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 42866 | . 2 ⊢ (𝜑 → (𝐴(𝐿‘𝑋)𝐵) = if(𝑋 = ∅, 0, ∏𝑘 ∈ 𝑋 (vol‘((𝐴‘𝑘)[,)(𝐵‘𝑘))))) |
6 | 0e0icopnf 12849 | . . . 4 ⊢ 0 ∈ (0[,)+∞) | |
7 | 6 | a1i 11 | . . 3 ⊢ (𝜑 → 0 ∈ (0[,)+∞)) |
8 | 0xr 10691 | . . . . 5 ⊢ 0 ∈ ℝ* | |
9 | 8 | a1i 11 | . . . 4 ⊢ (𝜑 → 0 ∈ ℝ*) |
10 | pnfxr 10698 | . . . . 5 ⊢ +∞ ∈ ℝ* | |
11 | 10 | a1i 11 | . . . 4 ⊢ (𝜑 → +∞ ∈ ℝ*) |
12 | 2 | ffvelrnda 6854 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → (𝐴‘𝑘) ∈ ℝ) |
13 | 3 | ffvelrnda 6854 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → (𝐵‘𝑘) ∈ ℝ) |
14 | volico 42275 | . . . . . . . 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 586 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → (vol‘((𝐴‘𝑘)[,)(𝐵‘𝑘))) = if((𝐴‘𝑘) < (𝐵‘𝑘), ((𝐵‘𝑘) − (𝐴‘𝑘)), 0)) |
16 | 13, 12 | resubcld 11071 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → ((𝐵‘𝑘) − (𝐴‘𝑘)) ∈ ℝ) |
17 | 0red 10647 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → 0 ∈ ℝ) | |
18 | 16, 17 | ifcld 4515 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → if((𝐴‘𝑘) < (𝐵‘𝑘), ((𝐵‘𝑘) − (𝐴‘𝑘)), 0) ∈ ℝ) |
19 | 15, 18 | eqeltrd 2916 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → (vol‘((𝐴‘𝑘)[,)(𝐵‘𝑘))) ∈ ℝ) |
20 | 4, 19 | fprodrecl 15310 | . . . . 5 ⊢ (𝜑 → ∏𝑘 ∈ 𝑋 (vol‘((𝐴‘𝑘)[,)(𝐵‘𝑘))) ∈ ℝ) |
21 | 20 | rexrd 10694 | . . . 4 ⊢ (𝜑 → ∏𝑘 ∈ 𝑋 (vol‘((𝐴‘𝑘)[,)(𝐵‘𝑘))) ∈ ℝ*) |
22 | nfv 1914 | . . . . 5 ⊢ Ⅎ𝑘𝜑 | |
23 | 13 | rexrd 10694 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → (𝐵‘𝑘) ∈ ℝ*) |
24 | icombl 24168 | . . . . . . 7 ⊢ (((𝐴‘𝑘) ∈ ℝ ∧ (𝐵‘𝑘) ∈ ℝ*) → ((𝐴‘𝑘)[,)(𝐵‘𝑘)) ∈ dom vol) | |
25 | 12, 23, 24 | syl2anc 586 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → ((𝐴‘𝑘)[,)(𝐵‘𝑘)) ∈ dom vol) |
26 | volge0 42252 | . . . . . 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 15350 | . . . 4 ⊢ (𝜑 → 0 ≤ ∏𝑘 ∈ 𝑋 (vol‘((𝐴‘𝑘)[,)(𝐵‘𝑘)))) |
29 | 20 | ltpnfd 12519 | . . . 4 ⊢ (𝜑 → ∏𝑘 ∈ 𝑋 (vol‘((𝐴‘𝑘)[,)(𝐵‘𝑘))) < +∞) |
30 | 9, 11, 21, 28, 29 | elicod 12790 | . . 3 ⊢ (𝜑 → ∏𝑘 ∈ 𝑋 (vol‘((𝐴‘𝑘)[,)(𝐵‘𝑘))) ∈ (0[,)+∞)) |
31 | 7, 30 | ifcld 4515 | . 2 ⊢ (𝜑 → if(𝑋 = ∅, 0, ∏𝑘 ∈ 𝑋 (vol‘((𝐴‘𝑘)[,)(𝐵‘𝑘)))) ∈ (0[,)+∞)) |
32 | 5, 31 | eqeltrd 2916 | 1 ⊢ (𝜑 → (𝐴(𝐿‘𝑋)𝐵) ∈ (0[,)+∞)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1536 ∈ wcel 2113 ∅c0 4294 ifcif 4470 class class class wbr 5069 ↦ cmpt 5149 dom cdm 5558 ⟶wf 6354 ‘cfv 6358 (class class class)co 7159 ∈ cmpo 7161 ↑m cmap 8409 Fincfn 8512 ℝcr 10539 0cc0 10540 +∞cpnf 10675 ℝ*cxr 10677 < clt 10678 ≤ cle 10679 − cmin 10873 [,)cico 12743 ∏cprod 15262 volcvol 24067 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2796 