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Theorem meacl 41189
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 41184 . . 3 (𝜑𝑀:𝑆⟶(0[,]+∞))
65ffvelrnda 6504 . 2 ((𝜑𝐴𝑆) → (𝑀𝐴) ∈ (0[,]+∞))
71, 2, 6syl2anc 573 1 (𝜑 → (𝑀𝐴) ∈ (0[,]+∞))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1631  wcel 2145  dom cdm 5250  cfv 6030  (class class class)co 6795  0cc0 10141  +∞cpnf 10276  [,]cicc 12382  Meascmea 41180
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-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-rep 4905  ax-sep 4916  ax-nul 4924  ax-pr 5035
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  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-reu 3068  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 4227  df-pw 4300  df-sn 4318  df-pr 4320  df-op 4324  df-uni 4576  df-iun 4657  df-br 4788  df-opab 4848  df-mpt 4865  df-id 5158  df-xp 5256  df-rel 5257  df-cnv 5258  df-co 5259  df-dm 5260  df-rn 5261  df-res 5262  df-ima 5263  df-iota 5993  df-fun 6032  df-fn 6033  df-f 6034  df-f1 6035  df-fo 6036  df-f1o 6037  df-fv 6038  df-mea 41181
This theorem is referenced by:  meaxrcl  41192  meassle  41194  meaiunlelem  41199  meage0  41206  voncl  41397
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