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Theorem limcmpted 13799
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 2319 . . . . . 6  |-  F/_ w D
2 nfcsb1v 3090 . . . . . 6  |-  F/_ z [_ w  /  z ]_ D
3 csbeq1a 3066 . . . . . 6  |-  ( z  =  w  ->  D  =  [_ w  /  z ]_ D )
41, 2, 3cbvmpt 4095 . . . . 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 5884 . . 3  |-  ( ph  ->  ( ( z  e.  A  |->  D ) lim CC  B )  =  ( ( w  e.  A  |-> 
[_ w  /  z ]_ D ) lim CC  B
) )
76eleq2d 2247 . 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 5667 . . . 4  |-  ( ph  ->  ( z  e.  A  |->  D ) : A --> CC )
104feq1i 5354 . . . 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 2319 . . . 4  |-  F/_ z A
1514, 2nfmpt 4092 . . 3  |-  F/_ z
( w  e.  A  |-> 
[_ w  /  z ]_ D )
1611, 12, 13, 15ellimc3apf 13796 . 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 2177 . . . . . . . . . 10  |-  ( w  e.  A  |->  [_ w  /  z ]_ D
)  =  ( w  e.  A  |->  [_ w  /  z ]_ D
)
18 eqcom 2179 . . . . . . . . . . 11  |-  ( z  =  w  <->  w  =  z )
19 eqcom 2179 . . . . . . . . . . 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 5605 . . . . . . . . 9  |-  ( (
ph  /\  z  e.  A )  ->  (
( w  e.  A  |-> 
[_ w  /  z ]_ D ) `  z
)  =  D )
2322fvoveq1d 5891 . . . . . . . 8  |-  ( (
ph  /\  z  e.  A )  ->  ( abs `  ( ( ( w  e.  A  |->  [_ w  /  z ]_ D
) `  z )  -  C ) )  =  ( abs `  ( D  -  C )
) )
2423breq1d 4010 . . . . . . 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 2473 . . . . 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 2478 . . . 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 2477 . . 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 1353    e. wcel 2148   A.wral 2455   E.wrex 2456   [_csb 3057    C_ wss 3129   class class class wbr 4000    |-> cmpt 4061   -->wf 5208   ` cfv 5212  (class class class)co 5869   CCcc 7800    < clt 7982    - cmin 8118   # cap 8528   RR+crp 9640   abscabs 10990   lim CC climc 13790
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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4118  ax-pow 4171  ax-pr 4206  ax-un 4430  ax-setind 4533  ax-cnex 7893
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-ral 2460  df-rex 2461  df-rab 2464  df-v 2739  df-sbc 2963  df-csb 3058  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-pw 3576  df-sn 3597  df-pr 3598  df-op 3600  df-uni 3808  df-br 4001  df-opab 4062  df-mpt 4063  df-id 4290  df-xp 4629  df-rel 4630  df-cnv 4631  df-co 4632  df-dm 4633  df-rn 4634  df-res 4635  df-ima 4636  df-iota 5174  df-fun 5214  df-fn 5215  df-f 5216  df-fv 5220  df-ov 5872  df-oprab 5873  df-mpo 5874  df-pm 6645  df-limced 13792
This theorem is referenced by:  limccnp2cntop  13813  limccoap  13814
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