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Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > measbasedom | Structured version Visualization version GIF version |
Description: The base set of a measure is its domain. (Contributed by Thierry Arnoux, 25-Dec-2016.) |
Ref | Expression |
---|---|
measbasedom | β’ (π β βͺ ran measures β π β (measuresβdom π)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | isrnmeas 33742 | . . . 4 β’ (π β βͺ ran measures β (dom π β βͺ ran sigAlgebra β§ (π:dom πβΆ(0[,]+β) β§ (πββ ) = 0 β§ βπ₯ β π« dom π((π₯ βΌ Ο β§ Disj π¦ β π₯ π¦) β (πββͺ π₯) = Ξ£*π¦ β π₯(πβπ¦))))) | |
2 | 1 | simprd 495 | . . 3 β’ (π β βͺ ran measures β (π:dom πβΆ(0[,]+β) β§ (πββ ) = 0 β§ βπ₯ β π« dom π((π₯ βΌ Ο β§ Disj π¦ β π₯ π¦) β (πββͺ π₯) = Ξ£*π¦ β π₯(πβπ¦)))) |
3 | dmmeas 33743 | . . . 4 β’ (π β βͺ ran measures β dom π β βͺ ran sigAlgebra) | |
4 | ismeas 33741 | . . . 4 β’ (dom π β βͺ ran sigAlgebra β (π β (measuresβdom π) β (π:dom πβΆ(0[,]+β) β§ (πββ ) = 0 β§ βπ₯ β π« dom π((π₯ βΌ Ο β§ Disj π¦ β π₯ π¦) β (πββͺ π₯) = Ξ£*π¦ β π₯(πβπ¦))))) | |
5 | 3, 4 | syl 17 | . . 3 β’ (π β βͺ ran measures β (π β (measuresβdom π) β (π:dom πβΆ(0[,]+β) β§ (πββ ) = 0 β§ βπ₯ β π« dom π((π₯ βΌ Ο β§ Disj π¦ β π₯ π¦) β (πββͺ π₯) = Ξ£*π¦ β π₯(πβπ¦))))) |
6 | 2, 5 | mpbird 257 | . 2 β’ (π β βͺ ran measures β π β (measuresβdom π)) |
7 | elfvunirn 6923 | . 2 β’ (π β (measuresβdom π) β π β βͺ ran measures) | |
8 | 6, 7 | impbii 208 | 1 β’ (π β βͺ ran measures β π β (measuresβdom π)) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β wb 205 β§ wa 395 β§ w3a 1085 = wceq 1534 β wcel 2099 βwral 3056 β c0 4318 π« cpw 4598 βͺ cuni 4903 Disj wdisj 5107 class class class wbr 5142 dom cdm 5672 ran crn 5673 βΆwf 6538 βcfv 6542 (class class class)co 7414 Οcom 7862 βΌ cdom 8951 0cc0 11124 +βcpnf 11261 [,]cicc 13345 Ξ£*cesum 33569 sigAlgebracsiga 33650 measurescmeas 33737 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2164 ax-ext 2698 ax-sep 5293 ax-nul 5300 ax-pow 5359 ax-pr 5423 ax-un 7732 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2705 df-cleq 2719 df-clel 2805 df-nfc 2880 df-ne 2936 df-ral 3057 df-rex 3066 df-rab 3428 df-v 3471 df-sbc 3775 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-br 5143 df-opab 5205 df-mpt 5226 df-id 5570 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-fv 6550 df-ov 7417 df-esum 33570 df-meas 33738 |
This theorem is referenced by: truae 33785 aean 33786 sibfinima 33882 sibfof 33883 domprobmeas 33953 probmeasd 33966 probfinmeasb 33971 probfinmeasbALTV 33972 probmeasb 33973 dstrvprob 34014 |
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