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Theorem elcncf1ii 20314
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 11 . . 3  |-  ( T. 
->  F : A --> B )
3 elcncf1i.2 . . . 4  |-  ( ( x  e.  A  /\  y  e.  RR+ )  ->  Z  e.  RR+ )
43a1i 11 . . 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 11 . . 3  |-  ( T. 
->  ( ( ( x  e.  A  /\  w  e.  A )  /\  y  e.  RR+ )  ->  (
( abs `  (
x  -  w ) )  <  Z  -> 
( abs `  (
( F `  x
)  -  ( F `
 w ) ) )  <  y ) ) )
72, 4, 6elcncf1di 20313 . 2  |-  ( T. 
->  ( ( A  C_  CC  /\  B  C_  CC )  ->  F  e.  ( A -cn-> B ) ) )
87trud 1371 1  |-  ( ( A  C_  CC  /\  B  C_  CC )  ->  F  e.  ( A -cn-> B ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369   T. wtru 1363    e. wcel 1755    C_ wss 3316   class class class wbr 4280   -->wf 5402   ` cfv 5406  (class class class)co 6080   CCcc 9268    < clt 9406    - cmin 9583   RR+crp 10979   abscabs 12707   -cn->ccncf 20294
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1594  ax-4 1605  ax-5 1669  ax-6 1707  ax-7 1727  ax-8 1757  ax-9 1759  ax-10 1774  ax-11 1779  ax-12 1791  ax-13 1942  ax-ext 2414  ax-sep 4401  ax-nul 4409  ax-pow 4458  ax-pr 4519  ax-un 6361  ax-cnex 9326
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 960  df-tru 1365  df-ex 1590  df-nf 1593  df-sb 1700  df-eu 2258  df-mo 2259  df-clab 2420  df-cleq 2426  df-clel 2429  df-nfc 2558  df-ne 2598  df-ral 2710  df-rex 2711  df-rab 2714  df-v 2964  df-sbc 3176  df-dif 3319  df-un 3321  df-in 3323  df-ss 3330  df-nul 3626  df-if 3780  df-pw 3850  df-sn 3866  df-pr 3868  df-op 3872  df-uni 4080  df-br 4281  df-opab 4339  df-id 4623  df-xp 4833  df-rel 4834  df-cnv 4835  df-co 4836  df-dm 4837  df-rn 4838  df-iota 5369  df-fun 5408  df-fn 5409  df-f 5410  df-fv 5414  df-ov 6083  df-oprab 6084  df-mpt2 6085  df-map 7204  df-cncf 20296
This theorem is referenced by:  logcnlem5  21976
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