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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 | ⊢ 𝐿 = (𝑥 ∈ Fin ↦ (𝑎 ∈ (ℝ ↑𝑚 𝑥), 𝑏 ∈ (ℝ ↑𝑚 𝑥) ↦ if(𝑥 = ∅, 0, ∏𝑘 ∈ 𝑥 (vol‘((𝑎‘𝑘)[,)(𝑏‘𝑘)))))) |
hoidmvcl.x | ⊢ (𝜑 → 𝑋 ∈ Fin) |
hoidmvcl.a | ⊢ (𝜑 → 𝐴:𝑋⟶ℝ) |
hoidmvcl.b | ⊢ (𝜑 → 𝐵:𝑋⟶ℝ) |
Ref | Expression |
---|---|
hoidmvcl | ⊢ (𝜑 → (𝐴(𝐿‘𝑋)𝐵) ∈ (0[,)+∞)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | hoidmvcl.l | . . 3 ⊢ 𝐿 = (𝑥 ∈ Fin ↦ (𝑎 ∈ (ℝ ↑𝑚 𝑥), 𝑏 ∈ (ℝ ↑𝑚 𝑥) ↦ if(𝑥 = ∅, 0, ∏𝑘 ∈ 𝑥 (vol‘((𝑎‘𝑘)[,)(𝑏‘𝑘)))))) | |
2 | hoidmvcl.a | . . 3 ⊢ (𝜑 → 𝐴:𝑋⟶ℝ) | |
3 | hoidmvcl.b | . . 3 ⊢ (𝜑 → 𝐵:𝑋⟶ℝ) | |
4 | hoidmvcl.x | . . 3 ⊢ (𝜑 → 𝑋 ∈ Fin) | |
5 | 1, 2, 3, 4 | hoidmvval 41585 | . 2 ⊢ (𝜑 → (𝐴(𝐿‘𝑋)𝐵) = if(𝑋 = ∅, 0, ∏𝑘 ∈ 𝑋 (vol‘((𝐴‘𝑘)[,)(𝐵‘𝑘))))) |
6 | 0e0icopnf 12572 | . . . 4 ⊢ 0 ∈ (0[,)+∞) | |
7 | 6 | a1i 11 | . . 3 ⊢ (𝜑 → 0 ∈ (0[,)+∞)) |
8 | 0xr 10403 | . . . . 5 ⊢ 0 ∈ ℝ* | |
9 | 8 | a1i 11 | . . . 4 ⊢ (𝜑 → 0 ∈ ℝ*) |
10 | pnfxr 10410 | . . . . 5 ⊢ +∞ ∈ ℝ* | |
11 | 10 | a1i 11 | . . . 4 ⊢ (𝜑 → +∞ ∈ ℝ*) |
12 | 2 | ffvelrnda 6608 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → (𝐴‘𝑘) ∈ ℝ) |
13 | 3 | ffvelrnda 6608 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → (𝐵‘𝑘) ∈ ℝ) |
14 | volico 40994 | . . . . . . . 8 ⊢ (((𝐴‘𝑘) ∈ ℝ ∧ (𝐵‘𝑘) ∈ ℝ) → (vol‘((𝐴‘𝑘)[,)(𝐵‘𝑘))) = if((𝐴‘𝑘) < (𝐵‘𝑘), ((𝐵‘𝑘) − (𝐴‘𝑘)), 0)) | |
15 | 12, 13, 14 | syl2anc 581 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → (vol‘((𝐴‘𝑘)[,)(𝐵‘𝑘))) = if((𝐴‘𝑘) < (𝐵‘𝑘), ((𝐵‘𝑘) − (𝐴‘𝑘)), 0)) |
16 | 13, 12 | resubcld 10782 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → ((𝐵‘𝑘) − (𝐴‘𝑘)) ∈ ℝ) |
17 | 0red 10360 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → 0 ∈ ℝ) | |
18 | 16, 17 | ifcld 4351 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → if((𝐴‘𝑘) < (𝐵‘𝑘), ((𝐵‘𝑘) − (𝐴‘𝑘)), 0) ∈ ℝ) |
19 | 15, 18 | eqeltrd 2906 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → (vol‘((𝐴‘𝑘)[,)(𝐵‘𝑘))) ∈ ℝ) |
20 | 4, 19 | fprodrecl 15056 | . . . . 5 ⊢ (𝜑 → ∏𝑘 ∈ 𝑋 (vol‘((𝐴‘𝑘)[,)(𝐵‘𝑘))) ∈ ℝ) |
21 | 20 | rexrd 10406 | . . . 4 ⊢ (𝜑 → ∏𝑘 ∈ 𝑋 (vol‘((𝐴‘𝑘)[,)(𝐵‘𝑘))) ∈ ℝ*) |
