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

Theorem lmle 19256
Description: If the distance from each member of a converging sequence to a given point is less than or equal to a given amount, so is the convergence value. (Contributed by NM, 23-Dec-2007.) (Proof shortened by Mario Carneiro, 1-May-2014.)
Hypotheses
Ref Expression
lmle.1  |-  Z  =  ( ZZ>= `  M )
lmle.3  |-  J  =  ( MetOpen `  D )
lmle.4  |-  ( ph  ->  D  e.  ( * Met `  X ) )
lmle.6  |-  ( ph  ->  M  e.  ZZ )
lmle.7  |-  ( ph  ->  F ( ~~> t `  J ) P )
lmle.8  |-  ( ph  ->  Q  e.  X )
lmle.9  |-  ( ph  ->  R  e.  RR* )
lmle.10  |-  ( (
ph  /\  k  e.  Z )  ->  ( Q D ( F `  k ) )  <_  R )
Assertion
Ref Expression
lmle  |-  ( ph  ->  ( Q D P )  <_  R )
Distinct variable groups:    D, k    k, J    ph, k    k, Z   
k, F    P, k    Q, k    R, k    k, X
Allowed substitution hint:    M( k)

Proof of Theorem lmle
Dummy variables  j  x are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 lmle.1 . . . 4  |-  Z  =  ( ZZ>= `  M )
2 lmle.4 . . . . 5  |-  ( ph  ->  D  e.  ( * Met `  X ) )
3 lmle.3 . . . . . 6  |-  J  =  ( MetOpen `  D )
43mopntopon 18471 . . . . 5  |-  ( D  e.  ( * Met `  X )  ->  J  e.  (TopOn `  X )
)
52, 4syl 16 . . . 4  |-  ( ph  ->  J  e.  (TopOn `  X ) )
6 lmle.6 . . . 4  |-  ( ph  ->  M  e.  ZZ )
7 lmrel 17296 . . . . 5  |-  Rel  ( ~~> t `  J )
8 lmle.7 . . . . 5  |-  ( ph  ->  F ( ~~> t `  J ) P )
9 releldm 5104 . . . . 5  |-  ( ( Rel  ( ~~> t `  J )  /\  F
( ~~> t `  J
) P )  ->  F  e.  dom  ( ~~> t `  J ) )
107, 8, 9sylancr 646 . . . 4  |-  ( ph  ->  F  e.  dom  ( ~~> t `  J )
)
111, 5, 6, 10lmff 17367 . . 3  |-  ( ph  ->  E. j  e.  Z  ( F  |`  ( ZZ>= `  j ) ) : ( ZZ>= `  j ) --> X )
12 eqid 2438 . . . 4  |-  ( ZZ>= `  j )  =  (
ZZ>= `  j )
135adantr 453 . . . 4  |-  ( (
ph  /\  ( j  e.  Z  /\  ( F  |`  ( ZZ>= `  j
) ) : (
ZZ>= `  j ) --> X ) )  ->  J  e.  (TopOn `  X )
)
14 simprl 734 . . . . . 6  |-  ( (
ph  /\  ( j  e.  Z  /\  ( F  |`  ( ZZ>= `  j
) ) : (
ZZ>= `  j ) --> X ) )  ->  j  e.  Z )
1514, 1syl6eleq 2528 . . . . 5  |-  ( (
ph  /\  ( j  e.  Z  /\  ( F  |`  ( ZZ>= `  j
) ) : (
ZZ>= `  j ) --> X ) )  ->  j  e.  ( ZZ>= `  M )
)
16 eluzelz 10498 . . . . 5  |-  ( j  e.  ( ZZ>= `  M
)  ->  j  e.  ZZ )
1715, 16syl 16 . . . 4  |-  ( (
ph  /\  ( j  e.  Z  /\  ( F  |`  ( ZZ>= `  j
) ) : (
ZZ>= `  j ) --> X ) )  ->  j  e.  ZZ )
188adantr 453 . . . 4  |-  ( (
ph  /\  ( j  e.  Z  /\  ( F  |`  ( ZZ>= `  j
) ) : (
ZZ>= `  j ) --> X ) )  ->  F
( ~~> t `  J
) P )
19 fvres 5747 . . . . . . 7  |-  ( k  e.  ( ZZ>= `  j
)  ->  ( ( F  |`  ( ZZ>= `  j
) ) `  k
)  =  ( F `
 k ) )
2019adantl 454 . . . . . 6  |-  ( ( ( ph  /\  (
j  e.  Z  /\  ( F  |`  ( ZZ>= `  j ) ) : ( ZZ>= `  j ) --> X ) )  /\  k  e.  ( ZZ>= `  j ) )  -> 
( ( F  |`  ( ZZ>= `  j )
) `  k )  =  ( F `  k ) )
21 simprr 735 . . . . . . 7  |-  ( (
ph  /\  ( j  e.  Z  /\  ( F  |`  ( ZZ>= `  j
) ) : (
ZZ>= `  j ) --> X ) )  ->  ( F  |`  ( ZZ>= `  j
) ) : (
ZZ>= `  j ) --> X )
2221ffvelrnda 5872 . . . . . 6  |-  ( ( ( ph  /\  (
j  e.  Z  /\  ( F  |`  ( ZZ>= `  j ) ) : ( ZZ>= `  j ) --> X ) )  /\  k  e.  ( ZZ>= `  j ) )  -> 
( ( F  |`  ( ZZ>= `  j )
) `  k )  e.  X )
2320, 22eqeltrrd 2513 . . . . 5  |-  ( ( ( ph  /\  (
j  e.  Z  /\  ( F  |`  ( ZZ>= `  j ) ) : ( ZZ>= `  j ) --> X ) )  /\  k  e.  ( ZZ>= `  j ) )  -> 
( F `  k
)  e.  X )
241uztrn2 10505 . . . . . . 7  |-  ( ( j  e.  Z  /\  k  e.  ( ZZ>= `  j ) )  -> 
k  e.  Z )
2514, 24sylan 459 . . . . . 6  |-  ( ( ( ph  /\  (
j  e.  Z  /\  ( F  |`  ( ZZ>= `  j ) ) : ( ZZ>= `  j ) --> X ) )  /\  k  e.  ( ZZ>= `  j ) )  -> 
k  e.  Z )
26 lmle.10 . . . . . . 7  |-  ( (
ph  /\  k  e.  Z )  ->  ( Q D ( F `  k ) )  <_  R )
2726adantlr 697 . . . . . 6  |-  ( ( ( ph  /\  (
j  e.  Z  /\  ( F  |`  ( ZZ>= `  j ) ) : ( ZZ>= `  j ) --> X ) )  /\  k  e.  Z )  ->  ( Q D ( F `  k ) )  <_  R )
2825, 27syldan 458 . . . . 5  |-  ( ( ( ph  /\  (
j  e.  Z  /\  ( F  |`  ( ZZ>= `  j ) ) : ( ZZ>= `  j ) --> X ) )  /\  k  e.  ( ZZ>= `  j ) )  -> 
( Q D ( F `  k ) )  <_  R )
29 oveq2 6091 . . . . . . 7  |-  ( x  =  ( F `  k )  ->  ( Q D x )  =  ( Q D ( F `  k ) ) )
3029breq1d 4224 . . . . . 6  |-  ( x  =  ( F `  k )  ->  (
( Q D x )  <_  R  <->  ( Q D ( F `  k ) )  <_  R ) )
3130elrab 3094 . . . . 5  |-  ( ( F `  k )  e.  { x  e.  X  |  ( Q D x )  <_  R }  <->  ( ( F `
 k )  e.  X  /\  ( Q D ( F `  k ) )  <_  R ) )
3223, 28, 31sylanbrc 647 . . . 4  |-  ( ( ( ph  /\  (
j  e.  Z  /\  ( F  |`  ( ZZ>= `  j ) ) : ( ZZ>= `  j ) --> X ) )  /\  k  e.  ( ZZ>= `  j ) )  -> 
( F `  k
)  e.  { x  e.  X  |  ( Q D x )  <_  R } )
33 lmle.8 . . . . . 6  |-  ( ph  ->  Q  e.  X )
34 lmle.9 . . . . . 6  |-  ( ph  ->  R  e.  RR* )
35 eqid 2438 . . . . . . 7  |-  { x  e.  X  |  ( Q D x )  <_  R }  =  {
x  e.  X  | 
( Q D x )  <_  R }
363, 35blcld 18537 . . . . . 6  |-  ( ( D  e.  ( * Met `  X )  /\  Q  e.  X  /\  R  e.  RR* )  ->  { x  e.  X  |  ( Q D x )  <_  R }  e.  ( Clsd `  J ) )
372, 33, 34, 36syl3anc 1185 . . . . 5  |-  ( ph  ->  { x  e.  X  |  ( Q D x )  <_  R }  e.  ( Clsd `  J ) )
3837adantr 453 . . . 4  |-  ( (
ph  /\  ( j  e.  Z  /\  ( F  |`  ( ZZ>= `  j
) ) : (
ZZ>= `  j ) --> X ) )  ->  { x  e.  X  |  ( Q D x )  <_  R }  e.  ( Clsd `  J ) )
3912, 13, 17, 18, 32, 38lmcld 17369 . . 3  |-  ( (
ph  /\  ( j  e.  Z  /\  ( F  |`  ( ZZ>= `  j
) ) : (
ZZ>= `  j ) --> X ) )  ->  P  e.  { x  e.  X  |  ( Q D x )  <_  R } )
