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Mathbox for Thierry Arnoux |
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| Mirrors > Home > MPE Home > Th. List > Mathboxes > sxuni | Structured version Visualization version GIF version | ||
| Description: The base set of a product sigma-algebra. (Contributed by Thierry Arnoux, 1-Jun-2017.) |
| Ref | Expression |
|---|---|
| sxuni | β’ ((π β βͺ ran sigAlgebra β§ π β βͺ ran sigAlgebra) β (βͺ π Γ βͺ π) = βͺ (π Γs π)) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | sxsigon 33185 | . 2 β’ ((π β βͺ ran sigAlgebra β§ π β βͺ ran sigAlgebra) β (π Γs π) β (sigAlgebraβ(βͺ π Γ βͺ π))) | |
| 2 | issgon 33116 | . . 3 β’ ((π Γs π) β (sigAlgebraβ(βͺ π Γ βͺ π)) β ((π Γs π) β βͺ ran sigAlgebra β§ (βͺ π Γ βͺ π) = βͺ (π Γs π))) | |
| 3 | 2 | simprbi 497 | . 2 β’ ((π Γs π) β (sigAlgebraβ(βͺ π Γ βͺ π)) β (βͺ π Γ βͺ π) = βͺ (π Γs π)) |
| 4 | 1, 3 | syl 17 | 1 β’ ((π β βͺ ran sigAlgebra β§ π β βͺ ran sigAlgebra) β (βͺ π Γ βͺ π) = βͺ (π Γs π)) |
| Colors of variables: wff setvar class |
| Syntax hints: β wi 4 β§ wa 396 = wceq 1541 β wcel 2106 βͺ cuni 4908 Γ cxp 5674 ran crn 5677 βcfv 6543 (class class class)co 7408 sigAlgebracsiga 33101 Γs csx 33181 |
| 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 |
| This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 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-ral 3062 df-rex 3071 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-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-br 5149 df-opab 5211 df-mpt 5232 df-id 5574 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-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-ov 7411 df-oprab 7412 df-mpo 7413 df-1st 7974 df-2nd 7975 df-siga 33102 df-sigagen 33132 df-sx 33182 |
| This theorem is referenced by: 1stmbfm 33254 2ndmbfm 33255 mbfmco2 33259 |
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