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Theorem cnmptc 23685
Description: A constant function is continuous. (Contributed by Mario Carneiro, 5-May-2014.) (Revised by Mario Carneiro, 22-Aug-2015.)
Hypotheses
Ref Expression
cnmptid.j (𝜑𝐽 ∈ (TopOn‘𝑋))
cnmptc.k (𝜑𝐾 ∈ (TopOn‘𝑌))
cnmptc.p (𝜑𝑃𝑌)
Assertion
Ref Expression
cnmptc (𝜑 → (𝑥𝑋𝑃) ∈ (𝐽 Cn 𝐾))
Distinct variable groups:   𝜑,𝑥   𝑥,𝐽   𝑥,𝑋   𝑥,𝑌   𝑥,𝐾   𝑥,𝑃

Proof of Theorem cnmptc
StepHypRef Expression
1 fconstmpt 5750 . 2 (𝑋 × {𝑃}) = (𝑥𝑋𝑃)
2 cnmptid.j . . 3 (𝜑𝐽 ∈ (TopOn‘𝑋))
3 cnmptc.k . . 3 (𝜑𝐾 ∈ (TopOn‘𝑌))
4 cnmptc.p . . 3 (𝜑𝑃𝑌)
5 cnconst2 23306 . . 3 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝑃𝑌) → (𝑋 × {𝑃}) ∈ (𝐽 Cn 𝐾))
62, 3, 4, 5syl3anc 1370 . 2 (𝜑 → (𝑋 × {𝑃}) ∈ (𝐽 Cn 𝐾))
71, 6eqeltrrid 2843 1 (𝜑 → (𝑥𝑋𝑃) ∈ (𝐽 Cn 𝐾))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2105  {csn 4630  cmpt 5230   × cxp 5686  cfv 6562  (class class class)co 7430  TopOnctopon 22931   Cn ccn 23247
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-sep 5301  ax-nul 5311  ax-pow 5370  ax-pr 5437  ax-un 7753
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-ne 2938  df-ral 3059  df-rex 3068  df-rab 3433  df-v 3479  df-sbc 3791  df-csb 3908  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-nul 4339  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4912  df-iun 4997  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5582  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-rn 5699  df-res 5700  df-ima 5701  df-iota 6515  df-fun 6564  df-fn 6565  df-f 6566  df-fv 6570  df-ov 7433  df-oprab 7434  df-mpo 7435  df-1st 8012  df-2nd 8013  df-map 8866  df-topgen 17489  df-top 22915  df-topon 22932  df-cn 23250  df-cnp 23251
This theorem is referenced by:  cnmpt2c  23693  xkoinjcn  23710  txconn  23712  imasnopn  23713  imasncld  23714  imasncls  23715  istgp2  24114  tmdmulg  24115  tmdgsum  24118  tmdlactcn  24125  clsnsg  24133  tgpt0  24142  tlmtgp  24219  nmcn  24879  fsumcn  24907  expcn  24909  divccn  24910  expcnOLD  24911  divccnOLD  24912  cncfmptc  24951  cdivcncf  24960  iirevcn  24970  iihalf1cn  24972  iihalf1cnOLD  24973  iihalf2cn  24975  iihalf2cnOLD  24976  icchmeo  24984  icchmeoOLD  24985  evth  25004  evth2  25005  pcocn  25063  pcopt  25068  pcopt2  25069  pcoass  25070  csscld  25296  clsocv  25297  dvcnvlem  26028  plycn  26314  plycnOLD  26315  psercn2  26480  psercn2OLD  26481  resqrtcn  26806  sqrtcn  26807  atansopn  26989  efrlim  27026  efrlimOLD  27027  ipasslem7  30864  occllem  31331  rmulccn  33888  cxpcncf1  34588  txsconnlem  35224  cvxpconn  35226  cvmlift2lem2  35288  cvmlift2lem3  35289  cvmliftphtlem  35301  sinccvglem  35656  knoppcnlem10  36484  areacirclem2  37695  fprodcn  45555
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