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Theorem elcncf1ii 18400
Description: Membership in the set of continuous complex functions from 
A to  B. (Contributed by Paul Chapman, 26-Nov-2007.)
Hypotheses
Ref Expression
elcncf1i.1  |-  F : A
--> B
elcncf1i.2  |-  ( ( x  e.  A  /\  y  e.  RR+ )  ->  Z  e.  RR+ )
elcncf1i.3  |-  ( ( ( x  e.  A  /\  w  e.  A
)  /\  y  e.  RR+ )  ->  ( ( abs `  ( x  -  w ) )  < 
Z  ->  ( abs `  ( ( F `  x )  -  ( F `  w )
) )  <  y
) )
Assertion
Ref Expression
elcncf1ii  |-  ( ( A  C_  CC  /\  B  C_  CC )  ->  F  e.  ( A -cn-> B ) )
Distinct variable groups:    x, w, y, A    w, B, x, y    w, F, x, y    w, Z
Allowed substitution hints:    Z( x, y)

Proof of Theorem elcncf1ii
StepHypRef Expression
1 elcncf1i.1 . . . 4  |-  F : A
--> B
21a1i 10 . . 3  |-  (  T. 
->  F : A --> B )
3 elcncf1i.2 . . . 4  |-  ( ( x  e.  A  /\  y  e.  RR+ )  ->  Z  e.  RR+ )
43a1i 10 . . 3  |-  (  T. 
->  ( ( x  e.  A  /\  y  e.  RR+ )  ->  Z  e.  RR+ ) )
5 elcncf1i.3 . . . 4  |-  ( ( ( x  e.  A  /\  w  e.  A
)  /\  y  e.  RR+ )  ->  ( ( abs `  ( x  -  w ) )  < 
Z  ->  ( abs `  ( ( F `  x )  -  ( F `  w )
) )  <  y
) )
65a1i 10 . . 3  |-  (  T. 
->  ( ( ( x  e.  A  /\  w  e.  A )  /\  y  e.  RR+ )  ->  (
( abs `  (
x  -  w ) )  <  Z  -> 
( abs `  (
( F `  x
)  -  ( F `
 w ) ) )  <  y ) ) )
72, 4, 6elcncf1di 18399 . 2  |-  (  T. 
->  ( ( A  C_  CC  /\  B  C_  CC )  ->  F  e.  ( A -cn-> B ) ) )
87trud 1314 1  |-  ( ( A  C_  CC  /\  B  C_  CC )  ->  F  e.  ( A -cn-> B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    T. wtru 1307    e. wcel 1684    C_ wss 3152   class class class wbr 4023   -->wf 5251   ` cfv 5255  (class class class)co 5858   CCcc 8735    < clt 8867    - cmin 9037   RR+crp 10354   abscabs 11719   -cn->ccncf 18380
This theorem is referenced by:  logcnlem5  19993
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-cnex 8793
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-map 6774  df-cncf 18382
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