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Theorem txcldOLD 26489
Description: The product of two closed sets is closed in the product topology. (Moved to txcld 17298 in main set.mm and may be deleted by mathbox owner, JM. --NM 15-Oct-2012.) (Contributed by Jeff Madsen, 2-Sep-2009.) (Proof modification is discouraged.) (New usage is discouraged.)
Hypothesis
Ref Expression
txcldOLD.1  |-  T  =  ( R  tX  S
)
Assertion
Ref Expression
txcldOLD  |-  ( ( ( R  e.  Top  /\  S  e.  Top )  /\  ( A  e.  (
Clsd `  R )  /\  B  e.  ( Clsd `  S ) ) )  ->  ( A  X.  B )  e.  (
Clsd `  T )
)

Proof of Theorem txcldOLD
StepHypRef Expression
1 txcld 17298 . . 3  |-  ( ( A  e.  ( Clsd `  R )  /\  B  e.  ( Clsd `  S
) )  ->  ( A  X.  B )  e.  ( Clsd `  ( R  tX  S ) ) )
2 txcldOLD.1 . . . 4  |-  T  =  ( R  tX  S
)
32fveq2i 5528 . . 3  |-  ( Clsd `  T )  =  (
Clsd `  ( R  tX  S ) )
41, 3syl6eleqr 2374 . 2  |-  ( ( A  e.  ( Clsd `  R )  /\  B  e.  ( Clsd `  S
) )  ->  ( A  X.  B )  e.  ( Clsd `  T
) )
54adantl 452 1  |-  ( ( ( R  e.  Top  /\  S  e.  Top )  /\  ( A  e.  (
Clsd `  R )  /\  B  e.  ( Clsd `  S ) ) )  ->  ( A  X.  B )  e.  (
Clsd `  T )
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1623    e. wcel 1684    X. cxp 4687   ` cfv 5255  (class class class)co 5858   Topctop 16631   Clsdccld 16753    tX ctx 17255
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-top 16636  df-bases 16638  df-topon 16639  df-cld 16756  df-tx 17257
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