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Theorem cncfi 18925
Description: Defining property of a continuous function. (Contributed by Mario Carneiro, 30-Apr-2014.) (Revised by Mario Carneiro, 25-Aug-2014.)
Assertion
Ref Expression
cncfi  |-  ( ( F  e.  ( A
-cn-> B )  /\  C  e.  A  /\  R  e.  RR+ )  ->  E. z  e.  RR+  A. w  e.  A  ( ( abs `  ( w  -  C
) )  <  z  ->  ( abs `  (
( F `  w
)  -  ( F `
 C ) ) )  <  R ) )
Distinct variable groups:    z, w, A    w, C, z    w, F, z    w, R, z   
w, B, z

Proof of Theorem cncfi
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 cncfrss 18922 . . . . . 6  |-  ( F  e.  ( A -cn-> B )  ->  A  C_  CC )
2 cncfrss2 18923 . . . . . 6  |-  ( F  e.  ( A -cn-> B )  ->  B  C_  CC )
3 elcncf2 18921 . . . . . 6  |-  ( ( A  C_  CC  /\  B  C_  CC )  ->  ( F  e.  ( A -cn-> B )  <->  ( F : A --> B  /\  A. x  e.  A  A. y  e.  RR+  E. z  e.  RR+  A. w  e.  A  ( ( abs `  ( w  -  x
) )  <  z  ->  ( abs `  (
( F `  w
)  -  ( F `
 x ) ) )  <  y ) ) ) )
41, 2, 3syl2anc 644 . . . . 5  |-  ( F  e.  ( A -cn-> B )  ->  ( F  e.  ( A -cn-> B )  <-> 
( F : A --> B  /\  A. x  e.  A  A. y  e.  RR+  E. z  e.  RR+  A. w  e.  A  ( ( abs `  (
w  -  x ) )  <  z  -> 
( abs `  (
( F `  w
)  -  ( F `
 x ) ) )  <  y ) ) ) )
54ibi 234 . . . 4  |-  ( F  e.  ( A -cn-> B )  ->  ( F : A --> B  /\  A. x  e.  A  A. y  e.  RR+  E. z  e.  RR+  A. w  e.  A  ( ( abs `  ( w  -  x
) )  <  z  ->  ( abs `  (
( F `  w
)  -  ( F `
 x ) ) )  <  y ) ) )
65simprd 451 . . 3  |-  ( F  e.  ( A -cn-> B )  ->  A. x  e.  A  A. y  e.  RR+  E. z  e.  RR+  A. w  e.  A  ( ( abs `  (
w  -  x ) )  <  z  -> 
( abs `  (
( F `  w
)  -  ( F `
 x ) ) )  <  y ) )
7 oveq2 6090 . . . . . . . 8  |-  ( x  =  C  ->  (
w  -  x )  =  ( w  -  C ) )
87fveq2d 5733 . . . . . . 7  |-  ( x  =  C  ->  ( abs `  ( w  -  x ) )  =  ( abs `  (
w  -  C ) ) )
98breq1d 4223 . . . . . 6  |-  ( x  =  C  ->  (
( abs `  (
w  -  x ) )  <  z  <->  ( abs `  ( w  -  C
) )  <  z
) )
10 fveq2 5729 . . . . . . . . 9  |-  ( x  =  C  ->  ( F `  x )  =  ( F `  C ) )
1110oveq2d 6098 . . . . . . . 8  |-  ( x  =  C  ->  (
( F `  w
)  -  ( F `
 x ) )  =  ( ( F `
 w )  -  ( F `  C ) ) )
1211fveq2d 5733 . . . . . . 7  |-  ( x  =  C  ->  ( abs `  ( ( F `
 w )  -  ( F `  x ) ) )  =  ( abs `  ( ( F `  w )  -  ( F `  C ) ) ) )
1312breq1d 4223 . . . . . 6  |-  ( x  =  C  ->  (
( abs `  (
( F `  w
)  -  ( F `
 x ) ) )  <  y  <->  ( abs `  ( ( F `  w )  -  ( F `  C )
) )  <  y
) )
149, 13imbi12d 313 . . . . 5  |-  ( x  =  C  ->  (
( ( abs `  (
w  -  x ) )  <  z  -> 
( abs `  (
( F `  w
)  -  ( F `
 x ) ) )  <  y )  <-> 
( ( abs `  (
w  -  C ) )  <  z  -> 
( abs `  (
( F `  w
)  -  ( F `
 C ) ) )  <  y ) ) )
1514rexralbidv 2750 . . . 4  |-  ( x  =  C  ->  ( E. z  e.  RR+  A. w  e.  A  ( ( abs `  ( w  -  x ) )  < 
z  ->  ( abs `  ( ( F `  w )  -  ( F `  x )
) )  <  y
)  <->  E. z  e.  RR+  A. w  e.  A  ( ( abs `  (
w  -  C ) )  <  z  -> 
( abs `  (
( F `  w
)  -  ( F `
 C ) ) )  <  y ) ) )
16 breq2 4217 . . . . . 6  |-  ( y  =  R  ->  (
( abs `  (
( F `  w
)  -  ( F `
 C ) ) )  <  y  <->  ( abs `  ( ( F `  w )  -  ( F `  C )
) )  <  R
) )
1716imbi2d 309 . . . . 5  |-  ( y  =  R  ->  (
( ( abs `  (
