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Theorem pttoponconst 21210
 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 2950 . . 3 (𝑅 ∈ (TopOn‘𝑋) → ∀𝑥𝐴 𝑅 ∈ (TopOn‘𝑋))
3 ptuniconst.2 . . . . 5 𝐽 = (∏t‘(𝐴 × {𝑅}))
4 fconstmpt 5085 . . . . . 6 (𝐴 × {𝑅}) = (𝑥𝐴𝑅)
54fveq2i 6106 . . . . 5 (∏t‘(𝐴 × {𝑅})) = (∏t‘(𝑥𝐴𝑅))
63, 5eqtri 2632 . . . 4 𝐽 = (∏t‘(𝑥𝐴𝑅))
76pttopon 21209 . . 3 ((𝐴𝑉 ∧ ∀𝑥𝐴 𝑅 ∈ (TopOn‘𝑋)) → 𝐽 ∈ (TopOn‘X𝑥𝐴 𝑋))
82, 7sylan2 490 . 2 ((𝐴𝑉𝑅 ∈ (TopOn‘𝑋)) → 𝐽 ∈ (TopOn‘X𝑥𝐴 𝑋))
9 toponmax 20543 . . . 4 (𝑅 ∈ (TopOn‘𝑋) → 𝑋𝑅)
10 ixpconstg 7803 . . . 4 ((𝐴𝑉𝑋𝑅) → X𝑥𝐴 𝑋 = (𝑋𝑚 𝐴))
119, 10sylan2 490 . . 3 ((𝐴𝑉𝑅 ∈ (TopOn‘𝑋)) → X𝑥𝐴 𝑋 = (𝑋𝑚 𝐴))
1211fveq2d 6107 . 2 ((𝐴𝑉𝑅 ∈ (TopOn‘𝑋)) → (TopOn‘X𝑥𝐴 𝑋) = (TopOn‘(𝑋𝑚 𝐴)))
138, 12eleqtrd 2690 1 ((𝐴𝑉𝑅 ∈ (TopOn‘𝑋)) → 𝐽 ∈ (TopOn‘(𝑋𝑚 𝐴)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383   = wceq 1475   ∈ wcel 1977  ∀wral 2896  {csn 4125   ↦ cmpt 4643   × cxp 5036  ‘cfv 5804  (class class class)co 6549   ↑𝑚 cmap 7744  Xcixp 7794  ∏tcpt 15922  TopOnctopon 20518 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-oadd 7451  df-er 7629  df-map 7746  df-ixp 7795  df-en 7842  df-fin 7845  df-fi 8200  df-topgen 15927  df-pt 15928  df-top 20521  df-bases 20522  df-topon 20523 This theorem is referenced by:  ptuniconst  21211  pt1hmeo  21419  tmdgsum  21709  symgtgp  21715  poimir  32612  broucube  32613
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