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Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > vonn0icc | Structured version Visualization version GIF version |
Description: The n-dimensional Lebesgue measure of a closed interval, when the dimension of the space is nonzero. This is the second statement in Proposition 115G (d) of [Fremlin1] p. 32. (Contributed by Glauco Siliprandi, 8-Apr-2021.) |
Ref | Expression |
---|---|
vonn0icc.x | β’ (π β π β Fin) |
vonn0icc.n | β’ (π β π β β ) |
vonn0icc.a | β’ (π β π΄:πβΆβ) |
vonn0icc.b | β’ (π β π΅:πβΆβ) |
vonn0icc.i | β’ πΌ = Xπ β π ((π΄βπ)[,](π΅βπ)) |
Ref | Expression |
---|---|
vonn0icc | β’ (π β ((volnβπ)βπΌ) = βπ β π (volβ((π΄βπ)[,](π΅βπ)))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | vonn0icc.x | . . . 4 β’ (π β π β Fin) | |
2 | vonn0icc.a | . . . 4 β’ (π β π΄:πβΆβ) | |
3 | vonn0icc.b | . . . 4 β’ (π β π΅:πβΆβ) | |
4 | vonn0icc.i | . . . 4 β’ πΌ = Xπ β π ((π΄βπ)[,](π΅βπ)) | |
5 | fveq2 6885 | . . . . . . . . . . 11 β’ (π = π β (πβπ) = (πβπ)) | |
6 | fveq2 6885 | . . . . . . . . . . 11 β’ (π = π β (πβπ) = (πβπ)) | |
7 | 5, 6 | oveq12d 7423 | . . . . . . . . . 10 β’ (π = π β ((πβπ)[,)(πβπ)) = ((πβπ)[,)(πβπ))) |
8 | 7 | fveq2d 6889 | . . . . . . . . 9 β’ (π = π β (volβ((πβπ)[,)(πβπ))) = (volβ((πβπ)[,)(πβπ)))) |
9 | 8 | cbvprodv 15866 | . . . . . . . 8 β’ βπ β π₯ (volβ((πβπ)[,)(πβπ))) = βπ β π₯ (volβ((πβπ)[,)(πβπ))) |
10 | ifeq2 4528 | . . . . . . . 8 β’ (βπ β π₯ (volβ((πβπ)[,)(πβπ))) = βπ β π₯ (volβ((πβπ)[,)(πβπ))) β if(π₯ = β , 0, βπ β π₯ (volβ((πβπ)[,)(πβπ)))) = if(π₯ = β , 0, βπ β π₯ (volβ((πβπ)[,)(πβπ))))) | |
11 | 9, 10 | ax-mp 5 | . . . . . . 7 β’ if(π₯ = β , 0, βπ β π₯ (volβ((πβπ)[,)(πβπ)))) = if(π₯ = β , 0, βπ β π₯ (volβ((πβπ)[,)(πβπ)))) |
12 | 11 | a1i 11 | . . . . . 6 β’ ((π β (β βm π₯) β§ π β (β βm π₯)) β if(π₯ = β , 0, βπ β π₯ (volβ((πβπ)[,)(πβπ)))) = if(π₯ = β , 0, βπ β π₯ (volβ((πβπ)[,)(πβπ))))) |
13 | 12 | mpoeq3ia 7483 | . . . . 5 β’ (π β (β βm π₯), π β (β βm π₯) β¦ if(π₯ = β , 0, βπ β π₯ (volβ((πβπ)[,)(πβπ))))) = (π β (β βm π₯), π β (β βm π₯) β¦ if(π₯ = β , 0, βπ β π₯ (volβ((πβπ)[,)(πβπ))))) |
14 | 13 | mpteq2i 5246 | . . . 4 β’ (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)))))) = (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)))))) |
15 | 1, 2, 3, 4, 14 | vonicc 45970 | . . 3 β’ (π β ((volnβπ)βπΌ) = (π΄((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))))))βπ)π΅)) |
16 | 14 | fveq1i 6886 | . . . . 5 β’ ((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))))))βπ) = ((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))))))βπ) |
17 | 16 | oveqi 7418 | . . . 4 β’ (π΄((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))))))βπ)π΅) = (π΄((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))))))βπ)π΅) |
