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Theorem limcdif 22015
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 5733 . . . . . . . 8  |-  ( F : A --> CC  ->  dom 
F  =  A )
31, 2syl 16 . . . . . . 7  |-  ( ph  ->  dom  F  =  A )
43adantr 465 . . . . . 6  |-  ( (
ph  /\  x  e.  ( F lim CC  B ) )  ->  dom  F  =  A )
5 limcrcl 22013 . . . . . . . 8  |-  ( x  e.  ( F lim CC  B )  ->  ( F : dom  F --> CC  /\  dom  F  C_  CC  /\  B  e.  CC ) )
65adantl 466 . . . . . . 7  |-  ( (
ph  /\  x  e.  ( F lim CC  B ) )  ->  ( F : dom  F --> CC  /\  dom  F  C_  CC  /\  B  e.  CC ) )
76simp2d 1009 . . . . . 6  |-  ( (
ph  /\  x  e.  ( F lim CC  B ) )  ->  dom  F  C_  CC )
84, 7eqsstr3d 3539 . . . . 5  |-  ( (
ph  /\  x  e.  ( F lim CC  B ) )  ->  A  C_  CC )
96simp3d 1010 . . . . 5  |-  ( (
ph  /\  x  e.  ( F lim CC  B ) )  ->  B  e.  CC )
108, 9jca 532 . . . 4  |-  ( (
ph  /\  x  e.  ( F lim CC  B ) )  ->  ( A  C_  CC  /\  B  e.  CC ) )
1110ex 434 . . 3  |-  ( ph  ->  ( x  e.  ( F lim CC  B )  ->  ( A  C_  CC  /\  B  e.  CC ) ) )
12 undif1 3902 . . . . . . 7  |-  ( ( A  \  { B } )  u.  { B } )  =  ( A  u.  { B } )
13 difss 3631 . . . . . . . . . . . 12  |-  ( A 
\  { B }
)  C_  A
14 fssres 5749 . . . . . . . . . . . 12  |-  ( ( F : A --> CC  /\  ( A  \  { B } )  C_  A
)  ->  ( F  |`  ( A  \  { B } ) ) : ( A  \  { B } ) --> CC )
151, 13, 14sylancl 662 . . . . . . . . . . 11  |-  ( ph  ->  ( F  |`  ( A  \  { B }
) ) : ( A  \  { B } ) --> CC )
16 fdm 5733 . . . . . . . . . . 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 465 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  ( ( F  |`  ( A  \  { B } ) ) lim CC  B ) )  ->  dom  ( F  |`  ( A  \  { B }
) )  =  ( A  \  { B } ) )
19 limcrcl 22013 . . . . . . . . . . 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 466 . . . . . . . . . 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 1009 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  ( ( F  |`  ( A  \  { B } ) ) lim CC  B ) )  ->  dom  ( F  |`  ( A  \  { B }
) )  C_  CC )
2218, 21eqsstr3d 3539 . . . . . . . 8  |-  ( (
ph  /\  x  e.  ( ( F  |`  ( A  \  { B } ) ) lim CC  B ) )  -> 
( A  \  { B } )  C_  CC )
2320simp3d 1010 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  ( ( F  |`  ( A  \  { B } ) ) lim CC  B ) )  ->  B  e.  CC )
2423snssd 4172 . . . . . . . 8  |-  ( (
ph  /\  x  e.  ( ( F  |`  ( A  \  { B } ) ) lim CC  B ) )  ->  { B }  C_  CC )
2522, 24unssd 3680 . . . . . . 7  |-  ( (
ph  /\  x  e.  ( ( F  |`  ( A  \  { B } ) ) lim CC  B ) )  -> 
( ( A  \  { B } )  u. 
