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| Mirrors > Home > MPE Home > Th. List > Mathboxes > carsgsiga | Structured version Visualization version GIF version | ||
| Description: The Caratheodory measurable sets constructed from outer measures form a Sigma-algebra. Statement (iii) of Theorem 1.11.4 of [Bogachev] p. 42. (Contributed by Thierry Arnoux, 17-May-2020.) |
| Ref | Expression |
|---|---|
| carsgval.1 | β’ (π β π β π) |
| carsgval.2 | β’ (π β π:π« πβΆ(0[,]+β)) |
| carsgsiga.1 | β’ (π β (πββ ) = 0) |
| carsgsiga.2 | β’ ((π β§ π₯ βΌ Ο β§ π₯ β π« π) β (πββͺ π₯) β€ Ξ£*π¦ β π₯(πβπ¦)) |
| carsgsiga.3 | β’ ((π β§ π₯ β π¦ β§ π¦ β π« π) β (πβπ₯) β€ (πβπ¦)) |
| Ref | Expression |
|---|---|
| carsgsiga | β’ (π β (toCaraSigaβπ) β (sigAlgebraβπ)) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | carsgval.1 | . . . 4 β’ (π β π β π) | |
| 2 | carsgval.2 | . . . 4 β’ (π β π:π« πβΆ(0[,]+β)) | |
| 3 | 1, 2 | carsgcl 33758 | . . 3 β’ (π β (toCaraSigaβπ) β π« π) |
| 4 | carsgsiga.1 | . . . . 5 β’ (π β (πββ ) = 0) | |
| 5 | 1, 2, 4 | baselcarsg 33760 | . . . 4 β’ (π β π β (toCaraSigaβπ)) |
| 6 | 1 | adantr 480 | . . . . . 6 β’ ((π β§ π β (toCaraSigaβπ)) β π β π) |
| 7 | 2 | adantr 480 | . . . . . 6 β’ ((π β§ π β (toCaraSigaβπ)) β π:π« πβΆ(0[,]+β)) |
| 8 | simpr 484 | . . . . . 6 β’ ((π β§ π β (toCaraSigaβπ)) β π β (toCaraSigaβπ)) | |
| 9 | 6, 7, 8 | difelcarsg 33764 | . . . . 5 β’ ((π β§ π β (toCaraSigaβπ)) β (π β π) β (toCaraSigaβπ)) |
| 10 | 9 | ralrimiva 3138 | . . . 4 β’ (π β βπ β (toCaraSigaβπ)(π β π) β (toCaraSigaβπ)) |
| 11 | 1 | ad2antrr 723 | . . . . . . 7 β’ (((π β§ π β π« (toCaraSigaβπ)) β§ π βΌ Ο) β π β π) |
| 12 | 2 | ad2antrr 723 | . . . . . . 7 β’ (((π β§ π β π« (toCaraSigaβπ)) β§ π βΌ Ο) β π:π« πβΆ(0[,]+β)) |
| 13 | 4 | ad2antrr 723 | . . . . . . 7 β’ (((π β§ π β π« (toCaraSigaβπ)) β§ π βΌ Ο) β (πββ ) = 0) |
| 14 | carsgsiga.2 | . . . . . . . . 9 β’ ((π β§ π₯ βΌ Ο β§ π₯ β π« π) β (πββͺ π₯) β€ Ξ£*π¦ β π₯(πβπ¦)) | |
| 15 | 14 | 3adant1r 1174 | . . . . . . . 8 β’ (((π β§ π β π« (toCaraSigaβπ)) β§ π₯ βΌ Ο β§ π₯ β π« π) β (πββͺ π₯) β€ Ξ£*π¦ β π₯(πβπ¦)) |
| 16 | 15 | 3adant1r 1174 | . . . . . . 7 β’ ((((π β§ π β π« (toCaraSigaβπ)) β§ π βΌ Ο) β§ π₯ βΌ Ο β§ π₯ β π« π) β (πββͺ π₯) β€ Ξ£*π¦ β π₯(πβπ¦)) |
| 17 | carsgsiga.3 | . . . . . . . . 9 β’ ((π β§ π₯ β π¦ β§ π¦ β π« π) β (πβπ₯) β€ (πβπ¦)) | |
| 18 | 17 | 3adant1r 1174 | . . . . . . . 8 β’ (((π β§ π β π« (toCaraSigaβπ)) β§ π₯ β π¦ β§ π¦ β π« π) β (πβπ₯) β€ (πβπ¦)) |
| 19 | 18 | 3adant1r 1174 | . . . . . . 7 β’ ((((π β§ π β π« (toCaraSigaβπ)) β§ π βΌ Ο) β§ π₯ β π¦ β§ π¦ β π« π) β (πβπ₯) β€ (πβπ¦)) |
| 20 | simpr 484 | . . . . . . 7 β’ (((π β§ π β π« (toCaraSigaβπ)) β§ π βΌ Ο) β π βΌ Ο) | |
| 21 | elpwi 4601 | . . . . . . . 8 β’ (π β π« (toCaraSigaβπ) β π β (toCaraSigaβπ)) | |
