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Theorem sxsigon 27803
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 27802 . 2  |-  ( ( S  e.  U. ran sigAlgebra  /\  T  e.  U. ran sigAlgebra )  -> 
( S ×s  T )  e.  U. ran sigAlgebra )
2 eqid 2467 . . . . . 6  |-  ran  (
x  e.  S , 
y  e.  T  |->  ( x  X.  y ) )  =  ran  (
x  e.  S , 
y  e.  T  |->  ( x  X.  y ) )
32sxval 27801 . . . . 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 4255 . . . 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 6856 . . . . 5  |-  ( ( S  e.  U. ran sigAlgebra  /\  T  e.  U. ran sigAlgebra )  -> 
( x  e.  S ,  y  e.  T  |->  ( x  X.  y
) )  e.  _V )
6 rnexg 6713 . . . . 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 27783 . . . . 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 2508 . . 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 2467 . . . 4  |-  U. S  =  U. S
11 eqid 2467 . . . 4  |-  U. T  =  U. T
122, 10, 11txuni2 19801 . . 3  |-  ( U. S  X.  U. T )  =  U. ran  (
x  e.  S , 
y  e.  T  |->  ( x  X.  y ) )
139, 12syl6reqr 2527 . 2  |-  ( ( S  e.  U. ran sigAlgebra  /\  T  e.  U. ran sigAlgebra )  -> 
( U. S  X.  U. T )  =  U. ( S ×s  T ) )
14 issgon 27763 . 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 664 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 369    = wceq 1379    e. wcel 1767   _Vcvv 3113   U.cuni 4245    X. cxp 4997   ran crn 5000   ` cfv 5586  (class class class)co 6282    |-> cmpt2 6284  sigAlgebracsiga 27747  sigaGencsigagen 27778   ×s csx 27799
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6574
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-fal 1385  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-reu 2821  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-int 4283  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fun 5588  df-fn 5589  df-f 5590  df-f1 5591  df-fo 5592  df-f1o 5593  df-fv 5594  df-ov 6285  df-oprab 6286  df-mpt2 6287  df-1st 6781  df-2nd 6782  df-siga 27748  df-sigagen 27779  df-sx 27800
This theorem is referenced by:  sxuni  27804
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