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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 | β’ (π β π β Fin) |
vonsn.2 | β’ (π β π΄ β (β βm π)) |
Ref | Expression |
---|---|
vonsn | β’ (π β ((volnβπ)β{π΄}) = 0) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fveq2 6882 | . . . . 5 β’ (π = β β (volnβπ) = (volnββ )) | |
2 | 1 | fveq1d 6884 | . . . 4 β’ (π = β β ((volnβπ)β{π΄}) = ((volnββ )β{π΄})) |
3 | 2 | adantl 481 | . . 3 β’ ((π β§ π = β ) β ((volnβπ)β{π΄}) = ((volnββ )β{π΄})) |
4 | 0fin 9168 | . . . . . 6 β’ β β Fin | |
5 | 4 | a1i 11 | . . . . 5 β’ ((π β§ π = β ) β β β Fin) |
6 | vonsn.2 | . . . . . . 7 β’ (π β π΄ β (β βm π)) | |
7 | 6 | adantr 480 | . . . . . 6 β’ ((π β§ π = β ) β π΄ β (β βm π)) |
8 | oveq2 7410 | . . . . . . 7 β’ (π = β β (β βm π) = (β βm β )) | |
9 | 8 | adantl 481 | . . . . . 6 β’ ((π β§ π = β ) β (β βm π) = (β βm β )) |
10 | 7, 9 | eleqtrd 2827 | . . . . 5 β’ ((π β§ π = β ) β π΄ β (β βm β )) |
11 | 5, 10 | snvonmbl 45948 | . . . 4 β’ ((π β§ π = β ) β {π΄} β dom (volnββ )) |
12 | 11 | von0val 45933 | . . 3 β’ ((π β§ π = β ) β ((volnββ )β{π΄}) = 0) |
13 | 3, 12 | eqtrd 2764 | . 2 β’ ((π β§ π = β ) β ((volnβπ)β{π΄}) = 0) |
14 | neqne 2940 | . . . 4 β’ (Β¬ π = β β π β β ) | |
15 | 14 | adantl 481 | . . 3 β’ ((π β§ Β¬ π = β ) β π β β ) |
16 | 6 | rrxsnicc 45562 | . . . . . . 7 β’ (π β Xπ β π ((π΄βπ)[,](π΄βπ)) = {π΄}) |
17 | 16 | eqcomd 2730 | . . . . . 6 β’ (π β {π΄} = Xπ β π ((π΄βπ)[,](π΄βπ))) |
18 | 17 | fveq2d 6886 | . . . . 5 β’ (π β ((volnβπ)β{π΄}) = ((volnβπ)βXπ β π ((π΄βπ)[,](π΄βπ)))) |
19 | 18 | adantr 480 | . . . 4 β’ ((π β§ π β β ) β ((volnβπ)β{π΄}) = ((volnβπ)βXπ β π ((π΄βπ)[,](π΄βπ)))) |
20 | vonsn.1 | . . . . . 6 β’ (π β π β Fin) | |
21 | 20 | adantr 480 | . . . . 5 β’ ((π β§ π β β ) β π β Fin) |
22 | simpr 484 | . . . . 5 β’ ((π β§ π β β ) β π β β ) | |
23 | elmapi 8840 | . . . . . . 7 β’ (π΄ β (β βm π) β π΄:πβΆβ) | |
24 | 6, 23 | syl 17 | . . . . . 6 β’ (π β π΄:πβΆβ) |
25 | 24 | adantr 480 | . . . . 5 β’ ((π β§ π β β ) β π΄:πβΆβ) |
26 | eqid 2724 | . . . . 5 β’ Xπ β π ((π΄βπ)[,](π΄βπ)) = Xπ β π ((π΄βπ)[,](π΄βπ)) | |
27 | 21, 22, 25, 25, 26 | vonn0icc 45950 | . . . 4 β’ ((π β§ π β β ) β ((volnβπ)βXπ β π ((π΄βπ)[,](π΄βπ))) = βπ β π (volβ((π΄βπ)[,](π΄βπ)))) |
28 | 24 | ffvelcdmda 7077 | . . . . . . . . . . 11 β’ ((π β§ π β π) β (π΄βπ) β β) |
29 | 28 | rexrd 11263 | . . . . . . . . . 10 β’ ((π β§ π β π) β (π΄βπ) β β*) |