| 22 | 21 | ad2antlr 724 | . . . . . . 7 β’ (((π β§ π β π« (toCaraSigaβπ)) β§ π βΌ Ο) β π β (toCaraSigaβπ)) |
| 23 | 11, 12, 13, 16, 19, 20, 22 | carsgclctun 33775 | . . . . . 6 β’ (((π β§ π β π« (toCaraSigaβπ)) β§ π βΌ Ο) β βͺ π β (toCaraSigaβπ)) |
| 24 | 23 | ex 412 | . . . . 5 β’ ((π β§ π β π« (toCaraSigaβπ)) β (π βΌ Ο β βͺ π β (toCaraSigaβπ))) |
| 25 | 24 | ralrimiva 3138 | . . . 4 β’ (π β βπ β π« (toCaraSigaβπ)(π βΌ Ο β βͺ π β (toCaraSigaβπ))) |
| 26 | 5, 10, 25 | 3jca 1125 | . . 3 β’ (π β (π β (toCaraSigaβπ) β§ βπ β (toCaraSigaβπ)(π β π) β (toCaraSigaβπ) β§ βπ β π« (toCaraSigaβπ)(π βΌ Ο β βͺ π β (toCaraSigaβπ)))) |
| 27 | 3, 26 | jca 511 | . 2 β’ (π β ((toCaraSigaβπ) β π« π β§ (π β (toCaraSigaβπ) β§ βπ β (toCaraSigaβπ)(π β π) β (toCaraSigaβπ) β§ βπ β π« (toCaraSigaβπ)(π βΌ Ο β βͺ π β (toCaraSigaβπ))))) |
| 28 | fvex 6894 | . . 3 β’ (toCaraSigaβπ) β V | |
| 29 | issiga 33565 | . . 3 β’ ((toCaraSigaβπ) β V β ((toCaraSigaβπ) β (sigAlgebraβπ) β ((toCaraSigaβπ) β π« π β§ (π β (toCaraSigaβπ) β§ βπ β (toCaraSigaβπ)(π β π) β (toCaraSigaβπ) β§ βπ β π« (toCaraSigaβπ)(π βΌ Ο β βͺ π β (toCaraSigaβπ)))))) | |
| 30 | 28, 29 | ax-mp 5 | . 2 β’ ((toCaraSigaβπ) β (sigAlgebraβπ) β ((toCaraSigaβπ) β π« π β§ (π β (toCaraSigaβπ) β§ βπ β (toCaraSigaβπ)(π β π) β (toCaraSigaβπ) β§ βπ β π« (toCaraSigaβπ)(π βΌ Ο β βͺ π β (toCaraSigaβπ))))) |
| 31 | 27, 30 | sylibr 233 | 1 β’ (π β (toCaraSigaβπ) β (sigAlgebraβπ)) |
| Colors of variables: wff setvar class |
| Syntax hints: β wi 4 β wb 205 β§ wa 395 β§ w3a 1084 = wceq 1533 β wcel 2098 βwral 3053 Vcvv 3466 β cdif 3937 β wss 3940 β c0 4314 π« cpw 4594 βͺ cuni 4899 class class class wbr 5138 βΆwf 6529 βcfv 6533 (class class class)co 7401 Οcom 7848 βΌ cdom 8932 0cc0 11105 +βcpnf 11241 β€ cle 11245 [,]cicc 13323 Ξ£*cesum 33480 sigAlgebracsiga 33561 toCaraSigaccarsg 33755 |
| 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 5275 ax-sep 5289 ax-nul 5296 ax-pow 5353 ax-pr 5417 ax-un 7718 ax-inf2 9631 ax-ac2 10453 ax-cnex 11161 ax-resscn 11162 ax-1cn 11163 ax-icn 11164 ax-addcl 11165 ax-addrcl 11166 ax-mulcl 11167 ax-mulrcl 11168 ax-mulcom 11169 ax-addass 11170 ax-mulass 11171 ax-distr 11172 ax-i2m1 11173 ax-1ne0 11174 ax-1rid 11175 ax-rnegex 11176 ax-rrecex 11177 ax-cnre 11178 ax-pre-lttri 11179 ax-pre-lttrn 11180 ax-pre-ltadd 11181 ax-pre-mulgt0 11182 ax-pre-sup 11183 ax-addf 11184 ax-mulf 11185 |
| 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 3770 df-csb 3886 df-dif 3943 df-un 3945 df-in 3947 df-ss 3957 df-pss 3959 df-nul 4315 df-if 4521 df-pw 4596 df-sn 4621 df-pr 4623 df-tp 4625 df-op 4627 df-uni 4900 df-int 4941 df-iun 4989 df-iin 4990 df-disj 5104 df-br 5139 df-opab 5201 df-mpt 5222 df-tr 5256 df-id 5564 df-eprel 5570 df-po 5578 df-so 5579 df-fr 5621 df-se 5622 df-we 5623 df-xp 5672 df-rel 5673 df-cnv 5674 df-co 5675 df-dm 5676 