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Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > caragenuni | Structured version Visualization version GIF version |
Description: The base set of the sigma-algebra generated by the Caratheodory's construction is the whole base set of the original outer measure. (Contributed by Glauco Siliprandi, 17-Aug-2020.) |
Ref | Expression |
---|---|
caragenuni.o | β’ (π β π β OutMeas) |
caragenuni.s | β’ π = (CaraGenβπ) |
Ref | Expression |
---|---|
caragenuni | β’ (π β βͺ π = βͺ dom π) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | caragenuni.o | . . . 4 β’ (π β π β OutMeas) | |
2 | caragenuni.s | . . . . 5 β’ π = (CaraGenβπ) | |
3 | 2 | caragenss 45954 | . . . 4 β’ (π β OutMeas β π β dom π) |
4 | 1, 3 | syl 17 | . . 3 β’ (π β π β dom π) |
5 | 4 | unissd 4913 | . 2 β’ (π β βͺ π β βͺ dom π) |
6 | eqid 2725 | . . . 4 β’ βͺ dom π = βͺ dom π | |
7 | 1, 6, 2 | caragenunidm 45958 | . . 3 β’ (π β βͺ dom π β π) |
8 | elssuni 4935 | . . 3 β’ (βͺ dom π β π β βͺ dom π β βͺ π) | |
9 | 7, 8 | syl 17 | . 2 β’ (π β βͺ dom π β βͺ π) |
10 | 5, 9 | eqssd 3990 | 1 β’ (π β βͺ π = βͺ dom π) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 = wceq 1533 β wcel 2098 β wss 3940 βͺ cuni 4903 dom cdm 5672 βcfv 6542 OutMeascome 45939 CaraGenccaragen 45941 |
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 2166 ax-ext 2696 ax-sep 5294 ax-nul 5301 ax-pow 5359 ax-pr 5423 ax-un 7737 ax-cnex 11192 ax-resscn 11193 ax-1cn 11194 ax-icn 11195 ax-addcl 11196 ax-addrcl 11197 ax-mulcl 11198 ax-mulrcl 11199 ax-mulcom 11200 ax-addass 11201 ax-mulass 11202 ax-distr 11203 ax-i2m1 11204 ax-1ne0 11205 ax-1rid 11206 ax-rnegex 11207 ax-rrecex 11208 ax-cnre 11209 ax-pre-lttri 11210 ax-pre-lttrn 11211 ax-pre-ltadd 11212 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-rab 3420 df-v 3465 df-sbc 3770 df-csb 3886 df-dif 3943 df-un 3945 df-in 3947 df-ss 3957 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 5144 df-opab 5206 df-mpt 5227 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 7418 df-oprab 7419 df-mpo 7420 df-1st 7989 df-2nd 7990 df-er 8721 df-en 8961 df-dom 8962 df-sdom 8963 df-pnf 11278 df-mnf 11279 df-xr 11280 df-ltxr 11281 df-xadd 13123 df-icc 13361 df-ome 45940 df-caragen 45942 |
This theorem is referenced by: caragendifcl 45964 carageniuncl 45973 unidmvon 46067 |
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