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Theorem measvxrge0 28001
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 27999 . 2  |-  ( M  e.  (measures `  S
)  ->  M : S
--> ( 0 [,] +oo ) )
21ffvelrnda 6032 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 1767   ` cfv 5594  (class class class)co 6295   0cc0 9504   +oocpnf 9637   [,]cicc 11544  measurescmeas 27991
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-un 6587
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-fal 1385  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2822  df-rex 2823  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-op 4040  df-uni 4252  df-br 4454  df-opab 4512  df-mpt 4513  df-id 4801  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-fv 5602  df-ov 6298  df-esum 27866  df-meas 27992
This theorem is referenced by:  measge0  28003  measle0  28004  measxun2  28006  measun  28007  measvunilem  28008  measvuni  28010  measssd  28011  measunl  28012  measiun  28014  meascnbl  28015  measinb  28017  measdivcstOLD  28020  measdivcst  28021  sibfinima  28106  prob01  28177  probmeasb  28194
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