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Theorem limcdif 22449
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 5717 . . . . . . . 8  |-  ( F : A --> CC  ->  dom 
F  =  A )
31, 2syl 16 . . . . . . 7  |-  ( ph  ->  dom  F  =  A )
43adantr 463 . . . . . 6  |-  ( (
ph  /\  x  e.  ( F lim CC  B ) )  ->  dom  F  =  A )
5 limcrcl 22447 . . . . . . . 8  |-  ( x  e.  ( F lim CC  B )  ->  ( F : dom  F --> CC  /\  dom  F  C_  CC  /\  B  e.  CC ) )
65adantl 464 . . . . . . 7  |-  ( (
ph  /\  x  e.  ( F lim CC  B ) )  ->  ( F : dom  F --> CC  /\  dom  F  C_  CC  /\  B  e.  CC ) )
76simp2d 1007 . . . . . 6  |-  ( (
ph  /\  x  e.  ( F lim CC  B ) )  ->  dom  F  C_  CC )
84, 7eqsstr3d 3524 . . . . 5  |-  ( (
ph  /\  x  e.  ( F lim CC  B ) )  ->  A  C_  CC )
96simp3d 1008 . . . . 5  |-  ( (
ph  /\  x  e.  ( F lim CC  B ) )  ->  B  e.  CC )
108, 9jca 530 . . . 4  |-  ( (
ph  /\  x  e.  ( F lim CC  B ) )  ->  ( A  C_  CC  /\  B  e.  CC ) )
1110ex 432 . . 3  |-  ( ph  ->  ( x  e.  ( F lim CC  B )  ->  ( A  C_  CC  /\  B  e.  CC ) ) )
12 undif1 3891 . . . . . . 7  |-  ( ( A  \  { B } )  u.  { B } )  =  ( A  u.  { B } )
13 difss 3617 . . . . . . . . . . . 12  |-  ( A 
\  { B }
)  C_  A
14 fssres 5733 . . . . . . . . . . . 12  |-  ( ( F : A --> CC  /\  ( A  \  { B } )  C_  A
)  ->  ( F  |`  ( A  \  { B } ) ) : ( A  \  { B } ) --> CC )
151, 13, 14sylancl 660 . . . . . . . . . . 11  |-  ( ph  ->  ( F  |`  ( A  \  { B }
) ) : ( A  \  { B } ) --> CC )
16 fdm 5717 . . . . . . . . . . 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 463 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  ( ( F  |`  ( A  \  { B } ) ) lim CC  B ) )  ->  dom  ( F  |`  ( A  \  { B }
) )  =  ( A  \  { B } ) )
19 limcrcl 22447 . . . . . . . . . . 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 464 . . . . . . . . . 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 1007 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  ( ( F  |`  ( A  \  { B } ) ) lim CC  B ) )  ->  dom  ( F  |`  ( A  \  { B }
) )  C_  CC )
2218, 21eqsstr3d 3524 . . . . . . . 8  |-  ( (
ph  /\  x  e.  ( ( F  |`  ( A  \  { B } ) ) lim CC  B ) )  -> 
( A  \  { B } )  C_  CC )
2320simp3d 1008 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  ( ( F  |`  ( A  \  { B } ) ) lim CC  B ) )  ->  B  e.  CC )
2423snssd 4161 . . . . . . . 8  |-  ( (
ph  /\  x  e.  ( ( F  |`  ( A  \  { B } ) ) lim CC  B ) )  ->  { B }  C_  CC )
2522, 24unssd 3666 . . . . . . 7  |-  ( (
ph  /\  x  e.  ( ( F  |`  ( A  \  { B } ) ) lim CC  B ) )  -> 
( ( A  \  { B } )  u. 
