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

Theorem cnmptc 21667
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 5320 . 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 21289 . . 3 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝑃𝑌) → (𝑋 × {𝑃}) ∈ (𝐽 Cn 𝐾))
62, 3, 4, 5syl3anc 1477 . 2 (𝜑 → (𝑋 × {𝑃}) ∈ (𝐽 Cn 𝐾))
71, 6syl5eqelr 2844 1 (𝜑 → (𝑥𝑋𝑃) ∈ (𝐽 Cn 𝐾))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2139  {csn 4321  cmpt 4881   × cxp 5264  cfv 6049  (class class class)co 6813  TopOnctopon 20917   Cn ccn 21230
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7114
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-ral 3055  df-rex 3056  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-iun 4674  df-br 4805  df-opab 4865  df-mpt 4882  df-id 5174  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-fv 6057  df-ov 6816  df-oprab 6817  df-mpt2 6818  df-1st 7333  df-2nd 7334  df-map 8025  df-topgen 16306  df-top 20901  df-topon 20918  df-cn 21233  df-cnp 21234
This theorem is referenced by:  cnmpt2c  21675  xkoinjcn  21692  txconn  21694  imasnopn  21695  imasncld  21696  imasncls  21697  istgp2  22096  tmdmulg  22097  tmdgsum  22100  tmdlactcn  22107  clsnsg  22114  tgpt0  22123  tlmtgp  22200  nmcn  22848  fsumcn  22874  expcn  22876  divccn  22877  cncfmptc  22915  cdivcncf  22921  iirevcn  22930  iihalf1cn  22932  iihalf2cn  22934  icchmeo  22941  evth  22959  evth2  22960  pcocn  23017  pcopt  23022  pcopt2  23023  pcoass  23024  csscld  23248  clsocv  23249  dvcnvlem  23938  plycn  24216  psercn2  24376  resqrtcn  24689  sqrtcn  24690  atansopn  24858  efrlim  24895  ipasslem7  28000  occllem  28471  rmulccn  30283  cxpcncf1  30982  txsconnlem  31529  cvxpconn  31531  cvmlift2lem2  31593  cvmlift2lem3  31594  cvmliftphtlem  31606  sinccvglem  31873  knoppcnlem10  32798  areacirclem2  33814  fprodcn  40335
  Copyright terms: Public domain W3C validator