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Theorem sxsiga 34155
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 2740 . . . 4 ran (𝑥𝑆, 𝑦𝑇 ↦ (𝑥 × 𝑦)) = ran (𝑥𝑆, 𝑦𝑇 ↦ (𝑥 × 𝑦))
21sxval 34154 . . 3 ((𝑆 ran sigAlgebra ∧ 𝑇 ran sigAlgebra) → (𝑆 ×s 𝑇) = (sigaGen‘ran (𝑥𝑆, 𝑦𝑇 ↦ (𝑥 × 𝑦))))
31txbasex 23595 . . . 4 ((𝑆 ran sigAlgebra ∧ 𝑇 ran sigAlgebra) → ran (𝑥𝑆, 𝑦𝑇 ↦ (𝑥 × 𝑦)) ∈ V)
4 sigagensiga 34105 . . . 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 2844 . 2 ((𝑆 ran sigAlgebra ∧ 𝑇 ran sigAlgebra) → (𝑆 ×s 𝑇) ∈ (sigAlgebra‘ ran (𝑥𝑆, 𝑦𝑇 ↦ (𝑥 × 𝑦))))
7 elrnsiga 34090 . 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 395  wcel 2108  Vcvv 3488   cuni 4931   × cxp 5698  ran crn 5701  cfv 6573  (class class class)co 7448  cmpo 7450  sigAlgebracsiga 34072  sigaGencsigagen 34102   ×s csx 34152
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pow 5383  ax-pr 5447  ax-un 7770
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-ral 3068  df-rex 3077  df-rab 3444  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-int 4971  df-iun 5017  df-br 5167  df-opab 5229  df-mpt 5250  df-id 5593  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-iota 6525  df-fun 6575  df-fn 6576  df-f 6577  df-fv 6581  df-ov 7451  df-oprab 7452  df-mpo 7453  df-1st 8030  df-2nd 8031  df-siga 34073  df-sigagen 34103  df-sx 34153
This theorem is referenced by:  sxsigon  34156  1stmbfm  34225  2ndmbfm  34226  rrvadd  34417
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