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Theorem cncfi 18398
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 18395 . . . . . 6  |-  ( F  e.  ( A -cn-> B )  ->  A  C_  CC )
2 cncfrss2 18396 . . . . . 6  |-  ( F  e.  ( A -cn-> B )  ->  B  C_  CC )
3 elcncf2 18394 . . . . . 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 642 . . . . 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 232 . . . 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 449 . . 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 5866 . . . . . . . 8  |-  ( x  =  C  ->  (
w  -  x )  =  ( w  -  C ) )
87fveq2d 5529 . . . . . . 7  |-  ( x  =  C  ->  ( abs `  ( w  -  x ) )  =  ( abs `  (
w  -  C ) ) )
98breq1d 4033 . . . . . 6  |-  ( x  =  C  ->  (
( abs `  (
w  -  x ) )  <  z  <->  ( abs `  ( w  -  C
) )  <  z
) )
10 fveq2 5525 . . . . . . . . 9  |-  ( x  =  C  ->  ( F `  x )  =  ( F `  C ) )
1110oveq2d 5874 . . . . . . . 8  |-  ( x  =  C  ->  (
( F `  w
)  -  ( F `
 x ) )  =  ( ( F `
 w )  -  ( F `  C ) ) )
1211fveq2d 5529 . . . . . . 7  |-  ( x  =  C  ->  ( abs `  ( ( F `
 w )  -  ( F `  x ) ) )  =  ( abs `  ( ( F `  w )  -  ( F `  C ) ) ) )
1312breq1d 4033 . . . . . 6  |-  ( x  =  C  ->  (
( abs `  (
( F `  w
)  -  ( F `
 x ) ) )  <  y  <->  ( abs `  ( ( F `  w )  -  ( F `  C )
) )  <  y
) )
149, 13imbi12d 311 . . . . 5  |-  ( x  =  C  ->  (
( ( abs `  (
w  -  x ) )  <  z  -> 
( abs `  (
( F `  w
)  -  ( F `
 x ) ) )  <  y )  <-> 
( ( abs `  (
w  -  C ) )  <  z  -> 
( abs `  (
( F `  w
)  -  ( F `
 C ) ) )  <  y ) ) )
1514rexralbidv 2587 . . . 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 4027 . . . . . 6  |-  ( y  =  R  ->  (
( abs `  (
( F `  w
)  -  ( F `
 C ) ) )  <  y  <->  ( abs `  ( ( F `  w )  -  ( F `  C )
) )  <  R
) )
1716imbi2d 307 . . . . 5  |-  ( y  =  R  ->  (
( ( abs `  (
w  -  C ) )  <  z  -> 
( abs `  (
( F `  w
)  -  ( F `
 C ) ) )  <  y )  <-> 
( ( abs `  (
w  -  C ) )  <  z  -> 
( abs `  (
( F `  w
)  -  ( F `
 C ) ) )  <  R ) ) )
1817rexralbidv 2587 . . . 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 2890 . . 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 455 . 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 1147 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 176    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684   A.wral 2543   E.wrex 2544    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:  cncffvrn  18402  climcncf  18404  cncfco  18411  ivthlem2  18812  ivthlem3  18813  ulmcn  19776  pntlem3  20758  sinccvglem  24005
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  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  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-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  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-mpt 4079  df-id 4309  df-po 4314  df-so 4315  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-riota 6304  df-er 6660  df-map 6774  df-en 6864  df-dom 6865  df-sdom 6866  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-div 9424  df-2 9804  df-cj 11584  df-re 11585  df-im 11586  df-abs 11721  df-cncf 18382
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