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Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > hoidmvcl | Structured version Visualization version GIF version |
Description: The dimensional volume of a multidimensional half-open interval is a nonnegative real. (Contributed by Glauco Siliprandi, 21-Nov-2020.) |
Ref | Expression |
---|---|
hoidmvcl.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)))))) |
hoidmvcl.x | β’ (π β π β Fin) |
hoidmvcl.a | β’ (π β π΄:πβΆβ) |
hoidmvcl.b | β’ (π β π΅:πβΆβ) |
Ref | Expression |
---|---|
hoidmvcl | β’ (π β (π΄(πΏβπ)π΅) β (0[,)+β)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | hoidmvcl.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)))))) | |
2 | hoidmvcl.a | . . 3 β’ (π β π΄:πβΆβ) | |
3 | hoidmvcl.b | . . 3 β’ (π β π΅:πβΆβ) | |
4 | hoidmvcl.x | . . 3 β’ (π β π β Fin) | |
5 | 1, 2, 3, 4 | hoidmvval 44908 | . 2 β’ (π β (π΄(πΏβπ)π΅) = if(π = β , 0, βπ β π (volβ((π΄βπ)[,)(π΅βπ))))) |
6 | 0e0icopnf 13384 | . . . 4 β’ 0 β (0[,)+β) | |
7 | 6 | a1i 11 | . . 3 β’ (π β 0 β (0[,)+β)) |
8 | 0xr 11210 | . . . . 5 β’ 0 β β* | |
9 | 8 | a1i 11 | . . . 4 β’ (π β 0 β β*) |
10 | pnfxr 11217 | . . . . 5 β’ +β β β* | |
11 | 10 | a1i 11 | . . . 4 β’ (π β +β β β*) |
12 | 2 | ffvelcdmda 7039 | . . . . . . . 8 β’ ((π β§ π β π) β (π΄βπ) β β) |
13 | 3 | ffvelcdmda 7039 | . . . . . . . 8 β’ ((π β§ π β π) β (π΅βπ) β β) |
14 | volico 44314 | . . . . . . . 8 β’ (((π΄βπ) β β β§ (π΅βπ) β β) β (volβ((π΄βπ)[,)(π΅βπ))) = if((π΄βπ) < (π΅βπ), ((π΅βπ) β (π΄βπ)), 0)) | |
15 | 12, 13, 14 | syl2anc 585 | . . . . . . 7 β’ ((π β§ π β π) β (volβ((π΄βπ)[,)(π΅βπ))) = if((π΄βπ) < (π΅βπ), ((π΅βπ) β (π΄βπ)), 0)) |
16 | 13, 12 | resubcld 11591 | . . . . . . . 8 β’ ((π β§ π β π) β ((π΅βπ) β (π΄βπ)) β β) |
17 | 0red 11166 | . . . . . . . 8 β’ ((π β§ π β π) β 0 β β) | |
18 | 16, 17 | ifcld 4536 | . . . . . . 7 β’ ((π β§ π β π) β if((π΄βπ) < (π΅βπ), ((π΅βπ) β (π΄βπ)), 0) β β) |
19 | 15, 18 | eqeltrd 2834 | . . . . . 6 β’ ((π β§ π β π) β (volβ((π΄βπ)[,)(π΅βπ))) β β) |
20 | 4, 19 | fprodrecl 15844 | . . . . 5 β’ (π β βπ β π (volβ((π΄βπ)[,)(π΅βπ))) β β) |
21 | 20 | rexrd 11213 | . . . 4 β’ (π β βπ β π (volβ((π΄βπ)[,)(π΅βπ))) β β*) |
22 | nfv 1918 | . . . . 5 β’ β²ππ | |
23 | 13 | rexrd 11213 | . . . . . . 7 β’ ((π β§ π β π) β (π΅βπ) β β*) |
24 | icombl 24951 | . . . . . . 7 β’ (((π΄βπ) β β β§ (π΅βπ) β β*) β ((π΄βπ)[,)(π΅βπ)) β dom vol) | |
25 | 12, 23, 24 | syl2anc 585 | . . . . . 6 β’ ((π β§ π β π) β ((π΄βπ)[,)(π΅βπ)) β dom vol) |
26 | volge0 44292 | . . . . . 6 β’ (((π΄βπ)[,)(π΅βπ)) β dom vol β 0 β€ (volβ((π΄βπ)[,)(π΅βπ)))) | |
27 | 25, 26 | syl 17 | . . . . 5 β’ ((π β§ π β π) β 0 β€ (volβ((π΄βπ)[,)(π΅βπ)))) |
28 | 22, 4, 19, 27 | fprodge0 15884 | . . . 4 β’ (π β 0 β€ βπ β π (volβ((π΄βπ)[,)(π΅βπ)))) |
29 | 20 | ltpnfd 13050 | . . . 4 β’ (π β βπ β π (volβ((π΄βπ)[,)(π΅βπ))) < +β) |
30 | 9, 11, 21, 28, 29 | elicod 13323 | . . 3 β’ (π β βπ β π (volβ((π΄βπ)[,)(π΅βπ))) β (0[,)+β)) |
