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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 4000 | . . . . . . . 8 β’ (π£ = π‘ β (π β π£ β π β π‘)) | |
5 | 4 | cbvrabv 3434 | . . . . . . 7 β’ {π£ β πΏ β£ π β π£} = {π‘ β πΏ β£ π β π‘} |
6 | 5 | inteqi 4944 | . . . . . 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 33653 | . . . . 5 β’ (π β β© {π£ β πΏ β£ π β π£} β π) |
9 | 1 | ispisys2 33640 | . . . . . . . . 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 497 | . . . . . . . 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 4603 | . . . . . 6 β’ (π β π β π« π) |
13 | 2, 3, 12 | ldsysgenld 33647 | . . . . 5 β’ (π β β© {π£ β πΏ β£ π β π£} β πΏ) |
14 | 8, 13 | elind 4186 | . . . 4 β’ (π β β© {π£ β πΏ β£ π β π£} β (π β© πΏ)) |
15 | 1, 2 | sigapildsys 33649 | . . . 4 β’ (sigAlgebraβπ) = (π β© πΏ) |
16 | 14, 15 | eleqtrrdi 2836 | . . 3 β’ (π β β© {π£ β πΏ β£ π β π£} β (sigAlgebraβπ)) |
17 | ssintub 4960 | . . . 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 4000 | . . . 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 4968 | . . 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 583 | . 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 4000 | . . . 4 β’ (π£ = π β (π β π£ β π β π)) | |
25 | 24 | intminss 4968 | . . 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 583 | . 2 β’ (π β β© {π£ β πΏ β£ π β π£} β π) |
27 | 21, 26 | sstrd 3984 | 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 395 β§ w3a 1084 = wceq 1533 β wcel 2098 βwral 3053 {crab 3424 β cdif 3937 β© cin 3939 β wss 3940 β c0 4314 π« cpw 4594 {csn 4620 βͺ cuni 4899 β© cint 4940 Disj wdisj 5103 class class class wbr 5138 βcfv 6533 Οcom 7848 βΌ cdom 8933 Fincfn 8935 ficfi 9401 sigAlgebracsiga 33595 |
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 2695 ax-rep 5275 ax-sep 5289 ax-nul 5296 ax-pow 5353 ax-pr 5417 ax-un 7718 ax-inf2 9632 ax-ac2 10454 ax-cnex 11162 ax-resscn 11163 ax-1cn 11164 ax-icn 11165 ax-addcl 11166 ax-addrcl 11167 ax-mulcl 11168 ax-mulrcl 11169 ax-mulcom 11170 ax-addass 11171 ax-mulass 11172 ax-distr 11173 ax-i2m1 11174 ax-1ne0 11175 ax-1rid 11176 ax-rnegex 11177 ax-rrecex 11178 ax-cnre 11179 ax-pre-lttri 11180 ax-pre-lttrn 11181 ax-pre-ltadd 11182 ax-pre-mulgt0 11183 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2526 df-eu 2555 df-clab 2702 df-cleq 2716 df-clel 2802 df-nfc 2877 df-ne 2933 df-nel 3039 df-ral 3054 df-rex 3063 df-rmo 3368 df-reu 3369 df-rab 3425 df-v 3468 df-sbc 3770 df-csb 3886 df-dif 3943 df-un 3945 df-in 3947 df-ss 3957 df-pss 3959 df-nul 4315 df-if 4521 df-pw 4596 df-sn 4621 df-pr 4623 df-op 4627 df-uni 4900 df-int 4941 df-iun 4989 df-iin 4990 df-disj 5104 df-br 5139 df-opab 5201 df-mpt 5222 df-tr 5256 df-id 5564 df-eprel 5570 df-po 5578 df-so 5579 df-fr 5621 df-se 5622 df-we 5623 df-xp 5672 df-rel 5673 df-cnv 5674 df-co 5675 df-dm 5676 df-rn 5677 df-res 5678 df-ima 5679 df-pred 6290 df-ord 6357 df-on 6358 df-lim 6359 df-suc 6360 df-iota 6485 df-fun 6535 df-fn 6536 df-f 6537 df-f1 6538 df-fo 6539 df-f1o 6540 df-fv 6541 df-isom 6542 df-riota 7357 df-ov 7404 df-oprab 7405 df-mpo 7406 df-om 7849 df-1st 7968 df-2nd 7969 df-frecs 8261 df-wrecs 8292 df-recs 8366 df-rdg 8405 df-1o 8461 df-2o 8462 df-er 8699 df-map 8818 df-en 8936 df-dom 8937 df-sdom 8938 df-fin 8939 df-fi 9402 df-sup 9433 df-inf 9434 df-oi 9501 df-dju 9892 df-card 9930 df-acn 9933 df-ac 10107 df-pnf 11247 df-mnf 11248 df-xr 11249 df-ltxr 11250 df-le 11251 df-sub 11443 df-neg 11444 df-nn 12210 df-n0 12470 df-z 12556 df-uz 12820 df-fz 13482 df-fzo 13625 df-siga 33596 |
This theorem is referenced by: (None) |
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