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Theorem lmconst 17279
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 958 . 2  |-  ( ( J  e.  (TopOn `  X )  /\  P  e.  X  /\  M  e.  ZZ )  ->  P  e.  X )
2 simp3 959 . . . . . 6  |-  ( ( J  e.  (TopOn `  X )  /\  P  e.  X  /\  M  e.  ZZ )  ->  M  e.  ZZ )
3 uzid 10456 . . . . . 6  |-  ( M  e.  ZZ  ->  M  e.  ( ZZ>= `  M )
)
42, 3syl 16 . . . . 5  |-  ( ( J  e.  (TopOn `  X )  /\  P  e.  X  /\  M  e.  ZZ )  ->  M  e.  ( ZZ>= `  M )
)
5 lmconst.2 . . . . 5  |-  Z  =  ( ZZ>= `  M )
64, 5syl6eleqr 2495 . . . 4  |-  ( ( J  e.  (TopOn `  X )  /\  P  e.  X  /\  M  e.  ZZ )  ->  M  e.  Z )
7 idd 22 . . . . 5  |-  ( ( ( J  e.  (TopOn `  X )  /\  P  e.  X  /\  M  e.  ZZ )  /\  k  e.  ( ZZ>= `  M )
)  ->  ( P  e.  u  ->  P  e.  u ) )
87ralrimdva 2756 . . . 4  |-  ( ( J  e.  (TopOn `  X )  /\  P  e.  X  /\  M  e.  ZZ )  ->  ( P  e.  u  ->  A. k  e.  ( ZZ>= `  M ) P  e.  u ) )
9 fveq2 5687 . . . . . 6  |-  ( j  =  M  ->  ( ZZ>=
`  j )  =  ( ZZ>= `  M )
)
109raleqdv 2870 . . . . 5  |-  ( j  =  M  ->  ( A. k  e.  ( ZZ>=
`  j ) P  e.  u  <->  A. k  e.  ( ZZ>= `  M ) P  e.  u )
)
1110rspcev 3012 . . . 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, 11ee12an 1369 . . 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 2750 . 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 957 . . 3  |-  ( ( J  e.  (TopOn `  X )  /\  P  e.  X  /\  M  e.  ZZ )  ->  J  e.  (TopOn `  X )
)
15 fconst6g 5591 . . . 4  |-  ( P  e.  X  ->  ( Z  X.  { P }
) : Z --> X )
161, 15syl 16 . . 3  |-  ( ( J  e.  (TopOn `  X )  /\  P  e.  X  /\  M  e.  ZZ )  ->  ( Z  X.  { P }
) : Z --> X )
17 fvconst2g 5904 . . . 4  |-  ( ( P  e.  X  /\  k  e.  Z )  ->  ( ( Z  X.  { P } ) `  k )  =  P )
181, 17sylan 458 . . 3  |-  ( ( ( J  e.  (TopOn `  X )  /\  P  e.  X  /\  M  e.  ZZ )  /\  k  e.  Z )  ->  (
( Z  X.  { P } ) `  k
)  =  P )
1914, 5, 2, 16, 18lmbrf 17278 . 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 889 1  |-  ( ( J  e.  (TopOn `  X )  /\  P  e.  X  /\  M  e.  ZZ )  ->  ( Z  X.  { P }
) ( ~~> t `  J ) P )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1721   A.wral 2666   E.wrex 2667   {csn 3774   class class class wbr 4172    X. cxp 4835   -->wf 5409   ` cfv 5413   ZZcz 10238   ZZ>=cuz 10444  TopOnctopon 16914   ~~> tclm 17244
This theorem is referenced by:  hlim0  22691  occllem  22758  nlelchi  23517  hmopidmchi  23607  esumcvg  24429
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 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660  ax-cnex 9002  ax-resscn 9003  ax-pre-lttri 9020  ax-pre-lttrn 9021
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 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-op 3783  df-uni 3976  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-id 4458  df-po 4463  df-so 4464  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-1st 6308  df-2nd 6309  df-er 6864  df-pm 6980  df-en 7069  df-dom 7070  df-sdom 7071  df-pnf 9078  df-mnf 9079  df-xr 9080  df-ltxr 9081  df-le 9082  df-neg 9250  df-z 10239  df-uz 10445  df-top 16918  df-topon 16921  df-lm 17247
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