MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  pttoponconst Structured version   Visualization version   GIF version

Theorem pttoponconst 23621
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 3148 . . 3 (𝑅 ∈ (TopOn‘𝑋) → ∀𝑥𝐴 𝑅 ∈ (TopOn‘𝑋))
3 ptuniconst.2 . . . . 5 𝐽 = (∏t‘(𝐴 × {𝑅}))
4 fconstmpt 5751 . . . . . 6 (𝐴 × {𝑅}) = (𝑥𝐴𝑅)
54fveq2i 6910 . . . . 5 (∏t‘(𝐴 × {𝑅})) = (∏t‘(𝑥𝐴𝑅))
63, 5eqtri 2763 . . . 4 𝐽 = (∏t‘(𝑥𝐴𝑅))
76pttopon 23620 . . 3 ((𝐴𝑉 ∧ ∀𝑥𝐴 𝑅 ∈ (TopOn‘𝑋)) → 𝐽 ∈ (TopOn‘X𝑥𝐴 𝑋))
82, 7sylan2 593 . 2 ((𝐴𝑉𝑅 ∈ (TopOn‘𝑋)) → 𝐽 ∈ (TopOn‘X𝑥𝐴 𝑋))
9 toponmax 22948 . . . 4 (𝑅 ∈ (TopOn‘𝑋) → 𝑋𝑅)
10 ixpconstg 8945 . . . 4 ((𝐴𝑉𝑋𝑅) → X𝑥𝐴 𝑋 = (𝑋m 𝐴))
119, 10sylan2 593 . . 3 ((𝐴𝑉𝑅 ∈ (TopOn‘𝑋)) → X𝑥𝐴 𝑋 = (𝑋m 𝐴))
1211fveq2d 6911 . 2 ((𝐴𝑉𝑅 ∈ (TopOn‘𝑋)) → (TopOn‘X𝑥𝐴 𝑋) = (TopOn‘(𝑋m 𝐴)))
138, 12eleqtrd 2841 1 ((𝐴𝑉𝑅 ∈ (TopOn‘𝑋)) → 𝐽 ∈ (TopOn‘(𝑋m 𝐴)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1537  wcel 2106  wral 3059  {csn 4631  cmpt 5231   × cxp 5687  cfv 6563  (class class class)co 7431  m cmap 8865  Xcixp 8936  tcpt 17485  TopOnctopon 22932
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-int 4952  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1o 8505  df-2o 8506  df-map 8867  df-ixp 8937  df-en 8985  df-fin 8988  df-fi 9449  df-topgen 17490  df-pt 17491  df-top 22916  df-topon 22933  df-bases 22969
This theorem is referenced by:  ptuniconst  23622  pt1hmeo  23830  tmdgsum  24119  efmndtmd  24125  symgtgp  24130  poimir  37640  broucube  37641
  Copyright terms: Public domain W3C validator