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Theorem limcdif 19242
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 5409 . . . . . . . 8  |-  ( F : A --> CC  ->  dom 
F  =  A )
31, 2syl 15 . . . . . . 7  |-  ( ph  ->  dom  F  =  A )
43adantr 451 . . . . . 6  |-  ( (
ph  /\  x  e.  ( F lim CC  B ) )  ->  dom  F  =  A )
5 limcrcl 19240 . . . . . . . 8  |-  ( x  e.  ( F lim CC  B )  ->  ( F : dom  F --> CC  /\  dom  F  C_  CC  /\  B  e.  CC ) )
65adantl 452 . . . . . . 7  |-  ( (
ph  /\  x  e.  ( F lim CC  B ) )  ->  ( F : dom  F --> CC  /\  dom  F  C_  CC  /\  B  e.  CC ) )
76simp2d 968 . . . . . 6  |-  ( (
ph  /\  x  e.  ( F lim CC  B ) )  ->  dom  F  C_  CC )
84, 7eqsstr3d 3226 . . . . 5  |-  ( (
ph  /\  x  e.  ( F lim CC  B ) )  ->  A  C_  CC )
96simp3d 969 . . . . 5  |-  ( (
ph  /\  x  e.  ( F lim CC  B ) )  ->  B  e.  CC )
108, 9jca 518 . . . 4  |-  ( (
ph  /\  x  e.  ( F lim CC  B ) )  ->  ( A  C_  CC  /\  B  e.  CC ) )
1110ex 423 . . 3  |-  ( ph  ->  ( x  e.  ( F lim CC  B )  ->  ( A  C_  CC  /\  B  e.  CC ) ) )
12 ssun1 3351 . . . . . 6  |-  A  C_  ( A  u.  { B } )
13 undif1 3542 . . . . . . 7  |-  ( ( A  \  { B } )  u.  { B } )  =  ( A  u.  { B } )
14 difss 3316 . . . . . . . . . . . 12  |-  ( A 
\  { B }
)  C_  A
15 fssres 5424 . . . . . . . . . . . 12  |-  ( ( F : A --> CC  /\  ( A  \  { B } )  C_  A
)  ->  ( F  |`  ( A  \  { B } ) ) : ( A  \  { B } ) --> CC )
161, 14, 15sylancl 643 . . . . . . . . . . 11  |-  ( ph  ->  ( F  |`  ( A  \  { B }
) ) : ( A  \  { B } ) --> CC )
17 fdm 5409 . . . . . . . . . . 11  |-  ( ( F  |`  ( A  \  { B } ) ) : ( A 
\  { B }
) --> CC  ->  dom  ( F  |`  ( A 
\  { B }
) )  =  ( A  \  { B } ) )
1816, 17syl 15 . . . . . . . . . 10  |-  ( ph  ->  dom  ( F  |`  ( A  \  { B } ) )  =  ( A  \  { B } ) )
1918adantr 451 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  ( ( F  |`  ( A  \  { B } ) ) lim CC  B ) )  ->  dom  ( F  |`  ( A  \  { B }
) )  =  ( A  \  { B } ) )
20 limcrcl 19240 . . . . . . . . . . 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 ) )
2120adantl 452 . . . . . . . . . 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 ) )
2221simp2d 968 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  ( ( F  |`  ( A  \  { B } ) ) lim CC  B ) )  ->  dom  ( F  |`  ( A  \  { B }
) )  C_  CC )
2319, 22eqsstr3d 3226 . . . . . . . 8  |-  ( (
ph  /\  x  e.  ( ( F  |`  ( A  \  { B } ) ) lim CC  B ) )  -> 
( A  \  { B } )  C_  CC )
2421simp3d 969 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  ( ( F  |`  ( A  \  { B } ) ) lim CC  B ) )  ->  B  e.  CC )
2524snssd 3776 . . . . . . . 8  |-  ( (
ph  /\  x  e.  ( ( F  |`  ( A  \  { B } ) ) lim CC  B ) )  ->  { B }  C_  CC )
2623, 25unssd 3364 . . . . . . 7  |-  ( (
ph  /\  x  e.  ( ( F  |`  ( A  \  { B } ) ) lim CC  B ) )  -> 
( ( A  \  { B } )  u. 
