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Theorem lmconst 18870
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 989 . 2  |-  ( ( J  e.  (TopOn `  X )  /\  P  e.  X  /\  M  e.  ZZ )  ->  P  e.  X )
2 simp3 990 . . . . . 6  |-  ( ( J  e.  (TopOn `  X )  /\  P  e.  X  /\  M  e.  ZZ )  ->  M  e.  ZZ )
3 uzid 10880 . . . . . 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 2534 . . . 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 2811 . . . 4  |-  ( ( J  e.  (TopOn `  X )  /\  P  e.  X  /\  M  e.  ZZ )  ->  ( P  e.  u  ->  A. k  e.  ( ZZ>= `  M ) P  e.  u ) )
9 fveq2 5696 . . . . . 6  |-  ( j  =  M  ->  ( ZZ>=
`  j )  =  ( ZZ>= `  M )
)
109raleqdv 2928 . . . . 5  |-  ( j  =  M  ->  ( A. k  e.  ( ZZ>=
`  j ) P  e.  u  <->  A. k  e.  ( ZZ>= `  M ) P  e.  u )
)
1110rspcev 3078 . . . 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 545 . . 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 2805 . 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 988 . . 3  |-  ( ( J  e.  (TopOn `  X )  /\  P  e.  X  /\  M  e.  ZZ )  ->  J  e.  (TopOn `  X )
)
15 fconst6g 5604 . . . 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 5936 . . . 4  |-  ( ( P  e.  X  /\  k  e.  Z )  ->  ( ( Z  X.  { P } ) `  k )  =  P )
181, 17sylan 471 . . 3  |-  ( ( ( J  e.  (TopOn `  X )  /\  P  e.  X  /\  M  e.  ZZ )  /\  k  e.  Z )  ->  (
( Z  X.  { P } ) `  k
)  =  P )
1914, 5, 2, 16, 18lmbrf 18869 . 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 913 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 369    /\ w3a 965    = wceq 1369    e. wcel 1756   A.wral 2720   E.wrex 2721   {csn 3882   class class class wbr 4297    X. cxp 4843   -->wf 5419   ` cfv 5423   ZZcz 10651   ZZ>=cuz 10866  TopOnctopon 18504   ~~> tclm 18835
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4418  ax-nul 4426  ax-pow 4475  ax-pr 4536  ax-un 6377  ax-cnex 9343  ax-resscn 9344  ax-pre-lttri 9361  ax-pre-lttrn 9362
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2573  df-ne 2613  df-nel 2614  df-ral 2725  df-rex 2726  df-rab 2729  df-v 2979  df-sbc 3192  df-csb 3294  df-dif 3336  df-un 3338  df-in 3340  df-ss 3347  df-nul 3643  df-if 3797  df-pw 3867  df-sn 3883  df-pr 3885  df-op 3889  df-uni 4097  df-iun 4178  df-br 4298  df-opab 4356  df-mpt 4357  df-id 4641  df-po 4646  df-so 4647  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5386  df-fun 5425  df-fn 5426  df-f 5427  df-f1 5428  df-fo 5429  df-f1o 5430  df-fv 5431  df-ov 6099  df-oprab 6100  df-mpt2 6101  df-1st 6582  df-2nd 6583  df-er 7106  df-pm 7222  df-en 7316  df-dom 7317  df-sdom 7318  df-pnf 9425  df-mnf 9426  df-xr 9427  df-ltxr 9428  df-le 9429  df-neg 9603  df-z 10652  df-uz 10867  df-top 18508  df-topon 18511  df-lm 18838
This theorem is referenced by:  hlim0  24643  occllem  24711  nlelchi  25470  hmopidmchi  25560  esumcvg  26540
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