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

Theorem lmflf 17951
Description: The topological limit relation on functions can be written in terms of the filter limit along the filter generated by the upper integer sets. (Contributed by Mario Carneiro, 13-Oct-2015.)
Hypotheses
Ref Expression
lmflf.1  |-  Z  =  ( ZZ>= `  M )
lmflf.2  |-  L  =  ( Z filGen ( ZZ>= " Z ) )
Assertion
Ref Expression
lmflf  |-  ( ( J  e.  (TopOn `  X )  /\  M  e.  ZZ  /\  F : Z
--> X )  ->  ( F ( ~~> t `  J ) P  <->  P  e.  ( ( J  fLimf  L ) `  F ) ) )

Proof of Theorem lmflf
Dummy variables  j 
k  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 uzf 10416 . . . . . . . 8  |-  ZZ>= : ZZ --> ~P ZZ
2 ffn 5524 . . . . . . . 8  |-  ( ZZ>= : ZZ --> ~P ZZ  ->  ZZ>=  Fn  ZZ )
31, 2ax-mp 8 . . . . . . 7  |-  ZZ>=  Fn  ZZ
4 lmflf.1 . . . . . . . 8  |-  Z  =  ( ZZ>= `  M )
5 uzssz 10430 . . . . . . . 8  |-  ( ZZ>= `  M )  C_  ZZ
64, 5eqsstri 3314 . . . . . . 7  |-  Z  C_  ZZ
7 imaeq2 5132 . . . . . . . . 9  |-  ( y  =  ( ZZ>= `  j
)  ->  ( F " y )  =  ( F " ( ZZ>= `  j ) ) )
87sseq1d 3311 . . . . . . . 8  |-  ( y  =  ( ZZ>= `  j
)  ->  ( ( F " y )  C_  x 
<->  ( F " ( ZZ>=
`  j ) ) 
C_  x ) )
98rexima 5909 . . . . . . 7  |-  ( (
ZZ>=  Fn  ZZ  /\  Z  C_  ZZ )  ->  ( E. y  e.  ( ZZ>=
" Z ) ( F " y ) 
C_  x  <->  E. j  e.  Z  ( F " ( ZZ>= `  j )
)  C_  x )
)
103, 6, 9mp2an 654 . . . . . 6  |-  ( E. y  e.  ( ZZ>= " Z ) ( F
" y )  C_  x 
<->  E. j  e.  Z  ( F " ( ZZ>= `  j ) )  C_  x )
11 simpl3 962 . . . . . . . . 9  |-  ( ( ( J  e.  (TopOn `  X )  /\  M  e.  ZZ  /\  F : Z
--> X )  /\  j  e.  Z )  ->  F : Z --> X )
12 ffun 5526 . . . . . . . . 9  |-  ( F : Z --> X  ->  Fun  F )
1311, 12syl 16 . . . . . . . 8  |-  ( ( ( J  e.  (TopOn `  X )  /\  M  e.  ZZ  /\  F : Z
--> X )  /\  j  e.  Z )  ->  Fun  F )
14 uzss 10431 . . . . . . . . . . 11  |-  ( j  e.  ( ZZ>= `  M
)  ->  ( ZZ>= `  j )  C_  ( ZZ>=
`  M ) )
1514, 4eleq2s 2472 . . . . . . . . . 10  |-  ( j  e.  Z  ->  ( ZZ>=
`  j )  C_  ( ZZ>= `  M )
)
1615adantl 453 . . . . . . . . 9  |-  ( ( ( J  e.  (TopOn `  X )  /\  M  e.  ZZ  /\  F : Z
--> X )  /\  j  e.  Z )  ->  ( ZZ>=
`  j )  C_  ( ZZ>= `  M )
)
17 fdm 5528 . . . . . . . . . . 11  |-  ( F : Z --> X  ->  dom  F  =  Z )
1811, 17syl 16 . . . . . . . . . 10  |-  ( ( ( J  e.  (TopOn `  X )  /\  M  e.  ZZ  /\  F : Z
--> X )  /\  j  e.  Z )  ->  dom  F  =  Z )
1918, 4syl6eq 2428 . . . . . . . . 9  |-  ( ( ( J  e.  (TopOn `  X )  /\  M  e.  ZZ  /\  F : Z
--> X )  /\  j  e.  Z )  ->  dom  F  =  ( ZZ>= `  M
) )
2016, 19sseqtr4d 3321 . . . . . . . 8  |-  ( ( ( J  e.  (TopOn `  X )  /\  M  e.  ZZ  /\  F : Z
--> X )  /\  j  e.  Z )  ->  ( ZZ>=