{ B } ) 
C_  CC )
2713, 26syl5eqssr 3236 . . . . . 6  |-  ( (
ph  /\  x  e.  ( ( F  |`  ( A  \  { B } ) ) lim CC  B ) )  -> 
( A  u.  { B } )  C_  CC )
2812, 27syl5ss 3203 . . . . 5  |-  ( (
ph  /\  x  e.  ( ( F  |`  ( A  \  { B } ) ) lim CC  B ) )  ->  A  C_  CC )
2928, 24jca 518 . . . 4  |-  ( (
ph  /\  x  e.  ( ( F  |`  ( A  \  { B } ) ) lim CC  B ) )  -> 
( A  C_  CC  /\  B  e.  CC ) )
3029ex 423 . . 3  |-  ( ph  ->  ( x  e.  ( ( F  |`  ( A  \  { B }
) ) lim CC  B
)  ->  ( A  C_  CC  /\  B  e.  CC ) ) )
31 eqid 2296 . . . . . 6  |-  ( (
TopOpen ` fld )t  ( A  u.  { B } ) )  =  ( ( TopOpen ` fld )t  ( A  u.  { B } ) )
32 eqid 2296 . . . . . 6  |-  ( TopOpen ` fld )  =  ( TopOpen ` fld )
33 eqid 2296 . . . . . 6  |-  ( z  e.  ( A  u.  { B } )  |->  if ( z  =  B ,  x ,  ( F `  z ) ) )  =  ( z  e.  ( A  u.  { B }
)  |->  if ( z  =  B ,  x ,  ( F `  z ) ) )
341adantr 451 . . . . . 6  |-  ( (
ph  /\  ( A  C_  CC  /\  B  e.  CC ) )  ->  F : A --> CC )
35 simprl 732 . . . . . 6  |-  ( (
ph  /\  ( A  C_  CC  /\  B  e.  CC ) )  ->  A  C_  CC )
36 simprr 733 . . . . . 6  |-  ( (
ph  /\  ( A  C_  CC  /\  B  e.  CC ) )  ->  B  e.  CC )
3731, 32, 33, 34, 35, 36ellimc 19239 . . . . 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
) ) )
3813eqcomi 2300 . . . . . . 7  |-  ( A  u.  { B }
)  =  ( ( A  \  { B } )  u.  { B } )
3938oveq2i 5885 . . . . . 6  |-  ( (
TopOpen ` fld )t  ( A  u.  { B } ) )  =  ( ( TopOpen ` fld )t  ( ( A 
\  { B }
)  u.  { B } ) )
40 eqid 2296 . . . . . . . 8  |-  if ( z  =  B ,  x ,  ( F `  z ) )  =  if ( z  =  B ,  x ,  ( F `  z
) )
4138, 40mpteq12i 4120 . . . . . . 7  |-  ( z  e.  ( A  u.  { B } )  |->  if ( z  =  B ,  x ,  ( F `  z ) ) )  =  ( z  e.  ( ( A  \  { B } )  u.  { B } )  |->  if ( z  =  B ,  x ,  ( F `  z ) ) )
42 elun 3329 . . . . . . . . 9  |-  ( z  e.  ( ( A 
\  { B }
)  u.  { B } )  <->  ( z  e.  ( A  \  { B } )  \/  z  e.  { B } ) )
43 elsn 3668 . . . . . . . . . . 11  |-  ( z  e.  { B }  <->  z  =  B )
4443orbi2i 505 . . . . . . . . . 10  |-  ( ( z  e.  ( A 
\  { B }
)  \/  z  e. 
{ B } )  <-> 
( z  e.  ( A  \  { B } )  \/  z  =  B ) )
45 pm5.61 693 . . . . . . . . . . . 12  |-  ( ( ( z  e.  ( A  \  { B } )  \/  z  =  B )  /\  -.  z  =  B )  <->  ( z  e.  ( A 
\  { B }
)  /\  -.  z  =  B ) )
46 fvres 5558 . . . . . . . . . . . . 13  |-  ( z  e.  ( A  \  { B } )  -> 
( ( F  |`  ( A  \  { B } ) ) `  z )  =  ( F `  z ) )
4746adantr 451 . . . . . . . . . . . 12  |-  ( ( z  e.  ( A 
\  { B }
)  /\  -.  z  =  B )  ->  (
( F  |`  ( A  \  { B }
) ) `  z
)  =  ( F `
 z ) )
4845, 47sylbi 187 . . . . . . . . . . 11  |-  ( ( ( z  e.  ( A  \  { B } )  \/  z  =  B )  /\  -.  z  =  B )  ->  ( ( F  |`  ( A  \  { B } ) ) `  z )  =  ( F `  z ) )
4948ifeq2da 3604 . . . . . . . . . 10  |-  ( ( z  e.  ( A 
\  { B }
)  \/  z  =  B )  ->  if ( z  =  B ,  x ,  ( ( F  |`  ( A  \  { B }
) ) `  z
) )  =  if ( z  =  B ,  x ,  ( F `  z ) ) )
5044, 49sylbi 187 . . . . . . . . 9  |-  ( ( z  e.  ( A 
\  { B }
)  \/  z  e. 
