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Theorem limcdif 19763
Description: It suffices to consider functions which are not defined at 
B to define the limit of a function. In particular, the value of the original function  F at  B does not affect the limit of  F. (Contributed by Mario Carneiro, 25-Dec-2016.)
Hypothesis
Ref Expression
limccl.f  |-  ( ph  ->  F : A --> CC )
Assertion
Ref Expression
limcdif  |-  ( ph  ->  ( F lim CC  B
)  =  ( ( F  |`  ( A  \  { B } ) ) lim CC  B ) )

Proof of Theorem limcdif
Dummy variables  x  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 limccl.f . . . . . . . 8  |-  ( ph  ->  F : A --> CC )
2 fdm 5595 . . . . . . . 8  |-  ( F : A --> CC  ->  dom 
F  =  A )
31, 2syl 16 . . . . . . 7  |-  ( ph  ->  dom  F  =  A )
43adantr 452 . . . . . 6  |-  ( (
ph  /\  x  e.  ( F lim CC  B ) )  ->  dom  F  =  A )
5 limcrcl 19761 . . . . . . . 8  |-  ( x  e.  ( F lim CC  B )  ->  ( F : dom  F --> CC  /\  dom  F  C_  CC  /\  B  e.  CC ) )
65adantl 453 . . . . . . 7  |-  ( (
ph  /\  x  e.  ( F lim CC  B ) )  ->  ( F : dom  F --> CC  /\  dom  F  C_  CC  /\  B  e.  CC ) )
76simp2d 970 . . . . . 6  |-  ( (
ph  /\  x  e.  ( F lim CC  B ) )  ->  dom  F  C_  CC )
84, 7eqsstr3d 3383 . . . . 5  |-  ( (
ph  /\  x  e.  ( F lim CC  B ) )  ->  A  C_  CC )
96simp3d 971 . . . . 5  |-  ( (
ph  /\  x  e.  ( F lim CC  B ) )  ->  B  e.  CC )
108, 9jca 519 . . . 4  |-  ( (
ph  /\  x  e.  ( F lim CC  B ) )  ->  ( A  C_  CC  /\  B  e.  CC ) )
1110ex 424 . . 3  |-  ( ph  ->  ( x  e.  ( F lim CC  B )  ->  ( A  C_  CC  /\  B  e.  CC ) ) )
12 undif1 3703 . . . . . . 7  |-  ( ( A  \  { B } )  u.  { B } )  =  ( A  u.  { B } )
13 difss 3474 . . . . . . . . . . . 12  |-  ( A 
\  { B }
)  C_  A
14 fssres 5610 . . . . . . . . . . . 12  |-  ( ( F : A --> CC  /\  ( A  \  { B } )  C_  A
)  ->  ( F  |`  ( A  \  { B } ) ) : ( A  \  { B } ) --> CC )
151, 13, 14sylancl 644 . . . . . . . . . . 11  |-  ( ph  ->  ( F  |`  ( A  \  { B }
) ) : ( A  \  { B } ) --> CC )
16 fdm 5595 . . . . . . . . . . 11  |-  ( ( F  |`  ( A  \  { B } ) ) : ( A 
\  { B }
) --> CC  ->  dom  ( F  |`  ( A 
\  { B }
) )  =  ( A  \  { B } ) )
1715, 16syl 16 . . . . . . . . . 10  |-  ( ph  ->  dom  ( F  |`  ( A  \  { B } ) )  =  ( A  \  { B } ) )
1817adantr 452 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  ( ( F  |`  ( A  \  { B } ) ) lim CC  B ) )  ->  dom  ( F  |`  ( A  \  { B }
) )  =  ( A  \  { B } ) )
19 limcrcl 19761 . . . . . . . . . . 11  |-  ( x  e.  ( ( F  |`  ( A  \  { B } ) ) lim CC  B )  ->  (
( F  |`  ( A  \  { B }
) ) : dom  ( F  |`  ( A 
\  { B }
) ) --> CC  /\  dom  ( F  |`  ( A  \  { B }
) )  C_  CC  /\  B  e.  CC ) )
2019adantl 453 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  ( ( F  |`  ( A  \  { B } ) ) lim CC  B ) )  -> 
( ( F  |`  ( A  \  { B } ) ) : dom  ( F  |`  ( A  \  { B } ) ) --> CC 
/\  dom  ( F  |`  ( A  \  { B } ) )  C_  CC  /\  B  e.  CC ) )
2120simp2d 970 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  ( ( F  |`  ( A  \  { B } ) ) lim CC  B ) )  ->  dom  ( F  |`  ( A  \  { B }
) )  C_  CC )
2218, 21eqsstr3d 3383 . . . . . . . 8  |-  ( (
ph  /\  x  e.  ( ( F  |`  ( A  \  { B } ) ) lim CC  B ) )  -> 
( A  \  { B } )  C_  CC )
2320simp3d 971 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  ( ( F  |`  ( A  \  { B } ) ) lim CC  B ) )  ->  B  e.  CC )
2423snssd 3943 . . . . . . . 8  |-  ( (
ph  /\  x  e.  ( ( F  |`  ( A  \  { B } ) ) lim CC  B ) )  ->  { B }  C_  CC )
2522, 24unssd 3523 . . . . . . 7  |-  ( (
ph  /\  x  e.  ( ( F  |`  ( A  \  { B } ) ) lim CC  B ) )  -> 
( ( A  \  { B } )  u. 
