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Theorem cncfi 20475
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 20472 . . . . . 6  |-  ( F  e.  ( A -cn-> B )  ->  A  C_  CC )
2 cncfrss2 20473 . . . . . 6  |-  ( F  e.  ( A -cn-> B )  ->  B  C_  CC )
3 elcncf2 20471 . . . . . 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 6104 . . . . . . . 8  |-  ( x  =  C  ->  (
w  -  x )  =  ( w  -  C ) )
87fveq2d 5700 . . . . . . 7  |-  ( x  =  C  ->  ( abs `  ( w  -  x ) )  =  ( abs `  (
w  -  C ) ) )
98breq1d 4307 . . . . . 6  |-  ( x  =  C  ->  (
( abs `  (
w  -  x ) )  <  z  <->  ( abs `  ( w  -  C
) )  <  z
) )
10 fveq2 5696 . . . . . . . . 9  |-  ( x  =  C  ->  ( F `  x )  =  ( F `  C ) )
1110oveq2d 6112 . . . . . . . 8  |-  ( x  =  C  ->  (
( F `  w
)  -  ( F `
 x ) )  =  ( ( F `
 w )  -  ( F `  C ) ) )
1211fveq2d 5700 . . . . . . 7  |-  ( x  =  C  ->  ( abs `  ( ( F `
 w )  -  ( F `  x ) ) )  =  ( abs `  ( ( F `  w )  -  ( F `  C ) ) ) )
1312breq1d 4307 . . . . . 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 2764 . . . 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 4301 . . . . . 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 2764 . . . 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 3084 . . 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 1183 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 965    = wceq 1369    e. wcel 1756   A.wral 2720   E.wrex 2721    C_ wss 3333   class class class wbr 4297   -->wf 5419   ` cfv 5423  (class class class)co 6096   CCcc 9285    < clt 9423    - cmin 9600   RR+crp 10996   abscabs 12728   -cn->ccncf 20457
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4418  ax-nul 4426  ax-pow 4475  ax-pr 4536  ax-un 6377  ax-cnex 9343  ax-resscn 9344  ax-1cn 9345  ax-icn 9346  ax-addcl 9347  ax-addrcl 9348  ax-mulcl 9349  ax-mulrcl 9350  ax-mulcom 9351  ax-addass 9352  ax-mulass 9353  ax-distr 9354  ax-i2m1 9355  ax-1ne0 9356  ax-1rid 9357  ax-rnegex 9358  ax-rrecex 9359  ax-cnre 9360  ax-pre-lttri 9361  ax-pre-lttrn 9362  ax-pre-ltadd 9363  ax-pre-mulgt0 9364
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2573  df-ne 2613  df-nel 2614  df-ral 2725  df-rex 2726  df-reu 2727  df-rmo 2728  df-rab 2729  df-v 2979  df-sbc 3192  df-csb 3294  df-dif 3336  df-un 3338  df-in 3340  df-ss 3347  df-nul 3643  df-if 3797  df-pw 3867  df-sn 3883  df-pr 3885  df-op 3889  df-uni 4097  df-br 4298  df-opab 4356  df-mpt 4357  df-id 4641  df-po 4646  df-so 4647  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5386  df-fun 5425  df-fn 5426  df-f 5427  df-f1 5428  df-fo 5429  df-f1o 5430  df-fv 5431  df-riota 6057  df-ov 6099  df-oprab 6100  df-mpt2 6101  df-er 7106  df-map 7221  df-en 7316  df-dom 7317  df-sdom 7318  df-pnf 9425  df-mnf 9426  df-xr 9427  df-ltxr 9428  df-le 9429  df-sub 9602  df-neg 9603  df-div 9999  df-2 10385  df-cj 12593  df-re 12594  df-im 12595  df-abs 12730  df-cncf 20459
This theorem is referenced by:  cncffvrn  20479  climcncf  20481  cncfco  20488  ivthlem2  20941  ivthlem3  20942  ulmcn  21869  pntlem3  22863  sinccvglem  27322  itg2gt0cn  28452
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