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Theorem limcmpted 14817
Description: Express the limit operator for a function defined by a mapping, via epsilon-delta. (Contributed by Jim Kingdon, 3-Nov-2023.)
Hypotheses
Ref Expression
limcmpted.a  |-  ( ph  ->  A  C_  CC )
limcmpted.b  |-  ( ph  ->  B  e.  CC )
limcmpted.f  |-  ( (
ph  /\  z  e.  A )  ->  D  e.  CC )
Assertion
Ref Expression
limcmpted  |-  ( ph  ->  ( C  e.  ( ( z  e.  A  |->  D ) lim CC  B
)  <->  ( C  e.  CC  /\  A. x  e.  RR+  E. y  e.  RR+  A. z  e.  A  ( ( z #  B  /\  ( abs `  (
z  -  B ) )  <  y )  ->  ( abs `  ( D  -  C )
)  <  x )
) ) )
Distinct variable groups:    x, A, y, z    x, B, y, z    x, C, y, z    x, D, y    ph, x, y, z
Allowed substitution hint:    D( z)

Proof of Theorem limcmpted
Dummy variable  w is distinct from all other variables.
StepHypRef Expression
1 nfcv 2336 . . . . . 6  |-  F/_ w D
2 nfcsb1v 3113 . . . . . 6  |-  F/_ z [_ w  /  z ]_ D
3 csbeq1a 3089 . . . . . 6  |-  ( z  =  w  ->  D  =  [_ w  /  z ]_ D )
41, 2, 3cbvmpt 4124 . . . . 5  |-  ( z  e.  A  |->  D )  =  ( w  e.  A  |->  [_ w  /  z ]_ D )
54a1i 9 . . . 4  |-  ( ph  ->  ( z  e.  A  |->  D )  =  ( w  e.  A  |->  [_ w  /  z ]_ D
) )
65oveq1d 5933 . . 3  |-  ( ph  ->  ( ( z  e.  A  |->  D ) lim CC  B )  =  ( ( w  e.  A  |-> 
[_ w  /  z ]_ D ) lim CC  B
) )
76eleq2d 2263 . 2  |-  ( ph  ->  ( C  e.  ( ( z  e.  A  |->  D ) lim CC  B
)  <->  C  e.  (
( w  e.  A  |-> 
[_ w  /  z ]_ D ) lim CC  B
) ) )
8 limcmpted.f . . . . 5  |-  ( (
ph  /\  z  e.  A )  ->  D  e.  CC )
98fmpttd 5713 . . . 4  |-  ( ph  ->  ( z  e.  A  |->  D ) : A --> CC )
104feq1i 5396 . . . 4  |-  ( ( z  e.  A  |->  D ) : A --> CC  <->  ( w  e.  A  |->  [_ w  /  z ]_ D
) : A --> CC )
119, 10sylib 122 . . 3  |-  ( ph  ->  ( w  e.  A  |-> 
[_ w  /  z ]_ D ) : A --> CC )
12 limcmpted.a . . 3  |-  ( ph  ->  A  C_  CC )
13 limcmpted.b . . 3  |-  ( ph  ->  B  e.  CC )
14 nfcv 2336 . . . 4  |-  F/_ z A
1514, 2nfmpt 4121 . . 3  |-  F/_ z
( w  e.  A  |-> 
[_ w  /  z ]_ D )
1611, 12, 13, 15ellimc3apf 14814 . 2  |-  ( ph  ->  ( C  e.  ( ( w  e.  A  |-> 
[_ w  /  z ]_ D ) lim CC  B
)  <->  ( C  e.  CC  /\  A. x  e.  RR+  E. y  e.  RR+  A. z  e.  A  ( ( z #  B  /\  ( abs `  (
z  -  B ) )  <  y )  ->  ( abs `  (
( ( w  e.  A  |->  [_ w  /  z ]_ D ) `  z
)  -  C ) )  <  x ) ) ) )
17 eqid 2193 . . . . . . . . . 10  |-  ( w  e.  A  |->  [_ w  /  z ]_ D
)  =  ( w  e.  A  |->  [_ w  /  z ]_ D
)
18 eqcom 2195 . . . . . . . . . . 11  |-  ( z  =  w  <->  w  =  z )
19 eqcom 2195 . . . . . . . . . . 11  |-  ( D  =  [_ w  / 
z ]_ D  <->  [_ w  / 
z ]_ D  =  D )
203, 18, 193imtr3i 200 . . . . . . . . . 10  |-  ( w  =  z  ->  [_ w  /  z ]_ D  =  D )
21 simpr 110 . . . . . . . . . 10  |-  ( (
ph  /\  z  e.  A )  ->  z  e.  A )
2217, 20, 21, 8fvmptd3 5651 . . . . . . . . 9  |-  ( (
ph  /\  z  e.  A )  ->  (
( w  e.  A  |-> 
[_ w  /  z ]_ D ) `  z
)  =  D )
2322fvoveq1d 5940 . . . . . . . 8  |-  ( (
ph  /\  z  e.  A )  ->  ( abs `  ( ( ( w  e.  A  |->  [_ w  /  z ]_ D
) `  z )  -  C ) )  =  ( abs `  ( D  -  C )
) )
2423breq1d 4039 . . . . . . 7  |-  ( (
