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Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > rrnmbl | Structured version Visualization version GIF version |
Description: The set of n-dimensional Real numbers is Lebesgue measurable. (Contributed by Glauco Siliprandi, 24-Dec-2020.) |
Ref | Expression |
---|---|
rrnmbl.1 | ⊢ (𝜑 → 𝑋 ∈ Fin) |
Ref | Expression |
---|---|
rrnmbl | ⊢ (𝜑 → (ℝ ↑m 𝑋) ∈ dom (voln‘𝑋)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rrnmbl.1 | . . . 4 ⊢ (𝜑 → 𝑋 ∈ Fin) | |
2 | 1 | ovnome 46429 | . . 3 ⊢ (𝜑 → (voln*‘𝑋) ∈ OutMeas) |
3 | eqid 2734 | . . 3 ⊢ ∪ dom (voln*‘𝑋) = ∪ dom (voln*‘𝑋) | |
4 | eqid 2734 | . . 3 ⊢ (CaraGen‘(voln*‘𝑋)) = (CaraGen‘(voln*‘𝑋)) | |
5 | 2, 3, 4 | caragenunidm 46364 | . 2 ⊢ (𝜑 → ∪ dom (voln*‘𝑋) ∈ (CaraGen‘(voln*‘𝑋))) |
6 | 1 | dmovn 46460 | . . . . 5 ⊢ (𝜑 → dom (voln*‘𝑋) = 𝒫 (ℝ ↑m 𝑋)) |
7 | 6 | unieqd 4944 | . . . 4 ⊢ (𝜑 → ∪ dom (voln*‘𝑋) = ∪ 𝒫 (ℝ ↑m 𝑋)) |
8 | unipw 5473 | . . . . 5 ⊢ ∪ 𝒫 (ℝ ↑m 𝑋) = (ℝ ↑m 𝑋) | |
9 | 8 | a1i 11 | . . . 4 ⊢ (𝜑 → ∪ 𝒫 (ℝ ↑m 𝑋) = (ℝ ↑m 𝑋)) |
10 | 7, 9 | eqtr2d 2775 | . . 3 ⊢ (𝜑 → (ℝ ↑m 𝑋) = ∪ dom (voln*‘𝑋)) |
11 | 1 | dmvon 46462 | . . 3 ⊢ (𝜑 → dom (voln‘𝑋) = (CaraGen‘(voln*‘𝑋))) |
12 | 10, 11 | eleq12d 2832 | . 2 ⊢ (𝜑 → ((ℝ ↑m 𝑋) ∈ dom (voln‘𝑋) ↔ ∪ dom (voln*‘𝑋) ∈ (CaraGen‘(voln*‘𝑋)))) |
13 | 5, 12 | mpbird 257 | 1 ⊢ (𝜑 → (ℝ ↑m 𝑋) ∈ dom (voln‘𝑋)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 ∈ wcel 2103 𝒫 cpw 4622 ∪ cuni 4931 dom cdm 5699 ‘cfv 6572 (class class class)co 7445 ↑m cmap 8880 Fincfn 8999 ℝcr 11179 CaraGenccaragen 46347 voln*covoln 46392 volncvoln 46394 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 ax-5 1909 ax-6 1967 ax-7 2007 ax-8 2105 ax-9 2113 ax-10 2136 ax-11 2153 ax-12 2173 ax-ext 2705 ax-rep 5306 ax-sep 5320 ax-nul 5327 ax-pow 5386 ax-pr 5450 ax-un 7766 ax-inf2 9706 ax-cc 10500 ax-ac2 10528 ax-cnex 11236 ax-resscn 11237 ax-1cn 11238 ax-icn 11239 ax-addcl 11240 ax-addrcl 11241 ax-mulcl 11242 ax-mulrcl 11243 ax-mulcom 11244 ax-addass 11245 ax-mulass 11246 ax-distr 11247 ax-i2m1 11248 ax-1ne0 11249 ax-1rid 11250 ax-rnegex 11251 ax-rrecex 11252 ax-cnre 11253 ax-pre-lttri 11254 ax-pre-lttrn 11255 ax-pre-ltadd 11256 ax-pre-mulgt0 11257 ax-pre-sup 11258 ax-addf 11259 ax-mulf 11260 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-3or 1088 df-3an 1089 df-tru 1540 df-fal 1550 df-ex 1778 df-nf 1782 df-sb 2065 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2726 df-clel 2813 df-nfc 