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Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > dynkin | Structured version Visualization version GIF version |
Description: Dynkin's lambda-pi theorem: if a lambda-system contains a pi-system, it also contains the sigma-algebra generated by that pi-system. (Contributed by Thierry Arnoux, 16-Jun-2020.) |
Ref | Expression |
---|---|
dynkin.p | β’ π = {π β π« π« π β£ (fiβπ ) β π } |
dynkin.l | β’ πΏ = {π β π« π« π β£ (β β π β§ βπ₯ β π (π β π₯) β π β§ βπ₯ β π« π ((π₯ βΌ Ο β§ Disj π¦ β π₯ π¦) β βͺ π₯ β π ))} |
dynkin.o | β’ (π β π β π) |
dynkin.1 | β’ (π β π β πΏ) |
dynkin.2 | β’ (π β π β π) |
dynkin.3 | β’ (π β π β π) |
Ref | Expression |
---|---|
dynkin | β’ (π β β© {π’ β (sigAlgebraβπ) β£ π β π’} β π) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dynkin.p | . . . . . 6 β’ π = {π β π« π« π β£ (fiβπ ) β π } | |
2 | dynkin.l | . . . . . 6 β’ πΏ = {π β π« π« π β£ (β β π β§ βπ₯ β π (π β π₯) β π β§ βπ₯ β π« π ((π₯ βΌ Ο β§ Disj π¦ β π₯ π¦) β βͺ π₯ β π ))} | |
3 | dynkin.o | . . . . . 6 β’ (π β π β π) | |
4 | sseq2 3974 | . . . . . . . 8 β’ (π£ = π‘ β (π β π£ β π β π‘)) | |
5 | 4 | cbvrabv 3416 | . . . . . . 7 β’ {π£ β πΏ β£ π β π£} = {π‘ β πΏ β£ π β π‘} |
6 | 5 | inteqi 4915 | . . . . . 6 β’ β© {π£ β πΏ β£ π β π£} = β© {π‘ β πΏ β£ π β π‘} |
7 | dynkin.2 | . . . . . 6 β’ (π β π β π) | |
8 | 1, 2, 3, 6, 7 | ldgenpisys 32829 | . . . . 5 β’ (π β β© {π£ β πΏ β£ π β π£} β π) |
9 | 1 | ispisys2 32816 | . . . . . . . . 9 β’ (π β π β (π β π« π« π β§ βπ₯ β ((π« π β© Fin) β {β })β© π₯ β π)) |
10 | 9 | simplbi 499 | . . . . . . . 8 β’ (π β π β π β π« π« π) |
11 | 7, 10 | syl 17 | . . . . . . 7 β’ (π β π β π« π« π) |
12 | 11 | elpwid 4573 | . . . . . 6 β’ (π β π β π« π) |
13 | 2, 3, 12 | ldsysgenld 32823 | . . . . 5 β’ (π β β© {π£ β πΏ β£ π β π£} β πΏ) |
14 | 8, 13 | elind 4158 | . . . 4 β’ (π β β© {π£ β πΏ β£ π β π£} β (π β© πΏ)) |
15 | 1, 2 | sigapildsys 32825 | . . . 4 β’ (sigAlgebraβπ) = (π β© πΏ) |
16 | 14, 15 | eleqtrrdi 2845 | . . 3 β’ (π β β© {π£ β πΏ β£ π β π£} β (sigAlgebraβπ)) |
17 | ssintub 4931 | . . . 4 β’ π β β© {π£ β πΏ β£ π β π£} | |
18 | 17 | a1i 11 | . . 3 β’ (π β π β β© {π£ β πΏ β£ π β π£}) |
19 | sseq2 3974 | . . . 4 β’ (π’ = β© {π£ β πΏ β£ π β π£} β (π β π’ β π β β© {π£ β πΏ β£ π β π£})) | |
20 | 19 | intminss 4939 | . . 3 β’ ((β© {π£ β πΏ β£ π β π£} β (sigAlgebraβπ) β§ π β β© {π£ β πΏ β£ π β π£}) β β© {π’ β (sigAlgebraβπ) β£ π β π’} β β© {π£ β πΏ β£ π β π£}) |
21 | 16, 18, 20 | syl2anc 585 | . 2 β’ (π β β© {π’ β (sigAlgebraβπ) β£ π β π’} β β© {π£ β πΏ β£ π β π£}) |
