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Theorem txopn 17297
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 2283 . . . . . 6  |-  ran  (
u  e.  R , 
v  e.  S  |->  ( u  X.  v ) )  =  ran  (
u  e.  R , 
v  e.  S  |->  ( u  X.  v ) )
21txbasex 17261 . . . . 5  |-  ( ( R  e.  V  /\  S  e.  W )  ->  ran  ( u  e.  R ,  v  e.  S  |->  ( u  X.  v ) )  e. 
_V )
3 bastg 16704 . . . . 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 2283 . . . . . 6  |-  ( A  X.  B )  =  ( A  X.  B
)
7 xpeq1 4703 . . . . . . . 8  |-  ( u  =  A  ->  (
u  X.  v )  =  ( A  X.  v ) )
87eqeq2d 2294 . . . . . . 7  |-  ( u  =  A  ->  (
( A  X.  B
)  =  ( u  X.  v )  <->  ( A  X.  B )  =  ( A  X.  v ) ) )
9 xpeq2 4704 . . . . . . . 8  |-  ( v  =  B  ->  ( A  X.  v )  =  ( A  X.  B
) )
109eqeq2d 2294 . . . . . . 7  |-  ( v  =  B  ->  (
( A  X.  B
)  =  ( A  X.  v )  <->  ( A  X.  B )  =  ( A  X.  B ) ) )
118, 10rspc2ev 2892 . . . . . 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 4800 . . . . . 6  |-  ( ( A  e.  R  /\  B  e.  S )  ->  ( A  X.  B
)  e.  _V )
14 eqid 2283 . . . . . . 7  |-  ( u  e.  R ,  v  e.  S  |->  ( u  X.  v ) )  =  ( u  e.  R ,  v  e.  S  |->  ( u  X.  v ) )
1514elrnmpt2g 5956 . . . . . 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 3181 . 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 17259 . . 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 2360 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 1623    e. wcel 1684   E.wrex 2544   _Vcvv 2788    C_ wss 3152    X. cxp 4687   ran crn 4690   ` cfv 5255  (class class class)co 5858    e. cmpt2 5860   topGenctg 13342    tX ctx 17255
This theorem is referenced by:  txcld  17298  txbasval  17301  tx1cn  17303  tx2cn  17304  txlly  17330  txnlly  17331  txhaus  17341  txlm  17342  tx1stc  17344  txkgen  17346  xkococnlem  17353  cxpcn3  20088  cvmlift2lem11  23255  cvmlift2lem12  23256  prdnei  24985  txopnOLD  25900
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-topgen 13344  df-tx 17257
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