31 | 7, 30 | ifcld 4536 | . 2 β’ (π β if(π = β , 0, βπ β π (volβ((π΄βπ)[,)(π΅βπ)))) β (0[,)+β)) |
32 | 5, 31 | eqeltrd 2834 | 1 β’ (π β (π΄(πΏβπ)π΅) β (0[,)+β)) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 397 = wceq 1542 β wcel 2107 β c0 4286 ifcif 4490 class class class wbr 5109 β¦ cmpt 5192 dom cdm 5637 βΆwf 6496 βcfv 6500 (class class class)co 7361 β cmpo 7363 βm cmap 8771 Fincfn 8889 βcr 11058 0cc0 11059 +βcpnf 11194 β*cxr 11196 < clt 11197 β€ cle 11198 β cmin 11393 [,)cico 13275 βcprod 15796 volcvol 24850 |
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 5246 ax-sep 5260 ax-nul 5267 ax-pow 5324 ax-pr 5388 ax-un 7676 ax-inf2 9585 ax-cnex 11115 ax-resscn 11116 ax-1cn 11117 ax-icn 11118 ax-addcl 11119 ax-addrcl 11120 ax-mulcl 11121 ax-mulrcl 11122 ax-mulcom 11123 ax-addass 11124 ax-mulass 11125 ax-distr 11126 ax-i2m1 11127 ax-1ne0 11128 ax-1rid 11129 ax-rnegex 11130 ax-rrecex 11131 ax-cnre 11132 ax-pre-lttri 11133 ax-pre-lttrn 11134 ax-pre-ltadd 11135 ax-pre-mulgt0 11136 ax-pre-sup 11137 |
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 3449 df-sbc 3744 df-csb 3860 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-pss 3933 df-nul 4287 df-if 4491 df-pw 4566 df-sn 4591 df-pr 4593 df-op 4597 df-uni 4870 df-int 4912 df-iun 4960 df-br 5110 df-opab 5172 df-mpt 5193 df-tr 5227 df-id 5535 df-eprel 5541 df-po 5549 df-so 5550 df-fr 5592 df-se 5593 df-we 5594 df-xp 5643 df-rel 5644 df-cnv 5645 df-co 5646 df-dm 5647 df-rn 5648 df-res 5649 df-ima 5650 df-pred 6257 df-ord 6324 df-on 6325 df-lim 6326 df-suc 6327 df-iota 6452 df-fun 6502 df-fn 6503 df-f 6504 df-f1 6505 df-fo 6506 df-f1o 6507 df-fv 6508 df-isom 6509 df-riota 7317 df-ov 7364 df-oprab 7365 df-mpo 7366 df-of 7621 df-om 7807 df-1st 7925 df-2nd 7926 df-frecs 8216 df-wrecs 8247 df-recs 8321 df-rdg 8360 df-1o 8416 df-2o 8417 df-er 8654 df-map 8773 df-pm 8774 df-en 8890 df-dom 8891 df-sdom 8892 df-fin 8893 df-fi 9355 df-sup 9386 df-inf 9387 df-oi 9454 df-dju 9845 df-card 9883 df-pnf 11199 df-mnf 11200 df-xr 11201 df-ltxr 11202 df-le 11203 df-sub 11395 df-neg 11396 df-div 11821 df-nn 12162 df-2 12224 df-3 12225 df-n0 12422 df-z 12508 df-uz 12772 df-q 12882 df-rp 12924 df-xneg 13041 df-xadd 13042 df-xmul 13043 df-ioo 13277 df-ico 13279 df-icc 13280 df-fz 13434 df-fzo 13577 df-fl 13706 df-seq 13916 df-exp 13977 df-hash 14240 df-cj 14993 df-re 14994 df-im 14995 df-sqrt 15129 df-abs 15130 df-clim 15379 df-rlim 15380 df-sum 15580 df-prod 15797 df-rest 17312 df-topgen 17333 df-psmet 20811 df-xmet 20812 df-met 20813 df-bl 20814 df-mopn 20815 df-top 22266 df-topon 22283 df-bases 22319 df-cmp 22761 df-ovol 24851 df-vol 24852 |
This theorem is referenced by: sge0hsphoire 44920 hoidmv1le 44925 hoidmvlelem1 44926 hoidmvlelem2 44927 hoidmvlelem3 44928 hoidmvlelem4 44929 hoidmvlelem5 44930 hoidmvle 44931 ovnhoilem2 44933 ovnhoi 44934 ovnlecvr2 44941 hspmbllem1 44957 hspmbllem2 44958 |
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