22 | nfv 2015 | . . . . 5 ⊢ Ⅎ𝑘𝜑 | |
23 | 13 | rexrd 10406 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → (𝐵‘𝑘) ∈ ℝ*) |
24 | icombl 23730 | . . . . . . 7 ⊢ (((𝐴‘𝑘) ∈ ℝ ∧ (𝐵‘𝑘) ∈ ℝ*) → ((𝐴‘𝑘)[,)(𝐵‘𝑘)) ∈ dom vol) | |
25 | 12, 23, 24 | syl2anc 581 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → ((𝐴‘𝑘)[,)(𝐵‘𝑘)) ∈ dom vol) |
26 | volge0 40971 | . . . . . 6 ⊢ (((𝐴‘𝑘)[,)(𝐵‘𝑘)) ∈ dom vol → 0 ≤ (vol‘((𝐴‘𝑘)[,)(𝐵‘𝑘)))) | |
27 | 25, 26 | syl 17 | . . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → 0 ≤ (vol‘((𝐴‘𝑘)[,)(𝐵‘𝑘)))) |
28 | 22, 4, 19, 27 | fprodge0 15096 | . . . 4 ⊢ (𝜑 → 0 ≤ ∏𝑘 ∈ 𝑋 (vol‘((𝐴‘𝑘)[,)(𝐵‘𝑘)))) |
29 | 20 | ltpnfd 12241 | . . . 4 ⊢ (𝜑 → ∏𝑘 ∈ 𝑋 (vol‘((𝐴‘𝑘)[,)(𝐵‘𝑘))) < +∞) |
30 | 9, 11, 21, 28, 29 | elicod 12512 | . . 3 ⊢ (𝜑 → ∏𝑘 ∈ 𝑋 (vol‘((𝐴‘𝑘)[,)(𝐵‘𝑘))) ∈ (0[,)+∞)) |
31 | 7, 30 | ifcld 4351 | . 2 ⊢ (𝜑 → if(𝑋 = ∅, 0, ∏𝑘 ∈ 𝑋 (vol‘((𝐴‘𝑘)[,)(𝐵‘𝑘)))) ∈ (0[,)+∞)) |
32 | 5, 31 | eqeltrd 2906 | 1 ⊢ (𝜑 → (𝐴(𝐿‘𝑋)𝐵) ∈ (0[,)+∞)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 386 = wceq 1658 ∈ wcel 2166 ∅c0 4144 ifcif 4306 class class class wbr 4873 ↦ cmpt 4952 dom cdm 5342 ⟶wf 6119 ‘cfv 6123 (class class class)co 6905 ↦ cmpt2 6907 ↑𝑚 cmap 8122 Fincfn 8222 ℝcr 10251 0cc0 10252 +∞cpnf 10388 ℝ*cxr 10390 < clt 10391 ≤ cle 10392 − cmin 10585 [,)cico 12465 ∏cprod 15008 volcvol 23629 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1896 ax-4 1910 ax-5 2011 ax-6 2077 ax-7 2114 ax-8 2168 ax-9 2175 ax-10 2194 ax-11 2209 ax-12 2222 ax-13 2391 ax-ext 2803 ax-rep 4994 ax-sep 5005 ax-nul 5013 ax-pow 5065 ax-pr 5127 ax-un 7209 ax-inf2 8815 ax-cnex 10308 ax-resscn 10309 ax-1cn 10310 ax-icn 10311 ax-addcl 10312 ax-addrcl 10313 ax-mulcl 10314 ax-mulrcl 10315 ax-mulcom 10316 ax-addass 10317 ax-mulass 10318 ax-distr 10319 ax-i2m1 10320 ax-1ne0 10321 ax-1rid 10322 ax-rnegex 10323 ax-rrecex 10324 ax-cnre 10325 ax-pre-lttri 10326 ax-pre-lttrn 10327 ax-pre-ltadd 10328 ax-pre-mulgt0 10329 ax-pre-sup 10330 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 881 df-3or 1114 df-3an 1115 df-tru 1662 df-fal 1672 df-ex 