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Theorem pttoponconst 21809
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 5411 . . . . . 6 (𝐴 × {𝑅}) = (𝑥𝐴𝑅)
54fveq2i 6449 . . . . 5 (∏t‘(𝐴 × {𝑅})) = (∏t‘(𝑥𝐴𝑅))
63, 5eqtri 2802 . . . 4 𝐽 = (∏t‘(𝑥𝐴𝑅))
76pttopon 21808 . . 3 ((𝐴𝑉 ∧ ∀𝑥𝐴 𝑅 ∈ (TopOn‘𝑋)) → 𝐽 ∈ (TopOn‘X𝑥𝐴 𝑋))
82, 7sylan2 586 . 2 ((𝐴𝑉𝑅 ∈ (TopOn‘𝑋)) → 𝐽 ∈ (TopOn‘X𝑥𝐴 𝑋))
9 toponmax 21138 . . . 4 (𝑅 ∈ (TopOn‘𝑋) → 𝑋𝑅)
10 ixpconstg 8203 . . . 4 ((𝐴𝑉𝑋𝑅) → X𝑥𝐴 𝑋 = (𝑋𝑚 𝐴))
119, 10sylan2 586 . . 3 ((𝐴𝑉𝑅 ∈ (TopOn‘𝑋)) → X𝑥𝐴 𝑋 = (𝑋𝑚 𝐴))
1211fveq2d 6450 . 2 ((𝐴𝑉𝑅 ∈ (TopOn‘𝑋)) → (TopOn‘X𝑥𝐴 𝑋) = (TopOn‘(𝑋𝑚 𝐴)))
138, 12eleqtrd 2861 1 ((𝐴𝑉𝑅 ∈ (TopOn‘𝑋)) → 𝐽 ∈ (TopOn‘(𝑋𝑚 𝐴)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 386   = wceq 1601  wcel 2107  wral 3090  {csn 4398  cmpt 4965   × cxp 5353  cfv 6135  (class class class)co 6922  𝑚 cmap 8140  Xcixp 8194  tcpt 16485  TopOnctopon 21122
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-8 2109  ax-9 2116  ax-10 2135  ax-11 2150  ax-12 2163  ax-13 2334  ax-ext 2754  ax-rep 5006  ax-sep 5017  ax-nul 5025  ax-pow 5077  ax-pr 5138  ax-un 7226
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3or 1072  df-3an 1073  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-mo 2551  df-eu 2587  df-clab 2764  df-cleq 2770  df-clel 2774  df-nfc 2921  df-ne 2970  df-ral 3095  df-rex 3096  df-reu 3097  df-rab 3099  df-v 3400  df-sbc 3653  df-csb 3752  df-dif 3795  df-un 3797  df-in 3799  df-ss 3806  df-pss 3808  df-nul 4142  df-if 4308  df-pw 4381  df-sn 4399  df-pr 4401  df-tp 4403  df-op 4405  df-uni 4672  df-int 4711  df-iun 4755  df-br 4887  df-opab 4949  df-mpt 4966  df-tr 4988  df-id 5261  df-eprel 5266  df-po 5274  df-so 5275  df-fr 5314  df-we 5316  df-xp 5361  df-rel 5362  df-cnv 5363  df-co 5364  df-dm 5365  df-rn 5366  df-res 5367  df-ima 5368  df-pred 5933  df-ord 5979  df-on 5980  df-lim 5981  df-suc 5982  df-iota 6099  df-fun 6137  df-fn 6138  df-f 6139  df-f1 6140  df-fo 6141  df-f1o 6142  df-fv 6143  df-ov 6925  df-oprab 6926  df-mpt2 6927  df-om 7344  df-wrecs 7689  df-recs 7751  df-rdg 7789  df-1o 7843  df-oadd 7847  df-er 8026  df-map 8142  df-ixp 8195  df-en 8242  df-fin 8245  df-fi 8605  df-topgen 16490  df-pt 16491  df-top 21106  df-topon 21123  df-bases 21158
This theorem is referenced by:  ptuniconst  21810  pt1hmeo  22018  tmdgsum  22307  symgtgp  22313  poimir  34068  broucube  34069
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