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

Theorem lmcls 19970
Description: Any convergent sequence of points in a subset of a topological space converges to a point in the closure of the subset. (Contributed by Mario Carneiro, 30-Dec-2013.) (Revised by Mario Carneiro, 1-May-2014.)
Hypotheses
Ref Expression
lmff.1  |-  Z  =  ( ZZ>= `  M )
lmff.3  |-  ( ph  ->  J  e.  (TopOn `  X ) )
lmff.4  |-  ( ph  ->  M  e.  ZZ )
lmcls.5  |-  ( ph  ->  F ( ~~> t `  J ) P )
lmcls.7  |-  ( (
ph  /\  k  e.  Z )  ->  ( F `  k )  e.  S )
lmcls.8  |-  ( ph  ->  S  C_  X )
Assertion
Ref Expression
lmcls  |-  ( ph  ->  P  e.  ( ( cls `  J ) `
 S ) )
Distinct variable groups:    k, F    k, J    k, M    P, k    S, k    ph, k    k, X    k, Z

Proof of Theorem lmcls
Dummy variables  j  u are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 lmcls.5 . . . . 5  |-  ( ph  ->  F ( ~~> t `  J ) P )
2 lmff.3 . . . . . 6  |-  ( ph  ->  J  e.  (TopOn `  X ) )
3 lmff.1 . . . . . 6  |-  Z  =  ( ZZ>= `  M )
4 lmff.4 . . . . . 6  |-  ( ph  ->  M  e.  ZZ )
52, 3, 4lmbr2 19927 . . . . 5  |-  ( ph  ->  ( F ( ~~> t `  J ) P  <->  ( F  e.  ( X  ^pm  CC )  /\  P  e.  X  /\  A. u  e.  J  ( P  e.  u  ->  E. j  e.  Z  A. k  e.  ( ZZ>=
`  j ) ( k  e.  dom  F  /\  ( F `  k
)  e.  u ) ) ) ) )
61, 5mpbid 210 . . . 4  |-  ( ph  ->  ( F  e.  ( X  ^pm  CC )  /\  P  e.  X  /\  A. u  e.  J  ( P  e.  u  ->  E. j  e.  Z  A. k  e.  ( ZZ>=
`  j ) ( k  e.  dom  F  /\  ( F `  k
)  e.  u ) ) ) )
76simp3d 1008 . . 3  |-  ( ph  ->  A. u  e.  J  ( P  e.  u  ->  E. j  e.  Z  A. k  e.  ( ZZ>=
`  j ) ( k  e.  dom  F  /\  ( F `  k
)  e.  u ) ) )
83r19.2uz 13266 . . . . . 6  |-  ( E. j  e.  Z  A. k  e.  ( ZZ>= `  j ) ( k  e.  dom  F  /\  ( F `  k )  e.  u )  ->  E. k  e.  Z  ( k  e.  dom  F  /\  ( F `  k )  e.  u
) )
9 lmcls.7 . . . . . . . . 9  |-  ( (
ph  /\  k  e.  Z )  ->  ( F `  k )  e.  S )
10 inelcm 3869 . . . . . . . . . 10  |-  ( ( ( F `  k
)  e.  u  /\  ( F `  k )  e.  S )  -> 
( u  i^i  S
)  =/=  (/) )
1110a1i 11 . . . . . . . . 9  |-  ( (
ph  /\  k  e.  Z )  ->  (
( ( F `  k )  e.  u  /\  ( F `  k
)  e.  S )  ->  ( u  i^i 
S )  =/=  (/) ) )
129, 11mpan2d 672 . . . . . . . 8  |-  ( (
ph  /\  k  e.  Z )  ->  (
( F `  k
)  e.  u  -> 
( u  i^i  S
)  =/=  (/) ) )
1312adantld 465 . . . . . . 7  |-  ( (
ph  /\  k  e.  Z )  ->  (
( k  e.  dom  F  /\  ( F `  k )  e.  u
)  ->  ( u  i^i  S )  =/=  (/) ) )
1413rexlimdva 2946 . . . . . 6  |-  ( ph  ->  ( E. k  e.  Z  ( k  e. 