{ B } )  ->  if ( z  =  B ,  x ,  ( ( F  |`  ( A  \  { B } ) ) `  z ) )  =  if ( z  =  B ,  x ,  ( F `  z
) ) )
5142, 50sylbi 187 . . . . . . . 8  |-  ( z  e.  ( ( A 
\  { B }
)  u.  { B } )  ->  if ( z  =  B ,  x ,  ( ( F  |`  ( A  \  { B }
) ) `  z
) )  =  if ( z  =  B ,  x ,  ( F `  z ) ) )
5251mpteq2ia 4118 . . . . . . 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 ) ) )
5341, 52eqtr4i 2319 . . . . . 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 ) ) )
5416adantr 451 . . . . . 6  |-  ( (
ph  /\  ( A  C_  CC  /\  B  e.  CC ) )  -> 
( F  |`  ( A  \  { B }
) ) : ( A  \  { B } ) --> CC )
5514, 35syl5ss 3203 . . . . . 6  |-  ( (
ph  /\  ( A  C_  CC  /\  B  e.  CC ) )  -> 
( A  \  { B } )  C_  CC )
5639, 32, 53, 54, 55, 36ellimc 19239 . . . . 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
) ) )
5737, 56bitr4d 247 . . . 4  |-  ( (
ph  /\  ( A  C_  CC  /\  B  e.  CC ) )  -> 
( x  e.  ( F lim CC  B )  <-> 
x  e.  ( ( F  |`  ( A  \  { B } ) ) lim CC  B ) ) )
5857ex 423 . . 3  |-  ( ph  ->  ( ( A  C_  CC  /\  B  e.  CC )  ->  ( x  e.  ( F lim CC  B
)  <->  x  e.  (
( F  |`  ( A  \  { B }
) ) lim CC  B
) ) ) )
5911, 30, 58pm5.21ndd 343 . 2  |-  ( ph  ->  ( x  e.  ( F lim CC  B )  <-> 
x  e.  ( ( F  |`  ( A  \  { B } ) ) lim CC  B ) ) )
6059eqrdv 2294 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 176    \/ wo 357    /\ wa 358    /\ w3a 934    = wceq 1632    e. wcel 1696    \ cdif 3162    u. cun 3163    C_ wss 3165   ifcif 3578   {csn 3653    e. cmpt 4093   dom cdm 4705    |` cres 4707   -->wf 5267   ` cfv 5271  (class class class)co 5874   CCcc 8751   ↾t crest 13341   TopOpenctopn 13342  ℂfldccnfld 16393    CnP ccnp 16971   lim CC climc 19228
This theorem is referenced by:  dvcnp2  19285  dvmulbr  19304  dvrec  19320
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830  ax-pre-sup 8831
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-int 3879  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-riota 6320  df-recs 6404  df-rdg 6439  df-1o 6495  df-oadd 6499  df-er 6676  df-map 6790  df-pm 6791  df-en 6880  df-dom 6881  df-sdom 6882  df-fin 6883  df-fi 7181  df-sup 7210  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-div 9440  df-nn 9763  df-2 9820  df-3 9821  df-4 9822  df-5 9823  df-6 9824  df-7 9825  df-8 9826  df-9 9827  df-10 9828  df-n0 9982  df-z 10041  df-dec 10141  df-uz 10247  df-q 10333  df-rp 10371  df-xneg 10468  df-xadd 10469  df-xmul 10470  df-fz 10799  df-seq 11063  df-exp 11121  df-cj 11600  df-re 11601  df-im 11602  df-sqr 11736  df-abs 11737  df-struct 13166  df-ndx 13167  df-slot 13168  df-base 13169  df-plusg 13237  df-mulr 13238  df-starv 13239  df-tset 13243  df-ple 13244  df-ds 13246  df-rest 13343  df-topn 13344  df-topgen 13360  df-xmet 16389  df-met 16390  df-bl 16391  df-mopn 16392  df-cnfld 16394  df-top 16652  df-bases 16654  df-topon 16655  df-topsp 16656  df-cnp 16974  df-xms 17901  df-ms 17902  df-limc 19232
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