4011, 39rexlimddv 2836 . 2  |-  ( ph  ->  P  e.  { x  e.  X  |  ( Q D x )  <_  R } )
41 oveq2 6091 . . . . 5  |-  ( x  =  P  ->  ( Q D x )  =  ( Q D P ) )
4241breq1d 4224 . . . 4  |-  ( x  =  P  ->  (
( Q D x )  <_  R  <->  ( Q D P )  <_  R
) )
4342elrab 3094 . . 3  |-  ( P  e.  { x  e.  X  |  ( Q D x )  <_  R }  <->  ( P  e.  X  /\  ( Q D P )  <_  R ) )
4443simprbi 452 . 2  |-  ( P  e.  { x  e.  X  |  ( Q D x )  <_  R }  ->  ( Q D P )  <_  R )
4540, 44syl 16 1  |-  ( ph  ->  ( Q D P )  <_  R )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 360    = wceq 1653    e. wcel 1726   {crab 2711   class class class wbr 4214   dom cdm 4880    |` cres 4882   Rel wrel 4885   -->wf 5452   ` cfv 5456  (class class class)co 6083   RR*cxr 9121    <_ cle 9123   ZZcz 10284   ZZ>=cuz 10490   * Metcxmt 16688   MetOpencmopn 16693  TopOnctopon 16961   Clsdccld 17082   ~~> tclm 17292
This theorem is referenced by:  nvlmle  22190  minvecolem4  22384
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4322  ax-sep 4332  ax-nul 4340  ax-pow 4379  ax-pr 4405  ax-un 4703  ax-cnex 9048  ax-resscn 9049  ax-1cn 9050  ax-icn 9051  ax-addcl 9052  ax-addrcl 9053  ax-mulcl 9054  ax-mulrcl 9055  ax-mulcom 9056  ax-addass 9057  ax-mulass 9058  ax-distr 9059  ax-i2m1 9060  ax-1ne0 9061  ax-1rid 9062  ax-rnegex 9063  ax-rrecex 9064  ax-cnre 9065  ax-pre-lttri 9066  ax-pre-lttrn 9067  ax-pre-ltadd 9068  ax-pre-mulgt0 9069  ax-pre-sup 9070
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2712  df-rex 2713  df-reu 2714  df-rmo 2715  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-pss 3338  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-tp 3824  df-op 3825  df-uni 4018  df-int 4053  df-iun 4097  df-iin 4098  df-br 4215  df-opab 4269  df-mpt 4270  df-tr 4305  df-eprel 4496  df-id 4500  df-po 4505  df-so 4506  df-fr 4543  df-we 4545  df-ord 4586  df-on 4587  df-lim 4588  df-suc 4589  df-om 4848  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-res 4892  df-ima 4893  df-iota 5420  df-fun 5458  df-fn 5459  df-f 5460  df-f1 5461  df-fo 5462  df-f1o 5463  df-fv 5464  df-ov 6086  df-oprab 6087  df-mpt2 6088  df-1st 6351  df-2nd 6352  df-riota 6551  df-recs 6635  df-rdg 6670  df-er 6907  df-map 7022  df-pm 7023  df-en 7112  df-dom 7113  df-sdom 7114  df-sup 7448  df-pnf 9124  df-mnf 9125  df-xr 9126  df-ltxr 9127  df-le 9128  df-sub 9295  df-neg 9296  df-div 9680  df-nn 10003  df-2 10060  df-n0 10224  df-z 10285  df-uz 10491  df-q 10577  df-rp 10615  df-xneg 10712  df-xadd 10713  df-xmul 10714  df-topgen 13669  df-psmet 16696  df-xmet 16697  df-bl 16699  df-mopn 16700  df-top 16965  df-bases 16967  df-topon 16968  df-cld 17085  df-ntr 17086  df-cls 17087  df-lm 17295
  Copyright terms: Public domain W3C validator