![]() |
Mathbox for Glauco Siliprandi |
< Previous
Next >
Nearby theorems |
|
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 6530 | . . . . 5 ⊢ (𝑋 = ∅ → (voln‘𝑋) = (voln‘∅)) | |
2 | 1 | fveq1d 6532 | . . . 4 ⊢ (𝑋 = ∅ → ((voln‘𝑋)‘{𝐴}) = ((voln‘∅)‘{𝐴})) |
3 | 2 | adantl 482 | . . 3 ⊢ ((𝜑 ∧ 𝑋 = ∅) → ((voln‘𝑋)‘{𝐴}) = ((voln‘∅)‘{𝐴})) |
4 | 0fin 8582 | . . . . . 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 7015 | . . . . . . 7 ⊢ (𝑋 = ∅ → (ℝ ↑𝑚 𝑋) = (ℝ ↑𝑚 ∅)) | |
9 | 8 | adantl 482 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑋 = ∅) → (ℝ ↑𝑚 𝑋) = (ℝ ↑𝑚 ∅)) |
10 | 7, 9 | eleqtrd 2883 | . . . . 5 ⊢ ((𝜑 ∧ 𝑋 = ∅) → 𝐴 ∈ (ℝ ↑𝑚 ∅)) |
11 | 5, 10 | snvonmbl 42464 | . . . 4 ⊢ ((𝜑 ∧ 𝑋 = ∅) → {𝐴} ∈ dom (voln‘∅)) |
12 | 11 | von0val 42449 | . . 3 ⊢ ((𝜑 ∧ 𝑋 = ∅) → ((voln‘∅)‘{𝐴}) = 0) |
13 | 3, 12 | eqtrd 2829 | . 2 ⊢ ((𝜑 ∧ 𝑋 = ∅) → ((voln‘𝑋)‘{𝐴}) = 0) |
14 | neqne 2990 | . . . 4 ⊢ (¬ 𝑋 = ∅ → 𝑋 ≠ ∅) | |
15 | 14 | adantl 482 | . . 3 ⊢ ((𝜑 ∧ ¬ 𝑋 = ∅) → 𝑋 ≠ ∅) |
16 | 6 | rrxsnicc 42081 | . . . . . . 7 ⊢ (𝜑 → X𝑘 ∈ 𝑋 ((𝐴‘𝑘)[,](𝐴‘𝑘)) = {𝐴}) |
17 | 16 | eqcomd 2799 | . . . . . 6 ⊢ (𝜑 → {𝐴} = X𝑘 ∈ 𝑋 ((𝐴‘𝑘)[,](𝐴‘𝑘))) |
18 | 17 | fveq2d 6534 | . . . . 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 485 | . . . . 5 ⊢ ((𝜑 ∧ 𝑋 ≠ ∅) → 𝑋 ≠ ∅) | |
23 | elmapi 8269 | . . . . . . 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 2793 | . . . . 5 ⊢ X𝑘 ∈ 𝑋 ((𝐴‘𝑘)[,](𝐴‘𝑘)) = X𝑘 ∈ 𝑋 ((𝐴‘𝑘)[,](𝐴‘𝑘)) | |
27 | 21, 22, 25, 25, 26 | vonn0icc 42466 | . . . 4 ⊢ ((𝜑 ∧ 𝑋 ≠ ∅) → ((voln‘𝑋)‘X𝑘 ∈ 𝑋 ((𝐴‘𝑘)[,](𝐴‘𝑘))) = ∏𝑘 ∈ 𝑋 (vol‘((𝐴‘𝑘)[,](𝐴‘𝑘)))) |
28 | 24 | ffvelrnda 6707 | . . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → (𝐴‘𝑘) ∈ ℝ) |
29 | 28 | rexrd 10526 | . . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → (𝐴‘𝑘) ∈ ℝ*) |
30 | iccid 12622 | . . . . . . . . . 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 6534 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → (vol‘((𝐴‘𝑘)[,](𝐴‘𝑘))) = (vol‘{(𝐴‘𝑘)})) |
33 | volsn 41747 | . . . . . . . . 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 2829 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → (vol‘((𝐴‘𝑘)[,](𝐴‘𝑘))) = 0) |
36 | 35 | prodeq2dv 15098 | . . . . . 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 10469 | . . . . . . 7 ⊢ (𝜑 → 0 ∈ ℂ) | |
39 | fprodconst 15153 | . . . . . . 7 ⊢ ((𝑋 ∈ Fin ∧ 0 ∈ ℂ) → ∏𝑘 ∈ 𝑋 0 = (0↑(♯‘𝑋))) | |
40 | 20, 38, 39 | syl2anc 584 | . . . . . 6 ⊢ (𝜑 → ∏𝑘 ∈ 𝑋 0 = (0↑(♯‘𝑋))) |
41 | 40 | adantr 481 | . . . . 5 ⊢ ((𝜑 ∧ 𝑋 ≠ ∅) → ∏𝑘 ∈ 𝑋 0 = (0↑(♯‘𝑋))) |
