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

Theorem limcfval 22566
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 22560 . . . 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 5858 . . . . . 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 762 . . . . . . . . . 10  |-  ( ( ( ( F : A
--> CC  /\  A  C_  CC  /\  B  e.  CC )  /\  ( f  =  F  /\  x  =  B ) )  /\  j  =  ( TopOpen ` fld )
)  ->  f  =  F )
65dmeqd 5025 . . . . . . . . 9  |-  ( ( ( ( F : A
--> CC  /\  A  C_  CC  /\  B  e.  CC )  /\  ( f  =  F  /\  x  =  B ) )  /\  j  =  ( TopOpen ` fld )
)  ->  dom  f  =  dom  F )
7 simpll1 1036 . . . . . . . . . 10  |-  ( ( ( ( F : A
--> CC  /\  A  C_  CC  /\  B  e.  CC )  /\  ( f  =  F  /\  x  =  B ) )  /\  j  =  ( TopOpen ` fld )
)  ->  F : A
--> CC )
8 fdm 5717 . . . . . . . . . 10  |-  ( F : A --> CC  ->  dom 
F  =  A )
97, 8syl 17 . . . . . . . . 9  |-  ( ( ( ( F : A
--> CC  /\  A  C_  CC  /\  B  e.  CC )  /\  ( f  =  F  /\  x  =  B ) )  /\  j  =  ( TopOpen ` fld )
)  ->  dom  F  =  A )
106, 9eqtrd 2443 . . . . . . . 8  |-  ( ( ( ( F : A
--> CC  /\  A  C_  CC  /\  B  e.  CC )  /\  ( f  =  F  /\  x  =  B ) )  /\  j  =  ( TopOpen ` fld )
)  ->  dom  f  =  A )
11 simplrr 763 . . . . . . . . 9  |-  ( ( ( ( F : A
--> CC  /\  A  C_  CC  /\  B  e.  CC )  /\  ( f  =  F  /\  x  =  B ) )  /\  j  =  ( TopOpen ` fld )
)  ->  x  =  B )
1211sneqd 3983 . . . . . . . 8  |-  ( ( ( ( F : A
--> CC  /\  A  C_  CC  /\  B  e.  CC )  /\  ( f  =  F  /\  x  =  B ) )  /\  j  =  ( TopOpen ` fld )
)  ->  { x }  =  { B } )
1310, 12uneq12d 3597 . . . . . . 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 2416 . . . . . . . 8  |-  ( ( ( ( F : A
--> CC  /\  A  C_  CC  /\  B  e.  CC )  /\  ( f  =  F  /\  x  =  B ) )  /\  j  =  ( TopOpen ` fld )
)  ->  ( z  =  x  <->  z  =  B ) )
155fveq1d 5850 . . . . . . . 8  |-  ( ( ( ( F : A
--> CC  /\  A  C_  CC  /\  B  e.  CC )  /\  ( f  =  F  /\  x  =  B ) )  /\  j  =  ( TopOpen ` fld )
)  ->  ( f `  z )  =  ( F `  z ) )
1614, 15ifbieq2d 3909 . . . . . . 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 4472 . . . . . 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 459 . . . . . . . . . . 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 2461 . . . . . . . . . 10  |-  ( ( ( ( F : A
--> CC  /\  A  C_  CC  /\  B  e.  CC )  /\  ( f  =  F  /\  x  =  B ) )  /\  j  =  ( TopOpen ` fld )
)  ->  j  =  K )
2120, 13oveq12d 6295 . . . . . . . . 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 2461 . . . . . . . 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 6295 . . . . . . 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 5854 . . . . . 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 2484 . . . . 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 3313 . . . 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 2538 . . 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 9602 . . . . 5  |-  CC  e.  _V
30 elpm2r 7473 . . . . 5  |-  ( ( ( CC  e.  _V  /\  CC  e.  _V )  /\  ( F : A --> CC  /\  A  C_  CC ) )  ->  F  e.  ( CC  ^pm  CC ) )
3129, 29, 30mpanl12 680 . . . 4  |-  ( ( F : A --> CC  /\  A  C_  CC )  ->  F  e.  ( CC  ^pm 
CC ) )
32313adant3 1017 . . 3  |-  ( ( F : A --> CC  /\  A  C_  CC  /\  B  e.  CC )  ->  F  e.  ( CC  ^pm  CC ) )
33 simp3 999 . . 3  |-  ( ( F : A --> CC  /\  A  C_  CC  /\  B  e.  CC )  ->  B  e.  CC )
