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Theorem sxsigon 28626
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  e.  U. ran sigAlgebra  /\  T  e.  U. ran sigAlgebra )  -> 
( S ×s  T )  e.  (sigAlgebra `  ( U. S  X.  U. T ) ) )

Proof of Theorem sxsigon
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 sxsiga 28625 . 2  |-  ( ( S  e.  U. ran sigAlgebra  /\  T  e.  U. ran sigAlgebra )  -> 
( S ×s  T )  e.  U. ran sigAlgebra )
2 eqid 2402 . . . . . 6  |-  ran  (
x  e.  S , 
y  e.  T  |->  ( x  X.  y ) )  =  ran  (
x  e.  S , 
y  e.  T  |->  ( x  X.  y ) )
32sxval 28624 . . . . 5  |-  ( ( S  e.  U. ran sigAlgebra  /\  T  e.  U. ran sigAlgebra )  -> 
( S ×s  T )  =  (sigaGen `  ran  ( x  e.  S ,  y  e.  T  |->  ( x  X.  y ) ) ) )
43unieqd 4200 . . . 4  |-  ( ( S  e.  U. ran sigAlgebra  /\  T  e.  U. ran sigAlgebra )  ->  U. ( S ×s  T )  =  U. (sigaGen `  ran  ( x  e.  S ,  y  e.  T  |->  ( x  X.  y ) ) ) )
5 mpt2exga 6859 . . . . 5  |-  ( ( S  e.  U. ran sigAlgebra  /\  T  e.  U. ran sigAlgebra )  -> 
( x  e.  S ,  y  e.  T  |->  ( x  X.  y
) )  e.  _V )
6 rnexg 6715 . . . . 5  |-  ( ( x  e.  S , 
y  e.  T  |->  ( x  X.  y ) )  e.  _V  ->  ran  ( x  e.  S ,  y  e.  T  |->  ( x  X.  y
) )  e.  _V )
7 unisg 28577 . . . . 5  |-  ( ran  ( x  e.  S ,  y  e.  T  |->  ( x  X.  y
) )  e.  _V  ->  U. (sigaGen `  ran  ( x  e.  S ,  y  e.  T  |->  ( x  X.  y
) ) )  = 
U. ran  ( x  e.  S ,  y  e.  T  |->  ( x  X.  y ) ) )
85, 6, 73syl 20 . . . 4  |-  ( ( S  e.  U. ran sigAlgebra  /\  T  e.  U. ran sigAlgebra )  ->  U. (sigaGen `  ran  ( x  e.  S ,  y  e.  T  |->  ( x  X.  y ) ) )  =  U. ran  ( x  e.  S ,  y  e.  T  |->  ( x  X.  y
) ) )
94, 8eqtrd 2443 . . 3  |-  ( ( S  e.  U. ran sigAlgebra  /\  T  e.  U. ran sigAlgebra )  ->  U. ( S ×s  T )  =  U. ran  ( x  e.  S ,  y  e.  T  |->  ( x  X.  y
) ) )
10 eqid 2402 . . . 4  |-  U. S  =  U. S
11 eqid 2402 . . . 4  |-  U. T  =  U. T
122, 10, 11txuni2 20356 . . 3  |-  ( U. S  X.  U. T )  =  U. ran  (
x  e.  S , 
y  e.  T  |->  ( x  X.  y ) )
139, 12syl6reqr 2462 . 2  |-  ( ( S  e.  U. ran sigAlgebra  /\  T  e.  U. ran sigAlgebra )  -> 
( U. S  X.  U. T )  =  U. ( S ×s  T ) )
14 issgon 28557 . 2  |-  ( ( S ×s  T )  e.  (sigAlgebra `  ( U. S  X.  U. T ) )  <->  ( ( S ×s  T )  e.  U. ran sigAlgebra  /\  ( U. S  X.  U. T )  =  U. ( S ×s  T ) ) )
151, 13, 14sylanbrc 662 1  |-  ( ( S  e.  U. ran sigAlgebra  /\  T  e.  U. ran sigAlgebra )  -> 
( S ×s  T )  e.  (sigAlgebra `  ( U. S  X.  U. T ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 367    = wceq 1405    e. wcel 1842   _Vcvv 3058   U.cuni 4190    X. cxp 4820   ran crn 4823   ` cfv 5568  (class class class)co 6277    |-> cmpt2 6279  sigAlgebracsiga 28541  sigaGencsigagen 28572   ×s csx 28622
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-rep 4506  ax-sep 4516  ax-nul 4524  ax-pow 4571  ax-pr 4629  ax-un 6573
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 976  df-tru 1408  df-fal 1411  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-ral 2758  df-rex 2759  df-reu 2760  df-rab 2762  df-v 3060  df-sbc 3277  df-csb 3373  df-dif 3416  df-un 3418  df-in 3420  df-ss 3427  df-nul 3738  df-if 3885  df-pw 3956  df-sn 3972  df-pr 3974  df-op 3978  df-uni 4191  df-int 4227  df-iun 4272  df-br 4395  df-opab 4453  df-mpt 4454  df-id 4737  df-xp 4828  df-rel 4829  df-cnv 4830  df-co 4831  df-dm 4832  df-rn 4833  df-res 4834  df-ima 4835  df-iota 5532  df-fun 5570  df-fn 5571  df-f 5572  df-f1 5573  df-fo 5574  df-f1o 5575  df-fv 5576  df-ov 6280  df-oprab 6281  df-mpt2 6282  df-1st 6783  df-2nd 6784  df-siga 28542  df-sigagen 28573  df-sx 28623
This theorem is referenced by:  sxuni  28627
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