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

Theorem lmconst 20053
Description: A constant sequence converges to its value. (Contributed by NM, 8-Nov-2007.) (Revised by Mario Carneiro, 14-Nov-2013.)
Hypothesis
Ref Expression
lmconst.2  |-  Z  =  ( ZZ>= `  M )
Assertion
Ref Expression
lmconst  |-  ( ( J  e.  (TopOn `  X )  /\  P  e.  X  /\  M  e.  ZZ )  ->  ( Z  X.  { P }
) ( ~~> t `  J ) P )

Proof of Theorem lmconst
Dummy variables  j 
k  u are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simp2 998 . 2  |-  ( ( J  e.  (TopOn `  X )  /\  P  e.  X  /\  M  e.  ZZ )  ->  P  e.  X )
2 simp3 999 . . . . . 6  |-  ( ( J  e.  (TopOn `  X )  /\  P  e.  X  /\  M  e.  ZZ )  ->  M  e.  ZZ )
3 uzid 11140 . . . . . 6  |-  ( M  e.  ZZ  ->  M  e.  ( ZZ>= `  M )
)
42, 3syl 17 . . . . 5  |-  ( ( J  e.  (TopOn `  X )  /\  P  e.  X  /\  M  e.  ZZ )  ->  M  e.  ( ZZ>= `  M )
)
5 lmconst.2 . . . . 5  |-  Z  =  ( ZZ>= `  M )
64, 5syl6eleqr 2501 . . . 4  |-  ( ( J  e.  (TopOn `  X )  /\  P  e.  X  /\  M  e.  ZZ )  ->  M  e.  Z )
7 idd 24 . . . . 5  |-  ( ( ( J  e.  (TopOn `  X )  /\  P  e.  X  /\  M  e.  ZZ )  /\  k  e.  ( ZZ>= `  M )
)  ->  ( P  e.  u  ->  P  e.  u ) )
87ralrimdva 2821 . . . 4  |-  ( ( J  e.  (TopOn `  X )  /\  P  e.  X  /\  M  e.  ZZ )  ->  ( P  e.  u  ->  A. k  e.  ( ZZ>= `  M ) P  e.  u ) )
9 fveq2 5848 . . . . . 6  |-  ( j  =  M  ->  ( ZZ>=
`  j )  =  ( ZZ>= `  M )
)
109raleqdv 3009 . . . . 5  |-  ( j  =  M  ->  ( A. k  e.  ( ZZ>=
`  j ) P  e.  u  <->  A. k  e.  ( ZZ>= `  M ) P  e.  u )
)
1110rspcev 3159 . . . 4  |-  ( ( M  e.  Z  /\  A. k  e.  ( ZZ>= `  M ) P  e.  u )  ->  E. j  e.  Z  A. k  e.  ( ZZ>= `  j ) P  e.  u )
126, 8, 11syl6an 543 . . 3  |-  ( ( J  e.  (TopOn `  X )  /\  P  e.  X  /\  M  e.  ZZ )  ->  ( P  e.  u  ->  E. j  e.  Z  A. k  e.  ( ZZ>= `  j ) P  e.  u ) )
1312ralrimivw 2818 . 2  |-  ( ( J  e.  (TopOn `  X )  /\  P  e.  X  /\  M  e.  ZZ )  ->  A. u  e.  J  ( P  e.  u  ->  E. j  e.  Z  A. k  e.  ( ZZ>= `  j ) P  e.  u )
)
14 simp1 997 . . 3  |-  ( ( J  e.  (TopOn `  X )  /\  P  e.  X  /\  M  e.  ZZ )  ->  J  e.  (TopOn `  X )
)
15 fconst6g 5756 . . . 4  |-  ( P  e.  X  ->  ( Z  X.  { P }
) : Z --> X )
161, 15syl 17 . . 3  |-  ( ( J  e.  (TopOn `  X )  /\  P  e.  X  /\  M  e.  ZZ )  ->  ( Z  X.  { P }
) : Z --> X )
17 fvconst2g 6104 . . . 4  |-  ( ( P  e.  X  /\  k  e.  Z )  ->  ( ( Z  X.  { P } ) `  k )  =  P )
181, 17sylan 469 . . 3  |-  ( ( ( J  e.  (TopOn `  X )  /\  P  e.  X  /\  M  e.  ZZ )  /\  k  e.  Z )  ->  (
( Z  X.  { P } ) `  k
)  =  P )
1914, 5, 2, 16, 18lmbrf 20052 . 2  |-  ( ( J  e.  (TopOn `  X )  /\  P  e.  X  /\  M  e.  ZZ )  ->  (
( Z  X.  { P } ) ( ~~> t `  J ) P  <->  ( P  e.  X  /\  A. u  e.  J  ( P  e.  u  ->  E. j  e.  Z  A. k  e.  ( ZZ>= `  j ) P  e.  u )
) ) )
201, 13, 19mpbir2and 923 1  |-  ( ( J  e.  (TopOn `  X )  /\  P  e.  X  /\  M  e.  ZZ )  ->  ( Z  X.  { P }
) ( ~~> t `  J ) P )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 367    /\ w3a 974    = wceq 1405    e. wcel 1842   A.wral 2753   E.wrex 2754   {csn 3971   class class class wbr 4394    X. cxp 4820   -->wf 5564   ` cfv 5568   ZZcz 10904   ZZ>=cuz 11126  TopOnctopon 19685   ~~> tclm 20018
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-sep 4516  ax-nul 4524  ax-pow 4571  ax-pr 4629  ax-un 6573  ax-cnex 9577  ax-resscn 9578  ax-pre-lttri 9595  ax-pre-lttrn 9596
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-rab 2762  df-v 3060  df-sbc 3277  df-csb 3373  df-dif 3416  df-un 3418  df-in 3420  df-ss 3427  df-nul 3738  df-if 3885  df-pw 3956  df-sn 3972  df-pr 3974  df-op 3978  df-uni 4191  df-iun 4272  df-br 4395  df-opab 4453  df-mpt 4454  df-id 4737  df-po 4743  df-so 4744  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-iota 5532  df-fun 5570  df-fn 5571  df-f 5572  df-f1 5573  df-fo 5574  df-f1o 5575  df-fv 5576  df-ov 6280  df-oprab 6281  df-mpt2 6282  df-1st 6783  df-2nd 6784  df-er 7347  df-pm 7459  df-en 7554  df-dom 7555  df-sdom 7556  df-pnf 9659  df-mnf 9660  df-xr 9661  df-ltxr 9662  df-le 9663  df-neg 9843  df-z 10905  df-uz 11127  df-top 19689  df-topon 19692  df-lm 20021
This theorem is referenced by:  hlim0  26553  occllem  26621  nlelchi  27379  hmopidmchi  27469  esumcvg  28519
  Copyright terms: Public domain W3C validator