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

Theorem limcfval 22011
Description: Value and set bounds on the limit operator. (Contributed by Mario Carneiro, 25-Dec-2016.)
Hypotheses
Ref Expression
limcval.j  |-  J  =  ( Kt  ( A  u.  { B } ) )
limcval.k  |-  K  =  ( TopOpen ` fld )
Assertion
Ref Expression
limcfval  |-  ( ( F : A --> CC  /\  A  C_  CC  /\  B  e.  CC )  ->  (
( F lim CC  B
)  =  { y  |  ( z  e.  ( A  u.  { B } )  |->  if ( z  =  B , 
y ,  ( F `
 z ) ) )  e.  ( ( J  CnP  K ) `
 B ) }  /\  ( F lim CC  B )  C_  CC ) )
Distinct variable groups:    y, z, A    y, B, z    y, F, z    y, K, z   
y, J
Allowed substitution hint:    J( z)

Proof of Theorem limcfval
Dummy variables  f 
j  x are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-limc 22005 . . . 4  |- lim CC  =  ( f  e.  ( CC  ^pm  CC ) ,  x  e.  CC  |->  { y  |  [. ( TopOpen ` fld )  /  j ]. ( z  e.  ( dom  f  u.  {
x } )  |->  if ( z  =  x ,  y ,  ( f `  z ) ) )  e.  ( ( ( jt  ( dom  f  u.  { x } ) )  CnP  j ) `  x
) } )
21a1i 11 . . 3  |-  ( ( F : A --> CC  /\  A  C_  CC  /\  B  e.  CC )  -> lim CC  =  ( f  e.  ( CC  ^pm  CC ) ,  x  e.  CC  |->  { y  |  [. ( TopOpen ` fld )  /  j ]. ( z  e.  ( dom  f  u.  {
x } )  |->  if ( z  =  x ,  y ,  ( f `  z ) ) )  e.  ( ( ( jt  ( dom  f  u.  { x } ) )  CnP  j ) `  x
) } ) )
3 fvex 5874 . . . . . 6  |-  ( TopOpen ` fld )  e.  _V
43a1i 11 . . . . 5  |-  ( ( ( F : A --> CC  /\  A  C_  CC  /\  B  e.  CC )  /\  ( f  =  F  /\  x  =  B ) )  -> 
( TopOpen ` fld )  e.  _V )
5 simplrl 759 . . . . . . . . . 10  |-  ( ( ( ( F : A
--> CC  /\  A  C_  CC  /\  B  e.  CC )  /\  ( f  =  F  /\  x  =  B ) )  /\  j  =  ( TopOpen ` fld )
)  ->  f  =  F )
65dmeqd 5203 . . . . . . . . 9  |-  ( ( ( ( F : A
--> CC  /\  A  C_  CC  /\  B  e.  CC )  /\  ( f  =  F  /\  x  =  B ) )  /\  j  =  ( TopOpen ` fld )
)  ->  dom  f  =  dom  F )
7 simpll1 1035 . . . . . . . . . 10  |-  ( ( ( ( F : A
--> CC  /\  A  C_  CC  /\  B  e.  CC )  /\  ( f  =  F  /\  x  =  B ) )  /\  j  =  ( TopOpen ` fld )
)  ->  F : A
--> CC )
8 fdm 5733 . . . . . . . . . 10  |-  ( F : A --> CC  ->  dom 
F  =  A )
97, 8syl 16 . . . . . . . . 9  |-  ( ( ( ( F : A
--> CC  /\  A  C_  CC  /\  B  e.  CC )  /\  ( f  =  F  /\  x  =  B ) )  /\  j  =  ( TopOpen ` fld )
)  ->  dom  F  =  A )
106, 9eqtrd 2508 . . . . . . . 8  |-  ( ( ( ( F : A
--> CC  /\  A  C_  CC  /\  B  e.  CC )  /\  ( f  =  F  /\  x  =  B ) )  /\  j  =  ( TopOpen ` fld )
)  ->  dom  f  =  A )
11 simplrr 760 . . . . . . . . 9  |-  ( ( ( ( F : A