ax-rep 5193 ax-sep 5206 ax-nul 5213 ax-pow 5269 ax-pr 5333 ax-un 7464 ax-inf2 9107 ax-cnex 10596 ax-resscn 10597 ax-1cn 10598 ax-icn 10599 ax-addcl 10600 ax-addrcl 10601 ax-mulcl 10602 ax-mulrcl 10603 ax-mulcom 10604 ax-addass 10605 ax-mulass 10606 ax-distr 10607 ax-i2m1 10608 ax-1ne0 10609 ax-1rid 10610 ax-rnegex 10611 ax-rrecex 10612 ax-cnre 10613 ax-pre-lttri 10614 ax-pre-lttrn 10615 ax-pre-ltadd 10616 ax-pre-mulgt0 10617 ax-pre-sup 10618 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1539 df-fal 1549 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2803 df-cleq 2817 df-clel 2896 df-nfc 2966 df-ne 3020 df-nel 3127 df-ral 3146 df-rex 3147 df-reu 3148 df-rmo 3149 df-rab 3150 df-v 3499 df-sbc 3776 df-csb 3887 df-dif 3942 df-un 3944 df-in 3946 df-ss 3955 df-pss 3957 df-nul 4295 df-if 4471 df-pw 4544 df-sn 4571 df-pr 4573 df-tp 4575 df-op 4577 df-uni 4842 df-int 4880 df-iun 4924 df-br 5070 df-opab 5132 df-mpt 5150 df-tr 5176 df-id 5463 df-eprel 5468 df-po 5477 df-so 5478 df-fr 5517 df-se 5518 df-we 5519 df-xp 5564 df-rel 5565 df-cnv 5566 df-co 5567 df-dm 5568 df-rn 5569 df-res 5570 df-ima 5571 df-pred 6151 df-ord 6197 df-on 6198 df-lim 6199 df-suc 6200 df-iota 6317 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-isom 6367 df-riota 7117 df-ov 7162 df-oprab 7163 df-mpo 7164 df-of 7412 df-om 7584 df-1st 7692 df-2nd 7693 df-wrecs 7950 df-recs 8011 df-rdg 8049 df-1o 8105 df-2o 8106 df-oadd 8109 df-er 8292 df-map 8411 df-pm 8412 df-en 8513 df-dom 8514 df-sdom 8515 df-fin 8516 df-fi 8878 df-sup 8909 df-inf 8910 df-oi 8977 df-dju 9333 df-card 9371 df-pnf 10680 df-mnf 10681 df-xr 10682 df-ltxr 10683 df-le 10684 df-sub 10875 df-neg 10876 df-div 11301 df-nn 11642 df-2 11703 df-3 11704 df-n0 11901 df-z 11985 df-uz 12247 df-q 12352 df-rp 12393 df-xneg 12510 df-xadd 12511 df-xmul 12512 df-ioo 12745 df-ico 12747 df-icc 12748 df-fz 12896 df-fzo 13037 df-fl 13165 df-seq 13373 df-exp 13433 df-hash 13694 df-cj 14461 df-re 14462 df-im 14463 df-sqrt 14597 df-abs 14598 df-clim 14848 df-rlim 14849 df-sum 15046 df-prod 15263 df-rest 16699 df-topgen 16720 df-psmet 20540 df-xmet 20541 df-met 20542 df-bl 20543 df-mopn 20544 df-top 21505 df-topon 21522 df-bases 21557 df-cmp 21998 df-ovol 24068 df-vol 24069 |
This theorem is referenced by: sge0hsphoire 42878 hoidmv1le 42883 hoidmvlelem1 42884 hoidmvlelem2 42885 hoidmvlelem3 42886 hoidmvlelem4 42887 hoidmvlelem5 42888 hoidmvle 42889 ovnhoilem2 42891 ovnhoi 42892 ovnlecvr2 42899 hspmbllem1 42915 hspmbllem2 42916 |
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