Users' Mathboxes Mathbox for Glauco Siliprandi < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  vonval Structured version   Visualization version   GIF version

Theorem vonval 45928
Description: Value of the Lebesgue measure for a given finite dimension. (Contributed by Glauco Siliprandi, 11-Oct-2020.)
Hypothesis
Ref Expression
vonval.1 (πœ‘ β†’ 𝑋 ∈ Fin)
Assertion
Ref Expression
vonval (πœ‘ β†’ (volnβ€˜π‘‹) = ((voln*β€˜π‘‹) β†Ύ (CaraGenβ€˜(voln*β€˜π‘‹))))

Proof of Theorem vonval
Dummy variable π‘₯ is distinct from all other variables.
StepHypRef Expression
1 df-voln 45927 . 2 voln = (π‘₯ ∈ Fin ↦ ((voln*β€˜π‘₯) β†Ύ (CaraGenβ€˜(voln*β€˜π‘₯))))
2 fveq2 6897 . . 3 (π‘₯ = 𝑋 β†’ (voln*β€˜π‘₯) = (voln*β€˜π‘‹))
3 2fveq3 6902 . . 3 (π‘₯ = 𝑋 β†’ (CaraGenβ€˜(voln*β€˜π‘₯)) = (CaraGenβ€˜(voln*β€˜π‘‹)))
42, 3reseq12d 5986 . 2 (π‘₯ = 𝑋 β†’ ((voln*β€˜π‘₯) β†Ύ (CaraGenβ€˜(voln*β€˜π‘₯))) = ((voln*β€˜π‘‹) β†Ύ (CaraGenβ€˜(voln*β€˜π‘‹))))
5 vonval.1 . 2 (πœ‘ β†’ 𝑋 ∈ Fin)
6 fvex 6910 . . . 4 (voln*β€˜π‘‹) ∈ V
76resex 6033 . . 3 ((voln*β€˜π‘‹) β†Ύ (CaraGenβ€˜(voln*β€˜π‘‹))) ∈ V
87a1i 11 . 2 (πœ‘ β†’ ((voln*β€˜π‘‹) β†Ύ (CaraGenβ€˜(voln*β€˜π‘‹))) ∈ V)
91, 4, 5, 8fvmptd3 7028 1 (πœ‘ β†’ (volnβ€˜π‘‹) = ((voln*β€˜π‘‹) β†Ύ (CaraGenβ€˜(voln*β€˜π‘‹))))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   = wceq 1534   ∈ wcel 2099  Vcvv 3471   β†Ύ cres 5680  β€˜cfv 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
  Copyright terms: Public domain W3C validator