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Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > ovncl | Structured version Visualization version GIF version |
Description: The Lebesgue outer measure of a set is a nonnegative extended real. (Contributed by Glauco Siliprandi, 11-Oct-2020.) |
Ref | Expression |
---|---|
ovncl.1 | ⊢ (𝜑 → 𝑋 ∈ Fin) |
ovncl.2 | ⊢ (𝜑 → 𝐴 ⊆ (ℝ ↑𝑚 𝑋)) |
Ref | Expression |
---|---|
ovncl | ⊢ (𝜑 → ((voln*‘𝑋)‘𝐴) ∈ (0[,]+∞)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ovncl.1 | . . 3 ⊢ (𝜑 → 𝑋 ∈ Fin) | |
2 | 1 | ovnf 41297 | . 2 ⊢ (𝜑 → (voln*‘𝑋):𝒫 (ℝ ↑𝑚 𝑋)⟶(0[,]+∞)) |
3 | ovncl.2 | . . 3 ⊢ (𝜑 → 𝐴 ⊆ (ℝ ↑𝑚 𝑋)) | |
4 | ovexd 6825 | . . . . 5 ⊢ (𝜑 → (ℝ ↑𝑚 𝑋) ∈ V) | |
5 | 4, 3 | ssexd 4939 | . . . 4 ⊢ (𝜑 → 𝐴 ∈ V) |
6 | elpwg 4305 | . . . 4 ⊢ (𝐴 ∈ V → (𝐴 ∈ 𝒫 (ℝ ↑𝑚 𝑋) ↔ 𝐴 ⊆ (ℝ ↑𝑚 𝑋))) | |
7 | 5, 6 | syl 17 | . . 3 ⊢ (𝜑 → (𝐴 ∈ 𝒫 (ℝ ↑𝑚 𝑋) ↔ 𝐴 ⊆ (ℝ ↑𝑚 𝑋))) |
8 | 3, 7 | mpbird 247 | . 2 ⊢ (𝜑 → 𝐴 ∈ 𝒫 (ℝ ↑𝑚 𝑋)) |
9 | 2, 8 | ffvelrnd 6503 | 1 ⊢ (𝜑 → ((voln*‘𝑋)‘𝐴) ∈ (0[,]+∞)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 196 ∈ wcel 2145 Vcvv 3351 ⊆ wss 3723 𝒫 cpw 4297 ‘cfv 6031 (class class class)co 6793 ↑𝑚 cmap 8009 Fincfn 8109 ℝcr 10137 0cc0 10138 +∞cpnf 10273 [,]cicc 12383 voln*covoln 41270 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1870 ax-4 1885 ax-5 1991 ax-6 2057 ax-7 2093 ax-8 2147 ax-9 2154 ax-10 2174 ax-11 2190 ax-12 2203 ax-13 2408 ax-ext 2751 ax-rep 4904 ax-sep 4915 ax-nul 4923 ax-pow 4974 ax-pr 5034 ax-un 7096 ax-inf2 8702 ax-cnex 10194 ax-resscn 10195 ax-1cn 10196 ax-icn 10197 ax-addcl 10198 ax-addrcl 10199 ax-mulcl 10200 ax-mulrcl 10201 ax-mulcom 10202 ax-addass 10203 ax-mulass 10204 ax-distr 10205 ax-i2m1 10206 ax-1ne0 10207 ax-1rid 10208 ax-rnegex 10209 ax-rrecex 10210 ax-cnre 10211 ax-pre-lttri 10212 ax-pre-lttrn 10213 ax-pre-ltadd 10214 ax-pre-mulgt0 10215 ax-pre-sup 10216 |
This theorem depends on definitions: df-bi 197 df-an 383 df-or 837 df-3or 1072 df-3an 1073 df-tru 1634 df-fal 1637 df-ex 1853 df-nf 1858 df-sb 2050 df-eu 2622 df-mo 2623 df-clab 2758 df-cleq 2764 df-clel 2767 df-nfc 2902 df-ne 2944 df-nel 3047 df-ral 3066 df-rex 3067 df-reu 3068 df-rmo 3069 df-rab 3070 df-v 3353 df-sbc 3588 df-csb 3683 df-dif 3726 df-un 3728 df-in 3730 df-ss 3737 df-pss 3739 df-nul 4064 df-if 4226 df-pw 4299 df-sn 4317 df-pr 4319 df-tp 4321 df-op 4323 df-uni 4575 df-int 4612 df-iun 4656 df-br 4787 df-opab 4847 df-mpt 4864 df-tr 4887 df-id 5157 df-eprel 5162 df-po 5170 df-so 5171 df-fr 5208 df-se 5209 df-we 5210 df-xp 5255 df-rel 5256 df-cnv 5257 df-co 5258 df-dm 5259 df-rn 5260 df-res 5261 df-ima 5262 df-pred 5823 df-ord 5869 df-on 5870 df-lim 5871 df-suc 5872 df-iota 5994 df-fun 6033 df-fn 6034 df-f 6035 df-f1 6036 df-fo 6037 df-f1o 6038 df-fv 6039 df-isom 6040 df-riota 6754 df-ov 6796 df-oprab 6797 df-mpt2 6798 df-of 7044 df-om 7213 df-1st 7315 df-2nd 7316 df-wrecs 7559 df-recs 7621 df-rdg 7659 df-1o 7713 df-2o 7714 df-oadd 7717 df-er 7896 df-map 8011 df-pm 8012 df-ixp 8063 df-en 8110 df-dom 8111 df-sdom 8112 df-fin 8113 df-fi 8473 df-sup 8504 df-inf 8505 df-oi 8571 df-card 8965 df-cda 9192 df-pnf 10278 df-mnf 10279 df-xr 10280 df-ltxr 10281 df-le 10282 df-sub 10470 df-neg 10471 df-div 10887 df-nn 11223 df-2 11281 df-3 11282 df-n0 11495 df-z 11580 df-uz 11889 df-q 11992 df-rp 12036 df-xneg 12151 df-xadd 12152 df-xmul 12153 df-ioo 12384 df-ico 12386 df-icc 12387 df-fz 12534 df-fzo 12674 df-fl 12801 df-seq 13009 df-exp 13068 df-hash 13322 df-cj 14047 df-re 14048 df-im 14049 df-sqrt 14183 df-abs 14184 df-clim 14427 df-rlim 14428 df-sum 14625 df-prod 14843 df-rest 16291 df-topgen 16312 df-psmet 19953 df-xmet 19954 df-met 19955 df-bl 19956 df-mopn 19957 df-top 20919 df-topon 20936 df-bases 20971 df-cmp 21411 df-ovol 23452 df-vol 23453 df-sumge0 41097 df-ovoln 41271 |
This theorem is referenced by: ovnxrcl 41303 ovnsubaddlem1 41304 ovnsubadd 41306 hspmbllem3 41362 hspmbl 41363 ovnsubadd2lem 41379 |
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