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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 6843 | . . . . . . . . . . 11 β’ (π = π β (πβπ) = (πβπ)) | |
6 | fveq2 6843 | . . . . . . . . . . 11 β’ (π = π β (πβπ) = (πβπ)) | |
7 | 5, 6 | oveq12d 7376 | . . . . . . . . . 10 β’ (π = π β ((πβπ)[,)(πβπ)) = ((πβπ)[,)(πβπ))) |
8 | 7 | fveq2d 6847 | . . . . . . . . 9 β’ (π = π β (volβ((πβπ)[,)(πβπ))) = (volβ((πβπ)[,)(πβπ)))) |
9 | 8 | cbvprodv 15804 | . . . . . . . 8 β’ βπ β π₯ (volβ((πβπ)[,)(πβπ))) = βπ β π₯ (volβ((πβπ)[,)(πβπ))) |
10 | ifeq2 4492 | . . . . . . . 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 7436 | . . . . 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 5211 | . . . 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 45012 | . . 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 6844 | . . . . 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 7371 | . . . 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 2773 | . 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 2733 | . . 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 44911 | . 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 7036 | . . . . 5 β’ ((π β§ π β π) β (π΄βπ) β β) |
24 | 3 | ffvelcdmda 7036 | . . . . 5 β’ ((π β§ π β π) β (π΅βπ) β β) |
25 | 23, 24 | voliccico 44326 | . . . 4 β’ ((π β§ π β π) β (volβ((π΄βπ)[,](π΅βπ))) = (volβ((π΄βπ)[,)(π΅βπ)))) |
26 | 25 | eqcomd 2739 | . . 3 β’ ((π β§ π β π) β (volβ((π΄βπ)[,)(π΅βπ))) = (volβ((π΄βπ)[,](π΅βπ)))) |
27 | 26 | prodeq2dv 15811 | . 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 2777 | 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 397 = wceq 1542 β wcel 2107 β wne 2940 β c0 4283 ifcif 4487 β¦ cmpt 5189 βΆwf 6493 βcfv 6497 (class class class)co 7358 β cmpo 7360 βm cmap 8768 Xcixp 8838 Fincfn 8886 βcr 11055 0cc0 11056 [,)cico 13272 [,]cicc 13273 βcprod 15793 volcvol 24843 volncvoln 44865 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5243 ax-sep 5257 ax-nul 5264 ax-pow 5321 ax-pr 5385 ax-un 7673 ax-inf2 9582 ax-cc 10376 ax-ac2 10404 ax-cnex 11112 ax-resscn 11113 ax-1cn 11114 ax-icn 11115 ax-addcl 11116 ax-addrcl 11117 ax-mulcl 11118 ax-mulrcl 11119 ax-mulcom 11120 ax-addass 11121 ax-mulass 11122 ax-distr 11123 ax-i2m1 11124 ax-1ne0 11125 ax-1rid 11126 ax-rnegex 11127 ax-rrecex 11128 ax-cnre 11129 ax-pre-lttri 11130 ax-pre-lttrn 11131 ax-pre-ltadd 11132 ax-pre-mulgt0 11133 ax-pre-sup 11134 ax-addf 11135 ax-mulf 11136 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3352 df-reu 3353 df-rab 3407 df-v 3446 df-sbc 3741 df-csb 3857 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-pss 3930 df-nul 4284 df-if 4488 df-pw 4563 df-sn 4588 df-pr 4590 df-tp 4592 df-op 4594 df-uni 4867 df-int 4909 df-iun 4957 df-iin 4958 df-disj 5072 df-br 5107 df-opab 5169 df-mpt 5190 df-tr 5224 df-id 5532 df-eprel 5538 df-po 5546 df-so 5547 df-fr 5589 