42 | hashnncl 13565 | . . . . . . . . 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 258 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑋 ≠ ∅) → (♯‘𝑋) ∈ ℕ) |
46 | 0exp 13302 | . . . . . 6 ⊢ ((♯‘𝑋) ∈ ℕ → (0↑(♯‘𝑋)) = 0) | |
47 | 45, 46 | syl 17 | . . . . 5 ⊢ ((𝜑 ∧ 𝑋 ≠ ∅) → (0↑(♯‘𝑋)) = 0) |
48 | 37, 41, 47 | 3eqtrd 2833 | . . . 4 ⊢ ((𝜑 ∧ 𝑋 ≠ ∅) → ∏𝑘 ∈ 𝑋 (vol‘((𝐴‘𝑘)[,](𝐴‘𝑘))) = 0) |
49 | 19, 27, 48 | 3eqtrd 2833 | . . 3 ⊢ ((𝜑 ∧ 𝑋 ≠ ∅) → ((voln‘𝑋)‘{𝐴}) = 0) |
50 | 15, 49 | syldan 591 | . 2 ⊢ ((𝜑 ∧ ¬ 𝑋 = ∅) → ((voln‘𝑋)‘{𝐴}) = 0) |
51 | 13, 50 | pm2.61dan 809 | 1 ⊢ (𝜑 → ((voln‘𝑋)‘{𝐴}) = 0) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 207 ∧ wa 396 = wceq 1520 ∈ wcel 2079 ≠ wne 2982 ∅c0 4206 {csn 4466 ⟶wf 6213 ‘cfv 6217 (class class class)co 7007 ↑𝑚 cmap 8247 Xcixp 8300 Fincfn 8347 ℂcc 10370 ℝcr 10371 0cc0 10372 ℝ*cxr 10509 ℕcn 11475 [,]cicc 12580 ↑cexp 13267 ♯chash 13528 ∏cprod 15080 volcvol 23735 volncvoln 42316 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1775 ax-4 1789 ax-5 1886 ax-6 1945 ax-7 1990 ax-8 2081 ax-9 2089 ax-10 2110 ax-11 2124 ax-12 2139 ax-13 2342 ax-ext 2767 ax-rep 5075 ax-sep 5088 ax-nul 5095 ax-pow 5150 ax-pr 5214 ax-un 7310 ax-inf2 8939 ax-cc 9692 ax-ac2 9720 ax-cnex 10428 ax-resscn 10429 ax-1cn 10430 ax-icn 10431 ax-addcl 10432 ax-addrcl 10433 ax-mulcl 10434 ax-mulrcl 10435 ax-mulcom 10436 ax-addass 10437 ax-mulass 10438 ax-distr 10439 ax-i2m1 10440 ax-1ne0 10441 ax-1rid 10442 ax-rnegex 10443 ax-rrecex 10444 ax-cnre 10445 ax-pre-lttri 10446 ax-pre-lttrn 10447 ax-pre-ltadd 10448 ax-pre-mulgt0 10449 ax-pre-sup 10450 ax-addf 10451 ax-mulf 10452 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 843 df-3or 1079 df-3an 1080 df-tru 1523 df-fal 1533 df-ex 1760 df-nf 1764 df-sb 2041 df-mo 2574 df-eu 2610 df-clab 2774 df-cleq 2786 df-clel 2861 df-nfc 2933 df-ne 2983 df-nel 3089 df-ral 3108 df-rex 3109 df-reu 3110 df-rmo 3111 df-rab 3112 df-v 3434 df-sbc 3702 df-csb 3807 df-dif 3857 df-un 3859 df-in 3861 df-ss 3869 df-pss 3871 df-nul 4207 df-if 4376 df-pw 4449 df-sn 4467 df-pr 4469 df-tp 4471 df-op 4473 df-uni 4740 df-int 4777 df-iun 4821 df-iin 4822 df-disj 4925 df-br 4957 df-opab 5019 df-mpt 5036 df-tr 5058 df-id 5340 df-eprel 5345 df-po 5354 df-so 5355 df-fr 5394 df-se 5395 df-we 5396 df-xp 5441 df-rel 5442 df-cnv 5443 df-co 5444 df-dm 5445 df-rn 5446 df-res 5447 df-ima 5448 df-pred 6015 df-ord 6061 df-on 6062 df-lim 6063 df-suc 6064 df-iota 6181 df-fun 6219 df-fn 6220 df-f 6221 df-f1 6222 df-fo 6223 df-f1o 6224 df-fv 6225 df-isom 6226 df-riota 6968 df-ov 7010 df-oprab 7011 