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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 5054 | . . . . 5 ⊢ ∪ 𝑥 ∈ 𝐴 {𝑥} = 𝐴 | |
2 | 1 | eqcomi 2733 | . . . 4 ⊢ 𝐴 = ∪ 𝑥 ∈ 𝐴 {𝑥} |
3 | 2 | fveq2i 6885 | . . 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 1909 | . . 3 ⊢ Ⅎ𝑥𝜑 | |
6 | vonct.1 | . . . 4 ⊢ (𝜑 → 𝑋 ∈ Fin) | |
7 | 6 | vonmea 45835 | . . 3 ⊢ (𝜑 → (voln‘𝑋) ∈ Meas) |
8 | eqid 2724 | . . 3 ⊢ dom (voln‘𝑋) = dom (voln‘𝑋) | |
9 | 6 | adantr 480 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑋 ∈ Fin) |
10 | vonct.2 | . . . . 5 ⊢ (𝜑 → 𝐴 ⊆ (ℝ ↑m 𝑋)) | |
11 | 10 | sselda 3975 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ (ℝ ↑m 𝑋)) |
12 | 9, 11 | snvonmbl 45947 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → {𝑥} ∈ dom (voln‘𝑋)) |
13 | vonct.3 | . . 3 ⊢ (𝜑 → 𝐴 ≼ ω) | |
14 | sndisj 5130 | . . . 4 ⊢ Disj 𝑥 ∈ 𝐴 {𝑥} | |
15 | 14 | a1i 11 | . . 3 ⊢ (𝜑 → Disj 𝑥 ∈ 𝐴 {𝑥}) |
16 | 5, 7, 8, 12, 13, 15 | meadjiun 45727 | . 2 ⊢ (𝜑 → ((voln‘𝑋)‘∪ 𝑥 ∈ 𝐴 {𝑥}) = (Σ^‘(𝑥 ∈ 𝐴 ↦ ((voln‘𝑋)‘{𝑥})))) |
17 | 9, 11 | vonsn 45952 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ((voln‘𝑋)‘{𝑥}) = 0) |
18 | 17 | mpteq2dva 5239 | . . . 4 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ ((voln‘𝑋)‘{𝑥})) = (𝑥 ∈ 𝐴 ↦ 0)) |
19 | 18 | fveq2d 6886 | . . 3 ⊢ (𝜑 → (Σ^‘(𝑥 ∈ 𝐴 ↦ ((voln‘𝑋)‘{𝑥}))) = (Σ^‘(𝑥 ∈ 𝐴 ↦ 0))) |
20 | 7, 8 | dmmeasal 45713 | . . . . . 6 ⊢ (𝜑 → dom (voln‘𝑋) ∈ SAlg) |
21 | 20, 13, 12 | saliuncl 45584 | . . . . 5 ⊢ (𝜑 → ∪ 𝑥 ∈ 𝐴 {𝑥} ∈ dom (voln‘𝑋)) |
22 | 1, 21 | eqeltrrid 2830 | . . . 4 ⊢ (𝜑 → 𝐴 ∈ dom (voln‘𝑋)) |
23 | 5, 22 | sge0z 45636 | . . 3 ⊢ (𝜑 → (Σ^‘(𝑥 ∈ 𝐴 ↦ 0)) = 0) |
24 | 19, 23 | eqtrd 2764 | . 2 ⊢ (𝜑 → (Σ^‘(𝑥 ∈ 𝐴 ↦ ((voln‘𝑋)‘{𝑥}))) = 0) |
25 | 4, 16, 24 | 3eqtrd 2768 | 1 ⊢ (𝜑 → ((voln‘𝑋)‘𝐴) = 0) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1533 ∈ wcel 2098 ⊆ wss 3941 {csn 4621 ∪ ciun 4988 Disj wdisj 5104 class class class wbr 5139 ↦ cmpt 5222 dom cdm 5667 ‘cfv 6534 (class class class)co 7402 ωcom 7849 ↑m cmap 8817 ≼ cdom 8934 Fincfn 8936 ℝcr 11106 0cc0 11107 Σ^csumge0 45623 volncvoln 45799 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2695 ax-rep 5276 ax-sep 5290 ax-nul 5297 ax-pow 5354 ax-pr 5418 ax-un 7719 ax-inf2 9633 ax-cc 10427 ax-ac2 10455 ax-cnex 11163 ax-resscn 11164 ax-1cn 11165 ax-icn 11166 ax-addcl 11167 ax-addrcl 11168 ax-mulcl 11169 ax-mulrcl 11170 ax-mulcom 11171 ax-addass 11172 ax-mulass 11173 ax-distr 11174 ax-i2m1 11175 ax-1ne0 11176 ax-1rid 11177 ax-rnegex 11178 ax-rrecex 11179 ax-cnre 11180 ax-pre-lttri 11181 ax-pre-lttrn 11182 ax-pre-ltadd 11183 ax-pre-mulgt0 11184 ax-pre-sup 11185 ax-addf 11186 ax-mulf 11187 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2526 df-eu 2555 df-clab 2702 df-cleq 2716 df-clel 2802 df-nfc 2877 df-ne 2933 df-nel 3039 df-ral 3054 df-rex 3063 df-rmo 3368 df-reu 3369 df-rab 3425 df-v 3468 df-sbc 3771 df-csb 3887 df-dif 3944 df-un 3946 df-in 3948 df-ss 3958 df-pss 3960 df-nul 4316 df-if 4522 df-pw 4597 df-sn 4622 df-pr 4624 df-tp 4626 df-op 4628 df-uni 4901 df-int 4942 df-iun 4990 df-iin 4991 df-disj 5105 df-br 5140 df-opab 5202 df-mpt 5223 df-tr 5257 df-id 5565 df-eprel 5571 df-po 5579 df-so 5580 df-fr 5622 df-se 5623 df-we 5624 df-xp 5673 df-rel 5674 df-cnv 5675 df-co 5676 df-dm 5677 df-rn 5678 df-res 5679 df-ima 5680 df-pred 6291 df-ord 6358 df-on 6359 df-lim 6360 df-suc 6361 df-iota 6486 df-fun 6536 df-fn 6537 df-f 6538 df-f1 6539 df-fo 6540 df-f1o 6541 df-fv 6542 df-isom 6543 df-riota 7358 df-ov 7405 df-oprab 7406 df-mpo 7407 df-of 7664 df-om 7850 df-1st 7969 df-2nd 7970 df-supp 8142 df-tpos 8207 df-frecs 8262 df-wrecs 8293 df-recs 8367 df-rdg 8406 df-1o 8462 df-2o 8463 df-oadd 8466 df-omul 8467 df-er 8700 df-map 8819 df-pm 8820 df-ixp 8889 df-en 8937 df-dom 8938 df-sdom 8939 df-fin 8940 df-fsupp 9359 df-fi 9403 df-sup 9434 df-inf 9435 df-oi 9502 df-dju 9893 df-card 9931 df-acn 9934 df-ac 10108 df-pnf 11249 df-mnf 11250 df-xr 11251 df-ltxr 11252 df-le 11253 df-sub 11445 df-neg 11446 df-div 11871 df-nn 12212 df-2 12274 df-3 12275 df-4 12276 df-5 12277 df-6 12278 df-7 12279 df-8 12280 df-9 12281 df-n0 12472 df-z 12558 df-dec 12677 df-uz 12822 df-q 12932 df-rp 12976 df-xneg 13093 df-xadd 13094 df-xmul 13095 df-ioo 13329 df-ico 13331 df-icc 13332 df-fz 13486 df-fzo 13629 df-fl 13758 df-seq 13968 df-exp 14029 df-hash 14292 df-cj 15048 df-re 15049 df-im 15050 df-sqrt 15184 df-abs 15185 df-clim 15434 df-rlim 15435 df-sum 15635 df-prod 15852 df-struct 17085 df-sets 17102 df-slot 17120 df-ndx 17132 df-base 17150 df-ress 17179 df-plusg 17215 df-mulr 17216 df-starv 17217 df-sca 17218 df-vsca 17219 df-ip 17220 df-tset 17221 df-ple 17222 df-ds 17224 df-unif 17225 df-hom 17226 df-cco 17227 df-rest 17373 df-topn 17374 df-0g 17392 df-gsum 17393 df-topgen 17394 df-pt 17395 df-prds 17398 df-pws 17400 df-xrs 17453 df-qtop 17458 df-imas 17459 df-xps 17461 df-mre 17535 df-mrc 17536 df-acs 17538 df-mgm 18569 df-sgrp 18648 df-mnd 18664 df-mhm 18709 df-submnd 18710 df-grp 18862 df-minusg 18863 df-sbg 18864 df-mulg 18992 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 20866 df-lvec 20947 df-sra 21017 df-rgmod 21018 df-psmet 21226 df-xmet 21227 df-met 21228 df-bl 21229 df-mopn 21230 df-cnfld 21235 df-refld 21487 df-phl 21508 df-dsmm 21616 df-frlm 21631 df-top 22740 df-topon 22757 df-topsp 22779 df-bases 22793 df-cn 23075 df-cnp 23076 df-cmp 23235 df-tx 23410 df-hmeo 23603 df-xms 24170 df-ms 24171 df-tms 24172 df-nm 24435 df-ngp 24436 df-tng 24437 df-nrg 24438 df-nlm 24439 df-cncf 24742 df-clm 24934 df-cph 25040 df-tcph 25041 df-rrx 25257 df-ovol 25337 df-vol 25338 df-salg 45570 df-sumge0 45624 df-mea 45711 df-ome 45751 df-caragen 45753 df-ovoln 45798 df-voln 45800 |
This theorem is referenced by: (None) |
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