Mathbox for Glauco Siliprandi |
< Previous
Next >
Nearby theorems |
||
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 4975 | . . . . 5 ⊢ ∪ 𝑥 ∈ 𝐴 {𝑥} = 𝐴 | |
2 | 1 | eqcomi 2827 | . . . 4 ⊢ 𝐴 = ∪ 𝑥 ∈ 𝐴 {𝑥} |
3 | 2 | fveq2i 6666 | . . 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 1906 | . . 3 ⊢ Ⅎ𝑥𝜑 | |
6 | vonct.1 | . . . 4 ⊢ (𝜑 → 𝑋 ∈ Fin) | |
7 | 6 | vonmea 42733 | . . 3 ⊢ (𝜑 → (voln‘𝑋) ∈ Meas) |
8 | eqid 2818 | . . 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 3964 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ (ℝ ↑m 𝑋)) |
12 | 9, 11 | snvonmbl 42845 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → {𝑥} ∈ dom (voln‘𝑋)) |
13 | vonct.3 | . . 3 ⊢ (𝜑 → 𝐴 ≼ ω) | |
14 | sndisj 5048 | . . . 4 ⊢ Disj 𝑥 ∈ 𝐴 {𝑥} | |
15 | 14 | a1i 11 | . . 3 ⊢ (𝜑 → Disj 𝑥 ∈ 𝐴 {𝑥}) |
16 | 5, 7, 8, 12, 13, 15 | meadjiun 42625 | . 2 ⊢ (𝜑 → ((voln‘𝑋)‘∪ 𝑥 ∈ 𝐴 {𝑥}) = (Σ^‘(𝑥 ∈ 𝐴 ↦ ((voln‘𝑋)‘{𝑥})))) |
17 | 9, 11 | vonsn 42850 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ((voln‘𝑋)‘{𝑥}) = 0) |
18 | 17 | mpteq2dva 5152 | . . . 4 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ ((voln‘𝑋)‘{𝑥})) = (𝑥 ∈ 𝐴 ↦ 0)) |
19 | 18 | fveq2d 6667 | . . 3 ⊢ (𝜑 → (Σ^‘(𝑥 ∈ 𝐴 ↦ ((voln‘𝑋)‘{𝑥}))) = (Σ^‘(𝑥 ∈ 𝐴 ↦ 0))) |
20 | 7, 8 | dmmeasal 42611 | . . . . . 6 ⊢ (𝜑 → dom (voln‘𝑋) ∈ SAlg) |
21 | 20, 13, 12 | saliuncl 42484 | . . . . 5 ⊢ (𝜑 → ∪ 𝑥 ∈ 𝐴 {𝑥} ∈ dom (voln‘𝑋)) |
22 | 1, 21 | eqeltrrid 2915 | . . . 4 ⊢ (𝜑 → 𝐴 ∈ dom (voln‘𝑋)) |
23 | 5, 22 | sge0z 42534 | . . 3 ⊢ (𝜑 → (Σ^‘(𝑥 ∈ 𝐴 ↦ 0)) = 0) |
24 | 19, 23 | eqtrd 2853 | . 2 ⊢ (𝜑 → (Σ^‘(𝑥 ∈ 𝐴 ↦ ((voln‘𝑋)‘{𝑥}))) = 0) |
25 | 4, 16, 24 | 3eqtrd 2857 | 1 ⊢ (𝜑 → ((voln‘𝑋)‘𝐴) = 0) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1528 ∈ wcel 2105 ⊆ wss 3933 {csn 4557 ∪ ciun 4910 Disj wdisj 5022 class class class wbr 5057 ↦ cmpt 5137 dom cdm 5548 ‘cfv 6348 (class class class)co 7145 ωcom 7569 ↑m cmap 8395 ≼ cdom 8495 Fincfn 8497 ℝcr 10524 0cc0 10525 Σ^csumge0 42521 volncvoln 42697 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-13 2381 ax-ext 2790 ax-rep 5181 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 ax-inf2 9092 ax-cc 9845 ax-ac2 9873 ax-cnex 10581 ax-resscn 10582 ax-1cn 10583 ax-icn 10584 ax-addcl 10585 ax-addrcl 10586 ax-mulcl 10587 ax-mulrcl 10588 ax-mulcom 10589 ax-addass 10590 ax-mulass 10591 ax-distr 10592 ax-i2m1 10593 ax-1ne0 10594 ax-1rid 10595 ax-rnegex 10596 ax-rrecex 10597 ax-cnre 10598 ax-pre-lttri 10599 ax-pre-lttrn 10600 ax-pre-ltadd 10601 ax-pre-mulgt0 10602 ax-pre-sup 10603 ax-addf 10604 ax-mulf 10605 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-fal 1541 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-nel 3121 df-ral 3140 df-rex 3141 df-reu 3142 df-rmo 3143 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-pss 3951 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-tp 4562 df-op 