--> CC  /\  A  C_  CC  /\  B  e.  CC )  /\  ( f  =  F  /\  x  =  B ) )  /\  j  =  ( TopOpen ` fld )
)  ->  x  =  B )
1211sneqd 4039 . . . . . . . 8  |-  ( ( ( ( F : A
--> CC  /\  A  C_  CC  /\  B  e.  CC )  /\  ( f  =  F  /\  x  =  B ) )  /\  j  =  ( TopOpen ` fld )
)  ->  { x }  =  { B } )
1310, 12uneq12d 3659 . . . . . . 7  |-  ( ( ( ( F : A
--> CC  /\  A  C_  CC  /\  B  e.  CC )  /\  ( f  =  F  /\  x  =  B ) )  /\  j  =  ( TopOpen ` fld )
)  ->  ( dom  f  u.  { x } )  =  ( A  u.  { B } ) )
1411eqeq2d 2481 . . . . . . . 8  |-  ( ( ( ( F : A
--> CC  /\  A  C_  CC  /\  B  e.  CC )  /\  ( f  =  F  /\  x  =  B ) )  /\  j  =  ( TopOpen ` fld )
)  ->  ( z  =  x  <->  z  =  B ) )
155fveq1d 5866 . . . . . . . 8  |-  ( ( ( ( F : A
--> CC  /\  A  C_  CC  /\  B  e.  CC )  /\  ( f  =  F  /\  x  =  B ) )  /\  j  =  ( TopOpen ` fld )
)  ->  ( f `  z )  =  ( F `  z ) )
1614, 15ifbieq2d 3964 . . . . . . 7  |-  ( ( ( ( F : A
--> CC  /\  A  C_  CC  /\  B  e.  CC )  /\  ( f  =  F  /\  x  =  B ) )  /\  j  =  ( TopOpen ` fld )
)  ->  if (
z  =  x ,  y ,  ( f `
 z ) )  =  if ( z  =  B ,  y ,  ( F `  z ) ) )
1713, 16mpteq12dv 4525 . . . . . 6  |-  ( ( ( ( F : A
--> CC  /\  A  C_  CC  /\  B  e.  CC )  /\  ( f  =  F  /\  x  =  B ) )  /\  j  =  ( TopOpen ` fld )
)  ->  ( z  e.  ( dom  f  u. 
{ x } ) 
|->  if ( z  =  x ,  y ,  ( f `  z
) ) )  =  ( z  e.  ( A  u.  { B } )  |->  if ( z  =  B , 
y ,  ( F `
 z ) ) ) )
18 simpr 461 . . . . . . . . . . 11  |-  ( ( ( ( F : A
--> CC  /\  A  C_  CC  /\  B  e.  CC )  /\  ( f  =  F  /\  x  =  B ) )  /\  j  =  ( TopOpen ` fld )
)  ->  j  =  ( TopOpen ` fld ) )
19 limcval.k . . . . . . . . . . 11  |-  K  =  ( TopOpen ` fld )
2018, 19syl6eqr 2526 . . . . . . . . . 10  |-  ( ( ( ( F : A
--> CC  /\  A  C_  CC  /\  B  e.  CC )  /\  ( f  =  F  /\  x  =  B ) )  /\  j  =  ( TopOpen ` fld )
)  ->  j  =  K )
2120, 13oveq12d 6300 . . . . . . . . 9  |-  ( ( ( ( F : A
--> CC  /\  A  C_  CC  /\  B  e.  CC )  /\  ( f  =  F  /\  x  =  B ) )  /\  j  =  ( TopOpen ` fld )
)  ->  ( jt  ( dom  f  u.  { x } ) )  =  ( Kt  ( A  u.  { B } ) ) )
22 limcval.j . . . . . . . . 9  |-  J  =  ( Kt  ( A  u.  { B } ) )
2321, 22syl6eqr 2526 . . . . . . . 8  |-  ( ( ( ( F : A
--> CC  /\  A  C_  CC  /\  B  e.  CC )  /\  ( f  =  F  /\  x  =  B ) )  /\  j  =  ( TopOpen ` fld )
)  ->  ( jt  ( dom  f  u.  { x } ) )  =  J )
2423, 20oveq12d 6300 . . . . . . 7  |-  ( ( ( ( F : A
--> CC  /\  A  C_  CC  /\  B  e.  CC )  /\  ( f  =  F  /\  x  =  B ) )  /\  j  =  ( TopOpen ` fld )
)  ->  ( (
jt  ( dom  f  u. 
