MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  cnmptc Structured version   Visualization version   GIF version

Theorem cnmptc 23610
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 5740 . 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 23231 . . 3 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝑃𝑌) → (𝑋 × {𝑃}) ∈ (𝐽 Cn 𝐾))
62, 3, 4, 5syl3anc 1368 . 2 (𝜑 → (𝑋 × {𝑃}) ∈ (𝐽 Cn 𝐾))
71, 6eqeltrrid 2830 1 (𝜑 → (𝑥𝑋𝑃) ∈ (𝐽 Cn 𝐾))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2098  {csn 4630  cmpt 5232   × cxp 5676  cfv 6549  (class class class)co 7419  TopOnctopon 22856   Cn ccn 23172
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-sep 5300  ax-nul 5307  ax-pow 5365  ax-pr 5429  ax-un 7741
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2930  df-ral 3051  df-rex 3060  df-rab 3419  df-v 3463  df-sbc 3774  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4323  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4910  df-iun 4999  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5576  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-res 5690  df-ima 5691  df-iota 6501  df-fun 6551  df-fn 6552  df-f 6553  df-fv 6557  df-ov 7422  df-oprab 7423  df-mpo 7424  df-1st 7994  df-2nd 7995  df-map 8847  df-topgen 17428  df-top 22840  df-topon 22857  df-cn 23175  df-cnp 23176
This theorem is referenced by:  cnmpt2c  23618  xkoinjcn  23635  txconn  23637  imasnopn  23638  imasncld  23639  imasncls  23640  istgp2  24039  tmdmulg  24040  tmdgsum  24043  tmdlactcn  24050  clsnsg  24058  tgpt0  24067  tlmtgp  24144  nmcn  24804  fsumcn  24832  expcn  24834  divccn  24835  expcnOLD  24836  divccnOLD  24837  cncfmptc  24876  cdivcncf  24885  iirevcn  24895  iihalf1cn  24897  iihalf1cnOLD  24898  iihalf2cn  24900  iihalf2cnOLD  24901  icchmeo  24909  icchmeoOLD  24910  evth  24929  evth2  24930  pcocn  24988  pcopt  24993  pcopt2  24994  pcoass  24995  csscld  25221  clsocv  25222  dvcnvlem  25952  plycn  26240  plycnOLD  26241  psercn2  26404  psercn2OLD  26405  resqrtcn  26729  sqrtcn  26730  atansopn  26909  efrlim  26946  efrlimOLD  26947  ipasslem7  30718  occllem  31185  rmulccn  33657  cxpcncf1  34355  txsconnlem  34978  cvxpconn  34980  cvmlift2lem2  35042  cvmlift2lem3  35043  cvmliftphtlem  35055  sinccvglem  35404  knoppcnlem10  36105  areacirclem2  37310  fprodcn  45123
  Copyright terms: Public domain W3C validator