Users' Mathboxes Mathbox for Thierry Arnoux < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  sxsiga Structured version   Visualization version   GIF version

Theorem sxsiga 32854
Description: A product sigma-algebra is a sigma-algebra. (Contributed by Thierry Arnoux, 1-Jun-2017.)
Assertion
Ref Expression
sxsiga ((𝑆 ∈ βˆͺ ran sigAlgebra ∧ 𝑇 ∈ βˆͺ ran sigAlgebra) β†’ (𝑆 Γ—s 𝑇) ∈ βˆͺ ran sigAlgebra)

Proof of Theorem sxsiga
Dummy variables π‘₯ 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2733 . . . 4 ran (π‘₯ ∈ 𝑆, 𝑦 ∈ 𝑇 ↦ (π‘₯ Γ— 𝑦)) = ran (π‘₯ ∈ 𝑆, 𝑦 ∈ 𝑇 ↦ (π‘₯ Γ— 𝑦))
21sxval 32853 . . 3 ((𝑆 ∈ βˆͺ ran sigAlgebra ∧ 𝑇 ∈ βˆͺ ran sigAlgebra) β†’ (𝑆 Γ—s 𝑇) = (sigaGenβ€˜ran (π‘₯ ∈ 𝑆, 𝑦 ∈ 𝑇 ↦ (π‘₯ Γ— 𝑦))))
31txbasex 22940 . . . 4 ((𝑆 ∈ βˆͺ ran sigAlgebra ∧ 𝑇 ∈ βˆͺ ran sigAlgebra) β†’ ran (π‘₯ ∈ 𝑆, 𝑦 ∈ 𝑇 ↦ (π‘₯ Γ— 𝑦)) ∈ V)
4 sigagensiga 32804 . . . 4 (ran (π‘₯ ∈ 𝑆, 𝑦 ∈ 𝑇 ↦ (π‘₯ Γ— 𝑦)) ∈ V β†’ (sigaGenβ€˜ran (π‘₯ ∈ 𝑆, 𝑦 ∈ 𝑇 ↦ (π‘₯ Γ— 𝑦))) ∈ (sigAlgebraβ€˜βˆͺ ran (π‘₯ ∈ 𝑆, 𝑦 ∈ 𝑇 ↦ (π‘₯ Γ— 𝑦))))
53, 4syl 17 . . 3 ((𝑆 ∈ βˆͺ ran sigAlgebra ∧ 𝑇 ∈ βˆͺ ran sigAlgebra) β†’ (sigaGenβ€˜ran (π‘₯ ∈ 𝑆, 𝑦 ∈ 𝑇 ↦ (π‘₯ Γ— 𝑦))) ∈ (sigAlgebraβ€˜βˆͺ ran (π‘₯ ∈ 𝑆, 𝑦 ∈ 𝑇 ↦ (π‘₯ Γ— 𝑦))))
62, 5eqeltrd 2834 . 2 ((𝑆 ∈ βˆͺ ran sigAlgebra ∧ 𝑇 ∈ βˆͺ ran sigAlgebra) β†’ (𝑆 Γ—s 𝑇) ∈ (sigAlgebraβ€˜βˆͺ ran (π‘₯ ∈ 𝑆, 𝑦 ∈ 𝑇 ↦ (π‘₯ Γ— 𝑦))))
7 elrnsiga 32789 . 2 ((𝑆 Γ—s 𝑇) ∈ (sigAlgebraβ€˜βˆͺ ran (π‘₯ ∈ 𝑆, 𝑦 ∈ 𝑇 ↦ (π‘₯ Γ— 𝑦))) β†’ (𝑆 Γ—s 𝑇) ∈ βˆͺ ran sigAlgebra)
86, 7syl 17 1 ((𝑆 ∈ βˆͺ ran sigAlgebra ∧ 𝑇 ∈ βˆͺ ran sigAlgebra) β†’ (𝑆 Γ—s 𝑇) ∈ βˆͺ ran sigAlgebra)
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 397   ∈ wcel 2107  Vcvv 3447  βˆͺ cuni 4869   Γ— cxp 5635  ran crn 5638  β€˜cfv 6500  (class class class)co 7361   ∈ cmpo 7363  sigAlgebracsiga 32771  sigaGencsigagen 32801   Γ—s 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
  Copyright terms: Public domain W3C validator