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Theorem sxsiga 31458
Description: A product sigma-algebra is a sigma-algebra. (Contributed by Thierry Arnoux, 1-Jun-2017.)
Assertion
Ref Expression
sxsiga ((𝑆 ran sigAlgebra ∧ 𝑇 ran sigAlgebra) → (𝑆 ×s 𝑇) ∈ ran sigAlgebra)

Proof of Theorem sxsiga
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2821 . . . 4 ran (𝑥𝑆, 𝑦𝑇 ↦ (𝑥 × 𝑦)) = ran (𝑥𝑆, 𝑦𝑇 ↦ (𝑥 × 𝑦))
21sxval 31457 . . 3 ((𝑆 ran sigAlgebra ∧ 𝑇 ran sigAlgebra) → (𝑆 ×s 𝑇) = (sigaGen‘ran (𝑥𝑆, 𝑦𝑇 ↦ (𝑥 × 𝑦))))
31txbasex 22150 . . . 4 ((𝑆 ran sigAlgebra ∧ 𝑇 ran sigAlgebra) → ran (𝑥𝑆, 𝑦𝑇 ↦ (𝑥 × 𝑦)) ∈ V)
4 sigagensiga 31408 . . . 4 (ran (𝑥𝑆, 𝑦𝑇 ↦ (𝑥 × 𝑦)) ∈ V → (sigaGen‘ran (𝑥𝑆, 𝑦𝑇 ↦ (𝑥 × 𝑦))) ∈ (sigAlgebra‘ ran (𝑥𝑆, 𝑦𝑇 ↦ (𝑥 × 𝑦))))
53, 4syl 17 . . 3 ((𝑆 ran sigAlgebra ∧ 𝑇 ran sigAlgebra) → (sigaGen‘ran (𝑥𝑆, 𝑦𝑇 ↦ (𝑥 × 𝑦))) ∈ (sigAlgebra‘ ran (𝑥𝑆, 𝑦𝑇 ↦ (𝑥 × 𝑦))))
62, 5eqeltrd 2912 . 2 ((𝑆 ran sigAlgebra ∧ 𝑇 ran sigAlgebra) → (𝑆 ×s 𝑇) ∈ (sigAlgebra‘ ran (𝑥𝑆, 𝑦𝑇 ↦ (𝑥 × 𝑦))))
7 elrnsiga 31393 . 2 ((𝑆 ×s 𝑇) ∈ (sigAlgebra‘ ran (𝑥𝑆, 𝑦𝑇 ↦ (𝑥 × 𝑦))) → (𝑆 ×s 𝑇) ∈ ran sigAlgebra)
86, 7syl 17 1 ((𝑆 ran sigAlgebra ∧ 𝑇 ran sigAlgebra) → (𝑆 ×s 𝑇) ∈ ran sigAlgebra)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399  wcel 2115  Vcvv 3471   cuni 4811   × cxp 5526  ran crn 5529  cfv 6328  (class class class)co 7130  cmpo 7132  sigAlgebracsiga 31375  sigaGencsigagen 31405   ×s csx 31455
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 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2178  ax-ext 2793  ax-sep 5176  ax-nul 5183  ax-pow 5239  ax-pr 5303  ax-un 7436
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2623  df-eu 2654  df-clab 2800  df-cleq 2814  df-clel 2892  df-nfc 2960  df-ne 3008  df-ral 3131  df-rex 3132  df-rab 3135  df-v 3473  df-sbc 3750  df-csb 3858  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4267  df-if 4441  df-pw 4514  df-sn 4541  df-pr 4543  df-op 4547  df-uni 4812  df-int 4850  df-iun 4894  df-br 5040  df-opab 5102  df-mpt 5120  df-id 5433  df-xp 5534  df-rel 5535  df-cnv 5536  df-co 5537  df-dm 5538  df-rn 5539  df-res 5540  df-ima 5541  df-iota 6287  df-fun 6330  df-fn 6331  df-f 6332  df-fv 6336  df-ov 7133  df-oprab 7134  df-mpo 7135  df-1st 7664  df-2nd 7665  df-siga 31376  df-sigagen 31406  df-sx 31456
This theorem is referenced by:  sxsigon  31459  1stmbfm  31526  2ndmbfm  31527  rrvadd  31718
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