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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 45927 | . 2 β’ voln = (π₯ β Fin β¦ ((voln*βπ₯) βΎ (CaraGenβ(voln*βπ₯)))) | |
2 | fveq2 6897 | . . 3 β’ (π₯ = π β (voln*βπ₯) = (voln*βπ)) | |
3 | 2fveq3 6902 | . . 3 β’ (π₯ = π β (CaraGenβ(voln*βπ₯)) = (CaraGenβ(voln*βπ))) | |
4 | 2, 3 | reseq12d 5986 | . 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 6910 | . . . 4 β’ (voln*βπ) β V | |
7 | 6 | resex 6033 | . . 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 7028 | 1 β’ (π β (volnβπ) = ((voln*βπ) βΎ (CaraGenβ(voln*βπ)))) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 = wceq 1534 β wcel 2099 Vcvv 3471 βΎ cres 5680 βcfv 6548 Fincfn 8964 CaraGenccaragen 45879 voln*covoln 45924 volncvoln 45926 |
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 2167 ax-ext 2699 ax-sep 5299 ax-nul 5306 ax-pr 5429 |
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 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-ral 3059 df-rex 3068 df-rab 3430 df-v 3473 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4324 df-if 4530 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4909 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5576 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-res 5690 df-iota 6500 df-fun 6550 df-fv 6556 df-voln 45927 |
This theorem is referenced by: vonmea 45962 dmvon 45994 voncmpl 46009 mblvon 46027 |
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