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Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > vonval | Structured version Visualization version GIF version |
Description: Value of the Lebesgue measure for a given finite dimension. (Contributed by Glauco Siliprandi, 11-Oct-2020.) |
Ref | Expression |
---|---|
vonval.1 | β’ (π β π β Fin) |
Ref | Expression |
---|---|
vonval | β’ (π β (volnβπ) = ((voln*βπ) βΎ (CaraGenβ(voln*βπ)))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-voln 45808 | . 2 β’ voln = (π₯ β Fin β¦ ((voln*βπ₯) βΎ (CaraGenβ(voln*βπ₯)))) | |
2 | fveq2 6884 | . . 3 β’ (π₯ = π β (voln*βπ₯) = (voln*βπ)) | |
3 | 2fveq3 6889 | . . 3 β’ (π₯ = π β (CaraGenβ(voln*βπ₯)) = (CaraGenβ(voln*βπ))) | |
4 | 2, 3 | reseq12d 5975 | . 2 β’ (π₯ = π β ((voln*βπ₯) βΎ (CaraGenβ(voln*βπ₯))) = ((voln*βπ) βΎ (CaraGenβ(voln*βπ)))) |
5 | vonval.1 | . 2 β’ (π β π β Fin) | |
6 | fvex 6897 | . . . 4 β’ (voln*βπ) β V | |
7 | 6 | resex 6022 | . . 3 β’ ((voln*βπ) βΎ (CaraGenβ(voln*βπ))) β V |
8 | 7 | a1i 11 | . 2 β’ (π β ((voln*βπ) βΎ (CaraGenβ(voln*βπ))) β V) |
9 | 1, 4, 5, 8 | fvmptd3 7014 | 1 β’ (π β (volnβπ) = ((voln*βπ) βΎ (CaraGenβ(voln*βπ)))) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 = wceq 1533 β wcel 2098 Vcvv 3468 βΎ cres 5671 βcfv 6536 Fincfn 8938 CaraGenccaragen 45760 voln*covoln 45805 volncvoln 45807 |
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 2163 ax-ext 2697 ax-sep 5292 ax-nul 5299 ax-pr 5420 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-ral 3056 df-rex 3065 df-rab 3427 df-v 3470 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-nul 4318 df-if 4524 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-br 5142 df-opab 5204 df-mpt 5225 df-id 5567 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-res 5681 df-iota 6488 df-fun 6538 df-fv 6544 df-voln 45808 |
This theorem is referenced by: vonmea 45843 dmvon 45875 voncmpl 45890 mblvon 45908 |
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