{ B } ) 
C_  CC )
2612, 25syl5eqssr 3393 . . . . . 6  |-  ( (
ph  /\  x  e.  ( ( F  |`  ( A  \  { B } ) ) lim CC  B ) )  -> 
( A  u.  { B } )  C_  CC )
2726unssad 3524 . . . . 5  |-  ( (
ph  /\  x  e.  ( ( F  |`  ( A  \  { B } ) ) lim CC  B ) )  ->  A  C_  CC )
2827, 23jca 519 . . . 4  |-  ( (
ph  /\  x  e.  ( ( F  |`  ( A  \  { B } ) ) lim CC  B ) )  -> 
( A  C_  CC  /\  B  e.  CC ) )
2928ex 424 . . 3  |-  ( ph  ->  ( x  e.  ( ( F  |`  ( A  \  { B }
) ) lim CC  B
)  ->  ( A  C_  CC  /\  B  e.  CC ) ) )
30 eqid 2436 . . . . . 6  |-  ( (
TopOpen ` fld )t  ( A  u.  { B } ) )  =  ( ( TopOpen ` fld )t  ( A  u.  { B } ) )
31 eqid 2436 . . . . . 6  |-  ( TopOpen ` fld )  =  ( TopOpen ` fld )
32 eqid 2436 . . . . . 6  |-  ( z  e.  ( A  u.  { B } )  |->  if ( z  =  B ,  x ,  ( F `  z ) ) )  =  ( z  e.  ( A  u.  { B }
)  |->  if ( z  =  B ,  x ,  ( F `  z ) ) )
331adantr 452 . . . . . 6  |-  ( (
ph  /\  ( A  C_  CC  /\  B  e.  CC ) )  ->  F : A --> CC )
34 simprl 733 . . . . . 6  |-  ( (
ph  /\  ( A  C_  CC  /\  B  e.  CC ) )  ->  A  C_  CC )
35 simprr 734 . . . . . 6  |-  ( (
ph  /\  ( A  C_  CC  /\  B  e.  CC ) )  ->  B  e.  CC )
3630, 31, 32, 33, 34, 35ellimc 19760 . . . . 5  |-  ( (
ph  /\  ( A  C_  CC  /\  B  e.  CC ) )  -> 
( x  e.  ( F lim CC  B )  <-> 
( z  e.  ( A  u.  { B } )  |->  if ( z  =  B ,  x ,  ( F `  z ) ) )  e.  ( ( ( ( TopOpen ` fld )t  ( A  u.  { B } ) )  CnP  ( TopOpen ` fld ) ) `  B
) ) )
3712eqcomi 2440 . . . . . . 7  |-  ( A  u.  { B }
)  =  ( ( A  \  { B } )  u.  { B } )
3837oveq2i 6092 . . . . . 6  |-  ( (
TopOpen ` fld )t  ( A  u.  { B } ) )  =  ( ( TopOpen ` fld )t  ( ( A 
\  { B }
)  u.  { B } ) )
39 eqid 2436 . . . . . . . 8  |-  if ( z  =  B ,  x ,  ( F `  z ) )  =  if ( z  =  B ,  x ,  ( F `  z
) )
4037, 39mpteq12i 4293 . . . . . . 7  |-  ( z  e.  ( A  u.  { B } )  |->  if ( z  =  B ,  x ,  ( F `  z ) ) )  =  ( z  e.  ( ( A  \  { B } )  u.  { B } )  |->  if ( z  =  B ,  x ,  ( F `  z ) ) )
41 elun 3488 . . . . . . . . 9  |-  ( z  e.  ( ( A 
\  { B }
)  u.  { B } )  <->  ( z  e.  ( A  \  { B } )  \/  z  e.  { B } ) )
42 elsn 3829 . . . . . . . . . . 11  |-  ( z  e.  { B }  <->  z  =  B )
4342orbi2i 506 . . . . . . . . . 10  |-  ( ( z  e.  ( A 
\  { B }
)  \/  z  e. 
