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

Theorem sxsigon 26746
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 26745 . 2  |-  ( ( S  e.  U. ran sigAlgebra  /\  T  e.  U. ran sigAlgebra )  -> 
( S ×s  T )  e.  U. ran sigAlgebra )
2 eqid 2452 . . . . . 6  |-  ran  (
x  e.  S , 
y  e.  T  |->  ( x  X.  y ) )  =  ran  (
x  e.  S , 
y  e.  T  |->  ( x  X.  y ) )
32sxval 26744 . . . . 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 4204 . . . 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 6754 . . . . 5  |-  ( ( S  e.  U. ran sigAlgebra  /\  T  e.  U. ran sigAlgebra )  -> 
( x  e.  S ,  y  e.  T  |->  ( x  X.  y
) )  e.  _V )
6 rnexg 6615 . . . . 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 26726 . . . . 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 2493 . . 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 2452 . . . 4  |-  U. S  =  U. S
11 eqid 2452 . . . 4  |-  U. T  =  U. T
122, 10, 11txuni2 19265 . . 3  |-  ( U. S  X.  U. T )  =  U. ran  (
x  e.  S , 
y  e.  T  |->  ( x  X.  y ) )
139, 12syl6reqr 2512 . 2  |-  ( ( S  e.  U. ran sigAlgebra  /\  T  e.  U. ran sigAlgebra )  -> 
( U. S  X.  U. T )  =  U. ( S ×s  T ) )
14 issgon 26706 . 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 1370    e. wcel 1758   _Vcvv 3072   U.cuni 4194    X. cxp 4941   ran crn 4944   ` cfv 5521  (class class class)co 6195    |-> cmpt2 6197  sigAlgebracsiga 26690  sigaGencsigagen 26721   ×s csx 26742
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1954  ax-ext 2431  ax-rep 4506  ax-sep 4516  ax-nul 4524  ax-pow 4573  ax-pr 4634  ax-un 6477
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-fal 1376  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2265  df-mo 2266  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2602  df-ne 2647  df-ral 2801  df-rex 2802  df-reu 2803  df-rab 2805  df-v 3074  df-sbc 3289  df-csb 3391  df-dif 3434  df-un 3436  df-in 3438  df-ss 3445  df-nul 3741  df-if 3895  df-pw 3965  df-sn 3981  df-pr 3983  df-op 3987  df-uni 4195  df-int 4232  df-iun 4276  df-br 4396  df-opab 4454  df-mpt 4455  df-id 4739  df-xp 4949  df-rel 4950  df-cnv 4951  df-co 4952  df-dm 4953  df-rn 4954  df-res 4955  df-ima 4956  df-iota 5484  df-fun 5523  df-fn 5524  df-f 5525  df-f1 5526  df-fo 5527  df-f1o 5528  df-fv 5529  df-ov 6198  df-oprab 6199  df-mpt2 6200  df-1st 6682  df-2nd 6683  df-siga 26691  df-sigagen 26722  df-sx 26743
This theorem is referenced by:  sxuni  26747
  Copyright terms: Public domain W3C validator