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Theorem pttoponconst 19145
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 22 . . . 4  |-  ( R  e.  (TopOn `  X
)  ->  R  e.  (TopOn `  X ) )
21ralrimivw 2795 . . 3  |-  ( R  e.  (TopOn `  X
)  ->  A. x  e.  A  R  e.  (TopOn `  X ) )
3 ptuniconst.2 . . . . 5  |-  J  =  ( Xt_ `  ( A  X.  { R }
) )
4 fconstmpt 4877 . . . . . 6  |-  ( A  X.  { R }
)  =  ( x  e.  A  |->  R )
54fveq2i 5689 . . . . 5  |-  ( Xt_ `  ( A  X.  { R } ) )  =  ( Xt_ `  (
x  e.  A  |->  R ) )
63, 5eqtri 2458 . . . 4  |-  J  =  ( Xt_ `  (
x  e.  A  |->  R ) )
76pttopon 19144 . . 3  |-  ( ( A  e.  V  /\  A. x  e.  A  R  e.  (TopOn `  X )
)  ->  J  e.  (TopOn `  X_ x  e.  A  X ) )
82, 7sylan2 474 . 2  |-  ( ( A  e.  V  /\  R  e.  (TopOn `  X
) )  ->  J  e.  (TopOn `  X_ x  e.  A  X ) )
9 toponmax 18508 . . . 4  |-  ( R  e.  (TopOn `  X
)  ->  X  e.  R )
10 ixpconstg 7264 . . . 4  |-  ( ( A  e.  V  /\  X  e.  R )  -> 
X_ x  e.  A  X  =  ( X  ^m  A ) )
119, 10sylan2 474 . . 3  |-  ( ( A  e.  V  /\  R  e.  (TopOn `  X
) )  ->  X_ x  e.  A  X  =  ( X  ^m  A ) )
1211fveq2d 5690 . 2  |-  ( ( A  e.  V  /\  R  e.  (TopOn `  X
) )  ->  (TopOn `  X_ x  e.  A  X )  =  (TopOn `  ( X  ^m  A
) ) )
138, 12eleqtrd 2514 1  |-  ( ( A  e.  V  /\  R  e.  (TopOn `  X
) )  ->  J  e.  (TopOn `  ( X  ^m  A ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1369    e. wcel 1756   A.wral 2710   {csn 3872    e. cmpt 4345    X. cxp 4833   ` cfv 5413  (class class class)co 6086    ^m cmap 7206   X_cixp 7255   Xt_cpt 14369  TopOnctopon 18474
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2419  ax-rep 4398  ax-sep 4408  ax-nul 4416  ax-pow 4465  ax-pr 4526  ax-un 6367
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2256  df-mo 2257  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2715  df-rex 2716  df-reu 2717  df-rab 2719  df-v 2969  df-sbc 3182  df-csb 3284  df-dif 3326  df-un 3328  df-in 3330  df-ss 3337  df-pss 3339  df-nul 3633  df-if 3787  df-pw 3857  df-sn 3873  df-pr 3875  df-tp 3877  df-op 3879  df-uni 4087  df-int 4124  df-iun 4168  df-br 4288  df-opab 4346  df-mpt 4347  df-tr 4381  df-eprel 4627  df-id 4631  df-po 4636  df-so 4637  df-fr 4674  df-we 4676  df-ord 4717  df-on 4718  df-lim 4719  df-suc 4720  df-xp 4841  df-rel 4842  df-cnv 4843  df-co 4844  df-dm 4845  df-rn 4846  df-res 4847  df-ima 4848  df-iota 5376  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-ov 6089  df-oprab 6090  df-mpt2 6091  df-om 6472  df-recs 6824  df-rdg 6858  df-1o 6912  df-oadd 6916  df-er 7093  df-map 7208  df-ixp 7256  df-en 7303  df-fin 7306  df-fi 7653  df-topgen 14374  df-pt 14375  df-top 18478  df-bases 18480  df-topon 18481
This theorem is referenced by:  ptuniconst  19146  pt1hmeo  19354  tmdgsum  19641  symgtgp  19647
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