Mathbox for Glauco Siliprandi |
< Previous
Next >
Nearby theorems |
||
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 42828 | . 2 ⊢ voln = (𝑥 ∈ Fin ↦ ((voln*‘𝑥) ↾ (CaraGen‘(voln*‘𝑥)))) | |
2 | fveq2 6672 | . . 3 ⊢ (𝑥 = 𝑋 → (voln*‘𝑥) = (voln*‘𝑋)) | |
3 | 2fveq3 6677 | . . 3 ⊢ (𝑥 = 𝑋 → (CaraGen‘(voln*‘𝑥)) = (CaraGen‘(voln*‘𝑋))) | |
4 | 2, 3 | reseq12d 5856 | . 2 ⊢ (𝑥 = 𝑋 → ((voln*‘𝑥) ↾ (CaraGen‘(voln*‘𝑥))) = ((voln*‘𝑋) ↾ (CaraGen‘(voln*‘𝑋)))) |
5 | vonval.1 | . 2 ⊢ (𝜑 → 𝑋 ∈ Fin) | |
6 | fvex 6685 | . . . 4 ⊢ (voln*‘𝑋) ∈ V | |
7 | 6 | resex 5901 | . . 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 6793 | 1 ⊢ (𝜑 → (voln‘𝑋) = ((voln*‘𝑋) ↾ (CaraGen‘(voln*‘𝑋)))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 ∈ wcel 2114 Vcvv 3496 ↾ cres 5559 ‘cfv 6357 Fincfn 8511 CaraGenccaragen 42780 voln*covoln 42825 volncvoln 42827 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-sep 5205 ax-nul 5212 ax-pr 5332 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ral 3145 df-rex 3146 df-rab 3149 df-v 3498 df-sbc 3775 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-sn 4570 df-pr 4572 df-op 4576 df-uni 4841 df-br 5069 df-opab 5131 df-mpt 5149 df-id 5462 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-res 5569 df-iota 6316 df-fun 6359 df-fv 6365 df-voln 42828 |
This theorem is referenced by: vonmea 42863 dmvon 42895 voncmpl 42910 mblvon 42928 |
Copyright terms: Public domain | W3C validator |