dom  F  /\  ( F `  k )  e.  u )  ->  (
u  i^i  S )  =/=  (/) ) )
158, 14syl5 32 . . . . 5  |-  ( ph  ->  ( E. j  e.  Z  A. k  e.  ( ZZ>= `  j )
( k  e.  dom  F  /\  ( F `  k )  e.  u
)  ->  ( u  i^i  S )  =/=  (/) ) )
1615imim2d 52 . . . 4  |-  ( ph  ->  ( ( P  e.  u  ->  E. j  e.  Z  A. k  e.  ( ZZ>= `  j )
( k  e.  dom  F  /\  ( F `  k )  e.  u
) )  ->  ( P  e.  u  ->  ( u  i^i  S )  =/=  (/) ) ) )
1716ralimdv 2864 . . 3  |-  ( ph  ->  ( A. u  e.  J  ( P  e.  u  ->  E. j  e.  Z  A. k  e.  ( ZZ>= `  j )
( k  e.  dom  F  /\  ( F `  k )  e.  u
) )  ->  A. u  e.  J  ( P  e.  u  ->  ( u  i^i  S )  =/=  (/) ) ) )
187, 17mpd 15 . 2  |-  ( ph  ->  A. u  e.  J  ( P  e.  u  ->  ( u  i^i  S
)  =/=  (/) ) )
19 topontop 19594 . . . 4  |-  ( J  e.  (TopOn `  X
)  ->  J  e.  Top )
202, 19syl 16 . . 3  |-  ( ph  ->  J  e.  Top )
21 lmcls.8 . . . 4  |-  ( ph  ->  S  C_  X )
22 toponuni 19595 . . . . 5  |-  ( J  e.  (TopOn `  X
)  ->  X  =  U. J )
232, 22syl 16 . . . 4  |-  ( ph  ->  X  =  U. J
)
2421, 23sseqtrd 3525 . . 3  |-  ( ph  ->  S  C_  U. J )
25 lmcl 19965 . . . . 5  |-  ( ( J  e.  (TopOn `  X )  /\  F
( ~~> t `  J
) P )  ->  P  e.  X )
262, 1, 25syl2anc 659 . . . 4  |-  ( ph  ->  P  e.  X )
2726, 23eleqtrd 2544 . . 3  |-  ( ph  ->  P  e.  U. J
)
28 eqid 2454 . . . 4  |-  U. J  =  U. J
2928elcls 19741 . . 3  |-  ( ( J  e.  Top  /\  S  C_  U. J  /\  P  e.  U. J )  ->  ( P  e.  ( ( cls `  J
) `  S )  <->  A. u  e.  J  ( P  e.  u  -> 
( u  i^i  S
)  =/=  (/) ) ) )
3020, 24, 27, 29syl3anc 1226 . 2  |-  ( ph  ->  ( P  e.  ( ( cls `  J
) `  S )  <->  A. u  e.  J  ( P  e.  u  -> 
( u  i^i  S
)  =/=  (/) ) ) )
3118, 30mpbird 232 1  |-  ( ph  ->  P  e.  ( ( cls `  J ) `
 S ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 367    /\ w3a 971    = wceq 1398    e. wcel 1823    =/= wne 2649   A.wral 2804   E.wrex 2805    i^i cin 3460    C_ wss 3461   (/)c0 3783   U.cuni 4235   class class class wbr 4439   dom cdm 4988   ` cfv 5570  (class class class)co 6270    ^pm cpm 7413   CCcc 9479   ZZcz 10860   ZZ>=cuz 11082   Topctop 19561  TopOnctopon 19562   clsccl 19686   ~~> tclm 19894
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-pre-lttri 9555  ax-pre-lttrn 9556
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-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-op 4023  df-uni 4236  df-int 4272  df-iun 4317  df-iin 4318  df-br 4440  df-opab 4498  df-mpt 4499  df-id 4784  df-po 4789  df-so 4790  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-ov 6273  df-oprab 6274  df-mpt2 6275  df-1st 6773  df-2nd 6774  df-er 7303  df-pm 7415  df-en 7510  df-dom 7511  df-sdom 7512  df-pnf 9619  df-mnf 9620  df-xr 9621  df-ltxr 9622  df-le 9623  df-neg 9799  df-z 10861  df-uz 11083  df-top 19566  df-topon 19569  df-cld 19687  df-ntr 19688  df-cls 19689  df-lm 19897
This theorem is referenced by:  lmcld  19971  1stcelcls  20128  caublcls  21913
  Copyright terms: Public domain W3C validator