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Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > hoiprodcl | Structured version Visualization version GIF version |
Description: The pre-measure of half-open intervals is a nonnegative real. (Contributed by Glauco Siliprandi, 11-Oct-2020.) |
Ref | Expression |
---|---|
hoiprodcl.1 | ⊢ Ⅎ𝑘𝜑 |
hoiprodcl.2 | ⊢ (𝜑 → 𝑋 ∈ Fin) |
hoiprodcl.3 | ⊢ (𝜑 → 𝐼:𝑋⟶(ℝ × ℝ)) |
Ref | Expression |
---|---|
hoiprodcl | ⊢ (𝜑 → ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ 𝐼)‘𝑘)) ∈ (0[,)+∞)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 0xr 11293 | . . 3 ⊢ 0 ∈ ℝ* | |
2 | 1 | a1i 11 | . 2 ⊢ (𝜑 → 0 ∈ ℝ*) |
3 | pnfxr 11300 | . . 3 ⊢ +∞ ∈ ℝ* | |
4 | 3 | a1i 11 | . 2 ⊢ (𝜑 → +∞ ∈ ℝ*) |
5 | hoiprodcl.1 | . . . 4 ⊢ Ⅎ𝑘𝜑 | |
6 | hoiprodcl.2 | . . . 4 ⊢ (𝜑 → 𝑋 ∈ Fin) | |
7 | hoiprodcl.3 | . . . . . . . . 9 ⊢ (𝜑 → 𝐼:𝑋⟶(ℝ × ℝ)) | |
8 | 7 | adantr 479 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → 𝐼:𝑋⟶(ℝ × ℝ)) |
9 | simpr 483 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → 𝑘 ∈ 𝑋) | |
10 | 8, 9 | fvovco 44705 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → (([,) ∘ 𝐼)‘𝑘) = ((1st ‘(𝐼‘𝑘))[,)(2nd ‘(𝐼‘𝑘)))) |
11 | 10 | fveq2d 6900 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → (vol‘(([,) ∘ 𝐼)‘𝑘)) = (vol‘((1st ‘(𝐼‘𝑘))[,)(2nd ‘(𝐼‘𝑘))))) |
12 | 7 | ffvelcdmda 7093 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → (𝐼‘𝑘) ∈ (ℝ × ℝ)) |
13 | xp1st 8026 | . . . . . . . 8 ⊢ ((𝐼‘𝑘) ∈ (ℝ × ℝ) → (1st ‘(𝐼‘𝑘)) ∈ ℝ) | |
14 | 12, 13 | syl 17 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → (1st ‘(𝐼‘𝑘)) ∈ ℝ) |
15 | xp2nd 8027 | . . . . . . . 8 ⊢ ((𝐼‘𝑘) ∈ (ℝ × ℝ) → (2nd ‘(𝐼‘𝑘)) ∈ ℝ) | |
16 | 12, 15 | syl 17 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → (2nd ‘(𝐼‘𝑘)) ∈ ℝ) |
17 | volico 45509 | . . . . . . 7 ⊢ (((1st ‘(𝐼‘𝑘)) ∈ ℝ ∧ (2nd ‘(𝐼‘𝑘)) ∈ ℝ) → (vol‘((1st ‘(𝐼‘𝑘))[,)(2nd ‘(𝐼‘𝑘)))) = if((1st ‘(𝐼‘𝑘)) < (2nd ‘(𝐼‘𝑘)), ((2nd ‘(𝐼‘𝑘)) − (1st ‘(𝐼‘𝑘))), 0)) | |
18 | 14, 16, 17 | syl2anc 582 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → (vol‘((1st ‘(𝐼‘𝑘))[,)(2nd ‘(𝐼‘𝑘)))) = if((1st ‘(𝐼‘𝑘)) < (2nd ‘(𝐼‘𝑘)), ((2nd ‘(𝐼‘𝑘)) − (1st ‘(𝐼‘𝑘))), 0)) |
19 | 11, 18 | eqtrd 2765 | . . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → (vol‘(([,) ∘ 𝐼)‘𝑘)) = if((1st ‘(𝐼‘𝑘)) < (2nd ‘(𝐼‘𝑘)), ((2nd ‘(𝐼‘𝑘)) − (1st ‘(𝐼‘𝑘))), 0)) |
20 | 16, 14 | resubcld 11674 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → ((2nd ‘(𝐼‘𝑘)) − (1st ‘(𝐼‘𝑘))) ∈ ℝ) |
21 | 0red 11249 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → 0 ∈ ℝ) | |
22 | 20, 21 | ifcld 4576 | . . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → if((1st ‘(𝐼‘𝑘)) < (2nd ‘(𝐼‘𝑘)), ((2nd ‘(𝐼‘𝑘)) − (1st ‘(𝐼‘𝑘))), 0) ∈ ℝ) |
23 | 19, 22 | eqeltrd 2825 | . . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → (vol‘(([,) ∘ 𝐼)‘𝑘)) ∈ ℝ) |
24 | 5, 6, 23 | fprodreclf 15939 | . . 3 ⊢ (𝜑 → ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ 𝐼)‘𝑘)) ∈ ℝ) |
25 | 24 | rexrd 11296 | . 2 ⊢ (𝜑 → ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ 𝐼)‘𝑘)) ∈ ℝ*) |
26 | 16 | rexrd 11296 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → (2nd ‘(𝐼‘𝑘)) ∈ ℝ*) |
27 | icombl 25537 | . . . . . 6 ⊢ (((1st ‘(𝐼‘𝑘)) ∈ ℝ ∧ (2nd ‘(𝐼‘𝑘)) ∈ ℝ*) → ((1st ‘(𝐼‘𝑘))[,)(2nd ‘(𝐼‘𝑘))) ∈ dom vol) | |
28 | 14, 26, 27 | syl2anc 582 | . . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → ((1st ‘(𝐼‘𝑘))[,)(2nd ‘(𝐼‘𝑘))) ∈ dom vol) |