df-mpo 7012 df-of 7258 df-om 7428 df-1st 7536 df-2nd 7537 df-supp 7673 df-tpos 7734 df-wrecs 7789 df-recs 7851 df-rdg 7889 df-1o 7944 df-2o 7945 df-oadd 7948 df-omul 7949 df-er 8130 df-map 8249 df-pm 8250 df-ixp 8301 df-en 8348 df-dom 8349 df-sdom 8350 df-fin 8351 df-fsupp 8670 df-fi 8711 df-sup 8742 df-inf 8743 df-oi 8810 df-dju 9165 df-card 9203 df-acn 9206 df-ac 9377 df-pnf 10512 df-mnf 10513 df-xr 10514 df-ltxr 10515 df-le 10516 df-sub 10708 df-neg 10709 df-div 11135 df-nn 11476 df-2 11537 df-3 11538 df-4 11539 df-5 11540 df-6 11541 df-7 11542 df-8 11543 df-9 11544 df-n0 11735 df-z 11819 df-dec 11937 df-uz 12083 df-q 12187 df-rp 12229 df-xneg 12346 df-xadd 12347 df-xmul 12348 df-ioo 12581 df-ico 12583 df-icc 12584 df-fz 12732 df-fzo 12873 df-fl 13000 df-seq 13208 df-exp 13268 df-hash 13529 df-cj 14280 df-re 14281 df-im 14282 df-sqrt 14416 df-abs 14417 df-clim 14667 df-rlim 14668 df-sum 14865 df-prod 15081 df-struct 16302 df-ndx 16303 df-slot 16304 df-base 16306 df-sets 16307 df-ress 16308 df-plusg 16395 df-mulr 16396 df-starv 16397 df-sca 16398 df-vsca 16399 df-ip 16400 df-tset 16401 df-ple 16402 df-ds 16404 df-unif 16405 df-hom 16406 df-cco 16407 df-rest 16513 df-topn 16514 df-0g 16532 df-gsum 16533 df-topgen 16534 df-pt 16535 df-prds 16538 df-pws 16540 df-xrs 16592 df-qtop 16597 df-imas 16598 df-xps 16600 df-mre 16674 df-mrc 16675 df-acs 16677 df-mgm 17669 df-sgrp 17711 df-mnd 17722 df-mhm 17762 df-submnd 17763 df-grp 17852 df-minusg 17853 df-sbg 17854 df-mulg 17970 df-subg 18018 df-ghm 18085 df-cntz 18176 df-cmn 18623 df-abl 18624 df-mgp 18918 df-ur 18930 df-ring 18977 df-cring 18978 df-oppr 19051 df-dvdsr 19069 df-unit 19070 df-invr 19100 df-dvr 19111 df-rnghom 19145 df-drng 19182 df-field 19183 df-subrg 19211 df-abv 19266 df-staf 19294 df-srng 19295 df-lmod 19314 df-lss 19382 df-lmhm 19472 df-lvec 19553 df-sra 19622 df-rgmod 19623 df-psmet 20207 df-xmet 20208 df-met 20209 df-bl 20210 df-mopn 20211 df-cnfld 20216 df-refld 20419 df-phl 20440 df-dsmm 20546 df-frlm 20561 df-top 21174 df-topon 21191 df-topsp 21213 df-bases 21226 df-cn 21507 df-cnp 21508 df-cmp 21667 df-tx 21842 df-hmeo 22035 df-xms 22601 df-ms 22602 df-tms 22603 df-nm 22863 df-ngp 22864 df-tng 22865 df-nrg 22866 df-nlm 22867 df-cncf 23157 df-clm 23338 df-cph 23443 df-tcph 23444 df-rrx 23659 df-ovol 23736 df-vol 23737 df-salg 42090 df-sumge0 42141 df-mea 42228 df-ome 42268 df-caragen 42270 df-ovoln 42315 df-voln 42317 |
This theorem is referenced by: vonct 42471 |
Copyright terms: Public domain | W3C validator |