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Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > caragenunidm | Structured version Visualization version GIF version |
Description: The base set of an outer measure belongs to the sigma-algebra generated by the Caratheodory's construction. (Contributed by Glauco Siliprandi, 17-Aug-2020.) |
Ref | Expression |
---|---|
caragenunidm.o | β’ (π β π β OutMeas) |
caragenunidm.x | β’ π = βͺ dom π |
caragenunidm.s | β’ π = (CaraGenβπ) |
Ref | Expression |
---|---|
caragenunidm | β’ (π β π β π) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | caragenunidm.o | . 2 β’ (π β π β OutMeas) | |
2 | caragenunidm.x | . 2 β’ π = βͺ dom π | |
3 | caragenunidm.s | . 2 β’ π = (CaraGenβπ) | |
4 | dmexg 7901 | . . . . 5 β’ (π β OutMeas β dom π β V) | |
5 | uniexg 7737 | . . . . 5 β’ (dom π β V β βͺ dom π β V) | |
6 | 1, 4, 5 | 3syl 18 | . . . 4 β’ (π β βͺ dom π β V) |
7 | 2, 6 | eqeltrid 2832 | . . 3 β’ (π β π β V) |
8 | pwidg 4618 | . . 3 β’ (π β V β π β π« π) | |
9 | 7, 8 | syl 17 | . 2 β’ (π β π β π« π) |
10 | elpwi 4605 | . . . . . . 7 β’ (π β π« π β π β π) | |
11 | df-ss 3961 | . . . . . . . 8 β’ (π β π β (π β© π) = π) | |
12 | 11 | biimpi 215 | . . . . . . 7 β’ (π β π β (π β© π) = π) |
13 | 10, 12 | syl 17 | . . . . . 6 β’ (π β π« π β (π β© π) = π) |
14 | 13 | fveq2d 6895 | . . . . 5 β’ (π β π« π β (πβ(π β© π)) = (πβπ)) |
15 | 14 | adantl 481 | . . . 4 β’ ((π β§ π β π« π) β (πβ(π β© π)) = (πβπ)) |
16 | ssdif0 4359 | . . . . . . . 8 β’ (π β π β (π β π) = β ) | |
17 | 10, 16 | sylib 217 | . . . . . . 7 β’ (π β π« π β (π β π) = β ) |
18 | 17 | fveq2d 6895 | . . . . . 6 β’ (π β π« π β (πβ(π β π)) = (πββ )) |
19 | 18 | adantl 481 | . . . . 5 β’ ((π β§ π β π« π) β (πβ(π β π)) = (πββ )) |
20 | 1 | ome0 45798 | . . . . . 6 β’ (π β (πββ ) = 0) |
21 | 20 | adantr 480 | . . . . 5 β’ ((π β§ π β π« π) β (πββ ) = 0) |
22 | 19, 21 | eqtrd 2767 | . . . 4 β’ ((π β§ π β π« π) β (πβ(π β π)) = 0) |
23 | 15, 22 | oveq12d 7432 | . . 3 β’ ((π β§ π β π« π) β ((πβ(π β© π)) +π (πβ(π β π))) = ((πβπ) +π 0)) |
24 | iccssxr 13425 | . . . . 5 β’ (0[,]+β) β β* | |
25 | 1 | adantr 480 | . . . . . 6 β’ ((π β§ π β π« π) β π β OutMeas) |
26 | 10 | adantl 481 | . . . . . 6 β’ ((π β§ π β π« π) β π β π) |
27 | 25, 2, 26 | omecl 45804 | . . . . 5 β’ ((π β§ π β π« π) β (πβπ) β (0[,]+β)) |
28 | 24, 27 | sselid 3976 | . . . 4 β’ ((π β§ π β π« π) β (πβπ) β β*) |
29 | 28 | xaddridd 13240 | . . 3 β’ ((π β§ π β π« π) β ((πβπ) +π 0) = (πβπ)) |
30 | eqidd 2728 | . . 3 β’ ((π β§ π β π« π) β (πβπ) = (πβπ)) | |
31 | 23, 29, 30 | 3eqtrd 2771 | . 2 β’ ((π β§ π β π« π) β ((πβ(π β© π)) +π (πβ(π β π))) = (πβπ)) |
32 | 1, 2, 3, 9, 31 | carageneld 45803 | 1 β’ (π β π β π) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 395 = wceq 1534 β wcel 2099 Vcvv 3469 β cdif 3941 β© cin 3943 β wss 3944 β c0 4318 π« cpw 4598 βͺ cuni 4903 dom cdm 5672 βcfv 6542 (class class class)co 7414 0cc0 11124 +βcpnf 11261 β*cxr 11263 +π cxad 13108 [,]cicc 13345 OutMeascome 45790 CaraGenccaragen 45792 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2164 ax-ext 2698 ax-sep 5293 ax-nul 5300 ax-pow 5359 ax-pr 5423 ax-un 7732 ax-cnex 11180 ax-resscn 11181 ax-1cn 11182 ax-icn 11183 ax-addcl 11184 ax-addrcl 11185 ax-mulcl 11186 ax-mulrcl 11187 ax-mulcom 11188 ax-addass 11189 ax-mulass 11190 ax-distr 11191 ax-i2m1 11192 ax-1ne0 11193 ax-1rid 11194 ax-rnegex 11195 ax-rrecex 11196 ax-cnre 11197 ax-pre-lttri 11198 ax-pre-lttrn 11199 ax-pre-ltadd 11200 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2705 df-cleq 2719 df-clel 2805 df-nfc 2880 df-ne 2936 df-nel 3042 df-ral 3057 df-rex 3066 df-rab 3428 df-v 3471 df-sbc 3775 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-iun 4993 df-br 5143 df-opab 5205 df-mpt 5226 df-id 5570 df-po 5584 df-so 5585 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-ov 7417 df-oprab 7418 df-mpo 7419 df-1st 7985 df-2nd 7986 df-er 8716 df-en 8954 df-dom 8955 df-sdom 8956 df-pnf 11266 df-mnf 11267 df-xr 11268 df-ltxr 11269 df-xadd 13111 df-icc 13349 df-ome 45791 df-caragen 45793 |
This theorem is referenced by: caragenuni 45812 rrnmbl 45915 |
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