ph  /\  z  e.  A )  ->  (
( abs `  (
( ( w  e.  A  |->  [_ w  /  z ]_ D ) `  z
)  -  C ) )  <  x  <->  ( abs `  ( D  -  C
) )  <  x
) )
2524imbi2d 230 . . . . . 6  |-  ( (
ph  /\  z  e.  A )  ->  (
( ( z #  B  /\  ( abs `  (
z  -  B ) )  <  y )  ->  ( abs `  (
( ( w  e.  A  |->  [_ w  /  z ]_ D ) `  z
)  -  C ) )  <  x )  <-> 
( ( z #  B  /\  ( abs `  (
z  -  B ) )  <  y )  ->  ( abs `  ( D  -  C )
)  <  x )
) )
2625ralbidva 2490 . . . . 5  |-  ( ph  ->  ( A. z  e.  A  ( ( z #  B  /\  ( abs `  ( z  -  B
) )  <  y
)  ->  ( abs `  ( ( ( w  e.  A  |->  [_ w  /  z ]_ D
) `  z )  -  C ) )  < 
x )  <->  A. z  e.  A  ( (
z #  B  /\  ( abs `  ( z  -  B ) )  < 
y )  ->  ( abs `  ( D  -  C ) )  < 
x ) ) )
2726rexbidv 2495 . . . 4  |-  ( ph  ->  ( E. y  e.  RR+  A. z  e.  A  ( ( z #  B  /\  ( abs `  (
z  -  B ) )  <  y )  ->  ( abs `  (
( ( w  e.  A  |->  [_ w  /  z ]_ D ) `  z
)  -  C ) )  <  x )  <->  E. y  e.  RR+  A. z  e.  A  ( (
z #  B  /\  ( abs `  ( z  -  B ) )  < 
y )  ->  ( abs `  ( D  -  C ) )  < 
x ) ) )
2827ralbidv 2494 . . 3  |-  ( ph  ->  ( A. x  e.  RR+  E. y  e.  RR+  A. z  e.  A  ( ( z #  B  /\  ( abs `  ( z  -  B ) )  <  y )  -> 
( abs `  (
( ( w  e.  A  |->  [_ w  /  z ]_ D ) `  z
)  -  C ) )  <  x )  <->  A. x  e.  RR+  E. y  e.  RR+  A. z  e.  A  ( ( z #  B  /\  ( abs `  ( z  -  B
) )  <  y
)  ->  ( abs `  ( D  -  C
) )  <  x
) ) )
2928anbi2d 464 . 2  |-  ( ph  ->  ( ( C  e.  CC  /\  A. x  e.  RR+  E. y  e.  RR+  A. z  e.  A  ( ( z #  B  /\  ( abs `  (
z  -  B ) )  <  y )  ->  ( abs `  (
( ( w  e.  A  |->  [_ w  /  z ]_ D ) `  z
)  -  C ) )  <  x ) )  <->  ( C  e.  CC  /\  A. x  e.  RR+  E. y  e.  RR+  A. z  e.  A  ( ( z #  B  /\  ( abs `  (
z  -  B ) )  <  y )  ->  ( abs `  ( D  -  C )
)  <  x )
) ) )
307, 16, 293bitrd 214 1  |-  ( ph  ->  ( C  e.  ( ( z  e.  A  |->  D ) lim CC  B
)  <->  ( C  e.  CC  /\  A. x  e.  RR+  E. y  e.  RR+  A. z  e.  A  ( ( z #  B  /\  ( abs `  (
z  -  B ) )  <  y )  ->  ( abs `  ( D  -  C )
)  <  x )
) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 104    <-> wb 105    = wceq 1364    e. wcel 2164   A.wral 2472   E.wrex 2473   [_csb 3080    C_ wss 3153   class class class wbr 4029    |-> cmpt 4090   -->wf 5250   ` cfv 5254  (class class class)co 5918   CCcc 7870    < clt 8054    - cmin 8190   # cap 8600   RR+crp 9719   abscabs 11141   lim CC climc 14808
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-sep 4147  ax-pow 4203  ax-pr 4238  ax-un 4464  ax-setind 4569  ax-cnex 7963
This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ne 2365  df-ral 2477  df-rex 2478  df-rab 2481  df-v 2762  df-sbc 2986  df-csb 3081  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-uni 3836  df-br 4030  df-opab 4091  df-mpt 4092  df-id 4324  df-xp 4665  df-rel 4666  df-cnv 4667  df-co 4668  df-dm 4669  df-rn 4670  df-res 4671  df-ima 4672  df-iota 5215  df-fun 5256  df-fn 5257  df-f 5258  df-fv 5262  df-ov 5921  df-oprab 5922  df-mpo 5923  df-pm 6705  df-limced 14810
This theorem is referenced by:  limccnp2cntop  14831  limccoap  14832
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