`  j )  C_  dom  F )
21 funimass4 5709 . . . . . . . 8  |-  ( ( Fun  F  /\  ( ZZ>=
`  j )  C_  dom  F )  ->  (
( F " ( ZZ>=
`  j ) ) 
C_  x  <->  A. k  e.  ( ZZ>= `  j )
( F `  k
)  e.  x ) )
2213, 20, 21syl2anc 643 . . . . . . 7  |-  ( ( ( J  e.  (TopOn `  X )  /\  M  e.  ZZ  /\  F : Z
--> X )  /\  j  e.  Z )  ->  (
( F " ( ZZ>=
`  j ) ) 
C_  x  <->  A. k  e.  ( ZZ>= `  j )
( F `  k
)  e.  x ) )
2322rexbidva 2659 . . . . . 6  |-  ( ( J  e.  (TopOn `  X )  /\  M  e.  ZZ  /\  F : Z
--> X )  ->  ( E. j  e.  Z  ( F " ( ZZ>= `  j ) )  C_  x 
<->  E. j  e.  Z  A. k  e.  ( ZZ>=
`  j ) ( F `  k )  e.  x ) )
2410, 23syl5rbb 250 . . . . 5  |-  ( ( J  e.  (TopOn `  X )  /\  M  e.  ZZ  /\  F : Z
--> X )  ->  ( E. j  e.  Z  A. k  e.  ( ZZ>=
`  j ) ( F `  k )  e.  x  <->  E. y  e.  ( ZZ>= " Z ) ( F " y ) 
C_  x ) )
2524imbi2d 308 . . . 4  |-  ( ( J  e.  (TopOn `  X )  /\  M  e.  ZZ  /\  F : Z
--> X )  ->  (
( P  e.  x  ->  E. j  e.  Z  A. k  e.  ( ZZ>=
`  j ) ( F `  k )  e.  x )  <->  ( P  e.  x  ->  E. y  e.  ( ZZ>= " Z ) ( F " y ) 
C_  x ) ) )
2625ralbidv 2662 . . 3  |-  ( ( J  e.  (TopOn `  X )  /\  M  e.  ZZ  /\  F : Z
--> X )  ->  ( A. x  e.  J  ( P  e.  x  ->  E. j  e.  Z  A. k  e.  ( ZZ>=
`  j ) ( F `  k )  e.  x )  <->  A. x  e.  J  ( P  e.  x  ->  E. y  e.  ( ZZ>= " Z ) ( F " y ) 
C_  x ) ) )
2726anbi2d 685 . 2  |-  ( ( J  e.  (TopOn `  X )  /\  M  e.  ZZ  /\  F : Z
--> X )  ->  (
( P  e.  X  /\  A. x  e.  J  ( P  e.  x  ->  E. j  e.  Z  A. k  e.  ( ZZ>=
`  j ) ( F `  k )  e.  x ) )  <-> 
( P  e.  X  /\  A. x  e.  J  ( P  e.  x  ->  E. y  e.  (
ZZ>= " Z ) ( F " y ) 
C_  x ) ) ) )
28 simp1 957 . . 3  |-  ( ( J  e.  (TopOn `  X )  /\  M  e.  ZZ  /\  F : Z
--> X )  ->  J  e.  (TopOn `  X )
)
29 simp2 958 . . 3  |-  ( ( J  e.  (TopOn `  X )  /\  M  e.  ZZ  /\  F : Z
--> X )  ->  M  e.  ZZ )
30 simp3 959 . . 3  |-  ( ( J  e.  (TopOn `  X )  /\  M  e.  ZZ  /\  F : Z
--> X )  ->  F : Z --> X )
31 eqidd 2381 . . 3  |-  ( ( ( J  e.  (TopOn `  X )  /\  M  e.  ZZ  /\  F : Z
--> X )  /\  k  e.  Z )  ->  ( F `  k )  =  ( F `  k ) )
3228, 4, 29, 30, 31lmbrf 17239 . 2  |-  ( ( J  e.  (TopOn `  X )  /\  M  e.  ZZ  /\  F : Z
--> X )  ->  ( F ( ~~> t `  J ) P  <->  ( P  e.  X  /\  A. x  e.  J  ( P  e.  x  ->  E. j  e.  Z  A. k  e.  ( ZZ>= `  j )
( F `  k
)  e.  x ) ) ) )
334uzfbas 17844 . . 3  |-  ( M  e.  ZZ  ->  ( ZZ>=
" Z )  e.  ( fBas `  Z
) )
34 lmflf.2 . . . 4  |-  L  =  ( Z filGen ( ZZ>= " Z ) )
3534flffbas 17941 . . 3  |-  ( ( J  e.  (TopOn `  X )  /\  ( ZZ>=
" Z )  e.  ( fBas `  Z
)  /\  F : Z
--> X )  ->  ( P  e.  ( ( J  fLimf  L ) `  F )  <->  ( P  e.  X  /\  A. x  e.  J  ( P  e.  x  ->  E. y  e.  ( ZZ>= " Z ) ( F " y ) 
