MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  htpycn Structured version   Unicode version

Theorem htpycn 21300
Description: A homotopy is a continuous function. (Contributed by Mario Carneiro, 22-Feb-2015.)
Hypotheses
Ref Expression
ishtpy.1  |-  ( ph  ->  J  e.  (TopOn `  X ) )
ishtpy.3  |-  ( ph  ->  F  e.  ( J  Cn  K ) )
ishtpy.4  |-  ( ph  ->  G  e.  ( J  Cn  K ) )
Assertion
Ref Expression
htpycn  |-  ( ph  ->  ( F ( J Htpy 
K ) G ) 
C_  ( ( J 
tX  II )  Cn  K ) )

Proof of Theorem htpycn
Dummy variables  s  h are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ishtpy.1 . . . 4  |-  ( ph  ->  J  e.  (TopOn `  X ) )
2 ishtpy.3 . . . 4  |-  ( ph  ->  F  e.  ( J  Cn  K ) )
3 ishtpy.4 . . . 4  |-  ( ph  ->  G  e.  ( J  Cn  K ) )
41, 2, 3ishtpy 21299 . . 3  |-  ( ph  ->  ( h  e.  ( F ( J Htpy  K
) G )  <->  ( h  e.  ( ( J  tX  II )  Cn  K
)  /\  A. s  e.  X  ( (
s h 0 )  =  ( F `  s )  /\  (
s h 1 )  =  ( G `  s ) ) ) ) )
5 simpl 457 . . 3  |-  ( ( h  e.  ( ( J  tX  II )  Cn  K )  /\  A. s  e.  X  ( ( s h 0 )  =  ( F `
 s )  /\  ( s h 1 )  =  ( G `
 s ) ) )  ->  h  e.  ( ( J  tX  II )  Cn  K
) )
64, 5syl6bi 228 . 2  |-  ( ph  ->  ( h  e.  ( F ( J Htpy  K
) G )  ->  h  e.  ( ( J  tX  II )  Cn  K ) ) )
76ssrdv 3510 1  |-  ( ph  ->  ( F ( J Htpy 
K ) G ) 
C_  ( ( J 
tX  II )  Cn  K ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1379    e. wcel 1767   A.wral 2814    C_ wss 3476   ` cfv 5588  (class class class)co 6285   0cc0 9493   1c1 9494  TopOnctopon 19202    Cn ccn 19531    tX ctx 19888   IIcii 21206   Htpy chtpy 21294
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6577
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fun 5590  df-fn 5591  df-f 5592  df-fv 5596  df-ov 6288  df-oprab 6289  df-mpt2 6290  df-1st 6785  df-2nd 6786  df-map 7423  df-top 19206  df-topon 19209  df-cn 19534  df-htpy 21297
This theorem is referenced by:  htpycom  21303  htpyco1  21305  htpyco2  21306  htpycc  21307  phtpycn  21310
  Copyright terms: Public domain W3C validator