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Theorem pttoponconst 20224
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 2872 . . 3  |-  ( R  e.  (TopOn `  X
)  ->  A. x  e.  A  R  e.  (TopOn `  X ) )
3 ptuniconst.2 . . . . 5  |-  J  =  ( Xt_ `  ( A  X.  { R }
) )
4 fconstmpt 5052 . . . . . 6  |-  ( A  X.  { R }
)  =  ( x  e.  A  |->  R )
54fveq2i 5875 . . . . 5  |-  ( Xt_ `  ( A  X.  { R } ) )  =  ( Xt_ `  (
x  e.  A  |->  R ) )
63, 5eqtri 2486 . . . 4  |-  J  =  ( Xt_ `  (
x  e.  A  |->  R ) )
76pttopon 20223 . . 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 19556 . . . 4  |-  ( R  e.  (TopOn `  X
)  ->  X  e.  R )
10 ixpconstg 7497 . . . 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 5876 . 2  |-  ( ( A  e.  V  /\  R  e.  (TopOn `  X
) )  ->  (TopOn `  X_ x  e.  A  X )  =  (TopOn `  ( X  ^m  A
) ) )
138, 12eleqtrd 2547 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 1395    e. wcel 1819   A.wral 2807   {csn 4032    |-> cmpt 4515    X. cxp 5006   ` cfv 5594  (class class class)co 6296    ^m cmap 7438   X_cixp 7488   Xt_cpt 14856  TopOnctopon 19522
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-rep 4568  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-reu 2814  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-tp 4037  df-op 4039  df-uni 4252  df-int 4289  df-iun 4334  df-br 4457  df-opab 4516  df-mpt 4517  df-tr 4551  df-eprel 4800  df-id 4804  df-po 4809  df-so 4810  df-fr 4847  df-we 4849  df-ord 4890  df-on 4891  df-lim 4892  df-suc 4893  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-om 6700  df-recs 7060  df-rdg 7094  df-1o 7148  df-oadd 7152  df-er 7329  df-map 7440  df-ixp 7489  df-en 7536  df-fin 7539  df-fi 7889  df-topgen 14861  df-pt 14862  df-top 19526  df-bases 19528  df-topon 19529
This theorem is referenced by:  ptuniconst  20225  pt1hmeo  20433  tmdgsum  20720  symgtgp  20726
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