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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 44854 | . 2 β’ voln = (π₯ β Fin β¦ ((voln*βπ₯) βΎ (CaraGenβ(voln*βπ₯)))) | |
2 | fveq2 6847 | . . 3 β’ (π₯ = π β (voln*βπ₯) = (voln*βπ)) | |
3 | 2fveq3 6852 | . . 3 β’ (π₯ = π β (CaraGenβ(voln*βπ₯)) = (CaraGenβ(voln*βπ))) | |
4 | 2, 3 | reseq12d 5943 | . 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 6860 | . . . 4 β’ (voln*βπ) β V | |
7 | 6 | resex 5990 | . . 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 6976 | 1 β’ (π β (volnβπ) = ((voln*βπ) βΎ (CaraGenβ(voln*βπ)))) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 = wceq 1542 β wcel 2107 Vcvv 3448 βΎ cres 5640 βcfv 6501 Fincfn 8890 CaraGenccaragen 44806 voln*covoln 44851 volncvoln 44853 |
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-pr 5389 |
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-dif 3918 df-un 3920 df-in 3922 df-ss 3932 df-nul 4288 df-if 4492 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-res 5650 df-iota 6453 df-fun 6503 df-fv 6509 df-voln 44854 |
This theorem is referenced by: vonmea 44889 dmvon 44921 voncmpl 44936 mblvon 44954 |
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