{ B } ) 
C_  CC )
2612, 25syl5eqssr 3549 . . . . . 6  |-  ( (
ph  /\  x  e.  ( ( F  |`  ( A  \  { B } ) ) lim CC  B ) )  -> 
( A  u.  { B } )  C_  CC )
2726unssad 3681 . . . . 5  |-  ( (
ph  /\  x  e.  ( ( F  |`  ( A  \  { B } ) ) lim CC  B ) )  ->  A  C_  CC )
2827, 23jca 532 . . . 4  |-  ( (
ph  /\  x  e.  ( ( F  |`  ( A  \  { B } ) ) lim CC  B ) )  -> 
( A  C_  CC  /\  B  e.  CC ) )
2928ex 434 . . 3  |-  ( ph  ->  ( x  e.  ( ( F  |`  ( A  \  { B }
) ) lim CC  B
)  ->  ( A  C_  CC  /\  B  e.  CC ) ) )
30 eqid 2467 . . . . . 6  |-  ( (
TopOpen ` fld )t  ( A  u.  { B } ) )  =  ( ( TopOpen ` fld )t  ( A  u.  { B } ) )
31 eqid 2467 . . . . . 6  |-  ( TopOpen ` fld )  =  ( TopOpen ` fld )
32 eqid 2467 . . . . . 6  |-  ( z  e.  ( A  u.  { B } )  |->  if ( z  =  B ,  x ,  ( F `  z ) ) )  =  ( z  e.  ( A  u.  { B }
)  |->  if ( z  =  B ,  x ,  ( F `  z ) ) )
331adantr 465 . . . . . 6  |-  ( (
ph  /\  ( A  C_  CC  /\  B  e.  CC ) )  ->  F : A --> CC )
34 simprl 755 . . . . . 6  |-  ( (
ph  /\  ( A  C_  CC  /\  B  e.  CC ) )  ->  A  C_  CC )
35 simprr 756 . . . . . 6  |-  ( (
ph  /\  ( A  C_  CC  /\  B  e.  CC ) )  ->  B  e.  CC )
3630, 31, 32, 33, 34, 35ellimc 22012 . . . . 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 2480 . . . . . . 7  |-  ( A  u.  { B }
)  =  ( ( A  \  { B } )  u.  { B } )
3837oveq2i 6293 . . . . . 6  |-  ( (
TopOpen ` fld )t  ( A  u.  { B } ) )  =  ( ( TopOpen ` fld )t  ( ( A 
\  { B }
)  u.  { B } ) )
39 eqid 2467 . . . . . . . 8  |-  if ( z  =  B ,  x ,  ( F `  z ) )  =  if ( z  =  B ,  x ,  ( F `  z
) )
4037, 39mpteq12i 4531 . . . . . . 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 3645 . . . . . . . . 9  |-  ( z  e.  ( ( A 
\  { B }
)  u.  { B } )  <->  ( z  e.  ( A  \  { B } )  \/  z  e.  { B } ) )
42 elsn 4041 . . . . . . . . . . 11  |-  ( z  e.  { B }  <->  z  =  B )
4342orbi2i 519 . . . . . . . . . 10  |-  ( ( z  e.  ( A 
\  { B }
)  \/  z  e. 
{ B } )  <-> 
( z  e.  ( A  \  { B } )  \/  z  =  B ) )
44 pm5.61 712 . . . . . . . . . . . 12  |-  ( ( ( z  e.  ( A  \  { B } )  \/  z  =  B )  /\  -.  z  =  B )  <->  ( z  e.  ( A 
\  { B }
)  /\  -.  z  =  B ) )
45 fvres 5878 . . . . . . . . . . . . 13  |-  ( z  e.  ( A  \  { B } )  -> 
( ( F  |`  ( A  \  { B } ) ) `  z )  =  ( F `  z ) )
4645adantr 465 . . . . . . . . . . . 12  |-  ( ( z  e.  ( A 
\  { B }
)  /\  -.  z  =  B )  ->  (
( F  |`  ( A  \  { B }
) ) `  z
)  =  ( F `
 z ) )
4744, 46sylbi 195 . . . . . . . . . . 11  |-  ( ( ( z  e.  ( A  \  { B } )  \/  z  =  B )  /\  -.  z  =  B )  ->  ( ( F  |`  ( A  \  { B } ) ) `  z )  =  ( F `  z ) )
4847ifeq2da 3970 . . . . . . . . . 10  |-  ( ( z  e.  ( A 
\  { B }
)  \/  z  =  B )  ->  if ( z  =  B ,  x ,  ( ( F  |`  ( A  \  { B }
) ) `  z
) )  =  if ( z  =  B ,  x ,  ( F `  z ) ) )
4943, 48sylbi 195 . . . . . . . . 9  |-  ( ( z  e.  ( A 
\  { B }
)  \/  z  e. 