4564 df-uni 4831 df-int 4868 df-iun 4912 df-iin 4913 df-disj 5023 df-br 5058 df-opab 5120 df-mpt 5138 df-tr 5164 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-se 5508 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6141 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-isom 6357 df-riota 7103 df-ov 7148 df-oprab 7149 df-mpo 7150 df-of 7398 df-om 7570 df-1st 7678 df-2nd 7679 df-supp 7820 df-tpos 7881 df-wrecs 7936 df-recs 7997 df-rdg 8035 df-1o 8091 df-2o 8092 df-oadd 8095 df-omul 8096 df-er 8278 df-map 8397 df-pm 8398 df-ixp 8450 df-en 8498 df-dom 8499 df-sdom 8500 df-fin 8501 df-fsupp 8822 df-fi 8863 df-sup 8894 df-inf 8895 df-oi 8962 df-dju 9318 df-card 9356 df-acn 9359 df-ac 9530 df-pnf 10665 df-mnf 10666 df-xr 10667 df-ltxr 10668 df-le 10669 df-sub 10860 df-neg 10861 df-div 11286 df-nn 11627 df-2 11688 df-3 11689 df-4 11690 df-5 11691 df-6 11692 df-7 11693 df-8 11694 df-9 11695 df-n0 11886 df-z 11970 df-dec 12087 df-uz 12232 df-q 12337 df-rp 12378 df-xneg 12495 df-xadd 12496 df-xmul 12497 df-ioo 12730 df-ico 12732 df-icc 12733 df-fz 12881 df-fzo 13022 df-fl 13150 df-seq 13358 df-exp 13418 df-hash 13679 df-cj 14446 df-re 14447 df-im 14448 df-sqrt 14582 df-abs 14583 df-clim 14833 df-rlim 14834 df-sum 15031 df-prod 15248 df-struct 16473 df-ndx 16474 df-slot 16475 df-base 16477 df-sets 16478 df-ress 16479 df-plusg 16566 df-mulr 16567 df-starv 16568 df-sca 16569 df-vsca 16570 df-ip 16571 df-tset 16572 df-ple 16573 df-ds 16575 df-unif 16576 df-hom 16577 df-cco 16578 df-rest 16684 df-topn 16685 df-0g 16703 df-gsum 16704 df-topgen 16705 df-pt 16706 df-prds 16709 df-pws 16711 df-xrs 16763 df-qtop 16768 df-imas 16769 df-xps 16771 df-mre 16845 df-mrc 16846 df-acs 16848 df-mgm 17840 df-sgrp 17889 df-mnd 17900 df-mhm 17944 df-submnd 17945 df-grp 18044 df-minusg 18045 df-sbg 18046 df-mulg 18163 df-subg 18214 df-ghm 18294 df-cntz 18385 df-cmn 18837 df-abl 18838 df-mgp 19169 df-ur 19181 df-ring 19228 df-cring 19229 df-oppr 19302 df-dvdsr 19320 df-unit 19321 df-invr 19351 df-dvr 19362 df-rnghom 19396 df-drng 19433 df-field 19434 df-subrg 19462 df-abv 19517 df-staf 19545 df-srng 19546 df-lmod 19565 df-lss 19633 df-lmhm 19723 df-lvec 19804 df-sra 19873 df-rgmod 19874 df-psmet 20465 df-xmet 20466 df-met 20467 df-bl 20468 df-mopn 20469 df-cnfld 20474 df-refld 20677 df-phl 20698 df-dsmm 20804 df-frlm 20819 df-top 21430 df-topon 21447 df-topsp 21469 df-bases 21482 df-cn 21763 df-cnp 21764 df-cmp 21923 df-tx 22098 df-hmeo 22291 df-xms 22857 df-ms 22858 df-tms 22859 df-nm 23119 df-ngp 23120 df-tng 23121 df-nrg 23122 df-nlm 23123 df-cncf 23413 df-clm 23594 df-cph 23699 df-tcph 23700 df-rrx 23915 df-ovol 23992 df-vol 23993 df-salg 42471 df-sumge0 42522 df-mea 42609 df-ome 42649 df-caragen 42651 df-ovoln 42696 df-voln 42698 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |