Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > vonsn | Structured version Visualization version GIF version |
Description: The n-dimensional Lebesgue measure of a singleton is zero. This is the first statement in Proposition 115G (e) of [Fremlin1] p. 32. (Contributed by Glauco Siliprandi, 8-Apr-2021.) |
Ref | Expression |
---|---|
vonsn.1 | ⊢ (𝜑 → 𝑋 ∈ Fin) |
vonsn.2 | ⊢ (𝜑 → 𝐴 ∈ (ℝ ↑𝑚 𝑋)) |
Ref | Expression |
---|---|
vonsn | ⊢ (𝜑 → ((voln‘𝑋)‘{𝐴}) = 0) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fveq2 6191 | . . . . 5 ⊢ (𝑋 = ∅ → (voln‘𝑋) = (voln‘∅)) | |
2 | 1 | fveq1d 6193 | . . . 4 ⊢ (𝑋 = ∅ → ((voln‘𝑋)‘{𝐴}) = ((voln‘∅)‘{𝐴})) |
3 | 2 | adantl 482 | . . 3 ⊢ ((𝜑 ∧ 𝑋 = ∅) → ((voln‘𝑋)‘{𝐴}) = ((voln‘∅)‘{𝐴})) |
4 | 0fin 8188 | . . . . . 6 ⊢ ∅ ∈ Fin | |
5 | 4 | a1i 11 | . . . . 5 ⊢ ((𝜑 ∧ 𝑋 = ∅) → ∅ ∈ Fin) |
6 | vonsn.2 | . . . . . . 7 ⊢ (𝜑 → 𝐴 ∈ (ℝ ↑𝑚 𝑋)) | |
7 | 6 | adantr 481 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑋 = ∅) → 𝐴 ∈ (ℝ ↑𝑚 𝑋)) |
8 | oveq2 6658 | . . . . . . 7 ⊢ (𝑋 = ∅ → (ℝ ↑𝑚 𝑋) = (ℝ ↑𝑚 ∅)) | |
9 | 8 | adantl 482 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑋 = ∅) → (ℝ ↑𝑚 𝑋) = (ℝ ↑𝑚 ∅)) |
10 | 7, 9 | eleqtrd 2703 | . . . . 5 ⊢ ((𝜑 ∧ 𝑋 = ∅) → 𝐴 ∈ (ℝ ↑𝑚 ∅)) |
11 | 5, 10 | snvonmbl 40900 | . . . 4 ⊢ ((𝜑 ∧ 𝑋 = ∅) → {𝐴} ∈ dom (voln‘∅)) |
12 | 11 | von0val 40885 | . . 3 ⊢ ((𝜑 ∧ 𝑋 = ∅) → ((voln‘∅)‘{𝐴}) = 0) |
13 | 3, 12 | eqtrd 2656 | . 2 ⊢ ((𝜑 ∧ 𝑋 = ∅) → ((voln‘𝑋)‘{𝐴}) = 0) |
14 | neqne 2802 | . . . 4 ⊢ (¬ 𝑋 = ∅ → 𝑋 ≠ ∅) | |
15 | 14 | adantl 482 | . . 3 ⊢ ((𝜑 ∧ ¬ 𝑋 = ∅) → 𝑋 ≠ ∅) |
16 | 6 | rrxsnicc 40520 | . . . . . . 7 ⊢ (𝜑 → X𝑘 ∈ 𝑋 ((𝐴‘𝑘)[,](𝐴‘𝑘)) = {𝐴}) |
17 | 16 | eqcomd 2628 | . . . . . 6 ⊢ (𝜑 → {𝐴} = X𝑘 ∈ 𝑋 ((𝐴‘𝑘)[,](𝐴‘𝑘))) |
18 | 17 | fveq2d 6195 | . . . . 5 ⊢ (𝜑 → ((voln‘𝑋)‘{𝐴}) = ((voln‘𝑋)‘X𝑘 ∈ 𝑋 ((𝐴‘𝑘)[,](𝐴‘𝑘)))) |
19 | 18 | adantr 481 | . . . 4 ⊢ ((𝜑 ∧ 𝑋 ≠ ∅) → ((voln‘𝑋)‘{𝐴}) = ((voln‘𝑋)‘X𝑘 ∈ 𝑋 ((𝐴‘𝑘)[,](𝐴‘𝑘)))) |
20 | vonsn.1 | . . . . . 6 ⊢ (𝜑 → 𝑋 ∈ Fin) | |
21 | 20 | adantr 481 | . . . . 5 ⊢ ((𝜑 ∧ 𝑋 ≠ ∅) → 𝑋 ∈ Fin) |
22 | simpr 477 | . . . . 5 ⊢ ((𝜑 ∧ 𝑋 ≠ ∅) → 𝑋 ≠ ∅) | |
23 | elmapi 7879 | . . . . . . 7 ⊢ (𝐴 ∈ (ℝ ↑𝑚 𝑋) → 𝐴:𝑋⟶ℝ) | |
24 | 6, 23 | syl 17 | . . . . . 6 ⊢ (𝜑 → 𝐴:𝑋⟶ℝ) |
25 | 24 | adantr 481 | . . . . 5 ⊢ ((𝜑 ∧ 𝑋 ≠ ∅) → 𝐴:𝑋⟶ℝ) |
26 | eqid 2622 | . . . . 5 ⊢ X𝑘 ∈ 𝑋 ((𝐴‘𝑘)[,](𝐴‘𝑘)) = X𝑘 ∈ 𝑋 ((𝐴‘𝑘)[,](𝐴‘𝑘)) | |
27 | 21, 22, 25, 25, 26 | vonn0icc 40902 | . . . 4 ⊢ ((𝜑 ∧ 𝑋 ≠ ∅) → ((voln‘𝑋)‘X𝑘 ∈ 𝑋 ((𝐴‘𝑘)[,](𝐴‘𝑘))) = ∏𝑘 ∈ 𝑋 (vol‘((𝐴‘𝑘)[,](𝐴‘𝑘)))) |
28 | 24 | ffvelrnda 6359 | . . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → (𝐴‘𝑘) ∈ ℝ) |
29 | 28 | rexrd 10089 | . . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → (𝐴‘𝑘) ∈ ℝ*) |
30 | iccid 12220 | . . . . . . . . . 10 ⊢ ((𝐴‘𝑘) ∈ ℝ* → ((𝐴‘𝑘)[,](𝐴‘𝑘)) = {(𝐴‘𝑘)}) | |
31 | 29, 30 | syl 17 | . . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → ((𝐴‘𝑘)[,](𝐴‘𝑘)) = {(𝐴‘𝑘)}) |
32 | 31 | fveq2d 6195 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → (vol‘((𝐴‘𝑘)[,](𝐴‘𝑘))) = (vol‘{(𝐴‘𝑘)})) |
33 | volsn 40183 | . . . . . . . . 9 ⊢ ((𝐴‘𝑘) ∈ ℝ → (vol‘{(𝐴‘𝑘)}) = 0) | |
34 | 28, 33 | syl 17 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → (vol‘{(𝐴‘𝑘)}) = 0) |
35 | 32, 34 | eqtrd 2656 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → (vol‘((𝐴‘𝑘)[,](𝐴‘𝑘))) = 0) |
36 | 35 | prodeq2dv 14653 | . . . . . 6 ⊢ (𝜑 → ∏𝑘 ∈ 𝑋 (vol‘((𝐴‘𝑘)[,](𝐴‘𝑘))) = ∏𝑘 ∈ 𝑋 0) |
37 | 36 | adantr 481 | . . . . 5 ⊢ ((𝜑 ∧ 𝑋 ≠ ∅) → ∏𝑘 ∈ 𝑋 (vol‘((𝐴‘𝑘)[,](𝐴‘𝑘))) = ∏𝑘 ∈ 𝑋 0) |
38 | 0cnd 10033 | . . . . . . 7 ⊢ (𝜑 → 0 ∈ ℂ) | |
39 | fprodconst 14708 | . . . . . . 7 ⊢ ((𝑋 ∈ Fin ∧ 0 ∈ ℂ) → ∏𝑘 ∈ 𝑋 0 = (0↑(#‘𝑋))) | |
40 | 20, 38, 39 | syl2anc 693 | . . . . . 6 ⊢ (𝜑 → ∏𝑘 ∈ 𝑋 0 = (0↑(#‘𝑋))) |
41 | 40 | adantr 481 | . . . . 5 ⊢ ((𝜑 ∧ 𝑋 ≠ ∅) → ∏𝑘 ∈ 𝑋 0 = (0↑(#‘𝑋))) |
42 | hashnncl 13157 | . . . . . . . . 9 ⊢ (𝑋 ∈ Fin → ((#‘𝑋) ∈ ℕ ↔ 𝑋 ≠ ∅)) | |
43 | 20, 42 | syl 17 | . . . . . . . 8 ⊢ (𝜑 → ((#‘𝑋) ∈ ℕ ↔ 𝑋 ≠ ∅)) |
44 | 43 | adantr 481 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑋 ≠ ∅) → ((#‘𝑋) ∈ ℕ ↔ 𝑋 ≠ ∅)) |
45 | 22, 44 | mpbird 247 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑋 ≠ ∅) → (#‘𝑋) ∈ ℕ) |
46 | 0exp 12895 | . . . . . 6 ⊢ ((#‘𝑋) ∈ ℕ → (0↑(#‘𝑋)) = 0) | |
47 | 45, 46 | syl 17 | . . . . 5 ⊢ ((𝜑 ∧ 𝑋 ≠ ∅) → (0↑(#‘𝑋)) = 0) |
48 | 37, 41, 47 | 3eqtrd 2660 | . . . 4 ⊢ ((𝜑 ∧ 𝑋 ≠ ∅) → ∏𝑘 ∈ 𝑋 (vol‘((𝐴‘𝑘)[,](𝐴‘𝑘))) = 0) |
49 | 19, 27, 48 | 3eqtrd 2660 | . . 3 ⊢ ((𝜑 ∧ 𝑋 ≠ ∅) → ((voln‘𝑋)‘{𝐴}) = 0) |
50 | 15, 49 | syldan 487 | . 2 ⊢ ((𝜑 ∧ ¬ 𝑋 = ∅) → ((voln‘𝑋)‘{𝐴}) = 0) |
51 | 13, 50 | pm2.61dan 832 | 1 ⊢ (𝜑 → ((voln‘𝑋)‘{𝐴}) = 0) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 196 ∧ wa 384 = wceq 1483 ∈ wcel 1990 ≠ wne 2794 ∅c0 3915 {csn 4177 ⟶wf 5884 ‘cfv 5888 (class class class)co 6650 ↑𝑚 cmap 7857 Xcixp 7908 Fincfn 7955 ℂcc 9934 ℝcr 9935 0cc0 9936 ℝ*cxr 10073 ℕcn 11020 [,]cicc 12178 ↑cexp 12860 #chash 13117 ∏cprod 14635 volcvol 23232 volncvoln 40752 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-inf2 8538 ax-cc 9257 ax-ac2 9285 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013 ax-pre-sup 10014 ax-addf 10015 ax-mulf 10016 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-fal 1489 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-int 4476 df-iun 4522 df-iin 4523 df-disj 4621 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-se 5074 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-isom 5897 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-of 6897 df-om 7066 df-1st 7168 df-2nd 7169 df-supp 7296 df-tpos 7352 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-1o 7560 df-2o 7561 df-oadd 7564 df-omul 7565 df-er 7742 df-map 7859 df-pm 7860 df-ixp 7909 df-en 7956 df-dom 7957 df-sdom 7958 df-fin 7959 df-fsupp 8276 df-fi 8317 df-sup 8348 df-inf 8349 df-oi 8415 df-card 8765 df-acn 8768 df-ac 8939 df-cda 8990 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-div 10685 df-nn 11021 df-2 11079 df-3 11080 df-4 11081 df-5 11082 df-6 11083 df-7 11084 df-8 11085 df-9 11086 df-n0 11293 df-z 11378 df-dec 11494 df-uz 11688 df-q 11789 df-rp 11833 df-xneg 11946 df-xadd 11947 df-xmul 11948 df-ioo 12179 df-ico 12181 df-icc 12182 df-fz 12327 df-fzo 12466 df-fl 12593 df-seq 12802 df-exp 12861 df-hash 13118 df-cj 13839 df-re 13840 df-im 13841 df-sqrt 13975 df-abs 13976 df-clim 14219 df-rlim 14220 df-sum 14417 df-prod 14636 df-struct 15859 df-ndx 15860 df-slot 15861 df-base 15863 df-sets 15864 df-ress 15865 df-plusg 15954 df-mulr 15955 df-starv 15956 df-sca 15957 df-vsca 15958 df-ip 15959 df-tset 15960 df-ple 15961 df-ds 15964 df-unif 15965 df-hom 15966 df-cco 15967 df-rest 16083 df-topn 16084 df-0g 16102 df-gsum 16103 df-topgen 16104 df-pt 16105 df-prds 16108 df-pws 16110 df-xrs 16162 df-qtop 16167 df-imas 16168 df-xps 16170 df-mre 16246 df-mrc 16247 df-acs 16249 df-mgm 17242 df-sgrp 17284 df-mnd 17295 df-mhm 17335 df-submnd 17336 df-grp 17425 df-minusg 17426 df-sbg 17427 df-mulg 17541 df-subg 17591 df-ghm 17658 df-cntz 17750 df-cmn 18195 df-abl 18196 df-mgp 18490 df-ur 18502 df-ring 18549 df-cring 18550 df-oppr 18623 df-dvdsr 18641 df-unit 18642 df-invr 18672 df-dvr 18683 df-rnghom 18715 df-drng 18749 df-field 18750 df-subrg 18778 df-abv 18817 df-staf 18845 df-srng 18846 df-lmod 18865 df-lss 18933 df-lmhm 19022 df-lvec 19103 df-sra 19172 df-rgmod 19173 df-psmet 19738 df-xmet 19739 df-met 19740 df-bl 19741 df-mopn 19742 df-cnfld 19747 df-refld 19951 df-phl 19971 df-dsmm 20076 df-frlm 20091 df-top 20699 df-topon 20716 df-topsp 20737 df-bases 20750 df-cn 21031 df-cnp 21032 df-cmp 21190 df-tx 21365 df-hmeo 21558 df-xms 22125 df-ms 22126 df-tms 22127 df-nm 22387 df-ngp 22388 df-tng 22389 df-nrg 22390 df-nlm 22391 df-cncf 22681 df-clm 22863 df-cph 22968 df-tch 22969 df-rrx 23173 df-ovol 23233 df-vol 23234 df-salg 40529 df-sumge0 40580 df-mea 40667 df-ome 40704 df-caragen 40706 df-ovoln 40751 df-voln 40753 |
This theorem is referenced by: vonct 40907 |
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