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Theorem sxsigon 26542
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 26541 . 2  |-  ( ( S  e.  U. ran sigAlgebra  /\  T  e.  U. ran sigAlgebra )  -> 
( S ×s  T )  e.  U. ran sigAlgebra )
2 eqid 2441 . . . . . 6  |-  ran  (
x  e.  S , 
y  e.  T  |->  ( x  X.  y ) )  =  ran  (
x  e.  S , 
y  e.  T  |->  ( x  X.  y ) )
32sxval 26540 . . . . 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 4098 . . . 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 6648 . . . . 5  |-  ( ( S  e.  U. ran sigAlgebra  /\  T  e.  U. ran sigAlgebra )  -> 
( x  e.  S ,  y  e.  T  |->  ( x  X.  y
) )  e.  _V )
6 rnexg 6509 . . . . 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 26522 . . . . 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 2473 . . 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 2441 . . . 4  |-  U. S  =  U. S
11 eqid 2441 . . . 4  |-  U. T  =  U. T
122, 10, 11txuni2 19097 . . 3  |-  ( U. S  X.  U. T )  =  U. ran  (
x  e.  S , 
y  e.  T  |->  ( x  X.  y ) )
139, 12syl6reqr 2492 . 2  |-  ( ( S  e.  U. ran sigAlgebra  /\  T  e.  U. ran sigAlgebra )  -> 
( U. S  X.  U. T )  =  U. ( S ×s  T ) )
14 issgon 26502 . 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 659 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 1364    e. wcel 1761   _Vcvv 2970   U.cuni 4088    X. cxp 4834   ran crn 4837   ` cfv 5415  (class class class)co 6090    e. cmpt2 6092  sigAlgebracsiga 26486  sigaGencsigagen 26517   ×s csx 26538
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1713  ax-7 1733  ax-8 1763  ax-9 1765  ax-10 1780  ax-11 1785  ax-12 1797  ax-13 1948  ax-ext 2422  ax-rep 4400  ax-sep 4410  ax-nul 4418  ax-pow 4467  ax-pr 4528  ax-un 6371
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 962  df-tru 1367  df-fal 1370  df-ex 1592  df-nf 1595  df-sb 1706  df-eu 2261  df-mo 2262  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-ral 2718  df-rex 2719  df-reu 2720  df-rab 2722  df-v 2972  df-sbc 3184  df-csb 3286  df-dif 3328  df-un 3330  df-in 3332  df-ss 3339  df-nul 3635  df-if 3789  df-pw 3859  df-sn 3875  df-pr 3877  df-op 3881  df-uni 4089  df-int 4126  df-iun 4170  df-br 4290  df-opab 4348  df-mpt 4349  df-id 4632  df-xp 4842  df-rel 4843  df-cnv 4844  df-co 4845  df-dm 4846  df-rn 4847  df-res 4848  df-ima 4849  df-iota 5378  df-fun 5417  df-fn 5418  df-f 5419  df-f1 5420  df-fo 5421  df-f1o 5422  df-fv 5423  df-ov 6093  df-oprab 6094  df-mpt2 6095  df-1st 6576  df-2nd 6577  df-siga 26487  df-sigagen 26518  df-sx 26539
This theorem is referenced by:  sxuni  26543
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