MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  limcdif Unicode version

Theorem limcdif 19226
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 5393 . . . . . . . 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 19224 . . . . . . . 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 3213 . . . . 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 3338 . . . . . 6  |-  A  C_  ( A  u.  { B } )
13 undif1 3529 . . . . . . 7  |-  ( ( A  \  { B } )  u.  { B } )  =  ( A  u.  { B } )
14 difss 3303 . . . . . . . . . . . 12  |-  ( A 
\  { B }
)  C_  A
15 fssres 5408 . . . . . . . . . . . 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 5393 . . . . . . . . . . 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 19224 . . . . . . . . . . 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 3213 . . . . . . . 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 3760 . . . . . . . 8  |-  ( (
ph  /\  x  e.  ( ( F  |`  ( A  \  { B } ) ) lim CC  B ) )  ->  { B }  C_  CC )
2623, 25unssd 3351 . . . . . . 7  |-  ( (
ph  /\  x  e.  ( ( F  |`  ( A  \  { B } ) ) lim CC  B ) )  -> 
( ( A  \  { B } )  u. 
{ B } ) 
C_  CC )
2713, 26syl5eqssr 3223 . . . . . 6  |-  ( (
ph  /\  x  e.  ( ( F  |`  ( A  \  { B } ) ) lim CC  B ) )  -> 
( A  u.  { B } )  C_  CC )
2812, 27syl5ss 3190 . . . . 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 2283 . . . . . 6  |-  ( (
TopOpen ` fld )t  ( A  u.  { B } ) )  =  ( ( TopOpen ` fld )t  ( A  u.  { B } ) )
32 eqid 2283 . . . . . 6  |-  ( TopOpen ` fld )  =  ( TopOpen ` fld )
33 eqid 2283 . . . . . 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 19223 . . . . 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 2287 . . . . . . 7  |-  ( A  u.  { B }
)  =  ( ( A  \  { B } )  u.  { B } )
3938oveq2i 5869 . . . . . 6  |-  ( (
TopOpen ` fld )t  ( A  u.  { B } ) )  =  ( ( TopOpen ` fld )t  ( ( A 
\  { B }
)  u.  { B } ) )
40 eqid 2283 . . . . . . . 8  |-  if ( z  =  B ,  x ,  ( F `  z ) )  =  if ( z  =  B ,  x ,  ( F `  z
) )
4138, 40mpteq12i 4104 . . . . . . 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 3316 . . . . . . . . 9  |-  ( z  e.  ( ( A 
\  { B }
)  u.  { B } )  <->  ( z  e.  ( A  \  { B } )  \/  z  e.  { B } ) )
43 elsn 3655 . . . . . . . . . . 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 5542 . . . . . . . . . . . . 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 3591 . . . . . . . . . 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 4102 . . . . . . 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 2306 . . . . . 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 3190 . . . . . 6  |-  ( (
ph  /\  ( A  C_  CC  /\  B  e.  CC ) )  -> 
( A  \  { B } )  C_  CC )
5639, 32, 53, 54, 55, 36ellimc 19223 . . . . 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 2281 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 1623    e. wcel 1684    \ cdif 3149    u. cun 3150    C_ wss 3152   ifcif 3565   {csn 3640    e. cmpt 4077   dom cdm 4689    |` cres 4691   -->wf 5251   ` cfv 5255  (class class class)co 5858   CCcc 8735   ↾t crest 13325   TopOpenctopn 13326  ℂfldccnfld 16377    CnP ccnp 16955   lim CC climc 19212
This theorem is referenced by:  dvcnp2  19269  dvmulbr  19288  dvrec  19304
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814  ax-pre-sup 8815
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 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-1o 6479  df-oadd 6483  df-er 6660  df-map 6774  df-pm 6775  df-en 6864  df-dom 6865  df-sdom 6866  df-fin 6867  df-fi 7165  df-sup 7194  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-div 9424  df-nn 9747  df-2 9804  df-3 9805  df-4 9806  df-5 9807  df-6 9808  df-7 9809  df-8 9810  df-9 9811  df-10 9812  df-n0 9966  df-z 10025  df-dec 10125  df-uz 10231  df-q 10317  df-rp 10355  df-xneg 10452  df-xadd 10453  df-xmul 10454  df-fz 10783  df-seq 11047  df-exp 11105  df-cj 11584  df-re 11585  df-im 11586  df-sqr 11720  df-abs 11721  df-struct 13150  df-ndx 13151  df-slot 13152  df-base 13153  df-plusg 13221  df-mulr 13222  df-starv 13223  df-tset 13227  df-ple 13228  df-ds 13230  df-rest 13327  df-topn 13328  df-topgen 13344  df-xmet 16373  df-met 16374  df-bl 16375  df-mopn 16376  df-cnfld 16378  df-top 16636  df-bases 16638  df-topon 16639  df-topsp 16640  df-cnp 16958  df-xms 17885  df-ms 17886  df-limc 19216
  Copyright terms: Public domain W3C validator