![]() |
Mathbox for Glauco Siliprandi |
< Previous
Next >
Nearby theorems |
|
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 45994 | . 2 β’ (π β (π΄(πΏβπ)π΅) = if(π = β , 0, βπ β π (volβ((π΄βπ)[,)(π΅βπ))))) |
6 | 0e0icopnf 13475 | . . . 4 β’ 0 β (0[,)+β) | |
7 | 6 | a1i 11 | . . 3 β’ (π β 0 β (0[,)+β)) |
8 | 0xr 11299 | . . . . 5 β’ 0 β β* | |
9 | 8 | a1i 11 | . . . 4 β’ (π β 0 β β*) |
10 | pnfxr 11306 | . . . . 5 β’ +β β β* | |
11 | 10 | a1i 11 | . . . 4 β’ (π β +β β β*) |
12 | 2 | ffvelcdmda 7099 | . . . . . . . 8 β’ ((π β§ π β π) β (π΄βπ) β β) |
13 | 3 | ffvelcdmda 7099 | . . . . . . . 8 β’ ((π β§ π β π) β (π΅βπ) β β) |
14 | volico 45400 | . . . . . . . 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 582 | . . . . . . 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 11680 | . . . . . . . 8 β’ ((π β§ π β π) β ((π΅βπ) β (π΄βπ)) β β) |
17 | 0red 11255 | . . . . . . . 8 β’ ((π β§ π β π) β 0 β β) | |
18 | 16, 17 | ifcld 4578 | . . . . . . 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 2829 | . . . . . 6 β’ ((π β§ π β π) β (volβ((π΄βπ)[,)(π΅βπ))) β β) |
20 | 4, 19 | fprodrecl 15937 | . . . . 5 β’ (π β βπ β π (volβ((π΄βπ)[,)(π΅βπ))) β β) |
21 | 20 | rexrd 11302 | . . . 4 β’ (π β βπ β π (volβ((π΄βπ)[,)(π΅βπ))) β β*) |
22 | nfv 1909 | . . . . 5 β’ β²ππ | |
23 | 13 | rexrd 11302 | . . . . . . 7 β’ ((π β§ π β π) β (π΅βπ) β β*) |
24 | icombl 25513 | . . . . . . 7 β’ (((π΄βπ) β β β§ (π΅βπ) β β*) β ((π΄βπ)[,)(π΅βπ)) β dom vol) | |
25 | 12, 23, 24 | syl2anc 582 | . . . . . 6 β’ ((π β§ π β π) β ((π΄βπ)[,)(π΅βπ)) β dom vol) |
26 | volge0 45378 | . . . . . 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 15977 | . . . 4 β’ (π β 0 β€ βπ β π (volβ((π΄βπ)[,)(π΅βπ)))) |
29 | 20 | ltpnfd 13141 | . . . 4 β’ (π β βπ β π (volβ((π΄βπ)[,)(π΅βπ))) < +β) |
30 | 9, 11, 21, 28, 29 | elicod 13414 | . . 3 β’ (π β βπ β π (volβ((π΄βπ)[,)(π΅βπ))) β (0[,)+β)) |
31 | 7, 30 | ifcld 4578 | . 2 β’ (π β if(π = β , 0, βπ β π (volβ((π΄βπ)[,)(π΅βπ)))) β (0[,)+β)) |
32 | 5, 31 | eqeltrd 2829 | 1 β’ (π β (π΄(πΏβπ)π΅) β (0[,)+β)) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 394 = wceq 1533 β wcel 2098 β c0 4326 ifcif 4532 class class class wbr 5152 β¦ cmpt 5235 dom cdm 5682 βΆwf 6549 βcfv 6553 (class class class)co 7426 β cmpo 7428 βm cmap 8851 Fincfn 8970 βcr 11145 0cc0 11146 +βcpnf 11283 β*cxr 11285 < clt 11286 β€ cle 11287 β cmin 11482 [,)cico 13366 βcprod 15889 volcvol 25412 |
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 2166 ax-ext 2699 ax-rep 5289 ax-sep 5303 ax-nul 5310 ax-pow 5369 ax-pr 5433 ax-un 7746 ax-inf2 9672 ax-cnex 11202 ax-resscn 11203 ax-1cn 11204 ax-icn 11205 ax-addcl 11206 ax-addrcl 11207 ax-mulcl 11208 ax-mulrcl 11209 ax-mulcom 11210 ax-addass 11211 ax-mulass 11212 ax-distr 11213 ax-i2m1 11214 ax-1ne0 11215 ax-1rid 11216 ax-rnegex 11217 ax-rrecex 11218 ax-cnre 11219 ax-pre-lttri 11220 ax-pre-lttrn 11221 ax-pre-ltadd 11222 ax-pre-mulgt0 11223 ax-pre-sup 11224 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3374 df-reu 3375 df-rab 3431 df-v 3475 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4327 df-if 4533 df-pw 4608 df-sn 4633 df-pr 4635 df-op 4639 df-uni 4913 df-int 4954 df-iun 5002 df-br 5153 df-opab 5215 df-mpt 5236 df-tr 5270 df-id 5580 df-eprel 5586 df-po 5594 df-so 5595 df-fr 5637 df-se 5638 df-we 5639 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-pred 6310 df-ord 6377 df-on 6378 df-lim 6379 df-suc 6380 df-iota 6505 df-fun 6555 df-fn 6556 df-f 6557 df-f1 6558 df-fo 6559 df-f1o 6560 df-fv 6561 df-isom 6562 df-riota 7382 df-ov 7429 df-oprab 7430 df-mpo 7431 df-of 7691 df-om 7877 df-1st 7999 df-2nd 8000 df-frecs 8293 df-wrecs 8324 df-recs 8398 df-rdg 8437 df-1o 8493 df-2o 8494 df-er 8731 df-map 8853 df-pm 8854 df-en 8971 df-dom 8972 df-sdom 8973 df-fin 8974 df-fi 9442 df-sup 9473 df-inf 9474 df-oi 9541 df-dju 9932 df-card 9970 df-pnf 11288 df-mnf 11289 df-xr 11290 df-ltxr 11291 df-le 11292 df-sub 11484 df-neg 11485 df-div 11910 df-nn 12251 df-2 12313 df-3 12314 df-n0 12511 df-z 12597 df-uz 12861 df-q 12971 df-rp 13015 df-xneg 13132 df-xadd 13133 df-xmul 13134 df-ioo 13368 df-ico 13370 df-icc 13371 df-fz 13525 df-fzo 13668 df-fl 13797 df-seq 14007 df-exp 14067 df-hash 14330 df-cj 15086 df-re 15087 df-im 15088 df-sqrt 15222 df-abs 15223 df-clim 15472 df-rlim 15473 df-sum 15673 df-prod 15890 df-rest 17411 df-topgen 17432 df-psmet 21278 df-xmet 21279 df-met 21280 df-bl 21281 df-mopn 21282 df-top 22816 df-topon 22833 df-bases 22869 df-cmp 23311 df-ovol 25413 df-vol 25414 |
This theorem is referenced by: sge0hsphoire 46006 hoidmv1le 46011 hoidmvlelem1 46012 hoidmvlelem2 46013 hoidmvlelem3 46014 hoidmvlelem4 46015 hoidmvlelem5 46016 hoidmvle 46017 ovnhoilem2 46019 ovnhoi 46020 ovnlecvr2 46027 hspmbllem1 46043 hspmbllem2 46044 |
Copyright terms: Public domain | W3C validator |