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Theorem cnmptc 23670
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 5747 . 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 23291 . . 3 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝑃𝑌) → (𝑋 × {𝑃}) ∈ (𝐽 Cn 𝐾))
62, 3, 4, 5syl3anc 1373 . 2 (𝜑 → (𝑋 × {𝑃}) ∈ (𝐽 Cn 𝐾))
71, 6eqeltrrid 2846 1 (𝜑 → (𝑥𝑋𝑃) ∈ (𝐽 Cn 𝐾))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2108  {csn 4626  cmpt 5225   × cxp 5683  cfv 6561  (class class class)co 7431  TopOnctopon 22916   Cn ccn 23232
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-fv 6569  df-ov 7434  df-oprab 7435  df-mpo 7436  df-1st 8014  df-2nd 8015  df-map 8868  df-topgen 17488  df-top 22900  df-topon 22917  df-cn 23235  df-cnp 23236
This theorem is referenced by:  cnmpt2c  23678  xkoinjcn  23695  txconn  23697  imasnopn  23698  imasncld  23699  imasncls  23700  istgp2  24099  tmdmulg  24100  tmdgsum  24103  tmdlactcn  24110  clsnsg  24118  tgpt0  24127  tlmtgp  24204  nmcn  24866  fsumcn  24894  expcn  24896  divccn  24897  expcnOLD  24898  divccnOLD  24899  cncfmptc  24938  cdivcncf  24947  iirevcn  24957  iihalf1cn  24959  iihalf1cnOLD  24960  iihalf2cn  24962  iihalf2cnOLD  24963  icchmeo  24971  icchmeoOLD  24972  evth  24991  evth2  24992  pcocn  25050  pcopt  25055  pcopt2  25056  pcoass  25057  csscld  25283  clsocv  25284  dvcnvlem  26014  plycn  26300  plycnOLD  26301  psercn2  26466  psercn2OLD  26467  resqrtcn  26792  sqrtcn  26793  atansopn  26975  efrlim  27012  efrlimOLD  27013  ipasslem7  30855  occllem  31322  rmulccn  33927  cxpcncf1  34610  txsconnlem  35245  cvxpconn  35247  cvmlift2lem2  35309  cvmlift2lem3  35310  cvmliftphtlem  35322  sinccvglem  35677  knoppcnlem10  36503  areacirclem2  37716  fprodcn  45615
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