{ B } ) 
C_  CC )
2612, 25syl5eqssr 3534 . . . . . 6  |-  ( (
ph  /\  x  e.  ( ( F  |`  ( A  \  { B } ) ) lim CC  B ) )  -> 
( A  u.  { B } )  C_  CC )
2726unssad 3667 . . . . 5  |-  ( (
ph  /\  x  e.  ( ( F  |`  ( A  \  { B } ) ) lim CC  B ) )  ->  A  C_  CC )
2827, 23jca 530 . . . 4  |-  ( (
ph  /\  x  e.  ( ( F  |`  ( A  \  { B } ) ) lim CC  B ) )  -> 
( A  C_  CC  /\  B  e.  CC ) )
2928ex 432 . . 3  |-  ( ph  ->  ( x  e.  ( ( F  |`  ( A  \  { B }
) ) lim CC  B
)  ->  ( A  C_  CC  /\  B  e.  CC ) ) )
30 eqid 2454 . . . . . 6  |-  ( (
TopOpen ` fld )t  ( A  u.  { B } ) )  =  ( ( TopOpen ` fld )t  ( A  u.  { B } ) )
31 eqid 2454 . . . . . 6  |-  ( TopOpen ` fld )  =  ( TopOpen ` fld )
32 eqid 2454 . . . . . 6  |-  ( z  e.  ( A  u.  { B } )  |->  if ( z  =  B ,  x ,  ( F `  z ) ) )  =  ( z  e.  ( A  u.  { B }
)  |->  if ( z  =  B ,  x ,  ( F `  z ) ) )
331adantr 463 . . . . . 6  |-  ( (
ph  /\  ( A  C_  CC  /\  B  e.  CC ) )  ->  F : A --> CC )
34 simprl 754 . . . . . 6  |-  ( (
ph  /\  ( A  C_  CC  /\  B  e.  CC ) )  ->  A  C_  CC )
35 simprr 755 . . . . . 6  |-  ( (
ph  /\  ( A  C_  CC  /\  B  e.  CC ) )  ->  B  e.  CC )
3630, 31, 32, 33, 34, 35ellimc 22446 . . . . 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 2467 . . . . . . 7  |-  ( A  u.  { B }
)  =  ( ( A  \  { B } )  u.  { B } )
3837oveq2i 6281 . . . . . 6  |-  ( (
TopOpen ` fld )t  ( A  u.  { B } ) )  =  ( ( TopOpen ` fld )t  ( ( A 
\  { B }
)  u.  { B } ) )
39 eqid 2454 . . . . . . . 8  |-  if ( z  =  B ,  x ,  ( F `  z ) )  =  if ( z  =  B ,  x ,  ( F `  z
) )
4037, 39mpteq12i 4523 . . . . . . 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 3631 . . . . . . . . 9  |-  ( z  e.  ( ( A 
\  { B }
)  u.  { B } )  <->  ( z  e.  ( A  \  { B } )  \/  z  e.  { B } ) )
42 elsn 4030 . . . . . . . . . . 11  |-  ( z  e.  { B }  <->  z  =  B )
4342orbi2i 517 . . . . . . . . . 10  |-  ( ( z  e.  ( A 
\  { B }
)  \/  z  e. 
