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Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > vonhoi | Structured version Visualization version GIF version |
Description: The Lebesgue outer measure of a multidimensional half-open interval is its dimensional volume (the product of its length in each dimension, when the dimension is nonzero). A direct consequence of Proposition 115D (b) of [Fremlin1] p. 30. (Contributed by Glauco Siliprandi, 8-Apr-2021.) |
Ref | Expression |
---|---|
vonhoi.x | β’ (π β π β Fin) |
vonhoi.a | β’ (π β π΄:πβΆβ) |
vonhoi.b | β’ (π β π΅:πβΆβ) |
vonhoi.c | β’ πΌ = Xπ β π ((π΄βπ)[,)(π΅βπ)) |
vonhoi.l | β’ πΏ = (π₯ β Fin β¦ (π β (β βm π₯), π β (β βm π₯) β¦ if(π₯ = β , 0, βπ β π₯ (volβ((πβπ)[,)(πβπ)))))) |
Ref | Expression |
---|---|
vonhoi | β’ (π β ((volnβπ)βπΌ) = (π΄(πΏβπ)π΅)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | vonhoi.x | . . 3 β’ (π β π β Fin) | |
2 | vonhoi.c | . . . 4 β’ πΌ = Xπ β π ((π΄βπ)[,)(π΅βπ)) | |
3 | eqid 2728 | . . . . 5 β’ dom (volnβπ) = dom (volnβπ) | |
4 | vonhoi.a | . . . . 5 β’ (π β π΄:πβΆβ) | |
5 | vonhoi.b | . . . . 5 β’ (π β π΅:πβΆβ) | |
6 | 1, 3, 4, 5 | hoimbl 46010 | . . . 4 β’ (π β Xπ β π ((π΄βπ)[,)(π΅βπ)) β dom (volnβπ)) |
7 | 2, 6 | eqeltrid 2833 | . . 3 β’ (π β πΌ β dom (volnβπ)) |
8 | 1, 7 | mblvon 46018 | . 2 β’ (π β ((volnβπ)βπΌ) = ((voln*βπ)βπΌ)) |
9 | vonhoi.l | . . 3 β’ πΏ = (π₯ β Fin β¦ (π β (β βm π₯), π β (β βm π₯) β¦ if(π₯ = β , 0, βπ β π₯ (volβ((πβπ)[,)(πβπ)))))) | |
10 | 1, 4, 5, 2, 9 | ovnhoi 45982 | . 2 β’ (π β ((voln*βπ)βπΌ) = (π΄(πΏβπ)π΅)) |
11 | 8, 10 | eqtrd 2768 | 1 β’ (π β ((volnβπ)βπΌ) = (π΄(πΏβπ)π΅)) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 = wceq 1534 β wcel 2099 β c0 4319 ifcif 4525 β¦ cmpt 5226 dom cdm 5673 βΆwf 6539 βcfv 6543 (class class class)co 7415 β cmpo 7417 βm cmap 8839 Xcixp 8910 Fincfn 8958 βcr 11132 0cc0 11133 [,)cico 13353 βcprod 15876 volcvol 25386 voln*covoln 45915 volncvoln 45917 |
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-rep 5280 ax-sep 5294 ax-nul 5301 ax-pow 5360 ax-pr 5424 ax-un 7735 ax-inf2 9659 ax-cc 10453 ax-ac2 10481 ax-cnex 11189 ax-resscn 11190 ax-1cn 11191 ax-icn 11192 ax-addcl 11193 ax-addrcl 11194 ax-mulcl 11195 ax-mulrcl 11196 ax-mulcom 11197 ax-addass 11198 ax-mulass 11199 ax-distr 11200 ax-i2m1 11201 ax-1ne0 11202 ax-1rid 11203 ax-rnegex 11204 ax-rrecex 11205 ax-cnre 11206 ax-pre-lttri 11207 ax-pre-lttrn 11208 ax-pre-ltadd 11209 ax-pre-mulgt0 11210 ax-pre-sup 11211 ax-addf 11212 ax-mulf 11213 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 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 2937 df-nel 3043 df-ral 3058 df-rex 3067 df-rmo 3372 df-reu 3373 df-rab 3429 df-v 3472 df-sbc 3776 df-csb 3891 df-dif 3948 df-un 3950 df-in 3952 df-ss 3962 df-pss 3964 df-nul 4320 df-if 4526 df-pw 4601 df-sn 4626 df-pr 4628 df-tp 4630 df-op 4632 df-uni 4905 df-int 4946 df-iun 4994 df-iin 4995 df-disj 5109 df-br 5144 df-opab 5206 df-mpt 5227 df-tr 5261 df-id 5571 df-eprel 5577 df-po 5585 df-so 5586 df-fr 5628 df-se 5629 df-we 5630 df-xp 5679 df-rel 5680 df-cnv 5681 df-co 5682 df-dm 5683 df-rn 5684 df-res 5685 df-ima 5686 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-isom 6552 df-riota 7371 df-ov 7418 df-oprab 7419 df-mpo 7420 df-of 7680 df-om 7866 df-1st 7988 df-2nd 7989 df-tpos 8226 df-frecs 8281 df-wrecs 8312 df-recs 8386 df-rdg 8425 df-1o 8481 df-2o 8482 df-oadd 8485 df-omul 8486 df-er 8719 df-map 8841 df-pm 8842 df-ixp 8911 df-en 8959 df-dom 8960 df-sdom 8961 df-fin 8962 df-fi 9429 df-sup 9460 df-inf 9461 df-oi 9528 df-dju 9919 df-card 9957 df-acn 9960 df-ac 10134 df-pnf 11275 df-mnf 11276 df-xr 11277 df-ltxr 11278 df-le 11279 df-sub 11471 df-neg 11472 df-div 11897 df-nn 12238 df-2 12300 df-3 12301 df-4 12302 df-5 12303 df-6 12304 df-7 12305 df-8 12306 df-9 12307 df-n0 12498 df-z 12584 df-dec 12703 df-uz 12848 df-q 12958 df-rp 13002 df-xneg 13119 df-xadd 13120 df-xmul 13121 df-ioo 13355 df-ico 13357 df-icc 13358 df-fz 13512 df-fzo 13655 df-fl 13784 df-seq 13994 df-exp 14054 df-hash 14317 df-cj 15073 df-re 15074 df-im 15075 df-sqrt 15209 df-abs 15210 df-clim 15459 df-rlim 15460 df-sum 15660 df-prod 15877 df-struct 17110 df-sets 17127 df-slot 17145 df-ndx 17157 df-base 17175 df-ress 17204 df-plusg 17240 df-mulr 17241 df-starv 17242 df-tset 17246 df-ple 17247 df-ds 17249 df-unif 17250 df-rest 17398 df-0g 17417 df-topgen 17419 df-mgm 18594 df-sgrp 18673 df-mnd 18689 df-grp 18887 df-minusg 18888 df-subg 19072 df-cmn 19731 df-abl 19732 df-mgp 20069 df-rng 20087 df-ur 20116 df-ring 20169 df-cring 20170 df-oppr 20267 df-dvdsr 20290 df-unit 20291 df-invr 20321 df-dvr 20334 df-drng 20620 df-psmet 21265 df-xmet 21266 df-met 21267 df-bl 21268 df-mopn 21269 df-cnfld 21274 df-top 22790 df-topon 22807 df-bases 22843 df-cmp 23285 df-ovol 25387 df-vol 25388 df-salg 45688 df-sumge0 45742 df-mea 45829 df-ome 45869 df-caragen 45871 df-ovoln 45916 df-voln 45918 |
This theorem is referenced by: vonn0hoi 46049 |
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