22 | dynkin.1 | . . 3 β’ (π β π β πΏ) | |
23 | dynkin.3 | . . 3 β’ (π β π β π) | |
24 | sseq2 3974 | . . . 4 β’ (π£ = π β (π β π£ β π β π)) | |
25 | 24 | intminss 4939 | . . 3 β’ ((π β πΏ β§ π β π) β β© {π£ β πΏ β£ π β π£} β π) |
26 | 22, 23, 25 | syl2anc 585 | . 2 β’ (π β β© {π£ β πΏ β£ π β π£} β π) |
27 | 21, 26 | sstrd 3958 | 1 β’ (π β β© {π’ β (sigAlgebraβπ) β£ π β π’} β π) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 397 β§ w3a 1088 = wceq 1542 β wcel 2107 βwral 3061 {crab 3406 β cdif 3911 β© cin 3913 β wss 3914 β c0 4286 π« cpw 4564 {csn 4590 βͺ cuni 4869 β© cint 4911 Disj wdisj 5074 class class class wbr 5109 βcfv 6500 Οcom 7806 βΌ cdom 8887 Fincfn 8889 ficfi 9354 sigAlgebracsiga 32771 |
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 2704 ax-rep 5246 ax-sep 5260 ax-nul 5267 ax-pow 5324 ax-pr 5388 ax-un 7676 ax-inf2 9585 ax-ac2 10407 ax-cnex 11115 ax-resscn 11116 ax-1cn 11117 ax-icn 11118 ax-addcl 11119 ax-addrcl 11120 ax-mulcl 11121 ax-mulrcl 11122 ax-mulcom 11123 ax-addass 11124 ax-mulass 11125 ax-distr 11126 ax-i2m1 11127 ax-1ne0 11128 ax-1rid 11129 ax-rnegex 11130 ax-rrecex 11131 ax-cnre 11132 ax-pre-lttri 11133 ax-pre-lttrn 11134 ax-pre-ltadd 11135 ax-pre-mulgt0 11136 |
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 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3352 df-reu 3353 df-rab 3407 df-v 3449 df-sbc 3744 df-csb 3860 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-pss 3933 df-nul 4287 df-if 4491 df-pw 4566 df-sn 4591 df-pr 4593 df-op 4597 df-uni 4870 df-int 4912 df-iun 4960 df-iin 4961 df-disj 5075 df-br 5110 df-opab 5172 df-mpt 5193 df-tr 5227 df-id 5535 df-eprel 5541 df-po 5549 df-so 5550 df-fr 5592 df-se 5593 df-we 5594 df-xp 5643 df-rel 5644 df-cnv 5645 df-co 5646 df-dm 5647 df-rn 5648 df-res 5649 df-ima 5650 df-pred 6257 df-ord 6324 df-on 6325 df-lim 6326 df-suc 6327 df-iota 6452 df-fun 6502 df-fn 6503 df-f 6504 df-f1 6505 df-fo 6506 df-f1o 6507 df-fv 6508 df-isom 6509 df-riota 7317 df-ov 7364 df-oprab 7365 df-mpo 7366 df-om 7807 df-1st 7925 df-2nd 7926 df-frecs 8216 df-wrecs 8247 df-recs 8321 df-rdg 8360 df-1o 8416 df-2o 8417 df-er 8654 df-map 8773 df-en 8890 df-dom 8891 df-sdom 8892 df-fin 8893 df-fi 9355 df-sup 9386 df-inf 9387 df-oi 9454 df-dju 9845 df-card 9883 df-acn 9886 df-ac 10060 df-pnf 11199 df-mnf 11200 df-xr 11201 df-ltxr 11202 df-le 11203 df-sub 11395 df-neg 11396 df-nn 12162 df-n0 12422 df-z 12508 df-uz 12772 df-fz 13434 df-fzo 13577 df-siga 32772 |
This theorem is referenced by: (None) |
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