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Theorem measvxrge0 26555
Description: The values of a measure are positive extended reals. (Contributed by Thierry Arnoux, 26-Dec-2016.)
Assertion
Ref Expression
measvxrge0  |-  ( ( M  e.  (measures `  S
)  /\  A  e.  S )  ->  ( M `  A )  e.  ( 0 [,] +oo ) )

Proof of Theorem measvxrge0
StepHypRef Expression
1 measfrge0 26553 . 2  |-  ( M  e.  (measures `  S
)  ->  M : S
--> ( 0 [,] +oo ) )
21ffvelrnda 5840 1  |-  ( ( M  e.  (measures `  S
)  /\  A  e.  S )  ->  ( M `  A )  e.  ( 0 [,] +oo ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    e. wcel 1761   ` cfv 5415  (class class class)co 6090   0cc0 9278   +oocpnf 9411   [,]cicc 11299  measurescmeas 26545
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1713  ax-7 1733  ax-8 1763  ax-9 1765  ax-10 1780  ax-11 1785  ax-12 1797  ax-13 1948  ax-ext 2422  ax-sep 4410  ax-nul 4418  ax-pow 4467  ax-pr 4528  ax-un 6371
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 962  df-tru 1367  df-fal 1370  df-ex 1592  df-nf 1595  df-sb 1706  df-eu 2261  df-mo 2262  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-ral 2718  df-rex 2719  df-rab 2722  df-v 2972  df-sbc 3184  df-csb 3286  df-dif 3328  df-un 3330  df-in 3332  df-ss 3339  df-nul 3635  df-if 3789  df-pw 3859  df-sn 3875  df-pr 3877  df-op 3881  df-uni 4089  df-br 4290  df-opab 4348  df-mpt 4349  df-id 4632  df-xp 4842  df-rel 4843  df-cnv 4844  df-co 4845  df-dm 4846  df-rn 4847  df-iota 5378  df-fun 5417  df-fn 5418  df-f 5419  df-fv 5423  df-ov 6093  df-esum 26420  df-meas 26546
This theorem is referenced by:  measge0  26557  measle0  26558  measxun2  26560  measun  26561  measvunilem  26562  measvuni  26564  measssd  26565  measunl  26566  measiun  26568  meascnbl  26569  measinb  26571  measdivcstOLD  26574  measdivcst  26575  sibfinima  26655  prob01  26726  probmeasb  26743
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