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Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > carsgcl | Structured version Visualization version GIF version |
Description: Closure of the Caratheodory measurable sets. (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)) |
Ref | Expression |
---|---|
carsgcl | β’ (π β (toCaraSigaβπ) β π« π) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | carsgval.1 | . . 3 β’ (π β π β π) | |
2 | carsgval.2 | . . 3 β’ (π β π:π« πβΆ(0[,]+β)) | |
3 | 1, 2 | carsgval 33832 | . 2 β’ (π β (toCaraSigaβπ) = {π β π« π β£ βπ β π« π((πβ(π β© π)) +π (πβ(π β π))) = (πβπ)}) |
4 | ssrab2 4072 | . 2 β’ {π β π« π β£ βπ β π« π((πβ(π β© π)) +π (πβ(π β π))) = (πβπ)} β π« π | |
5 | 3, 4 | eqsstrdi 4031 | 1 β’ (π β (toCaraSigaβπ) β π« π) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 = wceq 1533 β wcel 2098 βwral 3055 {crab 3426 β cdif 3940 β© cin 3942 β wss 3943 π« cpw 4597 βΆwf 6532 βcfv 6536 (class class class)co 7404 0cc0 11109 +βcpnf 11246 +π cxad 13093 [,]cicc 13330 toCaraSigaccarsg 33830 |
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 2697 ax-rep 5278 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 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 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-ral 3056 df-rex 3065 df-reu 3371 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-iun 4992 df-br 5142 df-opab 5204 df-mpt 5225 df-id 5567 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-iota 6488 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-ov 7407 df-carsg 33831 |
This theorem is referenced by: carsguni 33837 elcarsgss 33838 carsggect 33847 carsgsiga 33851 omsmeas 33852 |
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