df-se 5590 df-we 5591 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-rn 5645 df-res 5646 df-ima 5647 df-pred 6254 df-ord 6321 df-on 6322 df-lim 6323 df-suc 6324 df-iota 6449 df-fun 6499 df-fn 6500 df-f 6501 df-f1 6502 df-fo 6503 df-f1o 6504 df-fv 6505 df-isom 6506 df-riota 7314 df-ov 7361 df-oprab 7362 df-mpo 7363 df-of 7618 df-om 7804 df-1st 7922 df-2nd 7923 df-supp 8094 df-tpos 8158 df-frecs 8213 df-wrecs 8244 df-recs 8318 df-rdg 8357 df-1o 8413 df-2o 8414 df-oadd 8417 df-omul 8418 df-er 8651 df-map 8770 df-pm 8771 df-ixp 8839 df-en 8887 df-dom 8888 df-sdom 8889 df-fin 8890 df-fsupp 9309 df-fi 9352 df-sup 9383 df-inf 9384 df-oi 9451 df-dju 9842 df-card 9880 df-acn 9883 df-ac 10057 df-pnf 11196 df-mnf 11197 df-xr 11198 df-ltxr 11199 df-le 11200 df-sub 11392 df-neg 11393 df-div 11818 df-nn 12159 df-2 12221 df-3 12222 df-4 12223 df-5 12224 df-6 12225 df-7 12226 df-8 12227 df-9 12228 df-n0 12419 df-z 12505 df-dec 12624 df-uz 12769 df-q 12879 df-rp 12921 df-xneg 13038 df-xadd 13039 df-xmul 13040 df-ioo 13274 df-ico 13276 df-icc 13277 df-fz 13431 df-fzo 13574 df-fl 13703 df-seq 13913 df-exp 13974 df-hash 14237 df-cj 14990 df-re 14991 df-im 14992 df-sqrt 15126 df-abs 15127 df-clim 15376 df-rlim 15377 df-sum 15577 df-prod 15794 df-struct 17024 df-sets 17041 df-slot 17059 df-ndx 17071 df-base 17089 df-ress 17118 df-plusg 17151 df-mulr 17152 df-starv 17153 df-sca 17154 df-vsca 17155 df-ip 17156 df-tset 17157 df-ple 17158 df-ds 17160 df-unif 17161 df-hom 17162 df-cco 17163 df-rest 17309 df-topn 17310 df-0g 17328 df-gsum 17329 df-topgen 17330 df-pt 17331 df-prds 17334 df-pws 17336 df-xrs 17389 df-qtop 17394 df-imas 17395 df-xps 17397 df-mre 17471 df-mrc 17472 df-acs 17474 df-mgm 18502 df-sgrp 18551 df-mnd 18562 df-mhm 18606 df-submnd 18607 df-grp 18756 df-minusg 18757 df-sbg 18758 df-mulg 18878 df-subg 18930 df-ghm 19011 df-cntz 19102 df-cmn 19569 df-abl 19570 df-mgp 19902 df-ur 19919 df-ring 19971 df-cring 19972 df-oppr 20054 df-dvdsr 20075 df-unit 20076 df-invr 20106 df-dvr 20117 df-rnghom 20153 df-drng 20199 df-field 20200 df-subrg 20234 df-abv 20290 df-staf 20318 df-srng 20319 df-lmod 20338 df-lss 20408 df-lmhm 20498 df-lvec 20579 df-sra 20649 df-rgmod 20650 df-psmet 20804 df-xmet 20805 df-met 20806 df-bl 20807 df-mopn 20808 df-cnfld 20813 df-refld 21025 df-phl 21046 df-dsmm 21154 df-frlm 21169 df-top 22259 df-topon 22276 df-topsp 22298 df-bases 22312 df-cn 22594 df-cnp 22595 df-cmp 22754 df-tx 22929 df-hmeo 23122 df-xms 23689 df-ms 23690 df-tms 23691 df-nm 23954 df-ngp 23955 df-tng 23956 df-nrg 23957 df-nlm 23958 df-cncf 24257 df-clm 24442 df-cph 24548 df-tcph 24549 df-rrx 24765 df-ovol 24844 df-vol 24845 df-salg 44636 df-sumge0 44690 df-mea 44777 df-ome 44817 df-caragen 44819 df-ovoln 44864 df-voln 44866 |
This theorem is referenced by: vonsn 45018 vonn0icc2 45019 |
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