ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  lmconst Unicode version

Theorem lmconst 13349
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 9518 . . . . . 6  |-  ( M  e.  ZZ  ->  M  e.  ( ZZ>= `  M )
)
42, 3syl 14 . . . . 5  |-  ( ( J  e.  (TopOn `  X )  /\  P  e.  X  /\  M  e.  ZZ )  ->  M  e.  ( ZZ>= `  M )
)
5 lmconst.2 . . . . 5  |-  Z  =  ( ZZ>= `  M )
64, 5eleqtrrdi 2271 . . . 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 2557 . . . 4  |-  ( ( J  e.  (TopOn `  X )  /\  P  e.  X  /\  M  e.  ZZ )  ->  ( P  e.  u  ->  A. k  e.  ( ZZ>= `  M ) P  e.  u ) )
9 fveq2 5510 . . . . . 6  |-  ( j  =  M  ->  ( ZZ>=
`  j )  =  ( ZZ>= `  M )
)
109raleqdv 2678 . . . . 5  |-  ( j  =  M  ->  ( A. k  e.  ( ZZ>=
`  j ) P  e.  u  <->  A. k  e.  ( ZZ>= `  M ) P  e.  u )
)
1110rspcev 2841 . . . 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 1434 . . 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 2551 . 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 5409 . . . 4  |-  ( P  e.  X  ->  ( Z  X.  { P }
) : Z --> X )
161, 15syl 14 . . 3  |-  ( ( J  e.  (TopOn `  X )  /\  P  e.  X  /\  M  e.  ZZ )  ->  ( Z  X.  { P }
) : Z --> X )
17 fvconst2g 5725 . . . 4  |-  ( ( P  e.  X  /\  k  e.  Z )  ->  ( ( Z  X.  { P } ) `  k )  =  P )
181, 17sylan 283 . . 3  |-  ( ( ( J  e.  (TopOn `  X )  /\  P  e.  X  /\  M  e.  ZZ )  /\  k  e.  Z )  ->  (
( Z  X.  { P } ) `  k
)  =  P )
1914, 5, 2, 16, 18lmbrf 13348 . 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 944 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 104    /\ w3a 978    = wceq 1353    e. wcel 2148   A.wral 2455   E.wrex 2456   {csn 3591   class class class wbr 4000    X. cxp 4620   -->wf 5207   ` cfv 5211   ZZcz 9229   ZZ>=cuz 9504  TopOnctopon 13141   ~~> tclm 13320
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4118  ax-pow 4171  ax-pr 4205  ax-un 4429  ax-setind 4532  ax-cnex 7880  ax-resscn 7881  ax-1cn 7882  ax-1re 7883  ax-icn 7884  ax-addcl 7885  ax-addrcl 7886  ax-mulcl 7887  ax-addcom 7889  ax-addass 7891  ax-distr 7893  ax-i2m1 7894  ax-0lt1 7895  ax-0id 7897  ax-rnegex 7898  ax-cnre 7900  ax-pre-ltirr 7901  ax-pre-ltwlin 7902  ax-pre-lttrn 7903  ax-pre-apti 7904  ax-pre-ltadd 7905
This theorem depends on definitions:  df-bi 117  df-dc 835  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-nel 2443  df-ral 2460  df-rex 2461  df-reu 2462  df-rab 2464  df-v 2739  df-sbc 2963  df-csb 3058  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-if 3535  df-pw 3576  df-sn 3597  df-pr 3598  df-op 3600  df-uni 3808  df-int 3843  df-iun 3886  df-br 4001  df-opab 4062  df-mpt 4063  df-id 4289  df-xp 4628  df-rel 4629  df-cnv 4630  df-co 4631  df-dm 4632  df-rn 4633  df-res 4634  df-ima 4635  df-iota 5173  df-fun 5213  df-fn 5214  df-f 5215  df-fv 5219  df-riota 5824  df-ov 5871  df-oprab 5872  df-mpo 5873  df-1st 6134  df-2nd 6135  df-pm 6644  df-pnf 7971  df-mnf 7972  df-xr 7973  df-ltxr 7974  df-le 7975  df-sub 8107  df-neg 8108  df-inn 8896  df-n0 9153  df-z 9230  df-uz 9505  df-top 13129  df-topon 13142  df-lm 13323
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator