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Theorem meacl 42177
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 meacl.1 . . 3 (𝜑𝑀 ∈ Meas)
2 meacl.2 . . 3 𝑆 = dom 𝑀
31, 2meaf 42172 . 2 (𝜑𝑀:𝑆⟶(0[,]+∞))
4 meacl.3 . 2 (𝜑𝐴𝑆)
53, 4ffvelrnd 6677 1 (𝜑 → (𝑀𝐴) ∈ (0[,]+∞))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1507  wcel 2050  dom cdm 5407  cfv 6188  (class class class)co 6976  0cc0 10335  +∞cpnf 10471  [,]cicc 12557  Meascmea 42168
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1965  ax-8 2052  ax-9 2059  ax-10 2079  ax-11 2093  ax-12 2106  ax-13 2301  ax-ext 2750  ax-rep 5049  ax-sep 5060  ax-nul 5067  ax-pr 5186
This theorem depends on definitions:  df-bi 199  df-an 388  df-or 834  df-3an 1070  df-tru 1510  df-ex 1743  df-nf 1747  df-sb 2016  df-mo 2547  df-eu 2584  df-clab 2759  df-cleq 2771  df-clel 2846  df-nfc 2918  df-ne 2968  df-ral 3093  df-rex 3094  df-reu 3095  df-rab 3097  df-v 3417  df-sbc 3682  df-csb 3787  df-dif 3832  df-un 3834  df-in 3836  df-ss 3843  df-nul 4179  df-if 4351  df-pw 4424  df-sn 4442  df-pr 4444  df-op 4448  df-uni 4713  df-iun 4794  df-br 4930  df-opab 4992  df-mpt 5009  df-id 5312  df-xp 5413  df-rel 5414  df-cnv 5415  df-co 5416  df-dm 5417  df-rn 5418  df-res 5419  df-ima 5420  df-iota 6152  df-fun 6190  df-fn 6191  df-f 6192  df-f1 6193  df-fo 6194  df-f1o 6195  df-fv 6196  df-mea 42169
This theorem is referenced by:  meaxrcl  42180  meassle  42182  meaiunlelem  42187  meage0  42194  voncl  42385
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