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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 44589 | . 2 ⊢ (𝜑 → 𝑀:𝑆⟶(0[,]+∞)) |
4 | meacl.3 | . 2 ⊢ (𝜑 → 𝐴 ∈ 𝑆) | |
5 | 3, 4 | ffvelcdmd 7033 | 1 ⊢ (𝜑 → (𝑀‘𝐴) ∈ (0[,]+∞)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1542 ∈ wcel 2107 dom cdm 5632 ‘cfv 6494 (class class class)co 7352 0cc0 11010 +∞cpnf 11145 [,]cicc 13222 Meascmea 44585 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2709 ax-rep 5241 ax-sep 5255 ax-nul 5262 ax-pr 5383 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2888 df-ne 2943 df-ral 3064 df-rex 3073 df-reu 3353 df-rab 3407 df-v 3446 df-sbc 3739 df-csb 3855 df-dif 3912 df-un 3914 df-in 3916 df-ss 3926 df-nul 4282 df-if 4486 df-pw 4561 df-sn 4586 df-pr 4588 df-op 4592 df-uni 4865 df-iun 4955 df-br 5105 df-opab 5167 df-mpt 5188 df-id 5530 df-xp 5638 df-rel 5639 df-cnv 5640 df-co 5641 df-dm 5642 df-rn 5643 df-res 5644 df-ima 5645 df-iota 6446 df-fun 6496 df-fn 6497 df-f 6498 df-f1 6499 df-fo 6500 df-f1o 6501 df-fv 6502 df-mea 44586 |
This theorem is referenced by: meaxrcl 44597 meassle 44599 meaiunlelem 44604 meage0 44611 voncl 44802 |
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