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Theorem pttoponconst 22211
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 3178 . . 3 (𝑅 ∈ (TopOn‘𝑋) → ∀𝑥𝐴 𝑅 ∈ (TopOn‘𝑋))
3 ptuniconst.2 . . . . 5 𝐽 = (∏t‘(𝐴 × {𝑅}))
4 fconstmpt 5602 . . . . . 6 (𝐴 × {𝑅}) = (𝑥𝐴𝑅)
54fveq2i 6666 . . . . 5 (∏t‘(𝐴 × {𝑅})) = (∏t‘(𝑥𝐴𝑅))
63, 5eqtri 2847 . . . 4 𝐽 = (∏t‘(𝑥𝐴𝑅))
76pttopon 22210 . . 3 ((𝐴𝑉 ∧ ∀𝑥𝐴 𝑅 ∈ (TopOn‘𝑋)) → 𝐽 ∈ (TopOn‘X𝑥𝐴 𝑋))
82, 7sylan2 595 . 2 ((𝐴𝑉𝑅 ∈ (TopOn‘𝑋)) → 𝐽 ∈ (TopOn‘X𝑥𝐴 𝑋))
9 toponmax 21540 . . . 4 (𝑅 ∈ (TopOn‘𝑋) → 𝑋𝑅)
10 ixpconstg 8468 . . . 4 ((𝐴𝑉𝑋𝑅) → X𝑥𝐴 𝑋 = (𝑋m 𝐴))
119, 10sylan2 595 . . 3 ((𝐴𝑉𝑅 ∈ (TopOn‘𝑋)) → X𝑥𝐴 𝑋 = (𝑋m 𝐴))
1211fveq2d 6667 . 2 ((𝐴𝑉𝑅 ∈ (TopOn‘𝑋)) → (TopOn‘X𝑥𝐴 𝑋) = (TopOn‘(𝑋m 𝐴)))
138, 12eleqtrd 2918 1 ((𝐴𝑉𝑅 ∈ (TopOn‘𝑋)) → 𝐽 ∈ (TopOn‘(𝑋m 𝐴)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399   = wceq 1538  wcel 2115  wral 3133  {csn 4550  cmpt 5133   × cxp 5541  cfv 6345  (class class class)co 7151  m cmap 8404  Xcixp 8459  tcpt 16714  TopOnctopon 21524
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-rep 5177  ax-sep 5190  ax-nul 5197  ax-pow 5254  ax-pr 5318  ax-un 7457
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ne 3015  df-ral 3138  df-rex 3139  df-reu 3140  df-rab 3142  df-v 3482  df-sbc 3759  df-csb 3867  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-pss 3938  df-nul 4277  df-if 4451  df-pw 4524  df-sn 4551  df-pr 4553  df-tp 4555  df-op 4557  df-uni 4825  df-int 4863  df-iun 4907  df-br 5054  df-opab 5116  df-mpt 5134  df-tr 5160  df-id 5448  df-eprel 5453  df-po 5462  df-so 5463  df-fr 5502  df-we 5504  df-xp 5549  df-rel 5550  df-cnv 5551  df-co 5552  df-dm 5553  df-rn 5554  df-res 5555  df-ima 5556  df-pred 6137  df-ord 6183  df-on 6184  df-lim 6185  df-suc 6186  df-iota 6304  df-fun 6347  df-fn 6348  df-f 6349  df-f1 6350  df-fo 6351  df-f1o 6352  df-fv 6353  df-ov 7154  df-oprab 7155  df-mpo 7156  df-om 7577  df-wrecs 7945  df-recs 8006  df-rdg 8044  df-1o 8100  df-oadd 8104  df-er 8287  df-map 8406  df-ixp 8460  df-en 8508  df-fin 8511  df-fi 8874  df-topgen 16719  df-pt 16720  df-top 21508  df-topon 21525  df-bases 21560
This theorem is referenced by:  ptuniconst  22212  pt1hmeo  22420  tmdgsum  22709  efmndtmd  22715  symgtgp  22720  poimir  35062  broucube  35063
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