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Theorem cnplimc 19642
Description: A function is continuous at  B iff its limit at  B equals the value of the function there. (Contributed by Mario Carneiro, 28-Dec-2016.)
Hypotheses
Ref Expression
cnplimc.k  |-  K  =  ( TopOpen ` fld )
cnplimc.j  |-  J  =  ( Kt  A )
Assertion
Ref Expression
cnplimc  |-  ( ( A  C_  CC  /\  B  e.  A )  ->  ( F  e.  ( ( J  CnP  K ) `  B )  <->  ( F : A --> CC  /\  ( F `  B )  e.  ( F lim CC  B
) ) ) )

Proof of Theorem cnplimc
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 cnplimc.j . . . . 5  |-  J  =  ( Kt  A )
2 cnplimc.k . . . . . . 7  |-  K  =  ( TopOpen ` fld )
32cnfldtopon 18689 . . . . . 6  |-  K  e.  (TopOn `  CC )
4 simpl 444 . . . . . 6  |-  ( ( A  C_  CC  /\  B  e.  A )  ->  A  C_  CC )
5 resttopon 17148 . . . . . 6  |-  ( ( K  e.  (TopOn `  CC )  /\  A  C_  CC )  ->  ( Kt  A )  e.  (TopOn `  A ) )
63, 4, 5sylancr 645 . . . . 5  |-  ( ( A  C_  CC  /\  B  e.  A )  ->  ( Kt  A )  e.  (TopOn `  A ) )
71, 6syl5eqel 2472 . . . 4  |-  ( ( A  C_  CC  /\  B  e.  A )  ->  J  e.  (TopOn `  A )
)
8 cnpf2 17237 . . . . 5  |-  ( ( J  e.  (TopOn `  A )  /\  K  e.  (TopOn `  CC )  /\  F  e.  (
( J  CnP  K
) `  B )
)  ->  F : A
--> CC )
983expia 1155 . . . 4  |-  ( ( J  e.  (TopOn `  A )  /\  K  e.  (TopOn `  CC )
)  ->  ( F  e.  ( ( J  CnP  K ) `  B )  ->  F : A --> CC ) )
107, 3, 9sylancl 644 . . 3  |-  ( ( A  C_  CC  /\  B  e.  A )  ->  ( F  e.  ( ( J  CnP  K ) `  B )  ->  F : A --> CC ) )
1110pm4.71rd 617 . 2  |-  ( ( A  C_  CC  /\  B  e.  A )  ->  ( F  e.  ( ( J  CnP  K ) `  B )  <->  ( F : A --> CC  /\  F  e.  ( ( J  CnP  K ) `  B ) ) ) )
12 simpr 448 . . . . . . 7  |-  ( ( ( A  C_  CC  /\  B  e.  A )  /\  F : A --> CC )  ->  F : A
--> CC )
13 simplr 732 . . . . . . . . . 10  |-  ( ( ( A  C_  CC  /\  B  e.  A )  /\  F : A --> CC )  ->  B  e.  A )
1413snssd 3887 . . . . . . . . 9  |-  ( ( ( A  C_  CC  /\  B  e.  A )  /\  F : A --> CC )  ->  { B }  C_  A )
15 ssequn2 3464 . . . . . . . . 9  |-  ( { B }  C_  A  <->  ( A  u.  { B } )  =  A )
1614, 15sylib 189 . . . . . . . 8  |-  ( ( ( A  C_  CC  /\  B  e.  A )  /\  F : A --> CC )  ->  ( A  u.  { B }
)  =  A )
1716feq2d 5522 . . . . . . 7  |-  ( ( ( A  C_  CC  /\  B  e.  A )  /\  F : A --> CC )  ->  ( F : ( A  u.  { B } ) --> CC  <->  F : A --> CC ) )
1812, 17mpbird 224 . . . . . 6  |-  ( ( ( A  C_  CC  /\  B  e.  A )  /\  F : A --> CC )  ->  F :
( A  u.  { B } ) --> CC )
1918feqmptd 5719 . . . . 5  |-  ( ( ( A  C_  CC  /\  B  e.  A )  /\  F : A --> CC )  ->  F  =  ( x  e.  ( A  u.  { B } )  |->  ( F `
 x ) ) )
2016oveq2d 6037 . . . . . . . 8  |-  ( ( ( A  C_  CC  /\  B  e.  A )  /\  F : A --> CC )  ->  ( Kt  ( A  u.  { B } ) )  =  ( Kt  A ) )