df-rn 5677 df-res 5678 df-ima 5679 df-pred 6290 df-ord 6357 df-on 6358 df-lim 6359 df-suc 6360 df-iota 6485 df-fun 6535 df-fn 6536 df-f 6537 df-f1 6538 df-fo 6539 df-f1o 6540 df-fv 6541 df-isom 6542 df-riota 7357 df-ov 7404 df-oprab 7405 df-mpo 7406 df-of 7663 df-om 7849 df-1st 7968 df-2nd 7969 df-supp 8141 df-frecs 8261 df-wrecs 8292 df-recs 8366 df-rdg 8405 df-1o 8461 df-2o 8462 df-er 8698 df-map 8817 df-pm 8818 df-ixp 8887 df-en 8935 df-dom 8936 df-sdom 8937 df-fin 8938 df-fsupp 9357 df-fi 9401 df-sup 9432 df-inf 9433 df-oi 9500 df-dju 9891 df-card 9929 df-acn 9932 df-ac 10106 df-pnf 11246 df-mnf 11247 df-xr 11248 df-ltxr 11249 df-le 11250 df-sub 11442 df-neg 11443 df-div 11868 df-nn 12209 df-2 12271 df-3 12272 df-4 12273 df-5 12274 df-6 12275 df-7 12276 df-8 12277 df-9 12278 df-n0 12469 df-z 12555 df-dec 12674 df-uz 12819 df-q 12929 df-rp 12971 df-xneg 13088 df-xadd 13089 df-xmul 13090 df-ioo 13324 df-ioc 13325 df-ico 13326 df-icc 13327 df-fz 13481 df-fzo 13624 df-fl 13753 df-mod 13831 df-seq 13963 df-exp 14024 df-fac 14230 df-bc 14259 df-hash 14287 df-shft 15010 df-cj 15042 df-re 15043 df-im 15044 df-sqrt 15178 df-abs 15179 df-limsup 15411 df-clim 15428 df-rlim 15429 df-sum 15629 df-ef 16007 df-sin 16009 df-cos 16010 df-pi 16012 df-struct 17078 df-sets 17095 df-slot 17113 df-ndx 17125 df-base 17143 df-ress 17172 df-plusg 17208 df-mulr 17209 df-starv 17210 df-sca 17211 df-vsca 17212 df-ip 17213 df-tset 17214 df-ple 17215 df-ds 17217 df-unif 17218 df-hom 17219 df-cco 17220 df-rest 17366 df-topn 17367 df-0g 17385 df-gsum 17386 df-topgen 17387 df-pt 17388 df-prds 17391 df-ordt 17445 df-xrs 17446 df-qtop 17451 df-imas 17452 df-xps 17454 df-mre 17528 df-mrc 17529 df-acs 17531 df-ps 18520 df-tsr 18521 df-plusf 18561 df-mgm 18562 df-sgrp 18641 df-mnd 18657 df-mhm 18702 df-submnd 18703 df-grp 18855 df-minusg 18856 df-sbg 18857 df-mulg 18985 df-subg 19039 df-cntz 19222 df-cmn 19691 df-abl 19692 df-mgp 20029 df-rng 20047 df-ur 20076 df-ring 20129 df-cring 20130 df-subrng 20435 df-subrg 20460 df-abv 20649 df-lmod 20697 df-scaf 20698 df-sra 21010 df-rgmod 21011 df-psmet 21219 df-xmet 21220 df-met 21221 df-bl 21222 df-mopn 21223 df-fbas 21224 df-fg 21225 df-cnfld 21228 df-top 22717 df-topon 22734 df-topsp 22756 df-bases 22770 df-cld 22844 df-ntr 22845 df-cls 22846 df-nei 22923 df-lp 22961 df-perf 22962 df-cn 23052 df-cnp 23053 df-haus 23140 df-tx 23387 df-hmeo 23580 df-fil 23671 df-fm 23763 df-flim 23764 df-flf 23765 df-tmd 23897 df-tgp 23898 df-tsms 23952 df-trg 23985 df-xms 24147 df-ms 24148 df-tms 24149 df-nm 24412 df-ngp 24413 df-nrg 24415 df-nlm 24416 df-ii 24718 df-cncf 24719 df-limc 25716 df-dv 25717 df-log 26406 df-esum 33481 df-siga 33562 df-carsg 33756 |
| This theorem is referenced by: omsmeas 33777 |
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