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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 46408 | . 2 ⊢ (𝜑 → 𝑀:𝑆⟶(0[,]+∞)) |
4 | meacl.3 | . 2 ⊢ (𝜑 → 𝐴 ∈ 𝑆) | |
5 | 3, 4 | ffvelcdmd 7104 | 1 ⊢ (𝜑 → (𝑀‘𝐴) ∈ (0[,]+∞)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1536 ∈ wcel 2105 dom cdm 5688 ‘cfv 6562 (class class class)co 7430 0cc0 11152 +∞cpnf 11289 [,]cicc 13386 Meascmea 46404 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-10 2138 ax-11 2154 ax-12 2174 ax-ext 2705 ax-rep 5284 ax-sep 5301 ax-nul 5311 ax-pr 5437 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1539 df-fal 1549 df-ex 1776 df-nf 1780 df-sb 2062 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2726 df-clel 2813 df-nfc 2889 df-ne 2938 df-ral 3059 df-rex 3068 df-reu 3378 df-rab 3433 df-v 3479 df-sbc 3791 df-csb 3908 df-dif 3965 df-un 3967 df-in 3969 df-ss 3979 df-nul 4339 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4912 df-iun 4997 df-br 5148 df-opab 5210 df-mpt 5231 df-id 5582 df-xp 5694 df-rel 5695 df-cnv 5696 df-co 5697 df-dm 5698 df-rn 5699 df-res 5700 df-ima 5701 df-iota 6515 df-fun 6564 df-fn 6565 df-f 6566 df-f1 6567 df-fo 6568 df-f1o 6569 df-fv 6570 df-mea 46405 |
This theorem is referenced by: meaxrcl 46416 meassle 46418 meaiunlelem 46423 meage0 46430 voncl 46621 |
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