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Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > measvnul | Structured version Visualization version GIF version |
Description: The measure of the empty set is always zero. (Contributed by Thierry Arnoux, 26-Dec-2016.) |
Ref | Expression |
---|---|
measvnul | β’ (π β (measuresβπ) β (πββ ) = 0) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | measbase 33725 | . . . 4 β’ (π β (measuresβπ) β π β βͺ ran sigAlgebra) | |
2 | ismeas 33727 | . . . 4 β’ (π β βͺ ran sigAlgebra β (π β (measuresβπ) β (π:πβΆ(0[,]+β) β§ (πββ ) = 0 β§ βπ¦ β π« π((π¦ βΌ Ο β§ Disj π₯ β π¦ π₯) β (πββͺ π¦) = Ξ£*π₯ β π¦(πβπ₯))))) | |
3 | 1, 2 | syl 17 | . . 3 β’ (π β (measuresβπ) β (π β (measuresβπ) β (π:πβΆ(0[,]+β) β§ (πββ ) = 0 β§ βπ¦ β π« π((π¦ βΌ Ο β§ Disj π₯ β π¦ π₯) β (πββͺ π¦) = Ξ£*π₯ β π¦(πβπ₯))))) |
4 | 3 | ibi 267 | . 2 β’ (π β (measuresβπ) β (π:πβΆ(0[,]+β) β§ (πββ ) = 0 β§ βπ¦ β π« π((π¦ βΌ Ο β§ Disj π₯ β π¦ π₯) β (πββͺ π¦) = Ξ£*π₯ β π¦(πβπ₯)))) |
5 | 4 | simp2d 1140 | 1 β’ (π β (measuresβπ) β (πββ ) = 0) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β wb 205 β§ wa 395 β§ w3a 1084 = wceq 1533 β wcel 2098 βwral 3055 β c0 4317 π« cpw 4597 βͺ cuni 4902 Disj wdisj 5106 class class class wbr 5141 ran crn 5670 βΆwf 6533 βcfv 6537 (class class class)co 7405 Οcom 7852 βΌ cdom 8939 0cc0 11112 +βcpnf 11249 [,]cicc 13333 Ξ£*cesum 33555 sigAlgebracsiga 33636 measurescmeas 33723 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2697 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7722 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-ral 3056 df-rex 3065 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-br 5142 df-opab 5204 df-mpt 5225 df-id 5567 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-iota 6489 df-fun 6539 df-fn 6540 df-f 6541 df-fv 6545 df-ov 7408 df-esum 33556 df-meas 33724 |
This theorem is referenced by: measxun2 33738 measvunilem0 33741 measssd 33743 measinb 33749 measres 33750 measdivcst 33752 measdivcstALTV 33753 truae 33771 probnul 33943 |
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