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Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > meacl | Structured version Visualization version GIF version |
Description: The measure of a set is a nonnegative extended real. (Contributed by Glauco Siliprandi, 17-Aug-2020.) |
Ref | Expression |
---|---|
meacl.1 | ⊢ (𝜑 → 𝑀 ∈ Meas) |
meacl.2 | ⊢ 𝑆 = dom 𝑀 |
meacl.3 | ⊢ (𝜑 → 𝐴 ∈ 𝑆) |
Ref | Expression |
---|---|
meacl | ⊢ (𝜑 → (𝑀‘𝐴) ∈ (0[,]+∞)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | meacl.1 | . . 3 ⊢ (𝜑 → 𝑀 ∈ Meas) | |
2 | meacl.2 | . . 3 ⊢ 𝑆 = dom 𝑀 | |
3 | 1, 2 | meaf 42172 | . 2 ⊢ (𝜑 → 𝑀:𝑆⟶(0[,]+∞)) |
4 | meacl.3 | . 2 ⊢ (𝜑 → 𝐴 ∈ 𝑆) | |
5 | 3, 4 | ffvelrnd 6677 | 1 ⊢ (𝜑 → (𝑀‘𝐴) ∈ (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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