18 | 17 | a1i 11 | . . 3 β’ (π β (π΄((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))))))βπ)π΅) = (π΄((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))))))βπ)π΅)) |
19 | 15, 18 | eqtrd 2766 | . 2 β’ (π β ((volnβπ)βπΌ) = (π΄((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))))))βπ)π΅)) |
20 | eqid 2726 | . . 3 β’ (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)))))) = (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)))))) | |
21 | vonn0icc.n | . . 3 β’ (π β π β β ) | |
22 | 20, 1, 21, 2, 3 | hoidmvn0val 45869 | . 2 β’ (π β (π΄((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))))))βπ)π΅) = βπ β π (volβ((π΄βπ)[,)(π΅βπ)))) |
23 | 2 | ffvelcdmda 7080 | . . . . 5 β’ ((π β§ π β π) β (π΄βπ) β β) |
24 | 3 | ffvelcdmda 7080 | . . . . 5 β’ ((π β§ π β π) β (π΅βπ) β β) |
25 | 23, 24 | voliccico 45284 | . . . 4 β’ ((π β§ π β π) β (volβ((π΄βπ)[,](π΅βπ))) = (volβ((π΄βπ)[,)(π΅βπ)))) |
26 | 25 | eqcomd 2732 | . . 3 β’ ((π β§ π β π) β (volβ((π΄βπ)[,)(π΅βπ))) = (volβ((π΄βπ)[,](π΅βπ)))) |
27 | 26 | prodeq2dv 15873 | . 2 β’ (π β βπ β π (volβ((π΄βπ)[,)(π΅βπ))) = βπ β π (volβ((π΄βπ)[,](π΅βπ)))) |
28 | 19, 22, 27 | 3eqtrd 2770 | 1 β’ (π β ((volnβπ)βπΌ) = βπ β π (volβ((π΄βπ)[,](π΅βπ)))) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 395 = wceq 1533 β wcel 2098 β wne 2934 β c0 4317 ifcif 4523 β¦ cmpt 5224 βΆwf 6533 βcfv 6537 (class class class)co 7405 β cmpo 7407 βm cmap 8822 Xcixp 8893 Fincfn 8941 βcr 11111 0cc0 11112 [,)cico 13332 [,]cicc 13333 βcprod 15855 volcvol 25347 volncvoln 45823 |
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 2697 ax-rep 5278 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7722 ax-inf2 9638 ax-cc 10432 ax-ac2 10460 ax-cnex 11168 ax-resscn 11169 ax-1cn 11170 ax-icn 11171 ax-addcl 11172 ax-addrcl 11173 ax-mulcl 11174 ax-mulrcl 11175 ax-mulcom 11176 ax-addass 11177 ax-mulass 11178 ax-distr 11179 ax-i2m1 11180 ax-1ne0 11181 ax-1rid 11182 ax-rnegex 11183 ax-rrecex 11184 ax-cnre 11185 ax-pre-lttri 11186 ax-pre-lttrn 11187 ax-pre-ltadd 11188 ax-pre-mulgt0 11189 ax-pre-sup 11190 ax-addf 11191 ax-mulf 11192 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-nel 3041 df-ral 3056 df-rex 3065 df-rmo 3370 df-reu 3371 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-pss 3962 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-tp 4628 df-op 4630 df-uni 4903 df-int 4944 df-iun 4992 df-iin 4993 df-disj 5107 df-br 5142 df-opab 5204 df-mpt 5225 df-tr 5259 df-id 5567 df-eprel 5573 df-po 5581 df-so 5582 df-fr 5624 df-se 5625 df-we 5626 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-pred 6294 df-ord 6361 df-on 6362 df-lim 6363 df-suc 6364 