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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 32839 | . . . 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 497 | . . 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 32840 | . . . 4 β’ (π β βͺ ran measures β dom π β βͺ ran sigAlgebra) | |
4 | ismeas 32838 | . . . 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 6879 | . 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 397 β§ w3a 1088 = wceq 1542 β wcel 2107 βwral 3065 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-sbc 3745 df-csb 3861 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-res 5650 df-ima 5651 df-iota 6453 df-fun 6503 df-fn 6504 df-f 6505 df-fv 6509 df-ov 7365 df-esum 32667 df-meas 32835 |
This theorem is referenced by: truae 32882 aean 32883 sibfinima 32979 sibfof 32980 domprobmeas 33050 probmeasd 33063 probfinmeasb 33068 probfinmeasbALTV 33069 probmeasb 33070 dstrvprob 33111 |
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