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Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > vonct | Structured version Visualization version GIF version |
Description: The n-dimensional Lebesgue measure of any countable set is zero. This is the second statement in Proposition 115G (e) of [Fremlin1] p. 32. (Contributed by Glauco Siliprandi, 8-Apr-2021.) |
Ref | Expression |
---|---|
vonct.1 | ⊢ (𝜑 → 𝑋 ∈ Fin) |
vonct.2 | ⊢ (𝜑 → 𝐴 ⊆ (ℝ ↑m 𝑋)) |
vonct.3 | ⊢ (𝜑 → 𝐴 ≼ ω) |
Ref | Expression |
---|---|
vonct | ⊢ (𝜑 → ((voln‘𝑋)‘𝐴) = 0) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | iunid 5056 | . . . . 5 ⊢ ∪ 𝑥 ∈ 𝐴 {𝑥} = 𝐴 | |
2 | 1 | eqcomi 2740 | . . . 4 ⊢ 𝐴 = ∪ 𝑥 ∈ 𝐴 {𝑥} |
3 | 2 | fveq2i 6881 | . . 3 ⊢ ((voln‘𝑋)‘𝐴) = ((voln‘𝑋)‘∪ 𝑥 ∈ 𝐴 {𝑥}) |
4 | 3 | a1i 11 | . 2 ⊢ (𝜑 → ((voln‘𝑋)‘𝐴) = ((voln‘𝑋)‘∪ 𝑥 ∈ 𝐴 {𝑥})) |
5 | nfv 1917 | . . 3 ⊢ Ⅎ𝑥𝜑 | |
6 | vonct.1 | . . . 4 ⊢ (𝜑 → 𝑋 ∈ Fin) | |
7 | 6 | vonmea 45063 | . . 3 ⊢ (𝜑 → (voln‘𝑋) ∈ Meas) |
8 | eqid 2731 | . . 3 ⊢ dom (voln‘𝑋) = dom (voln‘𝑋) | |
9 | 6 | adantr 481 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑋 ∈ Fin) |
10 | vonct.2 | . . . . 5 ⊢ (𝜑 → 𝐴 ⊆ (ℝ ↑m 𝑋)) | |
11 | 10 | sselda 3978 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ (ℝ ↑m 𝑋)) |
12 | 9, 11 | snvonmbl 45175 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → {𝑥} ∈ dom (voln‘𝑋)) |
13 | vonct.3 | . . 3 ⊢ (𝜑 → 𝐴 ≼ ω) | |
14 | sndisj 5132 | . . . 4 ⊢ Disj 𝑥 ∈ 𝐴 {𝑥} | |
15 | 14 | a1i 11 | . . 3 ⊢ (𝜑 → Disj 𝑥 ∈ 𝐴 {𝑥}) |
16 | 5, 7, 8, 12, 13, 15 | meadjiun 44955 | . 2 ⊢ (𝜑 → ((voln‘𝑋)‘∪ 𝑥 ∈ 𝐴 {𝑥}) = (Σ^‘(𝑥 ∈ 𝐴 ↦ ((voln‘𝑋)‘{𝑥})))) |
17 | 9, 11 | vonsn 45180 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ((voln‘𝑋)‘{𝑥}) = 0) |
18 | 17 | mpteq2dva 5241 | . . . 4 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ ((voln‘𝑋)‘{𝑥})) = (𝑥 ∈ 𝐴 ↦ 0)) |
19 | 18 | fveq2d 6882 | . . 3 ⊢ (𝜑 → (Σ^‘(𝑥 ∈ 𝐴 ↦ ((voln‘𝑋)‘{𝑥}))) = (Σ^‘(𝑥 ∈ 𝐴 ↦ 0))) |
20 | 7, 8 | dmmeasal 44941 | . . . . . 6 ⊢ (𝜑 → dom (voln‘𝑋) ∈ SAlg) |
21 | 20, 13, 12 | saliuncl 44812 | . . . . 5 ⊢ (𝜑 → ∪ 𝑥 ∈ 𝐴 {𝑥} ∈ dom (voln‘𝑋)) |
22 | 1, 21 | eqeltrrid 2837 | . . . 4 ⊢ (𝜑 → 𝐴 ∈ dom (voln‘𝑋)) |
23 | 5, 22 | sge0z 44864 | . . 3 ⊢ (𝜑 → (Σ^‘(𝑥 ∈ 𝐴 ↦ 0)) = 0) |
24 | 19, 23 | eqtrd 2771 | . 2 ⊢ (𝜑 → (Σ^‘(𝑥 ∈ 𝐴 ↦ ((voln‘𝑋)‘{𝑥}))) = 0) |
25 | 4, 16, 24 | 3eqtrd 2775 | 1 ⊢ (𝜑 → ((voln‘𝑋)‘𝐴) = 0) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1541 ∈ wcel 2106 ⊆ wss 3944 {csn 4622 ∪ ciun 4990 Disj wdisj 5106 class class class wbr 5141 ↦ cmpt 5224 dom cdm 5669 ‘cfv 6532 (class class class)co 7393 ωcom 7838 ↑m cmap 8803 ≼ cdom 8920 Fincfn 8922 ℝcr 11091 0cc0 11092 Σ^csumge0 44851 volncvoln 45027 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2702 ax-rep 5278 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7708 ax-inf2 9618 ax-cc 10412 ax-ac2 10440 ax-cnex 11148 ax-resscn 11149 ax-1cn 11150 ax-icn 11151 ax-addcl 11152 ax-addrcl 11153 ax-mulcl 11154 ax-mulrcl 11155 ax-mulcom 11156 ax-addass 11157 ax-mulass 11158 ax-distr 11159 ax-i2m1 11160 ax-1ne0 11161 ax-1rid 11162 ax-rnegex 11163 ax-rrecex 11164 ax-cnre 11165 ax-pre-lttri 11166 ax-pre-lttrn 11167 ax-pre-ltadd 11168 ax-pre-mulgt0 11169 ax-pre-sup 11170 ax-addf 11171 ax-mulf 11172 