2890 df-ne 2943 df-nel 3049 df-ral 3064 df-rex 3073 df-rmo 3383 df-reu 3384 df-rab 3439 df-v 3484 df-sbc 3799 df-csb 3916 df-dif 3973 df-un 3975 df-in 3977 df-ss 3987 df-pss 3990 df-nul 4348 df-if 4549 df-pw 4624 df-sn 4649 df-pr 4651 df-tp 4653 df-op 4655 df-uni 4932 df-int 4973 df-iun 5021 df-disj 5137 df-br 5170 df-opab 5232 df-mpt 5253 df-tr 5287 df-id 5597 df-eprel 5603 df-po 5611 df-so 5612 df-fr 5654 df-se 5655 df-we 5656 df-xp 5705 df-rel 5706 df-cnv 5707 df-co 5708 df-dm 5709 df-rn 5710 df-res 5711 df-ima 5712 df-pred 6331 df-ord 6397 df-on 6398 df-lim 6399 df-suc 6400 df-iota 6524 df-fun 6574 df-fn 6575 df-f 6576 df-f1 6577 df-fo 6578 df-f1o 6579 df-fv 6580 df-isom 6581 df-riota 7401 df-ov 7448 df-oprab 7449 df-mpo 7450 df-of 7710 df-om 7900 df-1st 8026 df-2nd 8027 df-tpos 8263 df-frecs 8318 df-wrecs 8349 df-recs 8423 df-rdg 8462 df-1o 8518 df-2o 8519 df-er 8759 df-map 8882 df-pm 8883 df-ixp 8952 df-en 9000 df-dom 9001 df-sdom 9002 df-fin 9003 df-fi 9476 df-sup 9507 df-inf 9508 df-oi 9575 df-dju 9966 df-card 10004 df-acn 10007 df-ac 10181 df-pnf 11322 df-mnf 11323 df-xr 11324 df-ltxr 11325 df-le 11326 df-sub 11518 df-neg 11519 df-div 11944 df-nn 12290 df-2 12352 df-3 12353 df-4 12354 df-5 12355 df-6 12356 df-7 12357 df-8 12358 df-9 12359 df-n0 12550 df-z 12636 df-dec 12755 df-uz 12900 df-q 13010 df-rp 13054 df-xneg 13171 df-xadd 13172 df-xmul 13173 df-ioo 13407 df-ico 13409 df-icc 13410 df-fz 13564 df-fzo 13708 df-fl 13839 df-seq 14049 df-exp 14109 df-hash 14376 df-cj 15144 df-re 15145 df-im 15146 df-sqrt 15280 df-abs 15281 df-clim 15530 df-rlim 15531 df-sum 15731 df-prod 15948 df-struct 17189 df-sets 17206 df-slot 17224 df-ndx 17236 df-base 17254 df-ress 17283 df-plusg 17319 df-mulr 17320 df-starv 17321 df-tset 17325 df-ple 17326 df-ds 17328 df-unif 17329 df-rest 17477 df-0g 17496 df-topgen 17498 df-mgm 18673 df-sgrp 18752 df-mnd 18768 df-grp 18971 df-minusg 18972 df-subg 19158 df-cmn 19819 df-abl 19820 df-mgp 20157 df-rng 20175 df-ur 20204 df-ring 20257 df-cring 20258 df-oppr 20355 df-dvdsr 20378 df-unit 20379 df-invr 20409 df-dvr 20422 df-drng 20748 df-psmet 21374 df-xmet 21375 df-met 21376 df-bl 21377 df-mopn 21378 df-cnfld 21383 df-top 22914 df-topon 22931 df-bases 22967 df-cmp 23409 df-ovol 25511 df-vol 25512 df-sumge0 46219 df-ome 46346 df-caragen 46348 df-ovoln 46393 df-voln 46395 |
This theorem is referenced by: hoimbl 46487 |
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