w  -  C ) )  <  z  -> 
( abs `  (
( F `  w
)  -  ( F `
 C ) ) )  <  y )  <-> 
( ( abs `  (
w  -  C ) )  <  z  -> 
( abs `  (
( F `  w
)  -  ( F `
 C ) ) )  <  R ) ) )
1817rexralbidv 2750 . . . 4  |-  ( y  =  R  ->  ( E. z  e.  RR+  A. w  e.  A  ( ( abs `  ( w  -  C ) )  < 
z  ->  ( abs `  ( ( F `  w )  -  ( F `  C )
) )  <  y
)  <->  E. z  e.  RR+  A. w  e.  A  ( ( abs `  (
w  -  C ) )  <  z  -> 
( abs `  (
( F `  w
)  -  ( F `
 C ) ) )  <  R ) ) )
1915, 18rspc2v 3059 . . 3  |-  ( ( C  e.  A  /\  R  e.  RR+ )  -> 
( A. x  e.  A  A. y  e.  RR+  E. z  e.  RR+  A. w  e.  A  ( ( abs `  (
w  -  x ) )  <  z  -> 
( abs `  (
( F `  w
)  -  ( F `
 x ) ) )  <  y )  ->  E. z  e.  RR+  A. w  e.  A  ( ( abs `  (
w  -  C ) )  <  z  -> 
( abs `  (
( F `  w
)  -  ( F `
 C ) ) )  <  R ) ) )
206, 19mpan9 457 . 2  |-  ( ( F  e.  ( A
-cn-> B )  /\  ( C  e.  A  /\  R  e.  RR+ ) )  ->  E. z  e.  RR+  A. w  e.  A  ( ( abs `  (
w  -  C ) )  <  z  -> 
( abs `  (
( F `  w
)  -  ( F `
 C ) ) )  <  R ) )
21203impb 1150 1  |-  ( ( F  e.  ( A
-cn-> B )  /\  C  e.  A  /\  R  e.  RR+ )  ->  E. z  e.  RR+  A. w  e.  A  ( ( abs `  ( w  -  C
) )  <  z  ->  ( abs `  (
( F `  w
)  -  ( F `
 C ) ) )  <  R ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 178    /\ wa 360    /\ w3a 937    = wceq 1653    e. wcel 1726   A.wral 2706   E.wrex 2707    C_ wss 3321   class class class wbr 4213   -->wf 5451   ` cfv 5455  (class class class)co 6082   CCcc 8989    < clt 9121    - cmin 9292   RR+crp 10613   abscabs 12040   -cn->ccncf 18907
This theorem is referenced by:  cncffvrn  18929  climcncf  18931  cncfco  18938  ivthlem2  19350  ivthlem3  19351  ulmcn  20316  pntlem3  21304  sinccvglem  25110  itg2gt0cn  26261
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2418  ax-sep 4331  ax-nul 4339  ax-pow 4378  ax-pr 4404  ax-un 4702  ax-cnex 9047  ax-resscn 9048  ax-1cn 9049  ax-icn 9050  ax-addcl 9051  ax-addrcl 9052  ax-mulcl 9053  ax-mulrcl 9054  ax-mulcom 9055  ax-addass 9056  ax-mulass 9057  ax-distr 9058  ax-i2m1 9059  ax-1ne0 9060  ax-1rid 9061  ax-rnegex 9062  ax-rrecex 9063  ax-cnre 9064  ax-pre-lttri 9065  ax-pre-lttrn 9066  ax-pre-ltadd 9067  ax-pre-mulgt0 9068
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2286  df-mo 2287  df-clab 2424  df-cleq 2430  df-clel 2433  df-nfc 2562  df-ne 2602  df-nel 2603  df-ral 2711  df-rex 2712  df-reu 2713  df-rmo 2714  df-rab 2715  df-v 2959  df-sbc 3163  df-csb 3253  df-dif 3324  df-un 3326  df-in 3328  df-ss 3335  df-nul 3630  df-if 3741  df-pw 3802  df-sn 3821  df-pr 3822  df-op 3824  df-uni 4017  df-br 4214  df-opab 4268  df-mpt 4269  df-id 4499  df-po 4504  df-so 4505  df-xp 4885  df-rel 4886  df-cnv 4887  df-co 4888  df-dm 4889  df-rn 4890  df-res 4891  df-ima 4892  df-iota 5419  df-fun 5457  df-fn 5458  df-f 5459  df-f1 5460  df-fo 5461  df-f1o 5462  df-fv 5463  df-ov 6085  df-oprab 6086  df-mpt2 6087  df-riota 6550  df-er 6906  df-map 7021  df-en 7111  df-dom 7112  df-sdom 7113  df-pnf 9123  df-mnf 9124  df-xr 9125  df-ltxr 9126  df-le 9127  df-sub 9294  df-neg 9295  df-div 9679  df-2 10059  df-cj 11905  df-re 11906  df-im 11907  df-abs 12042  df-cncf 18909
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