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Theorem sxsigon 29582
Description: A product sigma-algebra is a sigma-algebra on the product of the bases. (Contributed by Thierry Arnoux, 1-Jun-2017.)
Assertion
Ref Expression
sxsigon ((𝑆 ran sigAlgebra ∧ 𝑇 ran sigAlgebra) → (𝑆 ×s 𝑇) ∈ (sigAlgebra‘( 𝑆 × 𝑇)))

Proof of Theorem sxsigon
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 sxsiga 29581 . 2 ((𝑆 ran sigAlgebra ∧ 𝑇 ran sigAlgebra) → (𝑆 ×s 𝑇) ∈ ran sigAlgebra)
2 eqid 2610 . . . . . 6 ran (𝑥𝑆, 𝑦𝑇 ↦ (𝑥 × 𝑦)) = ran (𝑥𝑆, 𝑦𝑇 ↦ (𝑥 × 𝑦))
32sxval 29580 . . . . 5 ((𝑆 ran sigAlgebra ∧ 𝑇 ran sigAlgebra) → (𝑆 ×s 𝑇) = (sigaGen‘ran (𝑥𝑆, 𝑦𝑇 ↦ (𝑥 × 𝑦))))
43unieqd 4382 . . . 4 ((𝑆 ran sigAlgebra ∧ 𝑇 ran sigAlgebra) → (𝑆 ×s 𝑇) = (sigaGen‘ran (𝑥𝑆, 𝑦𝑇 ↦ (𝑥 × 𝑦))))
5 mpt2exga 7135 . . . . 5 ((𝑆 ran sigAlgebra ∧ 𝑇 ran sigAlgebra) → (𝑥𝑆, 𝑦𝑇 ↦ (𝑥 × 𝑦)) ∈ V)
6 rnexg 6990 . . . . 5 ((𝑥𝑆, 𝑦𝑇 ↦ (𝑥 × 𝑦)) ∈ V → ran (𝑥𝑆, 𝑦𝑇 ↦ (𝑥 × 𝑦)) ∈ V)
7 unisg 29533 . . . . 5 (ran (𝑥𝑆, 𝑦𝑇 ↦ (𝑥 × 𝑦)) ∈ V → (sigaGen‘ran (𝑥𝑆, 𝑦𝑇 ↦ (𝑥 × 𝑦))) = ran (𝑥𝑆, 𝑦𝑇 ↦ (𝑥 × 𝑦)))
85, 6, 73syl 18 . . . 4 ((𝑆 ran sigAlgebra ∧ 𝑇 ran sigAlgebra) → (sigaGen‘ran (𝑥𝑆, 𝑦𝑇 ↦ (𝑥 × 𝑦))) = ran (𝑥𝑆, 𝑦𝑇 ↦ (𝑥 × 𝑦)))
94, 8eqtrd 2644 . . 3 ((𝑆 ran sigAlgebra ∧ 𝑇 ran sigAlgebra) → (𝑆 ×s 𝑇) = ran (𝑥𝑆, 𝑦𝑇 ↦ (𝑥 × 𝑦)))
10 eqid 2610 . . . 4 𝑆 = 𝑆
11 eqid 2610 . . . 4 𝑇 = 𝑇
122, 10, 11txuni2 21178 . . 3 ( 𝑆 × 𝑇) = ran (𝑥𝑆, 𝑦𝑇 ↦ (𝑥 × 𝑦))
139, 12syl6reqr 2663 . 2 ((𝑆 ran sigAlgebra ∧ 𝑇 ran sigAlgebra) → ( 𝑆 × 𝑇) = (𝑆 ×s 𝑇))
14 issgon 29513 . 2 ((𝑆 ×s 𝑇) ∈ (sigAlgebra‘( 𝑆 × 𝑇)) ↔ ((𝑆 ×s 𝑇) ∈ ran sigAlgebra ∧ ( 𝑆 × 𝑇) = (𝑆 ×s 𝑇)))
151, 13, 14sylanbrc 695 1 ((𝑆 ran sigAlgebra ∧ 𝑇 ran sigAlgebra) → (𝑆 ×s 𝑇) ∈ (sigAlgebra‘( 𝑆 × 𝑇)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383   = wceq 1475  wcel 1977  Vcvv 3173   cuni 4372   × cxp 5036  ran crn 5039  cfv 5804  (class class class)co 6549  cmpt2 6551  sigAlgebracsiga 29497  sigaGencsigagen 29528   ×s csx 29578
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-fal 1481  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-1st 7059  df-2nd 7060  df-siga 29498  df-sigagen 29529  df-sx 29579
This theorem is referenced by:  sxuni  29583
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