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Theorem cncfi 21268
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 21265 . . . . . 6  |-  ( F  e.  ( A -cn-> B )  ->  A  C_  CC )
2 cncfrss2 21266 . . . . . 6  |-  ( F  e.  ( A -cn-> B )  ->  B  C_  CC )
3 elcncf2 21264 . . . . . 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 661 . . . . 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 241 . . . 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 463 . . 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 6286 . . . . . . . 8  |-  ( x  =  C  ->  (
w  -  x )  =  ( w  -  C ) )
87fveq2d 5857 . . . . . . 7  |-  ( x  =  C  ->  ( abs `  ( w  -  x ) )  =  ( abs `  (
w  -  C ) ) )
98breq1d 4444 . . . . . 6  |-  ( x  =  C  ->  (
( abs `  (
w  -  x ) )  <  z  <->  ( abs `  ( w  -  C
) )  <  z
) )
10 fveq2 5853 . . . . . . . . 9  |-  ( x  =  C  ->  ( F `  x )  =  ( F `  C ) )
1110oveq2d 6294 . . . . . . . 8  |-  ( x  =  C  ->  (
( F `  w
)  -  ( F `
 x ) )  =  ( ( F `
 w )  -  ( F `  C ) ) )
1211fveq2d 5857 . . . . . . 7  |-  ( x  =  C  ->  ( abs `  ( ( F `
 w )  -  ( F `  x ) ) )  =  ( abs `  ( ( F `  w )  -  ( F `  C ) ) ) )
1312breq1d 4444 . . . . . 6  |-  ( x  =  C  ->  (
( abs `  (
( F `  w
)  -  ( F `
 x ) ) )  <  y  <->  ( abs `  ( ( F `  w )  -  ( F `  C )
) )  <  y
) )
149, 13imbi12d 320 . . . . 5  |-  ( x  =  C  ->  (
( ( abs `  (
w  -  x ) )  <  z  -> 
( abs `  (
( F `  w
)  -  ( F `
 x ) ) )  <  y )  <-> 
( ( abs `  (
w  -  C ) )  <  z  -> 
( abs `  (
( F `  w
)  -  ( F `
 C ) ) )  <  y ) ) )
1514rexralbidv 2960 . . . 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 4438 . . . . . 6  |-  ( y  =  R  ->  (
( abs `  (
( F `  w
)  -  ( F `
 C ) ) )  <  y  <->  ( abs `  ( ( F `  w )  -  ( F `  C )
) )  <  R
) )
1716imbi2d 316 . . . . 5  |-  ( y  =  R  ->  (
( ( abs `  (
w  -  C ) )  <  z  -> 
( abs `  (
( F `  w
)  -  ( F `
 C ) ) )  <  y )  <-> 
( ( abs `  (
w  -  C ) )  <  z  -> 
( abs `  (
( F `  w
)  -  ( F `
 C ) ) )  <  R ) ) )
1817rexralbidv 2960 . . . 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 3203 . . 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 469 . 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 1191 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 setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 972    = wceq 1381    e. wcel 1802   A.wral 2791   E.wrex 2792    C_ wss 3459   class class class wbr 4434   -->wf 5571   ` cfv 5575  (class class class)co 6278   CCcc 9490    < clt 9628    - cmin 9807   RR+crp 11226   abscabs 13043   -cn->ccncf 21250
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1603  ax-4 1616  ax-5 1689  ax-6 1732  ax-7 1774  ax-8 1804  ax-9 1806  ax-10 1821  ax-11 1826  ax-12 1838  ax-13 1983  ax-ext 2419  ax-sep 4555  ax-nul 4563  ax-pow 4612  ax-pr 4673  ax-un 6574  ax-cnex 9548  ax-resscn 9549  ax-1cn 9550  ax-icn 9551  ax-addcl 9552  ax-addrcl 9553  ax-mulcl 9554  ax-mulrcl 9555  ax-mulcom 9556  ax-addass 9557  ax-mulass 9558  ax-distr 9559  ax-i2m1 9560  ax-1ne0 9561  ax-1rid 9562  ax-rnegex 9563  ax-rrecex 9564  ax-cnre 9565  ax-pre-lttri 9566  ax-pre-lttrn 9567  ax-pre-ltadd 9568  ax-pre-mulgt0 9569
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 973  df-3an 974  df-tru 1384  df-ex 1598  df-nf 1602  df-sb 1725  df-eu 2270  df-mo 2271  df-clab 2427  df-cleq 2433  df-clel 2436  df-nfc 2591  df-ne 2638  df-nel 2639  df-ral 2796  df-rex 2797  df-reu 2798  df-rmo 2799  df-rab 2800  df-v 3095  df-sbc 3312  df-csb 3419  df-dif 3462  df-un 3464  df-in 3466  df-ss 3473  df-nul 3769  df-if 3924  df-pw 3996  df-sn 4012  df-pr 4014  df-op 4018  df-uni 4232  df-br 4435  df-opab 4493  df-mpt 4494  df-id 4782  df-po 4787  df-so 4788  df-xp 4992  df-rel 4993  df-cnv 4994  df-co 4995  df-dm 4996  df-rn 4997  df-res 4998  df-ima 4999  df-iota 5538  df-fun 5577  df-fn 5578  df-f 5579  df-f1 5580  df-fo 5581  df-f1o 5582  df-fv 5583  df-riota 6239  df-ov 6281  df-oprab 6282  df-mpt2 6283  df-er 7310  df-map 7421  df-en 7516  df-dom 7517  df-sdom 7518  df-pnf 9630  df-mnf 9631  df-xr 9632  df-ltxr 9633  df-le 9634  df-sub 9809  df-neg 9810  df-div 10210  df-2 10597  df-cj 12908  df-re 12909  df-im 12910  df-abs 13045  df-cncf 21252
This theorem is referenced by:  cncffvrn  21272  climcncf  21274  cncfco  21281  ivthlem2  21734  ivthlem3  21735  ulmcn  22663  pntlem3  23663  sinccvglem  28908  itg2gt0cn  30042
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