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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 7845 | . . . . 5 β’ (π β OutMeas β dom π β V) | |
5 | uniexg 7682 | . . . . 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 2842 | . . 3 β’ (π β π β V) |
8 | pwidg 4585 | . . 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 4572 | . . . . . . 7 β’ (π β π« π β π β π) | |
11 | df-ss 3932 | . . . . . . . 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 6851 | . . . . 5 β’ (π β π« π β (πβ(π β© π)) = (πβπ)) |
15 | 14 | adantl 483 | . . . 4 β’ ((π β§ π β π« π) β (πβ(π β© π)) = (πβπ)) |
16 | ssdif0 4328 | . . . . . . . 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 6851 | . . . . . 6 β’ (π β π« π β (πβ(π β π)) = (πββ )) |
19 | 18 | adantl 483 | . . . . 5 β’ ((π β§ π β π« π) β (πβ(π β π)) = (πββ )) |
20 | 1 | ome0 44812 | . . . . . 6 β’ (π β (πββ ) = 0) |
21 | 20 | adantr 482 | . . . . 5 β’ ((π β§ π β π« π) β (πββ ) = 0) |
22 | 19, 21 | eqtrd 2777 | . . . 4 β’ ((π β§ π β π« π) β (πβ(π β π)) = 0) |
23 | 15, 22 | oveq12d 7380 | . . 3 β’ ((π β§ π β π« π) β ((πβ(π β© π)) +π (πβ(π β π))) = ((πβπ) +π 0)) |
24 | iccssxr 13354 | . . . . 5 β’ (0[,]+β) β β* | |
25 | 1 | adantr 482 | . . . . . 6 β’ ((π β§ π β π« π) β π β OutMeas) |
26 | 10 | adantl 483 | . . . . . 6 β’ ((π β§ π β π« π) β π β π) |
27 | 25, 2, 26 | omecl 44818 | . . . . 5 β’ ((π β§ π β π« π) β (πβπ) β (0[,]+β)) |
28 | 24, 27 | sselid 3947 | . . . 4 β’ ((π β§ π β π« π) β (πβπ) β β*) |
29 | 28 | xaddid1d 13169 | . . 3 β’ ((π β§ π β π« π) β ((πβπ) +π 0) = (πβπ)) |
30 | eqidd 2738 | . . 3 β’ ((π β§ π β π« π) β (πβπ) = (πβπ)) | |
31 | 23, 29, 30 | 3eqtrd 2781 | . 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 44817 | 1 β’ (π β π β π) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 397 = wceq 1542 β wcel 2107 Vcvv 3448 β cdif 3912 β© cin 3914 β wss 3915 β c0 4287 π« cpw 4565 βͺ cuni 4870 dom cdm 5638 βcfv 6501 (class class class)co 7362 0cc0 11058 +βcpnf 11193 β*cxr 11195 +π cxad 13038 [,]cicc 13274 OutMeascome 44804 CaraGenccaragen 44806 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2708 ax-sep 5261 ax-nul 5268 ax-pow 5325 ax-pr 5389 ax-un 7677 ax-cnex 11114 ax-resscn 11115 ax-1cn 11116 ax-icn 11117 ax-addcl 11118 ax-addrcl 11119 ax-mulcl 11120 ax-mulrcl 11121 ax-mulcom 11122 ax-addass 11123 ax-mulass 11124 ax-distr 11125 ax-i2m1 11126 ax-1ne0 11127 ax-1rid 11128 ax-rnegex 11129 ax-rrecex 11130 ax-cnre 11131 ax-pre-lttri 11132 ax-pre-lttrn 11133 ax-pre-ltadd 11134 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2890 df-ne 2945 df-nel 3051 df-ral 3066 df-rex 3075 df-rab 3411 df-v 3450 df-sbc 3745 df-csb 3861 df-dif 3918 df-un 3920 df-in 3922 df-ss 3932 df-nul 4288 df-if 4492 df-pw 4567 df-sn 4592 df-pr 4594 df-op 4598 df-uni 4871 df-iun 4961 df-br 5111 df-opab 5173 df-mpt 5194 df-id 5536 df-po 5550 df-so 5551 df-xp 5644 df-rel 5645 df-cnv 5646 df-co 5647 df-dm 5648 df-rn 5649 df-res 5650 df-ima 5651 df-iota 6453 df-fun 6503 df-fn 6504 df-f 6505 df-f1 6506 df-fo 6507 df-f1o 6508 df-fv 6509 df-ov 7365 df-oprab 7366 df-mpo 7367 df-1st 7926 df-2nd 7927 df-er 8655 df-en 8891 df-dom 8892 df-sdom 8893 df-pnf 11198 df-mnf 11199 df-xr 11200 df-ltxr 11201 df-xadd 13041 df-icc 13278 df-ome 44805 df-caragen 44807 |
This theorem is referenced by: caragenuni 44826 rrnmbl 44929 |
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