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Theorem txopn 20207
Description: The product of two open sets is open in the product topology. (Contributed by Jeff Madsen, 2-Sep-2009.)
Assertion
Ref Expression
txopn  |-  ( ( ( R  e.  V  /\  S  e.  W
)  /\  ( A  e.  R  /\  B  e.  S ) )  -> 
( A  X.  B
)  e.  ( R 
tX  S ) )

Proof of Theorem txopn
Dummy variables  u  v are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2392 . . . . . 6  |-  ran  (
u  e.  R , 
v  e.  S  |->  ( u  X.  v ) )  =  ran  (
u  e.  R , 
v  e.  S  |->  ( u  X.  v ) )
21txbasex 20171 . . . . 5  |-  ( ( R  e.  V  /\  S  e.  W )  ->  ran  ( u  e.  R ,  v  e.  S  |->  ( u  X.  v ) )  e. 
_V )
3 bastg 19571 . . . . 5  |-  ( ran  ( u  e.  R ,  v  e.  S  |->  ( u  X.  v
) )  e.  _V  ->  ran  ( u  e.  R ,  v  e.  S  |->  ( u  X.  v ) )  C_  ( topGen `  ran  ( u  e.  R ,  v  e.  S  |->  ( u  X.  v ) ) ) )
42, 3syl 16 . . . 4  |-  ( ( R  e.  V  /\  S  e.  W )  ->  ran  ( u  e.  R ,  v  e.  S  |->  ( u  X.  v ) )  C_  ( topGen `  ran  ( u  e.  R ,  v  e.  S  |->  ( u  X.  v ) ) ) )
54adantr 463 . . 3  |-  ( ( ( R  e.  V  /\  S  e.  W
)  /\  ( A  e.  R  /\  B  e.  S ) )  ->  ran  ( u  e.  R ,  v  e.  S  |->  ( u  X.  v
) )  C_  ( topGen `
 ran  ( u  e.  R ,  v  e.  S  |->  ( u  X.  v ) ) ) )
6 eqid 2392 . . . . . 6  |-  ( A  X.  B )  =  ( A  X.  B
)
7 xpeq1 4940 . . . . . . . 8  |-  ( u  =  A  ->  (
u  X.  v )  =  ( A  X.  v ) )
87eqeq2d 2406 . . . . . . 7  |-  ( u  =  A  ->  (
( A  X.  B
)  =  ( u  X.  v )  <->  ( A  X.  B )  =  ( A  X.  v ) ) )
9 xpeq2 4941 . . . . . . . 8  |-  ( v  =  B  ->  ( A  X.  v )  =  ( A  X.  B
) )
109eqeq2d 2406 . . . . . . 7  |-  ( v  =  B  ->  (
( A  X.  B
)  =  ( A  X.  v )  <->  ( A  X.  B )  =  ( A  X.  B ) ) )
118, 10rspc2ev 3159 . . . . . 6  |-  ( ( A  e.  R  /\  B  e.  S  /\  ( A  X.  B
)  =  ( A  X.  B ) )  ->  E. u  e.  R  E. v  e.  S  ( A  X.  B
)  =  ( u  X.  v ) )
126, 11mp3an3 1311 . . . . 5  |-  ( ( A  e.  R  /\  B  e.  S )  ->  E. u  e.  R  E. v  e.  S  ( A  X.  B
)  =  ( u  X.  v ) )
13 xpexg 6519 . . . . . 6  |-  ( ( A  e.  R  /\  B  e.  S )  ->  ( A  X.  B
)  e.  _V )
14 eqid 2392 . . . . . . 7  |-  ( u  e.  R ,  v  e.  S  |->  ( u  X.  v ) )  =  ( u  e.  R ,  v  e.  S  |->  ( u  X.  v ) )
1514elrnmpt2g 6331 . . . . . 6  |-  ( ( A  X.  B )  e.  _V  ->  (
( A  X.  B
)  e.  ran  (
u  e.  R , 
v  e.  S  |->  ( u  X.  v ) )  <->  E. u  e.  R  E. v  e.  S  ( A  X.  B
)  =  ( u  X.  v ) ) )
1613, 15syl 16 . . . . 5  |-  ( ( A  e.  R  /\  B  e.  S )  ->  ( ( A  X.  B )  e.  ran  ( u  e.  R ,  v  e.  S  |->  ( u  X.  v
