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Mirrors > Home > MPE Home > Th. List > Mathboxes > sxsiga | Structured version Visualization version GIF version |
Description: A product sigma-algebra is a sigma-algebra. (Contributed by Thierry Arnoux, 1-Jun-2017.) |
Ref | Expression |
---|---|
sxsiga | β’ ((π β βͺ ran sigAlgebra β§ π β βͺ ran sigAlgebra) β (π Γs π) β βͺ ran sigAlgebra) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2733 | . . . 4 β’ ran (π₯ β π, π¦ β π β¦ (π₯ Γ π¦)) = ran (π₯ β π, π¦ β π β¦ (π₯ Γ π¦)) | |
2 | 1 | sxval 32853 | . . 3 β’ ((π β βͺ ran sigAlgebra β§ π β βͺ ran sigAlgebra) β (π Γs π) = (sigaGenβran (π₯ β π, π¦ β π β¦ (π₯ Γ π¦)))) |
3 | 1 | txbasex 22940 | . . . 4 β’ ((π β βͺ ran sigAlgebra β§ π β βͺ ran sigAlgebra) β ran (π₯ β π, π¦ β π β¦ (π₯ Γ π¦)) β V) |
4 | sigagensiga 32804 | . . . 4 β’ (ran (π₯ β π, π¦ β π β¦ (π₯ Γ π¦)) β V β (sigaGenβran (π₯ β π, π¦ β π β¦ (π₯ Γ π¦))) β (sigAlgebraββͺ ran (π₯ β π, π¦ β π β¦ (π₯ Γ π¦)))) | |
5 | 3, 4 | syl 17 | . . 3 β’ ((π β βͺ ran sigAlgebra β§ π β βͺ ran sigAlgebra) β (sigaGenβran (π₯ β π, π¦ β π β¦ (π₯ Γ π¦))) β (sigAlgebraββͺ ran (π₯ β π, π¦ β π β¦ (π₯ Γ π¦)))) |
6 | 2, 5 | eqeltrd 2834 | . 2 β’ ((π β βͺ ran sigAlgebra β§ π β βͺ ran sigAlgebra) β (π Γs π) β (sigAlgebraββͺ ran (π₯ β π, π¦ β π β¦ (π₯ Γ π¦)))) |
7 | elrnsiga 32789 | . 2 β’ ((π Γs π) β (sigAlgebraββͺ ran (π₯ β π, π¦ β π β¦ (π₯ Γ π¦))) β (π Γs π) β βͺ ran sigAlgebra) | |
8 | 6, 7 | syl 17 | 1 β’ ((π β βͺ ran sigAlgebra β§ π β βͺ ran sigAlgebra) β (π Γs π) β βͺ ran sigAlgebra) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 397 β wcel 2107 Vcvv 3447 βͺ cuni 4869 Γ cxp 5635 ran crn 5638 βcfv 6500 (class class class)co 7361 β cmpo 7363 sigAlgebracsiga 32771 sigaGencsigagen 32801 Γs csx 32851 |
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-sep 5260 ax-nul 5267 ax-pow 5324 ax-pr 5388 ax-un 7676 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 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-ral 3062 df-rex 3071 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-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-br 5110 df-opab 5172 df-mpt 5193 df-id 5535 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-iota 6452 df-fun 6502 df-fn 6503 df-f 6504 df-fv 6508 df-ov 7364 df-oprab 7365 df-mpo 7366 df-1st 7925 df-2nd 7926 df-siga 32772 df-sigagen 32802 df-sx 32852 |
This theorem is referenced by: sxsigon 32855 1stmbfm 32924 2ndmbfm 32925 rrvadd 33116 |
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