{ x } ) )  CnP  j )  =  ( J  CnP  K ) )
2524, 11fveq12d 5870 . . . . . 6  |-  ( ( ( ( F : A
--> CC  /\  A  C_  CC  /\  B  e.  CC )  /\  ( f  =  F  /\  x  =  B ) )  /\  j  =  ( TopOpen ` fld )
)  ->  ( (
( jt  ( dom  f  u.  { x } ) )  CnP  j ) `
 x )  =  ( ( J  CnP  K ) `  B ) )
2617, 25eleq12d 2549 . . . . 5  |-  ( ( ( ( F : A
--> CC  /\  A  C_  CC  /\  B  e.  CC )  /\  ( f  =  F  /\  x  =  B ) )  /\  j  =  ( TopOpen ` fld )
)  ->  ( (
z  e.  ( dom  f  u.  { x } )  |->  if ( z  =  x ,  y ,  ( f `
 z ) ) )  e.  ( ( ( jt  ( dom  f  u.  { x } ) )  CnP  j ) `
 x )  <->  ( z  e.  ( A  u.  { B } )  |->  if ( z  =  B , 
y ,  ( F `
 z ) ) )  e.  ( ( J  CnP  K ) `
 B ) ) )
274, 26sbcied 3368 . . . 4  |-  ( ( ( F : A --> CC  /\  A  C_  CC  /\  B  e.  CC )  /\  ( f  =  F  /\  x  =  B ) )  -> 
( [. ( TopOpen ` fld )  /  j ]. ( z  e.  ( dom  f  u.  {
x } )  |->  if ( z  =  x ,  y ,  ( f `  z ) ) )  e.  ( ( ( jt  ( dom  f  u.  { x } ) )  CnP  j ) `  x
)  <->  ( z  e.  ( A  u.  { B } )  |->  if ( z  =  B , 
y ,  ( F `
 z ) ) )  e.  ( ( J  CnP  K ) `
 B ) ) )
2827abbidv 2603 . . 3  |-  ( ( ( F : A --> CC  /\  A  C_  CC  /\  B  e.  CC )  /\  ( f  =  F  /\  x  =  B ) )  ->  { y  |  [. ( TopOpen ` fld )  /  j ]. ( z  e.  ( dom  f  u.  {
x } )  |->  if ( z  =  x ,  y ,  ( f `  z ) ) )  e.  ( ( ( jt  ( dom  f  u.  { x } ) )  CnP  j ) `  x
) }  =  {
y  |  ( z  e.  ( A  u.  { B } )  |->  if ( z  =  B ,  y ,  ( F `  z ) ) )  e.  ( ( J  CnP  K
) `  B ) } )
29 cnex 9569 . . . . 5  |-  CC  e.  _V
30 elpm2r 7433 . . . . 5  |-  ( ( ( CC  e.  _V  /\  CC  e.  _V )  /\  ( F : A --> CC  /\  A  C_  CC ) )  ->  F  e.  ( CC  ^pm  CC ) )
3129, 29, 30mpanl12 682 . . . 4  |-  ( ( F : A --> CC  /\  A  C_  CC )  ->  F  e.  ( CC  ^pm 
CC ) )
32313adant3 1016 . . 3  |-  ( ( F : A --> CC  /\  A  C_  CC  /\  B  e.  CC )  ->  F  e.  ( CC  ^pm  CC ) )
33 simp3 998 . . 3  |-  ( ( F : A --> CC  /\  A  C_  CC  /\  B  e.  CC )  ->  B  e.  CC )
