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Theorem elcncf1ii 12750
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 9 . . 3  |-  ( T. 
->  F : A --> B )
3 elcncf1i.2 . . . 4  |-  ( ( x  e.  A  /\  y  e.  RR+ )  ->  Z  e.  RR+ )
43a1i 9 . . 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 9 . . 3  |-  ( T. 
->  ( ( ( x  e.  A  /\  w  e.  A )  /\  y  e.  RR+ )  ->  (
( abs `  (
x  -  w ) )  <  Z  -> 
( abs `  (
( F `  x
)  -  ( F `
 w ) ) )  <  y ) ) )
72, 4, 6elcncf1di 12749 . 2  |-  ( T. 
->  ( ( A  C_  CC  /\  B  C_  CC )  ->  F  e.  ( A -cn-> B ) ) )
87mptru 1340 1  |-  ( ( A  C_  CC  /\  B  C_  CC )  ->  F  e.  ( A -cn-> B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103   T. wtru 1332    e. wcel 1480    C_ wss 3071   class class class wbr 3929   -->wf 5119   ` cfv 5123  (class class class)co 5774   CCcc 7630    < clt 7812    - cmin 7945   RR+crp 9453   abscabs 10781   -cn->ccncf 12740
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-sep 4046  ax-pow 4098  ax-pr 4131  ax-un 4355  ax-setind 4452  ax-cnex 7723
This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ne 2309  df-ral 2421  df-rex 2422  df-rab 2425  df-v 2688  df-sbc 2910  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-br 3930  df-opab 3990  df-id 4215  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-rn 4550  df-iota 5088  df-fun 5125  df-fn 5126  df-f 5127  df-fv 5131  df-ov 5777  df-oprab 5778  df-mpo 5779  df-map 6544  df-cncf 12741
This theorem is referenced by: (None)
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