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Mirrors > Home > MPE Home > Th. List > Mathboxes > ovnome | Structured version Visualization version GIF version |
Description: (voln*‘𝑋) is an outer measure on the space of multidimensional real numbers with dimension equal to the cardinality of the finite set 𝑋. Proposition 115D (a) of [Fremlin1] p. 30 . (Contributed by Glauco Siliprandi, 11-Oct-2020.) |
Ref | Expression |
---|---|
ovnome.1 | ⊢ (𝜑 → 𝑋 ∈ Fin) |
Ref | Expression |
---|---|
ovnome | ⊢ (𝜑 → (voln*‘𝑋) ∈ OutMeas) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ovexd 7208 | . 2 ⊢ (𝜑 → (ℝ ↑m 𝑋) ∈ V) | |
2 | ovnome.1 | . . 3 ⊢ (𝜑 → 𝑋 ∈ Fin) | |
3 | 2 | ovnf 43666 | . 2 ⊢ (𝜑 → (voln*‘𝑋):𝒫 (ℝ ↑m 𝑋)⟶(0[,]+∞)) |
4 | 2 | ovn0 43669 | . 2 ⊢ (𝜑 → ((voln*‘𝑋)‘∅) = 0) |
5 | 2 | 3ad2ant1 1134 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ⊆ (ℝ ↑m 𝑋) ∧ 𝑦 ⊆ 𝑥) → 𝑋 ∈ Fin) |
6 | simp3 1139 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ⊆ (ℝ ↑m 𝑋) ∧ 𝑦 ⊆ 𝑥) → 𝑦 ⊆ 𝑥) | |
7 | simp2 1138 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ⊆ (ℝ ↑m 𝑋) ∧ 𝑦 ⊆ 𝑥) → 𝑥 ⊆ (ℝ ↑m 𝑋)) | |
8 | 5, 6, 7 | ovnssle 43664 | . 2 ⊢ ((𝜑 ∧ 𝑥 ⊆ (ℝ ↑m 𝑋) ∧ 𝑦 ⊆ 𝑥) → ((voln*‘𝑋)‘𝑦) ≤ ((voln*‘𝑋)‘𝑥)) |
9 | 2 | adantr 484 | . . 3 ⊢ ((𝜑 ∧ 𝑎:ℕ⟶𝒫 (ℝ ↑m 𝑋)) → 𝑋 ∈ Fin) |
10 | simpr 488 | . . 3 ⊢ ((𝜑 ∧ 𝑎:ℕ⟶𝒫 (ℝ ↑m 𝑋)) → 𝑎:ℕ⟶𝒫 (ℝ ↑m 𝑋)) | |
11 | 9, 10 | ovnsubadd 43675 | . 2 ⊢ ((𝜑 ∧ 𝑎:ℕ⟶𝒫 (ℝ ↑m 𝑋)) → ((voln*‘𝑋)‘∪ 𝑛 ∈ ℕ (𝑎‘𝑛)) ≤ (Σ^‘(𝑛 ∈ ℕ ↦ ((voln*‘𝑋)‘(𝑎‘𝑛))))) |
12 | 1, 3, 4, 8, 11 | isomennd 43634 | 1 ⊢ (𝜑 → (voln*‘𝑋) ∈ OutMeas) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 ∧ w3a 1088 ∈ wcel 2114 Vcvv 3399 ⊆ wss 3844 𝒫 cpw 4489 ⟶wf 6336 ‘cfv 6340 (class class class)co 7173 ↑m cmap 8440 Fincfn 8558 ℝcr 10617 ℕcn 11719 OutMeascome 43592 voln*covoln 43639 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2020 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2162 ax-12 2179 ax-ext 2711 ax-rep 5155 ax-sep 5168 ax-nul 5175 ax-pow 5233 ax-pr 5297 ax-un 7482 ax-inf2 9180 ax-cc 9938 ax-ac2 9966 ax-cnex 10674 ax-resscn 10675 ax-1cn 10676 ax-icn 10677 ax-addcl 10678 ax-addrcl 10679 ax-mulcl 10680 ax-mulrcl 10681 ax-mulcom 10682 ax-addass 10683 ax-mulass 10684 ax-distr 10685 ax-i2m1 10686 ax-1ne0 10687 ax-1rid 10688 ax-rnegex 10689 ax-rrecex 10690 ax-cnre 10691 ax-pre-lttri 10692 ax-pre-lttrn 10693 ax-pre-ltadd 10694 ax-pre-mulgt0 10695 ax-pre-sup 10696 ax-addf 10697 ax-mulf 10698 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2075 df-mo 2541 df-eu 2571 df-clab 2718 df-cleq 