34 eqid 2402 . . . . . 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 22565 . . . . 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 3512 . . . 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 4537 . . . 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 17 . . 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 6410 . 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 3475 . 2  |-  ( ( F : A --> CC  /\  A  C_  CC  /\  B  e.  CC )  ->  ( F lim CC  B )  C_  CC )
4139, 40jca 530 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 367    /\ w3a 974    = wceq 1405    e. wcel 1842   {cab 2387   _Vcvv 3058   [.wsbc 3276    u. cun 3411    C_ wss 3413   ifcif 3884   {csn 3971    |-> cmpt 4452   dom cdm 4822   -->wf 5564   ` cfv 5568  (class class class)co 6277    |-> cmpt2 6279    ^pm cpm 7457   CCcc 9519   ↾t crest 15033   TopOpenctopn 15034  ℂfldccnfld 18738    CnP ccnp 20017   lim CC climc 22556
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-rep 4506  ax-sep 4516  ax-nul 4524  ax-pow 4571  ax-pr 4629  ax-un 6573  ax-cnex 9577  ax-resscn 9578  ax-1cn 9579  ax-icn 9580  ax-addcl 9581  ax-addrcl 9582  ax-mulcl 9583  ax-mulrcl 9584  ax-mulcom 9585  ax-addass 9586  ax-mulass 9587  ax-distr 9588  ax-i2m1 9589  ax-1ne0 9590  ax-1rid 9591  ax-rnegex 9592  ax-rrecex 9593  ax-cnre 9594  ax-pre-lttri 9595  ax-pre-lttrn 9596  ax-pre-ltadd 9597  ax-pre-mulgt0 9598  ax-pre-sup 9599
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 975  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-nel 2601  df-ral 2758  df-rex 2759  df-reu 2760  df-rmo 2761  df-rab 2762  df-v 3060  df-sbc 3277  df-csb 3373  df-dif 3416  df-un 3418  df-in 3420  df-ss 3427  df-pss 3429  df-nul 3738  df-if 3885  df-pw 3956  df-sn 3972  df-pr 3974  df-tp 3976  df-op 3978  df-uni 4191  df-int 4227  df-iun 4272  df-br 4395  df-opab 4453  df-mpt 4454  df-tr 4489  df-eprel 4733  df-id 4737  df-po 4743  df-so 4744  df-fr 4781  df-we 4783  df-xp 4828  df-rel 4829  df-cnv 4830  df-co 4831  df-dm 4832  df-rn 4833  df-res 4834  df-ima 4835  df-pred 5366  df-ord 5412  df-on 5413  df-lim 5414  df-suc 5415  df-iota 5532  df-fun 5570  df-fn 5571  df-f 5572  df-f1 5573  df-fo 5574  df-f1o 5575  df-fv 5576  df-riota 6239  df-ov 6280  df-oprab 6281  df-mpt2 6282  df-om 6683  df-1st 6783  df-2nd 6784  df-wrecs 7012  df-recs 7074  df-rdg 7112  df-1o 7166  df-oadd 7170  df-er 7347  df-map 7458  df-pm 7459  df-en 7554  df-dom 7555  df-sdom 7556  df-fin 7557  df-fi 7904  df-sup 7934  df-pnf 9659  df-mnf 9660  df-xr 9661  df-ltxr 9662  df-le 9663  df-sub 9842  df-neg 9843  df-div 10247  df-nn 10576  df-2 10634  df-3 10635  df-4 10636  df-5 10637  df-6 10638  df-7 10639  df-8 10640  df-9 10641  df-10 10642  df-n0 10836  df-z 10905  df-dec 11019  df-uz 11127  df-q 11227  df-rp 11265  df-xneg 11370  df-xadd 11371  df-xmul 11372  df-fz 11725  df-seq 12150  df-exp 12209  df-cj 13079  df-re 13080  df-im 13081  df-sqrt 13215  df-abs 13216  df-struct 14841  df-ndx 14842  df-slot 14843  df-base 14844  df-plusg 14920  df-mulr 14921  df-starv 14922  df-tset 14926  df-ple 14927  df-ds 14929  df-unif 14930  df-rest 15035  df-topn 15036  df-topgen 15056  df-psmet 18729  df-xmet 18730  df-met 18731  df-bl 18732  df-mopn 18733  df-cnfld 18739  df-top 19689  df-bases 19691  df-topon 19692  df-topsp 19693  df-cnp 20020  df-xms 21113  df-ms 21114  df-limc 22560
This theorem is referenced by:  ellimc  22567  limccl  22569
  Copyright terms: Public domain W3C validator