{ B } )  <-> 
( z  e.  ( A  \  { B } )  \/  z  =  B ) )
44 pm5.61 694 . . . . . . . . . . . 12  |-  ( ( ( z  e.  ( A  \  { B } )  \/  z  =  B )  /\  -.  z  =  B )  <->  ( z  e.  ( A 
\  { B }
)  /\  -.  z  =  B ) )
45 fvres 5745 . . . . . . . . . . . . 13  |-  ( z  e.  ( A  \  { B } )  -> 
( ( F  |`  ( A  \  { B } ) ) `  z )  =  ( F `  z ) )
4645adantr 452 . . . . . . . . . . . 12  |-  ( ( z  e.  ( A 
\  { B }
)  /\  -.  z  =  B )  ->  (
( F  |`  ( A  \  { B }
) ) `  z
)  =  ( F `
 z ) )
4744, 46sylbi 188 . . . . . . . . . . 11  |-  ( ( ( z  e.  ( A  \  { B } )  \/  z  =  B )  /\  -.  z  =  B )  ->  ( ( F  |`  ( A  \  { B } ) ) `  z )  =  ( F `  z ) )
4847ifeq2da 3765 . . . . . . . . . 10  |-  ( ( z  e.  ( A 
\  { B }
)  \/  z  =  B )  ->  if ( z  =  B ,  x ,  ( ( F  |`  ( A  \  { B }
) ) `  z
) )  =  if ( z  =  B ,  x ,  ( F `  z ) ) )
4943, 48sylbi 188 . . . . . . . . 9  |-  ( ( z  e.  ( A 
\  { B }
)  \/  z  e. 
{ B } )  ->  if ( z  =  B ,  x ,  ( ( F  |`  ( A  \  { B } ) ) `  z ) )  =  if ( z  =  B ,  x ,  ( F `  z
) ) )
5041, 49sylbi 188 . . . . . . . 8  |-  ( z  e.  ( ( A 
\  { B }
)  u.  { B } )  ->  if ( z  =  B ,  x ,  ( ( F  |`  ( A  \  { B }
) ) `  z
) )  =  if ( z  =  B ,  x ,  ( F `  z ) ) )
5150mpteq2ia 4291 . . . . . . 7  |-  ( z  e.  ( ( A 
\  { B }
)  u.  { B } )  |->  if ( z  =  B ,  x ,  ( ( F  |`  ( A  \  { B } ) ) `
 z ) ) )  =  ( z  e.  ( ( A 
\  { B }
)  u.  { B } )  |->  if ( z  =  B ,  x ,  ( F `  z ) ) )
5240, 51eqtr4i 2459 . . . . . 6  |-  ( z  e.  ( A  u.  { B } )  |->  if ( z  =  B ,  x ,  ( F `  z ) ) )  =  ( z  e.  ( ( A  \  { B } )  u.  { B } )  |->  if ( z  =  B ,  x ,  ( ( F  |`  ( A  \  { B } ) ) `
 z ) ) )
5315adantr 452 . . . . . 6  |-  ( (
ph  /\  ( A  C_  CC  /\  B  e.  CC ) )  -> 
( F  |`  ( A  \  { B }
) ) : ( A  \  { B } ) --> CC )
5434ssdifssd 3485 . . . . . 6  |-  ( (
ph  /\  ( A  C_  CC  /\  B  e.  CC ) )  -> 
( A  \  { B } )  C_  CC )
5538, 31, 52, 53, 54, 35ellimc 19760 . . . . 5  |-  ( (
ph  /\  ( A  C_  CC  /\  B  e.  CC ) )  -> 
( x  e.  ( ( F  |`  ( A  \  { B }
) ) lim CC  B
)  <->  ( z  e.  ( A  u.  { B } )  |->  if ( z  =  B ,  x ,  ( F `  z ) ) )  e.  ( ( ( ( TopOpen ` fld )t  ( A  u.  { B } ) )  CnP  ( TopOpen ` fld ) ) `  B
) ) )
5636, 55bitr4d 248 . . . 4  |-  ( (
ph  /\  ( A  C_  CC  /\  B  e.  CC ) )  -> 
( x  e.  ( F lim CC  B )  <-> 
x  e.  ( ( F  |`  ( A  \  { B } ) ) lim CC  B ) ) )
5756ex 424 . . 3  |-  ( ph  ->  ( ( A  C_  CC  /\  B  e.  CC )  ->  ( x  e.  ( F lim CC  B
)  <->  x  e.  (
( F  |`  ( A  \  { B }
) ) lim CC  B
) ) ) )
5811, 29, 57pm5.21ndd 344 . 2  |-  ( ph  ->  ( x  e.  ( F lim CC  B )  <-> 
x  e.  ( ( F  |`  ( A  \  { B } ) ) lim CC  B ) ) )
5958eqrdv 2434 1  |-  ( ph  ->  ( F lim CC  B
