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Theorem cncfi 12744
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 12741 . . . . . 6  |-  ( F  e.  ( A -cn-> B )  ->  A  C_  CC )
2 cncfrss2 12742 . . . . . 6  |-  ( F  e.  ( A -cn-> B )  ->  B  C_  CC )
3 elcncf2 12740 . . . . . 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 408 . . . . 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 175 . . . 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 113 . . 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 5782 . . . . . . . 8  |-  ( x  =  C  ->  (
w  -  x )  =  ( w  -  C ) )
87fveq2d 5425 . . . . . . 7  |-  ( x  =  C  ->  ( abs `  ( w  -  x ) )  =  ( abs `  (
w  -  C ) ) )
98breq1d 3939 . . . . . 6  |-  ( x  =  C  ->  (
( abs `  (
w  -  x ) )  <  z  <->  ( abs `  ( w  -  C
) )  <  z
) )
10 fveq2 5421 . . . . . . . . 9  |-  ( x  =  C  ->  ( F `  x )  =  ( F `  C ) )
1110oveq2d 5790 . . . . . . . 8  |-  ( x  =  C  ->  (
( F `  w
)  -  ( F `
 x ) )  =  ( ( F `
 w )  -  ( F `  C ) ) )
1211fveq2d 5425 . . . . . . 7  |-  ( x  =  C  ->  ( abs `  ( ( F `
 w )  -  ( F `  x ) ) )  =  ( abs `  ( ( F `  w )  -  ( F `  C ) ) ) )
1312breq1d 3939 . . . . . 6  |-  ( x  =  C  ->  (
( abs `  (
( F `  w
)  -  ( F `
 x ) ) )  <  y  <->  ( abs `  ( ( F `  w )  -  ( F `  C )
) )  <  y
) )
149, 13imbi12d 233 . . . . 5  |-  ( x  =  C  ->  (
( ( abs `  (
w  -  x ) )  <  z  -> 
( abs `  (
( F `  w
)  -  ( F `
 x ) ) )  <  y )  <-> 
( ( abs `  (
w  -  C ) )  <  z  -> 
( abs `  (
( F `  w
)  -  ( F `
 C ) ) )  <  y ) ) )
1514rexralbidv 2461 . . . 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 3933 . . . . . 6  |-  ( y  =  R  ->  (
( abs `  (
( F `  w
)  -  ( F `
 C ) ) )  <  y  <->  ( abs `  ( ( F `  w )  -  ( F `  C )
) )  <  R
) )
1716imbi2d 229 . . . . 5  |-  ( y  =  R  ->  (
( ( abs `  (
w  -  C ) )  <  z  -> 
( abs `  (
( F `  w
)  -  ( F `
 C ) ) )  <  y )  <-> 
( ( abs `  (
w  -  C ) )  <  z  -> 
( abs `  (
( F `  w
)  -  ( F `
 C ) ) )  <  R ) ) )
1817rexralbidv 2461 . . . 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 2802 . . 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 279 . 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 1177 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    /\ wa 103    <-> wb 104    /\ w3a 962    = wceq 1331    e. wcel 1480   A.wral 2416   E.wrex 2417    C_ wss 3071   class class class wbr 3929   -->wf 5119   ` cfv 5123  (class class class)co 5774   CCcc 7625    < clt 7807    - cmin 7940   RR+crp 9448   abscabs 10776   -cn->ccncf 12736
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-coll 4043  ax-sep 4046  ax-pow 4098  ax-pr 4131  ax-un 4355  ax-setind 4452  ax-cnex 7718  ax-resscn 7719  ax-1cn 7720  ax-1re 7721  ax-icn 7722  ax-addcl 7723  ax-addrcl 7724  ax-mulcl 7725  ax-mulrcl 7726  ax-addcom 7727  ax-mulcom 7728  ax-addass 7729  ax-mulass 7730  ax-distr 7731  ax-i2m1 7732  ax-0lt1 7733  ax-1rid 7734  ax-0id 7735  ax-rnegex 7736  ax-precex 7737  ax-cnre 7738  ax-pre-ltirr 7739  ax-pre-ltwlin 7740  ax-pre-lttrn 7741  ax-pre-apti 7742  ax-pre-ltadd 7743  ax-pre-mulgt0 7744  ax-pre-mulext 7745
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-nel 2404  df-ral 2421  df-rex 2422  df-reu 2423  df-rmo 2424  df-rab 2425  df-v 2688  df-sbc 2910  df-csb 3004  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-iun 3815  df-br 3930  df-opab 3990  df-mpt 3991  df-id 4215  df-po 4218  df-iso 4219  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-rn 4550  df-res 4551  df-ima 4552  df-iota 5088  df-fun 5125  df-fn 5126  df-f 5127  df-f1 5128  df-fo 5129  df-f1o 5130  df-fv 5131  df-riota 5730  df-ov 5777  df-oprab 5778  df-mpo 5779  df-map 6544  df-pnf 7809  df-mnf 7810  df-xr 7811  df-ltxr 7812  df-le 7813  df-sub 7942  df-neg 7943  df-reap 8344  df-ap 8351  df-div 8440  df-2 8786  df-cj 10621  df-re 10622  df-im 10623  df-rsqrt 10777  df-abs 10778  df-cncf 12737
This theorem is referenced by:  cncffvrn  12748  climcncf  12750  cncfco  12757  mulcncf  12770  ivthinclemlopn  12793  ivthinclemuopn  12795
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