30 | iccid 13370 | . . . . . . . . . 10 β’ ((π΄βπ) β β* β ((π΄βπ)[,](π΄βπ)) = {(π΄βπ)}) | |
31 | 29, 30 | syl 17 | . . . . . . . . 9 β’ ((π β§ π β π) β ((π΄βπ)[,](π΄βπ)) = {(π΄βπ)}) |
32 | 31 | fveq2d 6886 | . . . . . . . 8 β’ ((π β§ π β π) β (volβ((π΄βπ)[,](π΄βπ))) = (volβ{(π΄βπ)})) |
33 | volsn 45229 | . . . . . . . . 9 β’ ((π΄βπ) β β β (volβ{(π΄βπ)}) = 0) | |
34 | 28, 33 | syl 17 | . . . . . . . 8 β’ ((π β§ π β π) β (volβ{(π΄βπ)}) = 0) |
35 | 32, 34 | eqtrd 2764 | . . . . . . 7 β’ ((π β§ π β π) β (volβ((π΄βπ)[,](π΄βπ))) = 0) |
36 | 35 | prodeq2dv 15869 | . . . . . 6 β’ (π β βπ β π (volβ((π΄βπ)[,](π΄βπ))) = βπ β π 0) |
37 | 36 | adantr 480 | . . . . 5 β’ ((π β§ π β β ) β βπ β π (volβ((π΄βπ)[,](π΄βπ))) = βπ β π 0) |
38 | 0cnd 11206 | . . . . . . 7 β’ (π β 0 β β) | |
39 | fprodconst 15924 | . . . . . . 7 β’ ((π β Fin β§ 0 β β) β βπ β π 0 = (0β(β―βπ))) | |
40 | 20, 38, 39 | syl2anc 583 | . . . . . 6 β’ (π β βπ β π 0 = (0β(β―βπ))) |
41 | 40 | adantr 480 | . . . . 5 β’ ((π β§ π β β ) β βπ β π 0 = (0β(β―βπ))) |
42 | hashnncl 14327 | . . . . . . . . 9 β’ (π β Fin β ((β―βπ) β β β π β β )) | |
43 | 20, 42 | syl 17 | . . . . . . . 8 β’ (π β ((β―βπ) β β β π β β )) |
44 | 43 | adantr 480 | . . . . . . 7 β’ ((π β§ π β β ) β ((β―βπ) β β β π β β )) |
45 | 22, 44 | mpbird 257 | . . . . . 6 β’ ((π β§ π β β ) β (β―βπ) β β) |
46 | 0exp 14064 | . . . . . 6 β’ ((β―βπ) β β β (0β(β―βπ)) = 0) | |
47 | 45, 46 | syl 17 | . . . . 5 β’ ((π β§ π β β ) β (0β(β―βπ)) = 0) |
48 | 37, 41, 47 | 3eqtrd 2768 | . . . 4 β’ ((π β§ π β β ) β βπ β π (volβ((π΄βπ)[,](π΄βπ))) = 0) |
49 | 19, 27, 48 | 3eqtrd 2768 | . . 3 β’ ((π β§ π β β ) β ((volnβπ)β{π΄}) = 0) |
50 | 15, 49 | syldan 590 | . 2 β’ ((π β§ Β¬ π = β ) β ((volnβπ)β{π΄}) = 0) |
51 | 13, 50 | pm2.61dan 810 | 1 β’ (π β ((volnβπ)β{π΄}) = 0) |
Colors of variables: wff setvar class |
Syntax hints: Β¬ wn 3 β wi 4 β wb 205 β§ wa 395 = wceq 1533 β wcel 2098 β wne 2932 β c0 4315 {csn 4621 βΆwf 6530 βcfv 6534 (class class class)co 7402 βm cmap 8817 Xcixp 8888 Fincfn 8936 βcc 11105 βcr 11106 0cc0 11107 β*cxr 11246 βcn 12211 [,]cicc 13328 βcexp 14028 β―chash 14291 βcprod 15851 volcvol 25336 volncvoln 45800 |
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 45571 df-sumge0 45625 df-mea 45712 df-ome 45752 df-caragen 45754 df-ovoln 45799 df-voln 45801 |
This theorem is referenced by: vonct 45955 |
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