)  =  ( ( F  |`  ( A  \  { B } ) ) lim CC  B ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 177    \/ wo 358    /\ wa 359    /\ w3a 936    = wceq 1652    e. wcel 1725    \ cdif 3317    u. cun 3318    C_ wss 3320   ifcif 3739   {csn 3814    e. cmpt 4266   dom cdm 4878    |` cres 4880   -->wf 5450   ` cfv 5454  (class class class)co 6081   CCcc 8988   ↾t crest 13648   TopOpenctopn 13649  ℂfldccnfld 16703    CnP ccnp 17289   lim CC climc 19749
This theorem is referenced by:  dvcnp2  19806  dvmulbr  19825  dvrec  19841
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-rep 4320  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701  ax-cnex 9046  ax-resscn 9047  ax-1cn 9048  ax-icn 9049  ax-addcl 9050  ax-addrcl 9051  ax-mulcl 9052  ax-mulrcl 9053  ax-mulcom 9054  ax-addass 9055  ax-mulass 9056  ax-distr 9057  ax-i2m1 9058  ax-1ne0 9059  ax-1rid 9060  ax-rnegex 9061  ax-rrecex 9062  ax-cnre 9063  ax-pre-lttri 9064  ax-pre-lttrn 9065  ax-pre-ltadd 9066  ax-pre-mulgt0 9067  ax-pre-sup 9068
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-nel 2602  df-ral 2710  df-rex 2711  df-reu 2712  df-rmo 2713  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-pss 3336  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-tp 3822  df-op 3823  df-uni 4016  df-int 4051  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-tr 4303  df-eprel 4494  df-id 4498  df-po 4503  df-so 4504  df-fr 4541  df-we 4543  df-ord 4584  df-on 4585  df-lim 4586  df-suc 4587  df-om 4846  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-1st 6349  df-2nd 6350  df-riota 6549  df-recs 6633  df-rdg 6668  df-1o 6724  df-oadd 6728  df-er 6905  df-map 7020  df-pm 7021  df-en 7110  df-dom 7111  df-sdom 7112  df-fin 7113  df-fi 7416  df-sup 7446  df-pnf 9122  df-mnf 9123  df-xr 9124  df-ltxr 9125  df-le 9126  df-sub 9293  df-neg 9294  df-div 9678  df-nn 10001  df-2 10058  df-3 10059  df-4 10060  df-5 10061  df-6 10062  df-7 10063  df-8 10064  df-9 10065  df-10 10066  df-n0 10222  df-z 10283  df-dec 10383  df-uz 10489  df-q 10575  df-rp 10613  df-xneg 10710  df-xadd 10711  df-xmul 10712  df-fz 11044  df-seq 11324  df-exp 11383  df-cj 11904  df-re 11905  df-im 11906  df-sqr 12040  df-abs 12041  df-struct 13471  df-ndx 13472  df-slot 13473  df-base 13474  df-plusg 13542  df-mulr 13543  df-starv 13544  df-tset 13548  df-ple 13549  df-ds 13551  df-unif 13552  df-rest 13650  df-topn 13651  df-topgen 13667  df-psmet 16694  df-xmet 16695  df-met 16696  df-bl 16697  df-mopn 16698  df-cnfld 16704  df-top 16963  df-bases 16965  df-topon 16966  df-topsp 16967  df-cnp 17292  df-xms 18350  df-ms 18351  df-limc 19753
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