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Theorem pttoponconst 17543
Description: The base set for a product topology when all factors are the same. (Contributed by Mario Carneiro, 22-Aug-2015.)
Hypothesis
Ref Expression
ptuniconst.2  |-  J  =  ( Xt_ `  ( A  X.  { R }
) )
Assertion
Ref Expression
pttoponconst  |-  ( ( A  e.  V  /\  R  e.  (TopOn `  X
) )  ->  J  e.  (TopOn `  ( X  ^m  A ) ) )

Proof of Theorem pttoponconst
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 id 20 . . . 4  |-  ( R  e.  (TopOn `  X
)  ->  R  e.  (TopOn `  X ) )
21ralrimivw 2726 . . 3  |-  ( R  e.  (TopOn `  X
)  ->  A. x  e.  A  R  e.  (TopOn `  X ) )
3 ptuniconst.2 . . . . 5  |-  J  =  ( Xt_ `  ( A  X.  { R }
) )
4 fconstmpt 4854 . . . . . 6  |-  ( A  X.  { R }
)  =  ( x  e.  A  |->  R )
54fveq2i 5664 . . . . 5  |-  ( Xt_ `  ( A  X.  { R } ) )  =  ( Xt_ `  (
x  e.  A  |->  R ) )
63, 5eqtri 2400 . . . 4  |-  J  =  ( Xt_ `  (
x  e.  A  |->  R ) )
76pttopon 17542 . . 3  |-  ( ( A  e.  V  /\  A. x  e.  A  R  e.  (TopOn `  X )
)  ->  J  e.  (TopOn `  X_ x  e.  A  X ) )
82, 7sylan2 461 . 2  |-  ( ( A  e.  V  /\  R  e.  (TopOn `  X
) )  ->  J  e.  (TopOn `  X_ x  e.  A  X ) )
9 toponmax 16909 . . . 4  |-  ( R  e.  (TopOn `  X
)  ->  X  e.  R )
10 ixpconstg 7000 . . . 4  |-  ( ( A  e.  V  /\  X  e.  R )  -> 
X_ x  e.  A  X  =  ( X  ^m  A ) )
119, 10sylan2 461 . . 3  |-  ( ( A  e.  V  /\  R  e.  (TopOn `  X
) )  ->  X_ x  e.  A  X  =  ( X  ^m  A ) )
1211fveq2d 5665 . 2  |-  ( ( A  e.  V  /\  R  e.  (TopOn `  X
) )  ->  (TopOn `  X_ x  e.  A  X )  =  (TopOn `  ( X  ^m  A
) ) )
138, 12eleqtrd 2456 1  |-  ( ( A  e.  V  /\  R  e.  (TopOn `  X
) )  ->  J  e.  (TopOn `  ( X  ^m  A ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1649    e. wcel 1717   A.wral 2642   {csn 3750    e. cmpt 4200    X. cxp 4809   ` cfv 5387  (class class class)co 6013    ^m cmap 6947   X_cixp 6992   Xt_cpt 13586  TopOnctopon 16875
This theorem is referenced by:  ptuniconst  17544  pt1hmeo  17752  tmdgsum  18039  symgtgp  18045
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2361  ax-rep 4254  ax-sep 4264  ax-nul 4272  ax-pow 4311  ax-pr 4337  ax-un 4634
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2235  df-mo 2236  df-clab 2367  df-cleq 2373  df-clel 2376  df-nfc 2505  df-ne 2545  df-ral 2647  df-rex 2648  df-reu 2649  df-rab 2651  df-v 2894  df-sbc 3098  df-csb 3188  df-dif 3259  df-un 3261  df-in 3263  df-ss 3270  df-pss 3272  df-nul 3565  df-if 3676  df-pw 3737  df-sn 3756  df-pr 3757  df-tp 3758  df-op 3759  df-uni 3951  df-int 3986  df-iun 4030  df-br 4147  df-opab 4201  df-mpt 4202  df-tr 4237  df-eprel 4428  df-id 4432  df-po 4437  df-so 4438  df-fr 4475  df-we 4477  df-ord 4518  df-on 4519  df-lim 4520  df-suc 4521  df-om 4779  df-xp 4817  df-rel 4818  df-cnv 4819  df-co 4820  df-dm 4821  df-rn 4822  df-res 4823  df-ima 4824  df-iota 5351  df-fun 5389  df-fn 5390  df-f 5391  df-f1 5392  df-fo 5393  df-f1o 5394  df-fv 5395  df-ov 6016  df-oprab 6017  df-mpt2 6018  df-recs 6562  df-rdg 6597  df-1o 6653  df-oadd 6657  df-er 6834  df-map 6949  df-ixp 6993  df-en 7039  df-fin 7042  df-fi 7344  df-topgen 13587  df-pt 13588  df-top 16879  df-bases 16881  df-topon 16882
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