34 eqid 2467 . . . . . 6  |-  ( z  e.  ( A  u.  { B } )  |->  if ( z  =  B ,  y ,  ( F `  z ) ) )  =  ( z  e.  ( A  u.  { B }
)  |->  if ( z  =  B ,  y ,  ( F `  z ) ) )
3522, 19, 34limcvallem 22010 . . . . 5  |-  ( ( F : A --> CC  /\  A  C_  CC  /\  B  e.  CC )  ->  (
( z  e.  ( A  u.  { B } )  |->  if ( z  =  B , 
y ,  ( F `
 z ) ) )  e.  ( ( J  CnP  K ) `
 B )  -> 
y  e.  CC ) )
3635abssdv 3574 . . . 4  |-  ( ( F : A --> CC  /\  A  C_  CC  /\  B  e.  CC )  ->  { y  |  ( z  e.  ( A  u.  { B } )  |->  if ( z  =  B , 
y ,  ( F `
 z ) ) )  e.  ( ( J  CnP  K ) `
 B ) } 
C_  CC )
3729ssex 4591 . . . 4  |-  ( { y  |  ( z  e.  ( A  u.  { B } )  |->  if ( z  =  B ,  y ,  ( F `  z ) ) )  e.  ( ( J  CnP  K
) `  B ) }  C_  CC  ->  { y  |  ( z  e.  ( A  u.  { B } )  |->  if ( z  =  B , 
y ,  ( F `
 z ) ) )  e.  ( ( J  CnP  K ) `
 B ) }  e.  _V )
3836, 37syl 16 . . 3  |-  ( ( F : A --> CC  /\  A  C_  CC  /\  B  e.  CC )  ->  { y  |  ( z  e.  ( A  u.  { B } )  |->  if ( z  =  B , 
y ,  ( F `
 z ) ) )  e.  ( ( J  CnP  K ) `
 B ) }  e.  _V )
392, 28, 32, 33, 38ovmpt2d 6412 . 2  |-  ( ( F : A --> CC  /\  A  C_  CC  /\  B  e.  CC )  ->  ( F lim CC  B )  =  { y  |  ( z  e.  ( A  u.  { B }
)  |->  if ( z  =  B ,  y ,  ( F `  z ) ) )  e.  ( ( J  CnP  K ) `  B ) } )
4039, 36eqsstrd 3538 . 2  |-  ( ( F : A --> CC  /\  A  C_  CC  /\  B  e.  CC )  ->  ( F lim CC  B )  C_  CC )
4139, 40jca 532 1  |-  ( ( F : A --> CC  /\  A  C_  CC  /\  B  e.  CC )  ->  (
( F lim CC  B
)  =  { y  |  ( z  e.  ( A  u.  { B } )  |->  if ( z  =  B , 
y ,  ( F `
 z ) ) )  e.  ( ( J  CnP  K ) `
 B ) }  /\  ( F lim CC  B )  C_  CC ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 973    = wceq 1379    e. wcel 1767   {cab 2452   _Vcvv 3113   [.wsbc 3331    u. cun 3474    C_ wss 3476   ifcif 3939   {csn 4027    |-> cmpt 4505   dom cdm 4999   -->wf 5582   ` cfv 5586  (class class class)co 6282    |-> cmpt2 6284    ^pm cpm 7418   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:  ellimc  22012  limccl  22014
  Copyright terms: Public domain W3C validator