1881 df-nf 1885 df-sb 2070 df-mo 2605 df-eu 2640 df-clab 2812 df-cleq 2818 df-clel 2821 df-nfc 2958 df-ne 3000 df-nel 3103 df-ral 3122 df-rex 3123 df-reu 3124 df-rmo 3125 df-rab 3126 df-v 3416 df-sbc 3663 df-csb 3758 df-dif 3801 df-un 3803 df-in 3805 df-ss 3812 df-pss 3814 df-nul 4145 df-if 4307 df-pw 4380 df-sn 4398 df-pr 4400 df-tp 4402 df-op 4404 df-uni 4659 df-int 4698 df-iun 4742 df-br 4874 df-opab 4936 df-mpt 4953 df-tr 4976 df-id 5250 df-eprel 5255 df-po 5263 df-so 5264 df-fr 5301 df-se 5302 df-we 5303 df-xp 5348 df-rel 5349 df-cnv 5350 df-co 5351 df-dm 5352 df-rn 5353 df-res 5354 df-ima 5355 df-pred 5920 df-ord 5966 df-on 5967 df-lim 5968 df-suc 5969 df-iota 6086 df-fun 6125 df-fn 6126 df-f 6127 df-f1 6128 df-fo 6129 df-f1o 6130 df-fv 6131 df-isom 6132 df-riota 6866 df-ov 6908 df-oprab 6909 df-mpt2 6910 df-of 7157 df-om 7327 df-1st 7428 df-2nd 7429 df-wrecs 7672 df-recs 7734 df-rdg 7772 df-1o 7826 df-2o 7827 df-oadd 7830 df-er 8009 df-map 8124 df-pm 8125 df-en 8223 df-dom 8224 df-sdom 8225 df-fin 8226 df-fi 8586 df-sup 8617 df-inf 8618 df-oi 8684 df-card 9078 df-cda 9305 df-pnf 10393 df-mnf 10394 df-xr 10395 df-ltxr 10396 df-le 10397 df-sub 10587 df-neg 10588 df-div 11010 df-nn 11351 df-2 11414 df-3 11415 df-n0 11619 df-z 11705 df-uz 11969 df-q 12072 df-rp 12113 df-xneg 12232 df-xadd 12233 df-xmul 12234 df-ioo 12467 df-ico 12469 df-icc 12470 df-fz 12620 df-fzo 12761 df-fl 12888 df-seq 13096 df-exp 13155 df-hash 13411 df-cj 14216 df-re 14217 df-im 14218 df-sqrt 14352 df-abs 14353 df-clim 14596 df-rlim 14597 df-sum 14794 df-prod 15009 df-rest 16436 df-topgen 16457 df-psmet 20098 df-xmet 20099 df-met 20100 df-bl 20101 df-mopn 20102 df-top 21069 df-topon 21086 df-bases 21121 df-cmp 21561 df-ovol 23630 df-vol 23631 |
This theorem is referenced by: sge0hsphoire 41597 hoidmv1le 41602 hoidmvlelem1 41603 hoidmvlelem2 41604 hoidmvlelem3 41605 hoidmvlelem4 41606 hoidmvlelem5 41607 hoidmvle 41608 ovnhoilem2 41610 ovnhoi 41611 ovnlecvr2 41618 hspmbllem1 41634 hspmbllem2 41635 |
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