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Theorem pttoponconst 17292
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 19 . . . 4  |-  ( R  e.  (TopOn `  X
)  ->  R  e.  (TopOn `  X ) )
21ralrimivw 2627 . . 3  |-  ( R  e.  (TopOn `  X
)  ->  A. x  e.  A  R  e.  (TopOn `  X ) )
3 ptuniconst.2 . . . . 5  |-  J  =  ( Xt_ `  ( A  X.  { R }
) )
4 fconstmpt 4732 . . . . . 6  |-  ( A  X.  { R }
)  =  ( x  e.  A  |->  R )
54fveq2i 5528 . . . . 5  |-  ( Xt_ `  ( A  X.  { R } ) )  =  ( Xt_ `  (
x  e.  A  |->  R ) )
63, 5eqtri 2303 . . . 4  |-  J  =  ( Xt_ `  (
x  e.  A  |->  R ) )
76pttopon 17291 . . 3  |-  ( ( A  e.  V  /\  A. x  e.  A  R  e.  (TopOn `  X )
)  ->  J  e.  (TopOn `  X_ x  e.  A  X ) )
82, 7sylan2 460 . 2  |-  ( ( A  e.  V  /\  R  e.  (TopOn `  X
) )  ->  J  e.  (TopOn `  X_ x  e.  A  X ) )
9 toponmax 16666 . . . 4  |-  ( R  e.  (TopOn `  X
)  ->  X  e.  R )
10 ixpconstg 6825 . . . 4  |-  ( ( A  e.  V  /\  X  e.  R )  -> 
X_ x  e.  A  X  =  ( X  ^m  A ) )
119, 10sylan2 460 . . 3  |-  ( ( A  e.  V  /\  R  e.  (TopOn `  X
) )  ->  X_ x  e.  A  X  =  ( X  ^m  A ) )
1211fveq2d 5529 . 2  |-  ( ( A  e.  V  /\  R  e.  (TopOn `  X
) )  ->  (TopOn `  X_ x  e.  A  X )  =  (TopOn `  ( X  ^m  A
) ) )
138, 12eleqtrd 2359 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 358    = wceq 1623    e. wcel 1684   A.wral 2543   {csn 3640    e. cmpt 4077    X. cxp 4687   ` cfv 5255  (class class class)co 5858    ^m cmap 6772   X_cixp 6817   Xt_cpt 13343  TopOnctopon 16632
This theorem is referenced by:  ptuniconst  17293  pt1hmeo  17497  tmdgsum  17778  symgtgp  17784
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-rep 4131  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-3or 935  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-reu 2550  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-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  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-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-recs 6388  df-rdg 6423  df-1o 6479  df-oadd 6483  df-er 6660  df-map 6774  df-ixp 6818  df-en 6864  df-fin 6867  df-fi 7165  df-topgen 13344  df-pt 13345  df-top 16636  df-bases 16638  df-topon 16639
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