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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 | ⊢ (𝜑 → 𝑋 ∈ Fin) |
Ref | Expression |
---|---|
vonval | ⊢ (𝜑 → (voln‘𝑋) = ((voln*‘𝑋) ↾ (CaraGen‘(voln*‘𝑋)))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-voln 45740 | . 2 ⊢ voln = (𝑥 ∈ Fin ↦ ((voln*‘𝑥) ↾ (CaraGen‘(voln*‘𝑥)))) | |
2 | fveq2 6881 | . . 3 ⊢ (𝑥 = 𝑋 → (voln*‘𝑥) = (voln*‘𝑋)) | |
3 | 2fveq3 6886 | . . 3 ⊢ (𝑥 = 𝑋 → (CaraGen‘(voln*‘𝑥)) = (CaraGen‘(voln*‘𝑋))) | |
4 | 2, 3 | reseq12d 5972 | . 2 ⊢ (𝑥 = 𝑋 → ((voln*‘𝑥) ↾ (CaraGen‘(voln*‘𝑥))) = ((voln*‘𝑋) ↾ (CaraGen‘(voln*‘𝑋)))) |
5 | vonval.1 | . 2 ⊢ (𝜑 → 𝑋 ∈ Fin) | |
6 | fvex 6894 | . . . 4 ⊢ (voln*‘𝑋) ∈ V | |
7 | 6 | resex 6019 | . . 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 7011 | 1 ⊢ (𝜑 → (voln‘𝑋) = ((voln*‘𝑋) ↾ (CaraGen‘(voln*‘𝑋)))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1533 ∈ wcel 2098 Vcvv 3466 ↾ cres 5668 ‘cfv 6533 Fincfn 8935 CaraGenccaragen 45692 voln*covoln 45737 volncvoln 45739 |
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 2695 ax-sep 5289 ax-nul 5296 ax-pr 5417 |
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 2526 df-eu 2555 df-clab 2702 df-cleq 2716 df-clel 2802 df-nfc 2877 df-ne 2933 df-ral 3054 df-rex 3063 df-rab 3425 df-v 3468 df-dif 3943 df-un 3945 df-in 3947 df-ss 3957 df-nul 4315 df-if 4521 df-sn 4621 df-pr 4623 df-op 4627 df-uni 4900 df-br 5139 df-opab 5201 df-mpt 5222 df-id 5564 df-xp 5672 df-rel 5673 df-cnv 5674 df-co 5675 df-dm 5676 df-res 5678 df-iota 6485 df-fun 6535 df-fv 6541 df-voln 45740 |
This theorem is referenced by: vonmea 45775 dmvon 45807 voncmpl 45822 mblvon 45840 |
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