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Theorem meacl 39998
Description: The measure of a set is a nonnegative extended real. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
Hypotheses
Ref Expression
meacl.1 (𝜑𝑀 ∈ Meas)
meacl.2 𝑆 = dom 𝑀
meacl.3 (𝜑𝐴𝑆)
Assertion
Ref Expression
meacl (𝜑 → (𝑀𝐴) ∈ (0[,]+∞))

Proof of Theorem meacl
StepHypRef Expression
1 id 22 . 2 (𝜑𝜑)
2 meacl.3 . 2 (𝜑𝐴𝑆)
3 meacl.1 . . . 4 (𝜑𝑀 ∈ Meas)
4 meacl.2 . . . 4 𝑆 = dom 𝑀
53, 4meaf 39993 . . 3 (𝜑𝑀:𝑆⟶(0[,]+∞))
65ffvelrnda 6317 . 2 ((𝜑𝐴𝑆) → (𝑀𝐴) ∈ (0[,]+∞))
71, 2, 6syl2anc 692 1 (𝜑 → (𝑀𝐴) ∈ (0[,]+∞))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1480  wcel 1987  dom cdm 5076  cfv 5849  (class class class)co 6607  0cc0 9883  +∞cpnf 10018  [,]cicc 12123  Meascmea 39989
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4733  ax-sep 4743  ax-nul 4751  ax-pr 4869
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-reu 2914  df-rab 2916  df-v 3188  df-sbc 3419  df-csb 3516  df-dif 3559  df-un 3561  df-in 3563  df-ss 3570  df-nul 3894  df-if 4061  df-pw 4134  df-sn 4151  df-pr 4153  df-op 4157  df-uni 4405  df-iun 4489  df-br 4616  df-opab 4676  df-mpt 4677  df-id 4991  df-xp 5082  df-rel 5083  df-cnv 5084  df-co 5085  df-dm 5086  df-rn 5087  df-res 5088  df-ima 5089  df-iota 5812  df-fun 5851  df-fn 5852  df-f 5853  df-f1 5854  df-fo 5855  df-f1o 5856  df-fv 5857  df-mea 39990
This theorem is referenced by:  meaxrcl  40001  meassle  40003  meaiunlelem  40008  meage0  40015  voncl  40203
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