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Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > voncmpl | Structured version Visualization version GIF version |
Description: The Lebesgue measure is complete. See Definition 112Df of [Fremlin1] p. 19. This is an observation written after Definition 115E of [Fremlin1] p. 31. (Contributed by Glauco Siliprandi, 24-Dec-2020.) |
Ref | Expression |
---|---|
voncmpl.x | β’ (π β π β Fin) |
voncmpl.s | β’ π = dom (volnβπ) |
voncmpl.e | β’ (π β πΈ β dom (volnβπ)) |
voncmpl.z | β’ (π β ((volnβπ)βπΈ) = 0) |
voncmpl.f | β’ (π β πΉ β πΈ) |
Ref | Expression |
---|---|
voncmpl | β’ (π β πΉ β π) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | voncmpl.x | . . . 4 β’ (π β π β Fin) | |
2 | 1 | ovnome 44967 | . . 3 β’ (π β (voln*βπ) β OutMeas) |
3 | eqid 2731 | . . 3 β’ βͺ dom (voln*βπ) = βͺ dom (voln*βπ) | |
4 | voncmpl.f | . . . 4 β’ (π β πΉ β πΈ) | |
5 | 1 | dmvon 45000 | . . . . . . 7 β’ (π β dom (volnβπ) = (CaraGenβ(voln*βπ))) |
6 | eqid 2731 | . . . . . . . . 9 β’ (CaraGenβ(voln*βπ)) = (CaraGenβ(voln*βπ)) | |
7 | 6 | caragenss 44898 | . . . . . . . 8 β’ ((voln*βπ) β OutMeas β (CaraGenβ(voln*βπ)) β dom (voln*βπ)) |
8 | 2, 7 | syl 17 | . . . . . . 7 β’ (π β (CaraGenβ(voln*βπ)) β dom (voln*βπ)) |
9 | 5, 8 | eqsstrd 4000 | . . . . . 6 β’ (π β dom (volnβπ) β dom (voln*βπ)) |
10 | voncmpl.e | . . . . . 6 β’ (π β πΈ β dom (volnβπ)) | |
11 | 9, 10 | sseldd 3963 | . . . . 5 β’ (π β πΈ β dom (voln*βπ)) |
12 | elssuni 4918 | . . . . 5 β’ (πΈ β dom (voln*βπ) β πΈ β βͺ dom (voln*βπ)) | |
13 | 11, 12 | syl 17 | . . . 4 β’ (π β πΈ β βͺ dom (voln*βπ)) |
14 | 4, 13 | sstrd 3972 | . . 3 β’ (π β πΉ β βͺ dom (voln*βπ)) |
15 | voncmpl.z | . . . . . . 7 β’ (π β ((volnβπ)βπΈ) = 0) | |
16 | 15 | eqcomd 2737 | . . . . . 6 β’ (π β 0 = ((volnβπ)βπΈ)) |
17 | 1 | vonval 44934 | . . . . . . 7 β’ (π β (volnβπ) = ((voln*βπ) βΎ (CaraGenβ(voln*βπ)))) |
18 | 17 | fveq1d 6864 | . . . . . 6 β’ (π β ((volnβπ)βπΈ) = (((voln*βπ) βΎ (CaraGenβ(voln*βπ)))βπΈ)) |
19 | 16, 18 | eqtrd 2771 | . . . . 5 β’ (π β 0 = (((voln*βπ) βΎ (CaraGenβ(voln*βπ)))βπΈ)) |
20 | voncmpl.s | . . . . . . . . 9 β’ π = dom (volnβπ) | |
21 | 20 | a1i 11 | . . . . . . . 8 β’ (π β π = dom (volnβπ)) |
22 | 21, 5 | eqtr2d 2772 | . . . . . . 7 β’ (π β (CaraGenβ(voln*βπ)) = π) |
23 | 22 | reseq2d 5957 | . . . . . 6 β’ (π β ((voln*βπ) βΎ (CaraGenβ(voln*βπ))) = ((voln*βπ) βΎ π)) |
24 | 23 | fveq1d 6864 | . . . . 5 β’ (π β (((voln*βπ) βΎ (CaraGenβ(voln*βπ)))βπΈ) = (((voln*βπ) βΎ π)βπΈ)) |
25 | 10, 20 | eleqtrrdi 2843 | . . . . . 6 β’ (π β πΈ β π) |
26 | fvres 6881 | . . . . . 6 β’ (πΈ β π β (((voln*βπ) βΎ π)βπΈ) = ((voln*βπ)βπΈ)) | |
27 | 25, 26 | syl 17 | . . . . 5 β’ (π β (((voln*βπ) βΎ π)βπΈ) = ((voln*βπ)βπΈ)) |
28 | 19, 24, 27 | 3eqtrrd 2776 | . . . 4 β’ (π β ((voln*βπ)βπΈ) = 0) |
29 | 2, 3, 13, 28, 4 | omess0 44928 | . . 3 β’ (π β ((voln*βπ)βπΉ) = 0) |
30 | 2, 3, 14, 29, 6 | caragencmpl 44929 | . 2 β’ (π β πΉ β (CaraGenβ(voln*βπ))) |