C_  x ) ) ) )
3633, 35syl3an2 1218 . 2  |-  ( ( J  e.  (TopOn `  X )  /\  M  e.  ZZ  /\  F : Z
--> X )  ->  ( P  e.  ( ( J  fLimf  L ) `  F )  <->  ( P  e.  X  /\  A. x  e.  J  ( P  e.  x  ->  E. y  e.  ( ZZ>= " Z ) ( F " y ) 
C_  x ) ) ) )
3727, 32, 363bitr4d 277 1  |-  ( ( J  e.  (TopOn `  X )  /\  M  e.  ZZ  /\  F : Z
--> X )  ->  ( F ( ~~> t `  J ) P  <->  P  e.  ( ( J  fLimf  L ) `  F ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1717   A.wral 2642   E.wrex 2643    C_ wss 3256   ~Pcpw 3735   class class class wbr 4146   dom cdm 4811   "cima 4814   Fun wfun 5381    Fn wfn 5382   -->wf 5383   ` cfv 5387  (class class class)co 6013   ZZcz 10207   ZZ>=cuz 10413   fBascfbas 16608   filGencfg 16609  TopOnctopon 16875   ~~> tclm 17205    fLimf cflf 17881
This theorem is referenced by:  cmetcaulem  19105
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 2361  ax-rep 4254  ax-sep 4264  ax-nul 4272  ax-pow 4311  ax-pr 4337  ax-un 4634  ax-cnex 8972  ax-resscn 8973  ax-1cn 8974  ax-icn 8975  ax-addcl 8976  ax-addrcl 8977  ax-mulcl 8978  ax-mulrcl 8979  ax-mulcom 8980  ax-addass 8981  ax-mulass 8982  ax-distr 8983  ax-i2m1 8984  ax-1ne0 8985  ax-1rid 8986  ax-rnegex 8987  ax-rrecex 8988  ax-cnre 8989  ax-pre-lttri 8990  ax-pre-lttrn 8991  ax-pre-ltadd 8992  ax-pre-mulgt0 8993
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 2235  df-mo 2236  df-clab 2367  df-cleq 2373  df-clel 2376  df-nfc 2505  df-ne 2545  df-nel 2546  df-ral 2647  df-rex 2648  df-reu 2649  df-rab 2651  df-v 2894  df-sbc 3098  df-csb 3188  df-dif 3259  df-un 3261  df-in 3263  df-ss 3270  df-pss 3272  df-nul 3565  df-if 3676  df-pw 3737  df-sn 3756  df-pr 3757  df-tp 3758  df-op 3759  df-uni 3951  df-iun 4030  df-br 4147  df-opab 4201  df-mpt 4202  df-tr 4237  df-eprel 4428  df-id 4432  df-po 4437  df-so 4438  df-fr 4475  df-we 4477  df-ord 4518  df-on 4519  df-lim 4520  df-suc 4521  df-om 4779  df-xp 4817  df-rel 4818  df-cnv 4819  df-co 4820  df-dm 4821  df-rn 4822  df-res 4823  df-ima 4824  df-iota 5351  df-fun 5389  df-fn 5390  df-f 5391  df-f1 5392  df-fo 5393  df-f1o 5394  df-fv 5395  df-ov 6016  df-oprab 6017  df-mpt2 6018  df-1st 6281  df-2nd 6282  df-riota 6478  df-recs 6562  df-rdg 6597  df-er 6834  df-map 6949  df-pm 6950  df-en 7039  df-dom 7040  df-sdom 7041  df-pnf 9048  df-mnf 9049  df-xr 9050  df-ltxr 9051  df-le 9052  df-sub 9218  df-neg 9219  df-nn 9926  df-z 10208  df-uz 10414  df-rest 13570  df-fbas 16616  df-fg 16617  df-top 16879  df-topon 16882  df-ntr 17000  df-nei 17078  df-lm 17208  df-fil 17792  df-fm 17884  df-flim 17885  df-flf 17886
  Copyright terms: Public domain W3C validator