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Theorem lmconst 17007
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 956 . 2  |-  ( ( J  e.  (TopOn `  X )  /\  P  e.  X  /\  M  e.  ZZ )  ->  P  e.  X )
2 simp3 957 . . . . . 6  |-  ( ( J  e.  (TopOn `  X )  /\  P  e.  X  /\  M  e.  ZZ )  ->  M  e.  ZZ )
3 uzid 10258 . . . . . 6  |-  ( M  e.  ZZ  ->  M  e.  ( ZZ>= `  M )
)
42, 3syl 15 . . . . 5  |-  ( ( J  e.  (TopOn `  X )  /\  P  e.  X  /\  M  e.  ZZ )  ->  M  e.  ( ZZ>= `  M )
)
5 lmconst.2 . . . . 5  |-  Z  =  ( ZZ>= `  M )
64, 5syl6eleqr 2387 . . . 4  |-  ( ( J  e.  (TopOn `  X )  /\  P  e.  X  /\  M  e.  ZZ )  ->  M  e.  Z )
7 idd 21 . . . . 5  |-  ( ( ( J  e.  (TopOn `  X )  /\  P  e.  X  /\  M  e.  ZZ )  /\  k  e.  ( ZZ>= `  M )
)  ->  ( P  e.  u  ->  P  e.  u ) )
87ralrimdva 2646 . . . 4  |-  ( ( J  e.  (TopOn `  X )  /\  P  e.  X  /\  M  e.  ZZ )  ->  ( P  e.  u  ->  A. k  e.  ( ZZ>= `  M ) P  e.  u ) )
9 fveq2 5541 . . . . . 6  |-  ( j  =  M  ->  ( ZZ>=
`  j )  =  ( ZZ>= `  M )
)
109raleqdv 2755 . . . . 5  |-  ( j  =  M  ->  ( A. k  e.  ( ZZ>=
`  j ) P  e.  u  <->  A. k  e.  ( ZZ>= `  M ) P  e.  u )
)
1110rspcev 2897 . . . 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 1353 . . 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 2640 . 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 955 . . 3  |-  ( ( J  e.  (TopOn `  X )  /\  P  e.  X  /\  M  e.  ZZ )  ->  J  e.  (TopOn `  X )
)
15 fconst6g 5446 . . . 4  |-  ( P  e.  X  ->  ( Z  X.  { P }
) : Z --> X )
161, 15syl 15 . . 3  |-  ( ( J  e.  (TopOn `  X )  /\  P  e.  X  /\  M  e.  ZZ )  ->  ( Z  X.  { P }
) : Z --> X )
17 fvconst2g 5743 . . . 4  |-  ( ( P  e.  X  /\  k  e.  Z )  ->  ( ( Z  X.  { P } ) `  k )  =  P )
181, 17sylan 457 . . 3  |-  ( ( ( J  e.  (TopOn `  X )  /\  P  e.  X  /\  M  e.  ZZ )  /\  k  e.  Z )  ->  (
( Z  X.  { P } ) `  k
)  =  P )
1914, 5, 2, 16, 18lmbrf 17006 . 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 888 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 358    /\ w3a 934    = wceq 1632    e. wcel 1696   A.wral 2556   E.wrex 2557   {csn 3653   class class class wbr 4039    X. cxp 4703   -->wf 5267   ` cfv 5271   ZZcz 10040   ZZ>=cuz 10246  TopOnctopon 16648   ~~> tclm 16972
This theorem is referenced by:  hlim0  21831  occllem  21898  nlelchi  22657  hmopidmchi  22747  esumcvg  23469
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-cnex 8809  ax-resscn 8810  ax-pre-lttri 8827  ax-pre-lttrn 8828
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-po 4330  df-so 4331  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-er 6676  df-pm 6791  df-en 6880  df-dom 6881  df-sdom 6882  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-neg 9056  df-z 10041  df-uz 10247  df-top 16652  df-topon 16655  df-lm 16975
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