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Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > caragensal | Structured version Visualization version GIF version |
Description: Caratheodory's method generates a sigma-algebra. (Contributed by Glauco Siliprandi, 17-Aug-2020.) |
Ref | Expression |
---|---|
caragensal.o | β’ (π β π β OutMeas) |
caragensal.s | β’ π = (CaraGenβπ) |
Ref | Expression |
---|---|
caragensal | β’ (π β π β SAlg) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | caragensal.o | . . . 4 β’ (π β π β OutMeas) | |
2 | caragensal.s | . . . 4 β’ π = (CaraGenβπ) | |
3 | 1, 2 | caragen0 45956 | . . 3 β’ (π β β β π) |
4 | 1 | adantr 479 | . . . . 5 β’ ((π β§ π₯ β π) β π β OutMeas) |
5 | simpr 483 | . . . . 5 β’ ((π β§ π₯ β π) β π₯ β π) | |
6 | 4, 2, 5 | caragendifcl 45964 | . . . 4 β’ ((π β§ π₯ β π) β (βͺ π β π₯) β π) |
7 | 6 | ralrimiva 3136 | . . 3 β’ (π β βπ₯ β π (βͺ π β π₯) β π) |
8 | 1 | ad2antrr 724 | . . . . . 6 β’ (((π β§ π₯ β π« π) β§ π₯ βΌ Ο) β π β OutMeas) |
9 | elpwi 4605 | . . . . . . 7 β’ (π₯ β π« π β π₯ β π) | |
10 | 9 | ad2antlr 725 | . . . . . 6 β’ (((π β§ π₯ β π« π) β§ π₯ βΌ Ο) β π₯ β π) |
11 | simpr 483 | . . . . . 6 β’ (((π β§ π₯ β π« π) β§ π₯ βΌ Ο) β π₯ βΌ Ο) | |
12 | 8, 2, 10, 11 | caragenunicl 45974 | . . . . 5 β’ (((π β§ π₯ β π« π) β§ π₯ βΌ Ο) β βͺ π₯ β π) |
13 | 12 | ex 411 | . . . 4 β’ ((π β§ π₯ β π« π) β (π₯ βΌ Ο β βͺ π₯ β π)) |
14 | 13 | ralrimiva 3136 | . . 3 β’ (π β βπ₯ β π« π(π₯ βΌ Ο β βͺ π₯ β π)) |
15 | 3, 7, 14 | 3jca 1125 | . 2 β’ (π β (β β π β§ βπ₯ β π (βͺ π β π₯) β π β§ βπ₯ β π« π(π₯ βΌ Ο β βͺ π₯ β π))) |
16 | 2 | fvexi 6905 | . . . 4 β’ π β V |
17 | 16 | a1i 11 | . . 3 β’ (π β π β V) |
18 | issal 45764 | . . 3 β’ (π β V β (π β SAlg β (β β π β§ βπ₯ β π (βͺ π β π₯) β π β§ βπ₯ β π« π(π₯ βΌ Ο β βͺ π₯ β π)))) | |
19 | 17, 18 | syl 17 | . 2 β’ (π β (π β SAlg β (β β π β§ βπ₯ β π (βͺ π β π₯) β π β§ βπ₯ β π« π(π₯ βΌ Ο β βͺ π₯ β π)))) |
20 | 15, 19 | mpbird 256 | 1 β’ (π β π β SAlg) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β wb 205 β§ wa 394 β§ w3a 1084 = wceq 1533 β wcel 2098 βwral 3051 Vcvv 3463 β cdif 3937 β wss 3940 β c0 4318 π« cpw 4598 βͺ cuni 4903 class class class wbr 5143 βcfv 6542 Οcom 7867 βΌ cdom 8958 SAlgcsalg 45758 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-rep 5280 ax-sep 5294 ax-nul 5301 ax-pow 5359 ax-pr 5423 ax-un 7737 ax-inf2 9662 ax-ac2 10484 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 ax-pre-mulgt0 11213 ax-pre-sup 11214 |
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-rmo 3364 df-reu 3365 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-pss 3960 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-int 4945 df-iun 4993 df-disj 5109 df-br 5144 df-opab 5206 df-mpt 5227 df-tr 5261 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-se 5628 df-we 5629 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-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 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-isom 6551 df-riota 7371 df-ov 7418 df-oprab 7419 df-mpo 7420 df-om 7868 df-1st 7989 df-2nd 7990 df-frecs 8283 df-wrecs 8314 df-recs 8388 df-rdg 8427 df-1o 8483 df-oadd 8487 df-omul 8488 df-er 8721 df-map 8843 df-en 8961 df-dom 8962 df-sdom 8963 df-fin 8964 df-sup 9463 df-inf 9464 df-oi 9531 df-card 9960 df-acn 9963 df-ac 10137 df-pnf 11278 df-mnf 11279 df-xr 11280 df-ltxr 11281 df-le 11282 df-sub 11474 df-neg 11475 df-div 11900 df-nn 12241 df-2 12303 df-3 12304 df-n0 12501 df-z 12587 df-uz 12851 df-q 12961 df-rp 13005 df-xadd 13123 df-ico 13360 df-icc 13361 df-fz 13515 df-fzo 13658 df-seq 13997 df-exp 14057 df-hash 14320 df-cj 15076 df-re 15077 df-im 15078 df-sqrt 15212 df-abs 15213 df-clim 15462 df-sum 15663 df-salg 45759 df-sumge0 45813 df-ome 45940 df-caragen 45942 |
This theorem is referenced by: caratheodory 45978 |
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