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Theorem cnmptc 21370
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 5128 . 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 20992 . . 3 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝑃𝑌) → (𝑋 × {𝑃}) ∈ (𝐽 Cn 𝐾))
62, 3, 4, 5syl3anc 1323 . 2 (𝜑 → (𝑋 × {𝑃}) ∈ (𝐽 Cn 𝐾))
71, 6syl5eqelr 2709 1 (𝜑 → (𝑥𝑋𝑃) ∈ (𝐽 Cn 𝐾))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 1992  {csn 4153  cmpt 4678   × cxp 5077  cfv 5850  (class class class)co 6605  TopOnctopon 20613   Cn ccn 20933
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1841  ax-6 1890  ax-7 1937  ax-8 1994  ax-9 2001  ax-10 2021  ax-11 2036  ax-12 2049  ax-13 2250  ax-ext 2606  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6903
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1883  df-eu 2478  df-mo 2479  df-clab 2613  df-cleq 2619  df-clel 2622  df-nfc 2756  df-ne 2797  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3193  df-sbc 3423  df-csb 3520  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-op 4160  df-uni 4408  df-iun 4492  df-br 4619  df-opab 4679  df-mpt 4680  df-id 4994  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-iota 5813  df-fun 5852  df-fn 5853  df-f 5854  df-fv 5858  df-ov 6608  df-oprab 6609  df-mpt2 6610  df-1st 7116  df-2nd 7117  df-map 7805  df-topgen 16020  df-top 20616  df-topon 20618  df-cn 20936  df-cnp 20937
This theorem is referenced by:  cnmpt2c  21378  xkoinjcn  21395  txconn  21397  imasnopn  21398  imasncld  21399  imasncls  21400  istgp2  21800  tmdmulg  21801  tmdgsum  21804  tmdlactcn  21811  clsnsg  21818  tgpt0  21827  tlmtgp  21904  nmcn  22550  fsumcn  22576  expcn  22578  divccn  22579  cncfmptc  22617  cdivcncf  22623  iirevcn  22632  iihalf1cn  22634  iihalf2cn  22636  icchmeo  22643  evth  22661  evth2  22662  pcocn  22720  pcopt  22725  pcopt2  22726  pcoass  22727  csscld  22951  clsocv  22952  dvcnvlem  23638  plycn  23916  psercn2  24076  resqrtcn  24385  sqrtcn  24386  atansopn  24554  efrlim  24591  ipasslem7  27531  occllem  28002  rmulccn  29748  txsconnlem  30922  cvxpconn  30924  cvmlift2lem2  30986  cvmlift2lem3  30987  cvmliftphtlem  30999  sinccvglem  31266  knoppcnlem10  32126  areacirclem2  33119  fprodcn  39223
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