 df-iota 5994 df-fun 6033 df-fn 6034 df-f 6035 df-fv 6039 df-ov 6796 df-esum 30430 df-meas 30599 |
This theorem is referenced by: measge0 30610 measle0 30611 measxun2 30613 measun 30614 measvunilem 30615 measvuni 30617 measssd 30618 measunl 30619 measiun 30621 meascnbl 30622 measinb 30624 measdivcstOLD 30627 measdivcst 30628 sibfinima 30741 prob01 30815 probmeasb 30832 |
Copyright terms: Public domain | W3C validator |