df-iota 6489 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-isom 6546 df-riota 7361 df-ov 7408 df-oprab 7409 df-mpo 7410 df-of 7667 df-om 7853 df-1st 7974 df-2nd 7975 df-supp 8147 df-tpos 8212 df-frecs 8267 df-wrecs 8298 df-recs 8372 df-rdg 8411 df-1o 8467 df-2o 8468 df-oadd 8471 df-omul 8472 df-er 8705 df-map 8824 df-pm 8825 df-ixp 8894 df-en 8942 df-dom 8943 df-sdom 8944 df-fin 8945 df-fsupp 9364 df-fi 9408 df-sup 9439 df-inf 9440 df-oi 9507 df-dju 9898 df-card 9936 df-acn 9939 df-ac 10113 df-pnf 11254 df-mnf 11255 df-xr 11256 df-ltxr 11257 df-le 11258 df-sub 11450 df-neg 11451 df-div 11876 df-nn 12217 df-2 12279 df-3 12280 df-4 12281 df-5 12282 df-6 12283 df-7 12284 df-8 12285 df-9 12286 df-n0 12477 df-z 12563 df-dec 12682 df-uz 12827 df-q 12937 df-rp 12981 df-xneg 13098 df-xadd 13099 df-xmul 13100 df-ioo 13334 df-ico 13336 df-icc 13337 df-fz 13491 df-fzo 13634 df-fl 13763 df-seq 13973 df-exp 14033 df-hash 14296 df-cj 15052 df-re 15053 df-im 15054 df-sqrt 15188 df-abs 15189 df-clim 15438 df-rlim 15439 df-sum 15639 df-prod 15856 df-struct 17089 df-sets 17106 df-slot 17124 df-ndx 17136 df-base 17154 df-ress 17183 df-plusg 17219 df-mulr 17220 df-starv 17221 df-sca 17222 df-vsca 17223 df-ip 17224 df-tset 17225 df-ple 17226 df-ds 17228 df-unif 17229 df-hom 17230 df-cco 17231 df-rest 17377 df-topn 17378 df-0g 17396 df-gsum 17397 df-topgen 17398 df-pt 17399 df-prds 17402 df-pws 17404 df-xrs 17457 df-qtop 17462 df-imas 17463 df-xps 17465 df-mre 17539 df-mrc 17540 df-acs 17542 df-mgm 18573 df-sgrp 18652 df-mnd 18668 df-mhm 18713 df-submnd 18714 df-grp 18866 df-minusg 18867 df-sbg 18868 df-mulg 18996 df-subg 19050 df-ghm 19139 df-cntz 19233 df-cmn 19702 df-abl 19703 df-mgp 20040 df-rng 20058 df-ur 20087 df-ring 20140 df-cring 20141 df-oppr 20236 df-dvdsr 20259 df-unit 20260 df-invr 20290 df-dvr 20303 df-rhm 20374 df-subrng 20446 df-subrg 20471 df-drng 20589 df-field 20590 df-abv 20660 df-staf 20688 df-srng 20689 df-lmod 20708 df-lss 20779 df-lmhm 20870 df-lvec 20951 df-sra 21021 df-rgmod 21022 df-psmet 21232 df-xmet 21233 df-met 21234 df-bl 21235 df-mopn 21236 df-cnfld 21241 df-refld 21498 df-phl 21519 df-dsmm 21627 df-frlm 21642 df-top 22751 df-topon 22768 df-topsp 22790 df-bases 22804 df-cn 23086 df-cnp 23087 df-cmp 23246 df-tx 23421 df-hmeo 23614 df-xms 24181 df-ms 24182 df-tms 24183 df-nm 24446 df-ngp 24447 df-tng 24448 df-nrg 24449 df-nlm 24450 df-cncf 24753 df-clm 24945 df-cph 25051 df-tcph 25052 df-rrx 25268 df-ovol 25348 df-vol 25349 df-salg 45594 df-sumge0 45648 df-mea 45735 df-ome 45775 df-caragen 45777 df-ovoln 45822 df-voln 45824 |
This theorem is referenced by: vonsn 45976 vonn0icc2 45977 |
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