{ B } )  <-> 
( z  e.  ( A  \  { B } )  \/  z  =  B ) )
44 pm5.61 710 . . . . . . . . . . . 12  |-  ( ( ( z  e.  ( A  \  { B } )  \/  z  =  B )  /\  -.  z  =  B )  <->  ( z  e.  ( A 
\  { B }
)  /\  -.  z  =  B ) )
45 fvres 5862 . . . . . . . . . . . . 13  |-  ( z  e.  ( A  \  { B } )  -> 
( ( F  |`  ( A  \  { B } ) ) `  z )  =  ( F `  z ) )
4645adantr 463 . . . . . . . . . . . 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 3960 . . . . . . . . . 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 4521 . . . . . . 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 2486 . . . . . 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 463 . . . . . 6  |-  ( (
ph  /\  ( A  C_  CC  /\  B  e.  CC ) )  -> 
( F  |`  ( A  \  { B }
) ) : ( A  \  { B } ) --> CC )
5434ssdifssd 3628 . . . . . 6  |-  ( (
ph  /\  ( A  C_  CC  /\  B  e.  CC ) )  -> 
( A  \  { B } )  C_  CC )
5538, 31, 52, 53, 54, 35ellimc 22446 . . . . 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 432 . . 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 352 . 2  |-  ( ph  ->  ( x  e.  ( F lim CC  B )  <-> 
x  e.  ( ( F  |`  ( A  \  { B } ) ) lim CC  B ) ) )
5958eqrdv 2451 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 366    /\ wa 367    /\ w3a 971    = wceq 1398    e. wcel 1823    \ cdif 3458    u. cun 3459    C_ wss 3461   ifcif 3929   {csn 4016    |-> cmpt 4497   dom cdm 4988    |` cres 4990   -->wf 5566   ` cfv 5570  (class class class)co 6270   CCcc 9479   ↾t crest 14913   TopOpenctopn 14914  ℂfldccnfld 18618    CnP ccnp 19896   lim CC climc 22435
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-rep 4550  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565  ax-cnex 9537  ax-resscn 9538  ax-1cn 9539  ax-icn 9540  ax-addcl 9541  ax-addrcl 9542  ax-mulcl 9543  ax-mulrcl 9544  ax-mulcom 9545  ax-addass 9546  ax-mulass 9547  ax-distr 9548  ax-i2m1 9549  ax-1ne0 9550  ax-1rid 9551  ax-rnegex 9552  ax-rrecex 9553  ax-cnre 9554  ax-pre-lttri 9555  ax-pre-lttrn 9556  ax-pre-ltadd 9557  ax-pre-mulgt0 9558  ax-pre-sup 9559
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-nel 2652  df-ral 2809  df-rex 2810  df-reu 2811  df-rmo 2812  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-tp 4021  df-op 4023  df-uni 4236  df-int 4272  df-iun 4317  df-br 4440  df-opab 4498  df-mpt 4499  df-tr 4533  df-eprel 4780  df-id 4784  df-po 4789  df-so 4790  df-fr 4827  df-we 4829  df-ord 4870  df-on 4871  df-lim 4872  df-suc 4873  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-riota 6232  df-ov 6273  df-oprab 6274  df-mpt2 6275  df-om 6674  df-1st 6773  df-2nd 6774  df-recs 7034  df-rdg 7068  df-1o 7122  df-oadd 7126  df-er 7303  df-map 7414  df-pm 7415  df-en 7510  df-dom 7511  df-sdom 7512  df-fin 7513  df-fi 7863  df-sup 7893  df-pnf 9619  df-mnf 9620  df-xr 9621  df-ltxr 9622  df-le 9623  df-sub 9798  df-neg 9799  df-div 10203  df-nn 10532  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 10977  df-uz 11083  df-q 11184  df-rp 11222  df-xneg 11321  df-xadd 11322  df-xmul 11323  df-fz 11676  df-seq 12093  df-exp 12152  df-cj 13017  df-re 13018  df-im 13019  df-sqrt 13153  df-abs 13154  df-struct 14721  df-ndx 14722  df-slot 14723  df-base 14724  df-plusg 14800  df-mulr 14801  df-starv 14802  df-tset 14806  df-ple 14807  df-ds 14809  df-unif 14810  df-rest 14915  df-topn 14916  df-topgen 14936  df-psmet 18609  df-xmet 18610  df-met 18611  df-bl 18612  df-mopn 18613  df-cnfld 18619  df-top 19569  df-bases 19571  df-topon 19572  df-topsp 19573  df-cnp 19899  df-xms 20992  df-ms 20993  df-limc 22439
This theorem is referenced by:  dvcnp2  22492  dvmulbr  22511  dvrec  22527  fourierdlem62  32193
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