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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 45847 | . 2 β’ (π β (π΄(πΏβπ)π΅) = if(π = β , 0, βπ β π (volβ((π΄βπ)[,)(π΅βπ))))) |
6 | 0e0icopnf 13438 | . . . 4 β’ 0 β (0[,)+β) | |
7 | 6 | a1i 11 | . . 3 β’ (π β 0 β (0[,)+β)) |
8 | 0xr 11262 | . . . . 5 β’ 0 β β* | |
9 | 8 | a1i 11 | . . . 4 β’ (π β 0 β β*) |
10 | pnfxr 11269 | . . . . 5 β’ +β β β* | |
11 | 10 | a1i 11 | . . . 4 β’ (π β +β β β*) |
12 | 2 | ffvelcdmda 7079 | . . . . . . . 8 β’ ((π β§ π β π) β (π΄βπ) β β) |
13 | 3 | ffvelcdmda 7079 | . . . . . . . 8 β’ ((π β§ π β π) β (π΅βπ) β β) |
14 | volico 45253 | . . . . . . . 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 583 | . . . . . . 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 11643 | . . . . . . . 8 β’ ((π β§ π β π) β ((π΅βπ) β (π΄βπ)) β β) |
17 | 0red 11218 | . . . . . . . 8 β’ ((π β§ π β π) β 0 β β) | |
18 | 16, 17 | ifcld 4569 | . . . . . . 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 2827 | . . . . . 6 β’ ((π β§ π β π) β (volβ((π΄βπ)[,)(π΅βπ))) β β) |
20 | 4, 19 | fprodrecl 15900 | . . . . 5 β’ (π β βπ β π (volβ((π΄βπ)[,)(π΅βπ))) β β) |
21 | 20 | rexrd 11265 | . . . 4 β’ (π β βπ β π (volβ((π΄βπ)[,)(π΅βπ))) β β*) |
22 | nfv 1909 | . . . . 5 β’ β²ππ | |
23 | 13 | rexrd 11265 | . . . . . . 7 β’ ((π β§ π β π) β (π΅βπ) β β*) |
24 | icombl 25443 | . . . . . . 7 β’ (((π΄βπ) β β β§ (π΅βπ) β β*) β ((π΄βπ)[,)(π΅βπ)) β dom vol) | |
25 | 12, 23, 24 | syl2anc 583 | . . . . . 6 β’ ((π β§ π β π) β ((π΄βπ)[,)(π΅βπ)) β dom vol) |
26 | volge0 45231 | . . . . . 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 15940 | . . . 4 β’ (π β 0 β€ βπ β π (volβ((π΄βπ)[,)(π΅βπ)))) |
29 | 20 | ltpnfd 13104 | . . . 4 β’ (π β βπ β π (volβ((π΄βπ)[,)(π΅βπ))) < +β) |
30 | 9, 11, 21, 28, 29 | elicod 13377 | . . 3 β’ (π β βπ β π (volβ((π΄βπ)[,)(π΅βπ))) β (0[,)+β)) |
31 | 7, 30 | ifcld 4569 | . 2 β’ (π β if(π = β , 0, βπ β π (volβ((π΄βπ)[,)(π΅βπ)))) β (0[,)+β)) |
32 | 5, 31 | eqeltrd 2827 | 1 β’ (π β (π΄(πΏβπ)π΅) β (0[,)+β)) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 395 = wceq 1533 β wcel 2098 β c0 4317 ifcif 4523 class class class wbr 5141 β¦ cmpt 5224 dom cdm 5669 βΆwf 6532 βcfv 6536 (class class class)co 7404 β cmpo 7406 βm cmap 8819 Fincfn 8938 βcr 11108 0cc0 11109 +βcpnf 11246 β*cxr 11248 < clt 11249 β€ cle 11250 β cmin 11445 [,)cico 13329 βcprod 15852 volcvol 25342 |
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 7721 ax-inf2 9635 ax-cnex 11165 ax-resscn 11166 ax-1cn 11167 ax-icn 11168 ax-addcl 11169 ax-addrcl 11170 ax-mulcl 11171 ax-mulrcl 11172 ax-mulcom 11173 ax-addass 11174 ax-mulass 11175 ax-distr 11176 ax-i2m1 11177 ax-1ne0 11178 ax-1rid 11179 ax-rnegex 11180 ax-rrecex 11181 ax-cnre 11182 ax-pre-lttri 11183 ax-pre-lttrn 11184 ax-pre-ltadd 11185 ax-pre-mulgt0 11186 ax-pre-sup 11187 |
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-op 4630 df-uni 4903 df-int 4944 df-iun 4992 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 6293 df-ord 6360 df-on 6361 df-lim 6362 df-suc 6363 df-iota 6488 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-isom 6545 df-riota 7360 df-ov 7407 df-oprab 7408 df-mpo 7409 df-of 7666 df-om 7852 df-1st 7971 df-2nd 7972 df-frecs 8264 df-wrecs 8295 df-recs 8369 df-rdg 8408 df-1o 8464 df-2o 8465 df-er 8702 df-map 8821 df-pm 8822 df-en 8939 df-dom 8940 df-sdom 8941 df-fin 8942 df-fi 9405 df-sup 9436 df-inf 9437 df-oi 9504 df-dju 9895 df-card 9933 df-pnf 11251 df-mnf 11252 df-xr 11253 df-ltxr 11254 df-le 11255 df-sub 11447 df-neg 11448 df-div 11873 df-nn 12214 df-2 12276 df-3 12277 df-n0 12474 df-z 12560 df-uz 12824 df-q 12934 df-rp 12978 df-xneg 13095 df-xadd 13096 df-xmul 13097 df-ioo 13331 df-ico 13333 df-icc 13334 df-fz 13488 df-fzo 13631 df-fl 13760 df-seq 13970 df-exp 14030 df-hash 14293 df-cj 15049 df-re 15050 df-im 15051 df-sqrt 15185 df-abs 15186 df-clim 15435 df-rlim 15436 df-sum 15636 df-prod 15853 df-rest 17374 df-topgen 17395 df-psmet 21227 df-xmet 21228 df-met 21229 df-bl 21230 df-mopn 21231 df-top 22746 df-topon 22763 df-bases 22799 df-cmp 23241 df-ovol 25343 df-vol 25344 |
This theorem is referenced by: sge0hsphoire 45859 hoidmv1le 45864 hoidmvlelem1 45865 hoidmvlelem2 45866 hoidmvlelem3 45867 hoidmvlelem4 45868 hoidmvlelem5 45869 hoidmvle 45870 ovnhoilem2 45872 ovnhoi 45873 ovnlecvr2 45880 hspmbllem1 45896 hspmbllem2 45897 |
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