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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 33849 | . . . 4 β’ (π β (measuresβπ) β π β βͺ ran sigAlgebra) | |
2 | ismeas 33851 | . . . 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 266 | . 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 394 β§ w3a 1084 = wceq 1533 β wcel 2098 βwral 3058 β c0 4326 π« cpw 4606 βͺ cuni 4912 Disj wdisj 5117 class class class wbr 5152 ran crn 5683 βΆwf 6549 βcfv 6553 (class class class)co 7426 Οcom 7876 βΌ cdom 8968 0cc0 11146 +βcpnf 11283 [,]cicc 13367 Ξ£*cesum 33679 sigAlgebracsiga 33760 measurescmeas 33847 |
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 2166 ax-ext 2699 ax-sep 5303 ax-nul 5310 ax-pow 5369 ax-pr 5433 ax-un 7746 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-ral 3059 df-rex 3068 df-rab 3431 df-v 3475 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4327 df-if 4533 df-pw 4608 df-sn 4633 df-pr 4635 df-op 4639 df-uni 4913 df-br 5153 df-opab 5215 df-mpt 5236 df-id 5580 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-iota 6505 df-fun 6555 df-fn 6556 df-f 6557 df-fv 6561 df-ov 7429 df-esum 33680 df-meas 33848 |
This theorem is referenced by: measxun2 33862 measvunilem0 33865 measssd 33867 measinb 33873 measres 33874 measdivcst 33876 measdivcstALTV 33877 truae 33895 probnul 34067 |
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