2120, 1syl6reqr 2439 . . . . . . 7  |-  ( ( ( A  C_  CC  /\  B  e.  A )  /\  F : A --> CC )  ->  J  =  ( Kt  ( A  u.  { B } ) ) )
2221oveq1d 6036 . . . . . 6  |-  ( ( ( A  C_  CC  /\  B  e.  A )  /\  F : A --> CC )  ->  ( J  CnP  K )  =  ( ( Kt  ( A  u.  { B }
) )  CnP  K
) )
2322fveq1d 5671 . . . . 5  |-  ( ( ( A  C_  CC  /\  B  e.  A )  /\  F : A --> CC )  ->  ( ( J  CnP  K ) `
 B )  =  ( ( ( Kt  ( A  u.  { B } ) )  CnP 
K ) `  B
) )
2419, 23eleq12d 2456 . . . 4  |-  ( ( ( A  C_  CC  /\  B  e.  A )  /\  F : A --> CC )  ->  ( F  e.  ( ( J  CnP  K ) `  B )  <->  ( x  e.  ( A  u.  { B } )  |->  ( F `
 x ) )  e.  ( ( ( Kt  ( A  u.  { B } ) )  CnP 
K ) `  B
) ) )
25 eqid 2388 . . . . 5  |-  ( Kt  ( A  u.  { B } ) )  =  ( Kt  ( A  u.  { B } ) )
26 ifid 3715 . . . . . . 7  |-  if ( x  =  B , 
( F `  x
) ,  ( F `
 x ) )  =  ( F `  x )
27 fveq2 5669 . . . . . . . . 9  |-  ( x  =  B  ->  ( F `  x )  =  ( F `  B ) )
2827adantl 453 . . . . . . . 8  |-  ( ( x  e.  ( A  u.  { B }
)  /\  x  =  B )  ->  ( F `  x )  =  ( F `  B ) )
2928ifeq1da 3708 . . . . . . 7  |-  ( x  e.  ( A  u.  { B } )  ->  if ( x  =  B ,  ( F `  x ) ,  ( F `  x ) )  =  if ( x  =  B , 
( F `  B
) ,  ( F `
 x ) ) )
3026, 29syl5eqr 2434 . . . . . 6  |-  ( x  e.  ( A  u.  { B } )  -> 
( F `  x
)  =  if ( x  =  B , 
( F `  B
) ,  ( F `
 x ) ) )
3130mpteq2ia 4233 . . . . 5  |-  ( x  e.  ( A  u.  { B } )  |->  ( F `  x ) )  =  ( x  e.  ( A  u.  { B } )  |->  if ( x  =  B ,  ( F `  B ) ,  ( F `  x ) ) )
32 simpll 731 . . . . 5  |-  ( ( ( A  C_  CC  /\  B  e.  A )  /\  F : A --> CC )  ->  A  C_  CC )
3332, 13sseldd 3293 . . . . 5  |-  ( ( ( A  C_  CC  /\  B  e.  A )  /\  F : A --> CC )  ->  B  e.  CC )
3425, 2, 31, 12, 32, 33ellimc 19628 . . . 4  |-  ( ( ( A  C_  CC  /\  B  e.  A )  /\  F : A --> CC )  ->  ( ( F `  B )  e.  ( F lim CC  B )  <->  ( x  e.  ( A  u.  { B } )  |->  ( F `
 x ) )  e.  ( ( ( Kt  ( A  u.  { B } ) )  CnP 
K ) `  B
) ) )
3524, 34bitr4d 248 . . 3  |-  ( ( ( A  C_  CC  /\  B  e.  A )  /\  F : A --> CC )  ->  ( F  e.  ( ( J  CnP  K ) `  B )  <->  ( F `  B )  e.  ( F lim CC  B ) ) )
3635pm5.32da 623 . 2  |-  ( ( A  C_  CC  /\  B  e.  A )  ->  (
( F : A --> CC  /\  F  e.  ( ( J  CnP  K
) `  B )
)  <->  ( F : A
--> CC  /\  ( F `
 B )  e.  ( F lim CC  B
) ) ) )
3711, 36bitrd 245 1  |-  ( ( A  C_  CC  /\  B  e.  A )  ->  ( F  e.  ( ( J  CnP  K ) `  B )  <->  ( F : A --> CC  /\  ( F `  B )  e.  ( F lim CC  B
) ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    = wceq 1649    e. wcel 1717    u. cun 3262    C_ wss 3264   ifcif 3683   {csn 3758    e. cmpt 4208   -->wf 5391   ` cfv 5395  (class class class)co 6021   CCcc 8922   ↾t crest 13576   TopOpenctopn 13577  ℂfldccnfld 16627  TopOnctopon 16883    CnP ccnp 17212   lim CC climc 19617