31 | 30, 22 | eleqtrd 2834 | 1 β’ (π β πΉ β π) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 = wceq 1541 β wcel 2106 β wss 3928 βͺ cuni 4885 dom cdm 5653 βΎ cres 5655 βcfv 6516 Fincfn 8905 0cc0 11075 OutMeascome 44883 CaraGenccaragen 44885 voln*covoln 44930 volncvoln 44932 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2702 ax-rep 5262 ax-sep 5276 ax-nul 5283 ax-pow 5340 ax-pr 5404 ax-un 7692 ax-inf2 9601 ax-cc 10395 ax-ac2 10423 ax-cnex 11131 ax-resscn 11132 ax-1cn 11133 ax-icn 11134 ax-addcl 11135 ax-addrcl 11136 ax-mulcl 11137 ax-mulrcl 11138 ax-mulcom 11139 ax-addass 11140 ax-mulass 11141 ax-distr 11142 ax-i2m1 11143 ax-1ne0 11144 ax-1rid 11145 ax-rnegex 11146 ax-rrecex 11147 ax-cnre 11148 ax-pre-lttri 11149 ax-pre-lttrn 11150 ax-pre-ltadd 11151 ax-pre-mulgt0 11152 ax-pre-sup 11153 ax-addf 11154 ax-mulf 11155 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-rmo 3364 df-reu 3365 df-rab 3419 df-v 3461 df-sbc 3758 df-csb 3874 df-dif 3931 df-un 3933 df-in 3935 df-ss 3945 df-pss 3947 df-nul 4303 df-if 4507 df-pw 4582 df-sn 4607 df-pr 4609 df-tp 4611 df-op 4613 df-uni 4886 df-int 4928 df-iun 4976 df-disj 5091 df-br 5126 df-opab 5188 df-mpt 5209 df-tr 5243 df-id 5551 df-eprel 5557 df-po 5565 df-so 5566 df-fr 5608 df-se 5609 df-we 5610 df-xp 5659 df-rel 5660 df-cnv 5661 df-co 5662 df-dm 5663 df-rn 5664 df-res 5665 df-ima 5666 df-pred 6273 df-ord 6340 df-on 6341 df-lim 6342 df-suc 6343 df-iota 6468 df-fun 6518 df-fn 6519 df-f 6520 df-f1 6521 df-fo 6522 df-f1o 6523 df-fv 6524 df-isom 6525 df-riota 7333 df-ov 7380 df-oprab 7381 df-mpo 7382 df-of 7637 df-om 7823 df-1st 7941 df-2nd 7942 df-tpos 8177 df-frecs 8232 df-wrecs 8263 df-recs 8337 df-rdg 8376 df-1o 8432 df-2o 8433 df-er 8670 df-map 8789 df-pm 8790 df-ixp 8858 df-en 8906 df-dom 8907 df-sdom 8908 df-fin 8909 df-fi 9371 df-sup 9402 df-inf 9403 df-oi 9470 df-dju 9861 df-card 9899 df-acn 9902 df-ac 10076 df-pnf 11215 df-mnf 11216 df-xr 11217 df-ltxr 11218 df-le 11219 df-sub 11411 df-neg 11412 df-div 11837 df-nn 12178 df-2 12240 df-3 12241 df-4 12242 df-5 12243 df-6 12244 df-7 12245 df-8 12246 df-9 12247 df-n0 12438 df-z 12524 df-dec 12643 df-uz 12788 df-q 12898 df-rp 12940 df-xneg 13057 df-xadd 13058 df-xmul 13059 df-ioo 13293 df-ico 13295 df-icc 13296 df-fz 13450 df-fzo 13593 df-fl 13722 df-seq 13932 df-exp 13993 df-hash 14256 df-cj 15011 df-re 15012 df-im 15013 df-sqrt 15147 df-abs 15148 df-clim 15397 df-rlim 15398 df-sum 15598 df-prod 15815 df-struct 17045 df-sets 17062 df-slot 17080 df-ndx 17092 df-base 17110 df-ress 17139 df-plusg 17175 df-mulr 17176 df-starv 17177 df-tset 17181 df-ple 17182 df-ds 17184 df-unif 17185 df-rest 17333 df-0g 17352 df-topgen 17354 df-mgm 18526 df-sgrp 18575 df-mnd 18586 df-grp 18780 df-minusg 18781 df-subg 18954 df-cmn 19593 df-abl 19594 df-mgp 19926 df-ur 19943 df-ring 19995 df-cring 19996 df-oppr 20078 df-dvdsr 20099 df-unit 20100 df-invr 20130 df-dvr 20141 df-drng 20242 df-psmet 20840 df-xmet 20841 df-met 20842 df-bl 20843 df-mopn 20844 df-cnfld 20849 df-top 22295 df-topon 22312 df-bases 22348 df-cmp 22790 df-ovol 24880 df-vol 24881 df-sumge0 44757 df-ome 44884 df-caragen 44886 df-ovoln 44931 df-voln 44933 |
This theorem is referenced by: (None) |
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