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Theorem txopn 17313
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 2296 . . . . . 6  |-  ran  (
u  e.  R , 
v  e.  S  |->  ( u  X.  v ) )  =  ran  (
u  e.  R , 
v  e.  S  |->  ( u  X.  v ) )
21txbasex 17277 . . . . 5  |-  ( ( R  e.  V  /\  S  e.  W )  ->  ran  ( u  e.  R ,  v  e.  S  |->  ( u  X.  v ) )  e. 
_V )
3 bastg 16720 . . . . 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 15 . . . 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 451 . . 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 2296 . . . . . 6  |-  ( A  X.  B )  =  ( A  X.  B
)
7 xpeq1 4719 . . . . . . . 8  |-  ( u  =  A  ->  (
u  X.  v )  =  ( A  X.  v ) )
87eqeq2d 2307 . . . . . . 7  |-  ( u  =  A  ->  (
( A  X.  B
)  =  ( u  X.  v )  <->  ( A  X.  B )  =  ( A  X.  v ) ) )
9 xpeq2 4720 . . . . . . . 8  |-  ( v  =  B  ->  ( A  X.  v )  =  ( A  X.  B
) )
109eqeq2d 2307 . . . . . . 7  |-  ( v  =  B  ->  (
( A  X.  B
)  =  ( A  X.  v )  <->  ( A  X.  B )  =  ( A  X.  B ) ) )
118, 10rspc2ev 2905 . . . . . 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 1266 . . . . 5  |-  ( ( A  e.  R  /\  B  e.  S )  ->  E. u  e.  R  E. v  e.  S  ( A  X.  B
)  =  ( u  X.  v ) )
13 xpexg 4816 . . . . . 6  |-  ( ( A  e.  R  /\  B  e.  S )  ->  ( A  X.  B
)  e.  _V )
14 eqid 2296 . . . . . . 7  |-  ( u  e.  R ,  v  e.  S  |->  ( u  X.  v ) )  =  ( u  e.  R ,  v  e.  S  |->  ( u  X.  v ) )
1514elrnmpt2g 5972 . . . . . 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 15 . . . . 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 223 . . . 4  |-  ( ( A  e.  R  /\  B  e.  S )  ->  ( A  X.  B
)  e.  ran  (
u  e.  R , 
v  e.  S  |->  ( u  X.  v ) ) )
1817adantl 452 . . 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 3194 . 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 17275 . . 3  |-  ( ( R  e.  V  /\  S  e.  W )  ->  ( R  tX  S
)  =  ( topGen ` 
ran  ( u  e.  R ,  v  e.  S  |->  ( u  X.  v ) ) ) )
2120adantr 451 . 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 2373 1  |-  ( ( ( R  e.  V  /\  S  e.  W
)  /\  ( A  e.  R  /\  B  e.  S ) )  -> 
( A  X.  B
)  e.  ( R 
tX  S ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1632    e. wcel 1696   E.wrex 2557   _Vcvv 2801    C_ wss 3165    X. cxp 4703   ran crn 4706   ` cfv 5271  (class class class)co 5874    e. cmpt2 5876   topGenctg 13358    tX ctx 17271
This theorem is referenced by:  txcld  17314  txbasval  17317  tx1cn  17319  tx2cn  17320  txlly  17346  txnlly  17347  txhaus  17357  txlm  17358  tx1stc  17360  txkgen  17362  xkococnlem  17369  cxpcn3  20104  cvmlift2lem11  23859  cvmlift2lem12  23860  prdnei  25676  txopnOLD  26591
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-topgen 13360  df-tx 17273
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