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Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > snvonmbl | Structured version Visualization version GIF version |
Description: A n-dimensional singleton is Lebesgue measurable. This is the first statement in Proposition 115G (e) of [Fremlin1] p. 32. (Contributed by Glauco Siliprandi, 8-Apr-2021.) |
Ref | Expression |
---|---|
snvonmbl.1 | ⊢ (𝜑 → 𝑋 ∈ Fin) |
snvonmbl.2 | ⊢ (𝜑 → 𝐴 ∈ (ℝ ↑m 𝑋)) |
Ref | Expression |
---|---|
snvonmbl | ⊢ (𝜑 → {𝐴} ∈ dom (voln‘𝑋)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | snvonmbl.2 | . . . 4 ⊢ (𝜑 → 𝐴 ∈ (ℝ ↑m 𝑋)) | |
2 | 1 | rrxsnicc 45474 | . . 3 ⊢ (𝜑 → X𝑘 ∈ 𝑋 ((𝐴‘𝑘)[,](𝐴‘𝑘)) = {𝐴}) |
3 | 2 | eqcomd 2737 | . 2 ⊢ (𝜑 → {𝐴} = X𝑘 ∈ 𝑋 ((𝐴‘𝑘)[,](𝐴‘𝑘))) |
4 | snvonmbl.1 | . . 3 ⊢ (𝜑 → 𝑋 ∈ Fin) | |
5 | eqid 2731 | . . 3 ⊢ dom (voln‘𝑋) = dom (voln‘𝑋) | |
6 | elmapi 8849 | . . . 4 ⊢ (𝐴 ∈ (ℝ ↑m 𝑋) → 𝐴:𝑋⟶ℝ) | |
7 | 1, 6 | syl 17 | . . 3 ⊢ (𝜑 → 𝐴:𝑋⟶ℝ) |
8 | 4, 5, 7, 7 | iccvonmbl 45853 | . 2 ⊢ (𝜑 → X𝑘 ∈ 𝑋 ((𝐴‘𝑘)[,](𝐴‘𝑘)) ∈ dom (voln‘𝑋)) |
9 | 3, 8 | eqeltrd 2832 | 1 ⊢ (𝜑 → {𝐴} ∈ dom (voln‘𝑋)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2105 {csn 4628 dom cdm 5676 ⟶wf 6539 ‘cfv 6543 (class class class)co 7412 ↑m cmap 8826 Xcixp 8897 Fincfn 8945 ℝcr 11115 [,]cicc 13334 volncvoln 45712 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2702 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7729 ax-inf2 9642 ax-cc 10436 ax-ac2 10464 ax-cnex 11172 ax-resscn 11173 ax-1cn 11174 ax-icn 11175 ax-addcl 11176 ax-addrcl 11177 ax-mulcl 11178 ax-mulrcl 11179 ax-mulcom 11180 ax-addass 11181 ax-mulass 11182 ax-distr 11183 ax-i2m1 11184 ax-1ne0 11185 ax-1rid 11186 ax-rnegex 11187 ax-rrecex 11188 ax-cnre 11189 ax-pre-lttri 11190 ax-pre-lttrn 11191 ax-pre-ltadd 11192 ax-pre-mulgt0 11193 ax-pre-sup 11194 ax-addf 11195 ax-mulf 11196 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-rmo 3375 df-reu 3376 df-rab 3432 df-v 3475 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-tp 4633 df-op 4635 df-uni 4909 df-int 4951 df-iun 4999 df-iin 5000 df-disj 5114 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-se 5632 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-isom 6552 df-riota 7368 df-ov 7415 df-oprab 7416 df-mpo 7417 df-of 7674 df-om 7860 df-1st 7979 df-2nd 7980 df-supp 8152 df-tpos 8217 df-frecs 8272 df-wrecs 8303 df-recs 8377 df-rdg 8416 df-1o 8472 df-2o 8473 df-oadd 8476 df-omul 8477 df-er 8709 df-map 8828 df-pm 8829 df-ixp 8898 df-en 8946 df-dom 8947 df-sdom 8948 df-fin 8949 df-fsupp 9368 df-fi 9412 df-sup 9443 df-inf 9444 df-oi 9511 df-dju 9902 df-card 9940 df-acn 9943 df-ac 10117 df-pnf 11257 df-mnf 11258 df-xr 11259 df-ltxr 11260 df-le 11261 df-sub 11453 df-neg 11454 df-div 11879 df-nn 12220 df-2 12282 df-3 12283 df-4 12284 df-5 12285 df-6 12286 df-7 12287 df-8 12288 df-9 12289 df-n0 12480 df-z 12566 df-dec 12685 df-uz 12830 df-q 12940 df-rp 12982 df-xneg 13099 df-xadd 13100 df-xmul 13101 df-ioo 13335 df-ico 13337 df-icc 13338 df-fz 13492 df-fzo 13635 df-fl 13764 df-seq 13974 df-exp 14035 df-hash 14298 df-cj 15053 df-re 15054 df-im 15055 df-sqrt 15189 df-abs 15190 df-clim 15439 df-rlim 15440 df-sum 15640 df-prod 15857 df-struct 17087 df-sets 17104 df-slot 17122 df-ndx 17134 df-base 17152 df-ress 17181 df-plusg 17217 df-mulr 17218 df-starv 17219 df-sca 17220 df-vsca 17221 df-ip 17222 df-tset 17223 df-ple 17224 df-ds 17226 df-unif 17227 df-hom 17228 df-cco 17229 df-rest 17375 df-topn 17376 df-0g 17394 df-gsum 17395 df-topgen 17396 df-prds 17400 df-pws 17402 df-mgm 18571 df-sgrp 18650 df-mnd 18666 df-mhm 18711 df-submnd 18712 df-grp 18864 df-minusg 18865 df-sbg 18866 df-subg 19046 df-ghm 19135 df-cntz 19229 df-cmn 19698 df-abl 19699 df-mgp 20036 df-rng 20054 df-ur 20083 df-ring 20136 df-cring 20137 df-oppr 20232 df-dvdsr 20255 df-unit 20256 df-invr 20286 df-dvr 20299 df-rhm 20370 df-subrng 20442 df-subrg 20467 df-drng 20585 df-field 20586 df-abv 20656 df-staf 20684 df-srng 20685 df-lmod 20704 df-lss 20775 df-lmhm 20865 df-lvec 20946 df-sra 21018 df-rgmod 21019 df-psmet 21224 df-xmet 21225 df-met 21226 df-bl 21227 df-mopn 21228 df-cnfld 21233 df-refld 21467 df-phl 21488 df-dsmm 21596 df-frlm 21611 df-top 22715 df-topon 22732 df-topsp 22754 df-bases 22768 df-cmp 23210 df-xms 24145 df-ms 24146 df-nm 24410 df-ngp 24411 df-tng 24412 df-nrg 24413 df-nlm 24414 df-clm 24909 df-cph 25015 df-tcph 25016 df-rrx 25232 df-ovol 25312 df-vol 25313 df-salg 45483 df-sumge0 45537 df-mea 45624 df-ome 45664 df-caragen 45666 df-ovoln 45711 df-voln 45713 |
This theorem is referenced by: ctvonmbl 45863 vonsn 45865 vonct 45867 |
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