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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 6928 | . 2 β’ (π β (measuresβπ) β π β dom measures) | |
2 | vex 3478 | . . . . 5 β’ π β V | |
3 | ovex 7444 | . . . . 5 β’ (0[,]+β) β V | |
4 | mapex 8828 | . . . . 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 690 | . . . 4 β’ {π β£ π:π βΆ(0[,]+β)} β V |
6 | simp1 1136 | . . . . 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 4063 | . . . 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 5322 | . . 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 33263 | . . 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 6694 | . 2 β’ dom measures = βͺ ran sigAlgebra |
11 | 1, 10 | eleqtrdi 2843 | 1 β’ (π β (measuresβπ) β π β βͺ ran sigAlgebra) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 396 β§ w3a 1087 = wceq 1541 β wcel 2106 {cab 2709 βwral 3061 Vcvv 3474 β c0 4322 π« cpw 4602 βͺ cuni 4908 Disj wdisj 5113 class class class wbr 5148 dom cdm 5676 ran crn 5677 βΆwf 6539 βcfv 6543 (class class class)co 7411 Οcom 7857 βΌ cdom 8939 0cc0 11112 +βcpnf 11247 [,]cicc 13329 Ξ£*cesum 33094 sigAlgebracsiga 33175 measurescmeas 33262 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7727 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3433 df-v 3476 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5574 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-fv 6551 df-ov 7414 df-meas 33263 |
This theorem is referenced by: measfrge0 33270 measvnul 33273 measvun 33276 measxun2 33277 measun 33278 measvuni 33281 measssd 33282 measunl 33283 measiuns 33284 measiun 33285 meascnbl 33286 measinblem 33287 measinb 33288 measinb2 33290 measdivcst 33291 measdivcstALTV 33292 aean 33311 domprobsiga 33479 prob01 33481 probfinmeasb 33496 probfinmeasbALTV 33497 probmeasb 33498 |
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