) )  <->  E. u  e.  R  E. v  e.  S  ( A  X.  B )  =  ( u  X.  v ) ) )
1712, 16mpbird 232 . . . 4  |-  ( ( A  e.  R  /\  B  e.  S )  ->  ( A  X.  B
)  e.  ran  (
u  e.  R , 
v  e.  S  |->  ( u  X.  v ) ) )
1817adantl 464 . . 3  |-  ( ( ( R  e.  V  /\  S  e.  W
)  /\  ( A  e.  R  /\  B  e.  S ) )  -> 
( A  X.  B
)  e.  ran  (
u  e.  R , 
v  e.  S  |->  ( u  X.  v ) ) )
195, 18sseldd 3431 . 2  |-  ( ( ( R  e.  V  /\  S  e.  W
)  /\  ( A  e.  R  /\  B  e.  S ) )  -> 
( A  X.  B
)  e.  ( topGen ` 
ran  ( u  e.  R ,  v  e.  S  |->  ( u  X.  v ) ) ) )
201txval 20169 . . 3  |-  ( ( R  e.  V  /\  S  e.  W )  ->  ( R  tX  S
)  =  ( topGen ` 
ran  ( u  e.  R ,  v  e.  S  |->  ( u  X.  v ) ) ) )
2120adantr 463 . 2  |-  ( ( ( R  e.  V  /\  S  e.  W
)  /\  ( A  e.  R  /\  B  e.  S ) )  -> 
( R  tX  S
)  =  ( topGen ` 
ran  ( u  e.  R ,  v  e.  S  |->  ( u  X.  v ) ) ) )
2219, 21eleqtrrd 2483 1  |-  ( ( ( R  e.  V  /\  S  e.  W
)  /\  ( A  e.  R  /\  B  e.  S ) )  -> 
( A  X.  B
)  e.  ( R 
tX  S ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 367    = wceq 1399    e. wcel 1836   E.wrex 2743   _Vcvv 3047    C_ wss 3402    X. cxp 4924   ran crn 4927   ` cfv 5509  (class class class)co 6214    |-> cmpt2 6216   topGenctg 14864    tX ctx 20165
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1633  ax-4 1646  ax-5 1719  ax-6 1765  ax-7 1808  ax-8 1838  ax-9 1840  ax-10 1855  ax-11 1860  ax-12 1872  ax-13 2016  ax-ext 2370  ax-sep 4501  ax-nul 4509  ax-pow 4556  ax-pr 4614  ax-un 6509
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1402  df-ex 1628  df-nf 1632  df-sb 1758  df-eu 2232  df-mo 2233  df-clab 2378  df-cleq 2384  df-clel 2387  df-nfc 2542  df-ne 2589  df-ral 2747  df-rex 2748  df-rab 2751  df-v 3049  df-sbc 3266  df-csb 3362  df-dif 3405  df-un 3407  df-in 3409  df-ss 3416  df-nul 3725  df-if 3871  df-pw 3942  df-sn 3958  df-pr 3960  df-op 3964  df-uni 4177  df-iun 4258  df-br 4381  df-opab 4439  df-mpt 4440  df-id 4722  df-xp 4932  df-rel 4933  df-cnv 4934  df-co 4935  df-dm 4936  df-rn 4937  df-res 4938  df-ima 4939  df-iota 5473  df-fun 5511  df-fn 5512  df-f 5513  df-fv 5517  df-ov 6217  df-oprab 6218  df-mpt2 6219  df-1st 6717  df-2nd 6718  df-topgen 14870  df-tx 20167
This theorem is referenced by:  txcld  20208  txbasval  20211  neitx  20212  tx1cn  20214  tx2cn  20215  txlly  20241  txnlly  20242  txhaus  20252  txlm  20253  tx1stc  20255  txkgen  20257  xkococnlem  20264  cxpcn3  23228  cvmlift2lem11  28983  cvmlift2lem12  28984
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