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 | ⊢ (𝜑 → 𝑋 ∈ Fin) |
Ref | Expression |
---|---|
vonval | ⊢ (𝜑 → (voln‘𝑋) = ((voln*‘𝑋) ↾ (CaraGen‘(voln*‘𝑋)))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-voln 42698 | . 2 ⊢ voln = (𝑥 ∈ Fin ↦ ((voln*‘𝑥) ↾ (CaraGen‘(voln*‘𝑥)))) | |
2 | fveq2 6663 | . . 3 ⊢ (𝑥 = 𝑋 → (voln*‘𝑥) = (voln*‘𝑋)) | |
3 | 2fveq3 6668 | . . 3 ⊢ (𝑥 = 𝑋 → (CaraGen‘(voln*‘𝑥)) = (CaraGen‘(voln*‘𝑋))) | |
4 | 2, 3 | reseq12d 5847 | . 2 ⊢ (𝑥 = 𝑋 → ((voln*‘𝑥) ↾ (CaraGen‘(voln*‘𝑥))) = ((voln*‘𝑋) ↾ (CaraGen‘(voln*‘𝑋)))) |
5 | vonval.1 | . 2 ⊢ (𝜑 → 𝑋 ∈ Fin) | |
6 | fvex 6676 | . . . 4 ⊢ (voln*‘𝑋) ∈ V | |
7 | 6 | resex 5892 | . . 3 ⊢ ((voln*‘𝑋) ↾ (CaraGen‘(voln*‘𝑋))) ∈ V |
8 | 7 | a1i 11 | . 2 ⊢ (𝜑 → ((voln*‘𝑋) ↾ (CaraGen‘(voln*‘𝑋))) ∈ V) |
9 | 1, 4, 5, 8 | fvmptd3 6783 | 1 ⊢ (𝜑 → (voln‘𝑋) = ((voln*‘𝑋) ↾ (CaraGen‘(voln*‘𝑋)))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1528 ∈ wcel 2105 Vcvv 3492 ↾ cres 5550 ‘cfv 6348 Fincfn 8497 CaraGenccaragen 42650 voln*covoln 42695 volncvoln 42697 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pr 5320 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ral 3140 df-rex 3141 df-rab 3144 df-v 3494 df-sbc 3770 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-sn 4558 df-pr 4560 df-op 4564 df-uni 4831 df-br 5058 df-opab 5120 df-mpt 5138 df-id 5453 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-res 5560 df-iota 6307 df-fun 6350 df-fv 6356 df-voln 42698 |
This theorem is referenced by: vonmea 42733 dmvon 42765 voncmpl 42780 mblvon 42798 |
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