This theorem is referenced by:  cnlimc  19643  dvcnp2  19674  dvmulbr  19693  dvcobr  19700
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2369  ax-rep 4262  ax-sep 4272  ax-nul 4280  ax-pow 4319  ax-pr 4345  ax-un 4642  ax-cnex 8980  ax-resscn 8981  ax-1cn 8982  ax-icn 8983  ax-addcl 8984  ax-addrcl 8985  ax-mulcl 8986  ax-mulrcl 8987  ax-mulcom 8988  ax-addass 8989  ax-mulass 8990  ax-distr 8991  ax-i2m1 8992  ax-1ne0 8993  ax-1rid 8994  ax-rnegex 8995  ax-rrecex 8996  ax-cnre 8997  ax-pre-lttri 8998  ax-pre-lttrn 8999  ax-pre-ltadd 9000  ax-pre-mulgt0 9001  ax-pre-sup 9002
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2243  df-mo 2244  df-clab 2375  df-cleq 2381  df-clel 2384  df-nfc 2513  df-ne 2553  df-nel 2554  df-ral 2655  df-rex 2656  df-reu 2657  df-rmo 2658  df-rab 2659  df-v 2902  df-sbc 3106  df-csb 3196  df-dif 3267  df-un 3269  df-in 3271  df-ss 3278  df-pss 3280  df-nul 3573  df-if 3684  df-pw 3745  df-sn 3764  df-pr 3765  df-tp 3766  df-op 3767  df-uni 3959  df-int 3994  df-iun 4038  df-br 4155  df-opab 4209  df-mpt 4210  df-tr 4245  df-eprel 4436  df-id 4440  df-po 4445  df-so 4446  df-fr 4483  df-we 4485  df-ord 4526  df-on 4527  df-lim 4528  df-suc 4529  df-om 4787  df-xp 4825  df-rel 4826  df-cnv 4827  df-co 4828  df-dm 4829  df-rn 4830  df-res 4831  df-ima 4832  df-iota 5359  df-fun 5397  df-fn 5398  df-f 5399  df-f1 5400  df-fo 5401  df-f1o 5402  df-fv 5403  df-ov 6024  df-oprab 6025  df-mpt2 6026  df-1st 6289  df-2nd 6290  df-riota 6486  df-recs 6570  df-rdg 6605  df-1o 6661  df-oadd 6665  df-er 6842  df-map 6957  df-pm 6958  df-en 7047  df-dom 7048  df-sdom 7049  df-fin 7050  df-fi 7352  df-sup 7382  df-pnf 9056  df-mnf 9057  df-xr 9058  df-ltxr 9059  df-le 9060  df-sub 9226  df-neg 9227  df-div 9611  df-nn 9934  df-2 9991  df-3 9992  df-4 9993  df-5 9994  df-6 9995  df-7 9996  df-8 9997  df-9 9998  df-10 9999  df-n0 10155  df-z 10216  df-dec 10316  df-uz 10422  df-q 10508  df-rp 10546  df-xneg 10643  df-xadd 10644  df-xmul 10645  df-fz 10977  df-seq 11252  df-exp 11311  df-cj 11832  df-re 11833  df-im 11834  df-sqr 11968  df-abs 11969  df-struct 13399  df-ndx 13400  df-slot 13401  df-base 13402  df-plusg 13470  df-mulr 13471  df-starv 13472  df-tset 13476  df-ple 13477  df-ds 13479  df-unif 13480  df-rest 13578  df-topn 13579  df-topgen 13595  df-xmet 16620  df-met 16621  df-bl 16622  df-mopn 16623  df-cnfld 16628  df-top 16887  df-bases 16889  df-topon 16890  df-topsp 16891  df-cnp 17215  df-xms 18260  df-ms 18261  df-limc 19621
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