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 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 3774 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3963 df-nul 4319 df-if 4523 df-pw 4598 df-sn 4623 df-pr 4625 df-tp 4627 df-op 4629 df-uni 4902 df-int 4944 df-iun 4992 df-iin 4993 df-disj 5107 df-br 5142 df-opab 5204 df-mpt 5225 df-tr 5259 df-id 5567 df-eprel 5573 df-po 5581 df-so 5582 df-fr 5624 df-se 5625 df-we 5626 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-pred 6289 df-ord 6356 df-on 6357 df-lim 6358 df-suc 6359 df-iota 6484 df-fun 6534 df-fn 6535 df-f 6536 df-f1 6537 df-fo 6538 df-f1o 6539 df-fv 6540 df-isom 6541 df-riota 7349 df-ov 7396 df-oprab 7397 df-mpo 7398 df-of 7653 df-om 7839 df-1st 7957 df-2nd 7958 df-supp 8129 df-tpos 8193 df-frecs 8248 df-wrecs 8279 df-recs 8353 df-rdg 8392 df-1o 8448 df-2o 8449 df-oadd 8452 df-omul 8453 df-er 8686 df-map 8805 df-pm 8806 df-ixp 8875 df-en 8923 df-dom 8924 df-sdom 8925 df-fin 8926 df-fsupp 9345 df-fi 9388 df-sup 9419 df-inf 9420 df-oi 9487 df-dju 9878 df-card 9916 df-acn 9919 df-ac 10093 df-pnf 11232 df-mnf 11233 df-xr 11234 df-ltxr 11235 df-le 11236 df-sub 11428 df-neg 11429 df-div 11854 df-nn 12195 df-2 12257 df-3 12258 df-4 12259 df-5 12260 df-6 12261 df-7 12262 df-8 12263 df-9 12264 df-n0 12455 df-z 12541 df-dec 12660 df-uz 12805 df-q 12915 df-rp 12957 df-xneg 13074 df-xadd 13075 df-xmul 13076 df-ioo 13310 df-ico 13312 df-icc 13313 df-fz 13467 df-fzo 13610 df-fl 13739 df-seq 13949 df-exp 14010 df-hash 14273 df-cj 15028 df-re 15029 df-im 15030 df-sqrt 15164 df-abs 15165 df-clim 15414 df-rlim 15415 df-sum 15615 df-prod 15832 df-struct 17062 df-sets 17079 df-slot 17097 df-ndx 17109 df-base 17127 df-ress 17156 df-plusg 17192 df-mulr 17193 df-starv 17194 df-sca 17195 df-vsca 17196 df-ip 17197 df-tset 17198 df-ple 17199 df-ds 17201 df-unif 17202 df-hom 17203 df-cco 17204 df-rest 17350 df-topn 17351 df-0g 17369 df-gsum 17370 df-topgen 17371 df-pt 17372 df-prds 17375 df-pws 17377 df-xrs 17430 df-qtop 17435 df-imas 17436 df-xps 17438 df-mre 17512 df-mrc 17513 df-acs 17515 df-mgm 18543 df-sgrp 18592 df-mnd 18603 df-mhm 18647 df-submnd 18648 df-grp 18797 df-minusg 18798 df-sbg 18799 df-mulg 18923 df-subg 18975 df-ghm 19056 df-cntz 19147 df-cmn 19614 df-abl 19615 df-mgp 19947 df-ur 19964 df-ring 20016 df-cring 20017 df-oppr 20102 df-dvdsr 20123 df-unit 20124 df-invr 20154 df-dvr 20165 df-rnghom 20201 df-drng 20267 df-field 20268 df-subrg 20310 df-abv 20374 df-staf 20402 df-srng 20403 df-lmod 20422 df-lss 20492 df-lmhm 20582 df-lvec 20663 df-sra 20734 df-rgmod 20735 df-psmet 20870 df-xmet 20871 df-met 20872 df-bl 20873 df-mopn 20874 df-cnfld 20879 df-refld 21091 df-phl 21112 df-dsmm 21220 df-frlm 21235 df-top 22325 df-topon 22342 df-topsp 22364 df-bases 22378 df-cn 22660 df-cnp 22661 df-cmp 22820 df-tx 22995 df-hmeo 23188 df-xms 23755 df-ms 23756 df-tms 23757 df-nm 24020 df-ngp 24021 df-tng 24022 df-nrg 24023 df-nlm 24024 df-cncf 24323 df-clm 24508 df-cph 24614 df-tcph 24615 df-rrx 24831 df-ovol 24910 df-vol 24911 df-salg 44798 df-sumge0 44852 df-mea 44939 df-ome 44979 df-caragen 44981 df-ovoln 45026 df-voln 45028 |
This theorem is referenced by: (None) |
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