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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 4008 | . . . . . . . 8 β’ (π£ = π‘ β (π β π£ β π β π‘)) | |
5 | 4 | cbvrabv 3442 | . . . . . . 7 β’ {π£ β πΏ β£ π β π£} = {π‘ β πΏ β£ π β π‘} |
6 | 5 | inteqi 4954 | . . . . . 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 33159 | . . . . 5 β’ (π β β© {π£ β πΏ β£ π β π£} β π) |
9 | 1 | ispisys2 33146 | . . . . . . . . 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 498 | . . . . . . . 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 4611 | . . . . . 6 β’ (π β π β π« π) |
13 | 2, 3, 12 | ldsysgenld 33153 | . . . . 5 β’ (π β β© {π£ β πΏ β£ π β π£} β πΏ) |
14 | 8, 13 | elind 4194 | . . . 4 β’ (π β β© {π£ β πΏ β£ π β π£} β (π β© πΏ)) |
15 | 1, 2 | sigapildsys 33155 | . . . 4 β’ (sigAlgebraβπ) = (π β© πΏ) |
16 | 14, 15 | eleqtrrdi 2844 | . . 3 β’ (π β β© {π£ β πΏ β£ π β π£} β (sigAlgebraβπ)) |
17 | ssintub 4970 | . . . 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 4008 | . . . 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 4978 | . . 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 584 | . 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 4008 | . . . 4 β’ (π£ = π β (π β π£ β π β π)) | |
25 | 24 | intminss 4978 | . . 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 584 | . 2 β’ (π β β© {π£ β πΏ β£ π β π£} β π) |
27 | 21, 26 | sstrd 3992 | 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 396 β§ w3a 1087 = wceq 1541 β wcel 2106 βwral 3061 {crab 3432 β cdif 3945 β© cin 3947 β wss 3948 β c0 4322 π« cpw 4602 {csn 4628 βͺ cuni 4908 β© cint 4950 Disj wdisj 5113 class class class wbr 5148 βcfv 6543 Οcom 7854 βΌ cdom 8936 Fincfn 8938 ficfi 9404 sigAlgebracsiga 33101 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7724 ax-inf2 9635 ax-ac2 10457 ax-cnex 11165 ax-resscn 11166 ax-1cn 11167 ax-icn 11168 ax-addcl 11169 ax-addrcl 11170 ax-mulcl 11171 ax-mulrcl 11172 ax-mulcom 11173 ax-addass 11174 ax-mulass 11175 ax-distr 11176 ax-i2m1 11177 ax-1ne0 11178 ax-1rid 11179 ax-rnegex 11180 ax-rrecex 11181 ax-cnre 11182 ax-pre-lttri 11183 ax-pre-lttrn 11184 ax-pre-ltadd 11185 ax-pre-mulgt0 11186 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3376 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-int 4951 df-iun 4999 df-iin 5000 df-disj 5114 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-se 5632 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-isom 6552 df-riota 7364 df-ov 7411 df-oprab 7412 df-mpo 7413 df-om 7855 df-1st 7974 df-2nd 7975 df-frecs 8265 df-wrecs 8296 df-recs 8370 df-rdg 8409 df-1o 8465 df-2o 8466 df-er 8702 df-map 8821 df-en 8939 df-dom 8940 df-sdom 8941 df-fin 8942 df-fi 9405 df-sup 9436 df-inf 9437 df-oi 9504 df-dju 9895 df-card 9933 df-acn 9936 df-ac 10110 df-pnf 11249 df-mnf 11250 df-xr 11251 df-ltxr 11252 df-le 11253 df-sub 11445 df-neg 11446 df-nn 12212 df-n0 12472 df-z 12558 df-uz 12822 df-fz 13484 df-fzo 13627 df-siga 33102 |
This theorem is referenced by: (None) |
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