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Theorem elcncf1ii 20614
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 20613 . 2  |-  ( T. 
->  ( ( A  C_  CC  /\  B  C_  CC )  ->  F  e.  ( A -cn-> B ) ) )
87trud 1379 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 1371    e. wcel 1758    C_ wss 3439   class class class wbr 4403   -->wf 5525   ` cfv 5529  (class class class)co 6203   CCcc 9395    < clt 9533    - cmin 9710   RR+crp 11106   abscabs 12845   -cn->ccncf 20594
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-sep 4524  ax-nul 4532  ax-pow 4581  ax-pr 4642  ax-un 6485  ax-cnex 9453
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-ral 2804  df-rex 2805  df-rab 2808  df-v 3080  df-sbc 3295  df-dif 3442  df-un 3444  df-in 3446  df-ss 3453  df-nul 3749  df-if 3903  df-pw 3973  df-sn 3989  df-pr 3991  df-op 3995  df-uni 4203  df-br 4404  df-opab 4462  df-id 4747  df-xp 4957  df-rel 4958  df-cnv 4959  df-co 4960  df-dm 4961  df-rn 4962  df-iota 5492  df-fun 5531  df-fn 5532  df-f 5533  df-fv 5537  df-ov 6206  df-oprab 6207  df-mpt2 6208  df-map 7329  df-cncf 20596
This theorem is referenced by:  logcnlem5  22234
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