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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 42734 | . 2 ⊢ (𝜑 → 𝑀:𝑆⟶(0[,]+∞)) |
4 | meacl.3 | . 2 ⊢ (𝜑 → 𝐴 ∈ 𝑆) | |
5 | 3, 4 | ffvelrnd 6851 | 1 ⊢ (𝜑 → (𝑀‘𝐴) ∈ (0[,]+∞)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1533 ∈ wcel 2110 dom cdm 5554 ‘cfv 6354 (class class class)co 7155 0cc0 10536 +∞cpnf 10671 [,]cicc 12740 Meascmea 42730 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-rep 5189 ax-sep 5202 ax-nul 5209 ax-pr 5329 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4567 df-pr 4569 df-op 4573 df-uni 4838 df-iun 4920 df-br 5066 df-opab 5128 df-mpt 5146 df-id 5459 df-xp 5560 df-rel 5561 df-cnv 5562 df-co 5563 df-dm 5564 df-rn 5565 df-res 5566 df-ima 5567 df-iota 6313 df-fun 6356 df-fn 6357 df-f 6358 df-f1 6359 df-fo 6360 df-f1o 6361 df-fv 6362 df-mea 42731 |
This theorem is referenced by: meaxrcl 42742 meassle 42744 meaiunlelem 42749 meage0 42756 voncl 42947 |
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