{ B } )  ->  if ( z  =  B ,  x ,  ( ( F  |`  ( A  \  { B } ) ) `  z ) )  =  if ( z  =  B ,  x ,  ( F `  z
) ) )
5041, 49sylbi 195 . . . . . . . 8  |-  ( z  e.  ( ( A 
\  { B }
)  u.  { B } )  ->  if ( z  =  B ,  x ,  ( ( F  |`  ( A  \  { B }
) ) `  z
) )  =  if ( z  =  B ,  x ,  ( F `  z ) ) )
5150mpteq2ia 4529 . . . . . . 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 2499 . . . . . 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 465 . . . . . 6  |-  ( (
ph  /\  ( A  C_  CC  /\  B  e.  CC ) )  -> 
( F  |`  ( A  \  { B }
) ) : ( A  \  { B } ) --> CC )
5434ssdifssd 3642 . . . . . 6  |-  ( (
ph  /\  ( A  C_  CC  /\  B  e.  CC ) )  -> 
( A  \  { B } )  C_  CC )
5538, 31, 52, 53, 54, 35ellimc 22012 . . . . 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 256 . . . 4  |-  ( (
ph  /\  ( A  C_  CC  /\  B  e.  CC ) )  -> 
( x  e.  ( F lim CC  B )  <-> 
x  e.  ( ( F  |`  ( A  \  { B } ) ) lim CC  B ) ) )
5756ex 434 . . 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 354 . 2  |-  ( ph  ->  ( x  e.  ( F lim CC  B )  <-> 
x  e.  ( ( F  |`  ( A  \  { B } ) ) lim CC  B ) ) )
5958eqrdv 2464 1  |-  ( ph  ->  ( F lim CC  B
)  =  ( ( F  |`  ( A  \  { B } ) ) lim CC  B ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    \/ wo 368    /\ wa 369    /\ w3a 973    = wceq 1379    e. wcel 1767    \ cdif 3473    u. cun 3474    C_ wss 3476   ifcif 3939   {csn 4027    |-> cmpt 4505   dom cdm 4999    |` cres 5001   -->wf 5582   ` cfv 5586  (class class class)co 6282   CCcc 9486   ↾t crest 14672   TopOpenctopn 14673  ℂfldccnfld 18191    CnP ccnp 19492   lim CC climc 22001
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6574  ax-cnex 9544  ax-resscn 9545  ax-1cn 9546  ax-icn 9547  ax-addcl 9548  ax-addrcl 9549  ax-mulcl 9550  ax-mulrcl 9551  ax-mulcom 9552  ax-addass 9553  ax-mulass 9554  ax-distr 9555  ax-i2m1 9556  ax-1ne0 9557  ax-1rid 9558  ax-rnegex 9559  ax-rrecex 9560  ax-cnre 9561  ax-pre-lttri 9562  ax-pre-lttrn 9563  ax-pre-ltadd 9564  ax-pre-mulgt0 9565  ax-pre-sup 9566
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-int 4283  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fun 5588  df-fn 5589  df-f 5590  df-f1 5591  df-fo 5592  df-f1o 5593  df-fv 5594  df-riota 6243  df-ov 6285  df-oprab 6286  df-mpt2 6287  df-om 6679  df-1st 6781  df-2nd 6782  df-recs 7039  df-rdg 7073  df-1o 7127  df-oadd 7131  df-er 7308  df-map 7419  df-pm 7420  df-en 7514  df-dom 7515  df-sdom 7516  df-fin 7517  df-fi 7867  df-sup 7897  df-pnf 9626  df-mnf 9627  df-xr 9628  df-ltxr 9629  df-le 9630  df-sub 9803  df-neg 9804  df-div 10203  df-nn 10533  df-2 10590  df-3 10591  df-4 10592  df-5 10593  df-6 10594  df-7 10595  df-8 10596  df-9 10597  df-10 10598  df-n0 10792  df-z 10861  df-dec 10973  df-uz 11079  df-q 11179  df-rp 11217  df-xneg 11314  df-xadd 11315  df-xmul 11316  df-fz 11669  df-seq 12072  df-exp 12131  df-cj 12891  df-re 12892  df-im 12893  df-sqrt 13027  df-abs 13028  df-struct 14488  df-ndx 14489  df-slot 14490  df-base 14491  df-plusg 14564  df-mulr 14565  df-starv 14566  df-tset 14570  df-ple 14571  df-ds 14573  df-unif 14574  df-rest 14674  df-topn 14675  df-topgen 14695  df-psmet 18182  df-xmet 18183  df-met 18184  df-bl 18185  df-mopn 18186  df-cnfld 18192  df-top 19166  df-bases 19168  df-topon 19169  df-topsp 19170  df-cnp 19495  df-xms 20558  df-ms 20559  df-limc 22005
This theorem is referenced by:  dvcnp2  22058  dvmulbr  22077  dvrec  22093  fourierdlem62  31469
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