29 | 10, 28 | eqeltrd 2825 | . . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → (([,) ∘ 𝐼)‘𝑘) ∈ dom vol) |
30 | volge0 45487 | . . . 4 ⊢ ((([,) ∘ 𝐼)‘𝑘) ∈ dom vol → 0 ≤ (vol‘(([,) ∘ 𝐼)‘𝑘))) | |
31 | 29, 30 | syl 17 | . . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → 0 ≤ (vol‘(([,) ∘ 𝐼)‘𝑘))) |
32 | 5, 6, 23, 31 | fprodge0 15973 | . 2 ⊢ (𝜑 → 0 ≤ ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ 𝐼)‘𝑘))) |
33 | 24 | ltpnfd 13136 | . 2 ⊢ (𝜑 → ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ 𝐼)‘𝑘)) < +∞) |
34 | 2, 4, 25, 32, 33 | elicod 13409 | 1 ⊢ (𝜑 → ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ 𝐼)‘𝑘)) ∈ (0[,)+∞)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 394 = wceq 1533 Ⅎwnf 1777 ∈ wcel 2098 ifcif 4530 class class class wbr 5149 × cxp 5676 dom cdm 5678 ∘ ccom 5682 ⟶wf 6545 ‘cfv 6549 (class class class)co 7419 1st c1st 7992 2nd c2nd 7993 Fincfn 8964 ℝcr 11139 0cc0 11140 +∞cpnf 11277 ℝ*cxr 11279 < clt 11280 ≤ cle 11281 − cmin 11476 [,)cico 13361 ∏cprod 15885 volcvol 25436 |
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 2696 ax-rep 5286 ax-sep 5300 ax-nul 5307 ax-pow 5365 ax-pr 5429 ax-un 7741 ax-inf2 9666 ax-cnex 11196 ax-resscn 11197 ax-1cn 11198 ax-icn 11199 ax-addcl 11200 ax-addrcl 11201 ax-mulcl 11202 ax-mulrcl 11203 ax-mulcom 11204 ax-addass 11205 ax-mulass 11206 ax-distr 11207 ax-i2m1 11208 ax-1ne0 11209 ax-1rid 11210 ax-rnegex 11211 ax-rrecex 11212 ax-cnre 11213 ax-pre-lttri 11214 ax-pre-lttrn 11215 ax-pre-ltadd 11216 ax-pre-mulgt0 11217 ax-pre-sup 11218 |
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 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2930 df-nel 3036 df-ral 3051 df-rex 3060 df-rmo 3363 df-reu 3364 df-rab 3419 df-v 3463 df-sbc 3774 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3964 df-nul 4323 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4910 df-int 4951 df-iun 4999 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-id 5576 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5633 df-se 5634 df-we 5635 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-pred 6307 df-ord 6374 df-on 6375 df-lim 6376 df-suc 6377 df-iota 6501 df-fun 6551 df-fn 6552 df-f 6553 df-f1 6554 df-fo 6555 df-f1o 6556 df-fv 6557 df-isom 6558 df-riota 7375 df-ov 7422 df-oprab 7423 df-mpo 7424 df-of 7685 df-om 7872 df-1st 7994 df-2nd 7995 df-frecs 8287 df-wrecs 8318 df-recs 8392 df-rdg 8431 df-1o 8487 df-2o 8488 df-er 8725 df-map 8847 df-pm 8848 df-en 8965 df-dom 8966 df-sdom 8967 df-fin 8968 df-fi 9436 df-sup 9467 df-inf 9468 df-oi 9535 df-dju 9926 df-card 9964 df-pnf 11282 df-mnf 11283 df-xr 11284 df-ltxr 11285 df-le 11286 df-sub 11478 df-neg 11479 df-div 11904 df-nn 12246 df-2 12308 df-3 12309 df-n0 12506 df-z 12592 df-uz 12856 df-q 12966 df-rp 13010 df-xneg 13127 df-xadd 13128 df-xmul 13129 df-ioo 13363 df-ico 13365 df-icc 13366 df-fz 13520 df-fzo 13663 df-fl 13793 df-seq 14003 df-exp 14063 df-hash 14326 df-cj 15082 df-re 15083 df-im 15084 df-sqrt 15218 df-abs 15219 df-clim 15468 df-rlim 15469 df-sum 15669 df-prod 15886 df-rest 17407 df-topgen 17428 df-psmet 21288 df-xmet 21289 df-met 21290 df-bl 21291 df-mopn 21292 df-top 22840 df-topon 22857 df-bases 22893 df-cmp 23335 df-ovol 25437 df-vol 25438 |
This theorem is referenced by: ovnprodcl 46080 hoiprodcl2 46081 ovnhoilem1 46127 |
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