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Theorem pttoponconst 23606
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 = (∏t‘(𝐴 × {𝑅}))
Assertion
Ref Expression
pttoponconst ((𝐴𝑉𝑅 ∈ (TopOn‘𝑋)) → 𝐽 ∈ (TopOn‘(𝑋m 𝐴)))

Proof of Theorem pttoponconst
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 id 22 . . . 4 (𝑅 ∈ (TopOn‘𝑋) → 𝑅 ∈ (TopOn‘𝑋))
21ralrimivw 3149 . . 3 (𝑅 ∈ (TopOn‘𝑋) → ∀𝑥𝐴 𝑅 ∈ (TopOn‘𝑋))
3 ptuniconst.2 . . . . 5 𝐽 = (∏t‘(𝐴 × {𝑅}))
4 fconstmpt 5746 . . . . . 6 (𝐴 × {𝑅}) = (𝑥𝐴𝑅)
54fveq2i 6908 . . . . 5 (∏t‘(𝐴 × {𝑅})) = (∏t‘(𝑥𝐴𝑅))
63, 5eqtri 2764 . . . 4 𝐽 = (∏t‘(𝑥𝐴𝑅))
76pttopon 23605 . . 3 ((𝐴𝑉 ∧ ∀𝑥𝐴 𝑅 ∈ (TopOn‘𝑋)) → 𝐽 ∈ (TopOn‘X𝑥𝐴 𝑋))
82, 7sylan2 593 . 2 ((𝐴𝑉𝑅 ∈ (TopOn‘𝑋)) → 𝐽 ∈ (TopOn‘X𝑥𝐴 𝑋))
9 toponmax 22933 . . . 4 (𝑅 ∈ (TopOn‘𝑋) → 𝑋𝑅)
10 ixpconstg 8947 . . . 4 ((𝐴𝑉𝑋𝑅) → X𝑥𝐴 𝑋 = (𝑋m 𝐴))
119, 10sylan2 593 . . 3 ((𝐴𝑉𝑅 ∈ (TopOn‘𝑋)) → X𝑥𝐴 𝑋 = (𝑋m 𝐴))
1211fveq2d 6909 . 2 ((𝐴𝑉𝑅 ∈ (TopOn‘𝑋)) → (TopOn‘X𝑥𝐴 𝑋) = (TopOn‘(𝑋m 𝐴)))
138, 12eleqtrd 2842 1 ((𝐴𝑉𝑅 ∈ (TopOn‘𝑋)) → 𝐽 ∈ (TopOn‘(𝑋m 𝐴)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1539  wcel 2107  wral 3060  {csn 4625  cmpt 5224   × cxp 5682  cfv 6560  (class class class)co 7432  m cmap 8867  Xcixp 8938  tcpt 17484  TopOnctopon 22917
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-rep 5278  ax-sep 5295  ax-nul 5305  ax-pow 5364  ax-pr 5431  ax-un 7756
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ne 2940  df-ral 3061  df-rex 3070  df-reu 3380  df-rab 3436  df-v 3481  df-sbc 3788  df-csb 3899  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-pss 3970  df-nul 4333  df-if 4525  df-pw 4601  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-int 4946  df-iun 4992  df-br 5143  df-opab 5205  df-mpt 5225  df-tr 5259  df-id 5577  df-eprel 5583  df-po 5591  df-so 5592  df-fr 5636  df-we 5638  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-rn 5695  df-res 5696  df-ima 5697  df-ord 6386  df-on 6387  df-lim 6388  df-suc 6389  df-iota 6513  df-fun 6562  df-fn 6563  df-f 6564  df-f1 6565  df-fo 6566  df-f1o 6567  df-fv 6568  df-ov 7435  df-oprab 7436  df-mpo 7437  df-om 7889  df-1o 8507  df-2o 8508  df-map 8869  df-ixp 8939  df-en 8987  df-fin 8990  df-fi 9452  df-topgen 17489  df-pt 17490  df-top 22901  df-topon 22918  df-bases 22954
This theorem is referenced by:  ptuniconst  23607  pt1hmeo  23815  tmdgsum  24104  efmndtmd  24110  symgtgp  24115  poimir  37661  broucube  37662
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