2731 df-clel 2812 df-nfc 2882 df-ne 2936 df-nel 3040 df-ral 3059 df-rex 3060 df-reu 3061 df-rmo 3062 df-rab 3063 df-v 3401 df-sbc 3682 df-csb 3792 df-dif 3847 df-un 3849 df-in 3851 df-ss 3861 df-pss 3863 df-nul 4213 df-if 4416 df-pw 4491 df-sn 4518 df-pr 4520 df-tp 4522 df-op 4524 df-uni 4798 df-int 4838 df-iun 4884 df-disj 4997 df-br 5032 df-opab 5094 df-mpt 5112 df-tr 5138 df-id 5430 df-eprel 5435 df-po 5443 df-so 5444 df-fr 5484 df-se 5485 df-we 5486 df-xp 5532 df-rel 5533 df-cnv 5534 df-co 5535 df-dm 5536 df-rn 5537 df-res 5538 df-ima 5539 df-pred 6130 df-ord 6176 df-on 6177 df-lim 6178 df-suc 6179 df-iota 6298 df-fun 6342 df-fn 6343 df-f 6344 df-f1 6345 df-fo 6346 df-f1o 6347 df-fv 6348 df-isom 6349 df-riota 7130 df-ov 7176 df-oprab 7177 df-mpo 7178 df-of 7428 df-om 7603 df-1st 7717 df-2nd 7718 df-tpos 7924 df-wrecs 7979 df-recs 8040 df-rdg 8078 df-1o 8134 df-2o 8135 df-er 8323 df-map 8442 df-pm 8443 df-ixp 8511 df-en 8559 df-dom 8560 df-sdom 8561 df-fin 8562 df-fi 8951 df-sup 8982 df-inf 8983 df-oi 9050 df-dju 9406 df-card 9444 df-acn 9447 df-ac 9619 df-pnf 10758 df-mnf 10759 df-xr 10760 df-ltxr 10761 df-le 10762 df-sub 10953 df-neg 10954 df-div 11379 df-nn 11720 df-2 11782 df-3 11783 df-4 11784 df-5 11785 df-6 11786 df-7 11787 df-8 11788 df-9 11789 df-n0 11980 df-z 12066 df-dec 12183 df-uz 12328 df-q 12434 df-rp 12476 df-xneg 12593 df-xadd 12594 df-xmul 12595 df-ioo 12828 df-ico 12830 df-icc 12831 df-fz 12985 df-fzo 13128 df-fl 13256 df-seq 13464 df-exp 13525 df-hash 13786 df-cj 14551 df-re 14552 df-im 14553 df-sqrt 14687 df-abs 14688 df-clim 14938 df-rlim 14939 df-sum 15139 df-prod 15355 df-struct 16591 df-ndx 16592 df-slot 16593 df-base 16595 df-sets 16596 df-ress 16597 df-plusg 16684 df-mulr 16685 df-starv 16686 df-tset 16690 df-ple 16691 df-ds 16693 df-unif 16694 df-rest 16802 df-0g 16821 df-topgen 16823 df-mgm 17971 df-sgrp 18020 df-mnd 18031 df-grp 18225 df-minusg 18226 df-subg 18397 df-cmn 19029 df-abl 19030 df-mgp 19362 df-ur 19374 df-ring 19421 df-cring 19422 df-oppr 19498 df-dvdsr 19516 df-unit 19517 df-invr 19547 df-dvr 19558 df-drng 19626 df-psmet 20212 df-xmet 20213 df-met 20214 df-bl 20215 df-mopn 20216 df-cnfld 20221 df-top 21648 df-topon 21665 df-bases 21700 df-cmp 22141 df-ovol 24219 df-vol 24220 df-sumge0 43466 df-ome 43593 df-ovoln 43640 |
This theorem is referenced by: vonmea 43677 dmvon 43709 rrnmbl 43717 unidmvon 43720 voncmpl 43724 hspmbl 43732 isvonmbl 43741 vonmblss 43743 |
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