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Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > measbase | Structured version Visualization version GIF version |
Description: The base set of a measure is a sigma-algebra. (Contributed by Thierry Arnoux, 25-Dec-2016.) |
Ref | Expression |
---|---|
measbase | β’ (π β (measuresβπ) β π β βͺ ran sigAlgebra) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elfvdm 6884 | . 2 β’ (π β (measuresβπ) β π β dom measures) | |
2 | vex 3452 | . . . . 5 β’ π β V | |
3 | ovex 7395 | . . . . 5 β’ (0[,]+β) β V | |
4 | mapex 8778 | . . . . 5 β’ ((π β V β§ (0[,]+β) β V) β {π β£ π:π βΆ(0[,]+β)} β V) | |
5 | 2, 3, 4 | mp2an 691 | . . . 4 β’ {π β£ π:π βΆ(0[,]+β)} β V |
6 | simp1 1137 | . . . . 5 β’ ((π:π βΆ(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¦))) β π:π βΆ(0[,]+β)) | |
7 | 6 | ss2abi 4028 | . . . 4 β’ {π β£ (π:π βΆ(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¦)))} β {π β£ π:π βΆ(0[,]+β)} |
8 | 5, 7 | ssexi 5284 | . . 3 β’ {π β£ (π:π βΆ(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¦)))} β V |
9 | df-meas 32835 | . . 3 β’ measures = (π β βͺ ran sigAlgebra β¦ {π β£ (π:π βΆ(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¦)))}) | |
10 | 8, 9 | dmmpti 6650 | . 2 β’ dom measures = βͺ ran sigAlgebra |
11 | 1, 10 | eleqtrdi 2848 | 1 β’ (π β (measuresβπ) β π β βͺ ran sigAlgebra) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 397 β§ w3a 1088 = wceq 1542 β wcel 2107 {cab 2714 βwral 3065 Vcvv 3448 β c0 4287 π« cpw 4565 βͺ cuni 4870 Disj wdisj 5075 class class class wbr 5110 dom cdm 5638 ran crn 5639 βΆwf 6497 βcfv 6501 (class class class)co 7362 Οcom 7807 βΌ cdom 8888 0cc0 11058 +βcpnf 11193 [,]cicc 13274 Ξ£*cesum 32666 sigAlgebracsiga 32747 measurescmeas 32834 |
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 2708 ax-sep 5261 ax-nul 5268 ax-pow 5325 ax-pr 5389 ax-un 7677 |
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 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2890 df-ne 2945 df-ral 3066 df-rex 3075 df-rab 3411 df-v 3450 df-dif 3918 df-un 3920 df-in 3922 df-ss 3932 df-nul 4288 df-if 4492 df-pw 4567 df-sn 4592 df-pr 4594 df-op 4598 df-uni 4871 df-br 5111 df-opab 5173 df-mpt 5194 df-id 5536 df-xp 5644 df-rel 5645 df-cnv 5646 df-co 5647 df-dm 5648 df-rn 5649 df-iota 6453 df-fun 6503 df-fn 6504 df-f 6505 df-fv 6509 df-ov 7365 df-meas 32835 |
This theorem is referenced by: measfrge0 32842 measvnul 32845 measvun 32848 measxun2 32849 measun 32850 measvuni 32853 measssd 32854 measunl 32855 measiuns 32856 measiun 32857 meascnbl 32858 measinblem 32859 measinb 32860 measinb2 32862 measdivcst 32863 measdivcstALTV 32864 aean 32883 domprobsiga 33051 prob01 33053 probfinmeasb 33068 probfinmeasbALTV 33069 probmeasb 33070 |
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