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Theorem lmle 20831
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 20033 . . . . 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 18853 . . . . 5  |-  Rel  ( ~~> t `  J )
8 lmle.7 . . . . 5  |-  ( ph  ->  F ( ~~> t `  J ) P )
9 releldm 5091 . . . . 5  |-  ( ( Rel  ( ~~> t `  J )  /\  F
( ~~> t `  J
) P )  ->  F  e.  dom  ( ~~> t `  J ) )
107, 8, 9sylancr 663 . . . 4  |-  ( ph  ->  F  e.  dom  ( ~~> t `  J )
)
111, 5, 6, 10lmff 18924 . . 3  |-  ( ph  ->  E. j  e.  Z  ( F  |`  ( ZZ>= `  j ) ) : ( ZZ>= `  j ) --> X )
12 eqid 2443 . . . 4  |-  ( ZZ>= `  j )  =  (
ZZ>= `  j )
135adantr 465 . . . 4  |-  ( (
ph  /\  ( j  e.  Z  /\  ( F  |`  ( ZZ>= `  j
) ) : (
ZZ>= `  j ) --> X ) )  ->  J  e.  (TopOn `  X )
)
14 simprl 755 . . . . . 6  |-  ( (
ph  /\  ( j  e.  Z  /\  ( F  |`  ( ZZ>= `  j
) ) : (
ZZ>= `  j ) --> X ) )  ->  j  e.  Z )
1514, 1syl6eleq 2533 . . . . 5  |-  ( (
ph  /\  ( j  e.  Z  /\  ( F  |`  ( ZZ>= `  j
) ) : (
ZZ>= `  j ) --> X ) )  ->  j  e.  ( ZZ>= `  M )
)
16 eluzelz 10889 . . . . 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 465 . . . 4  |-  ( (
ph  /\  ( j  e.  Z  /\  ( F  |`  ( ZZ>= `  j
) ) : (
ZZ>= `  j ) --> X ) )  ->  F
( ~~> t `  J
) P )
19 fvres 5723 . . . . . . 7  |-  ( k  e.  ( ZZ>= `  j
)  ->  ( ( F  |`  ( ZZ>= `  j
) ) `  k
)  =  ( F `
 k ) )
2019adantl 466 . . . . . 6  |-  ( ( ( ph  /\  (
j  e.  Z  /\  ( F  |`  ( ZZ>= `  j ) ) : ( ZZ>= `  j ) --> X ) )  /\  k  e.  ( ZZ>= `  j ) )  -> 
( ( F  |`  ( ZZ>= `  j )
) `  k )  =  ( F `  k ) )
21 simprr 756 . . . . . . 7  |-  ( (
ph  /\  ( j  e.  Z  /\  ( F  |`  ( ZZ>= `  j
) ) : (
ZZ>= `  j ) --> X ) )  ->  ( F  |`  ( ZZ>= `  j
) ) : (
ZZ>= `  j ) --> X )
2221ffvelrnda 5862 . . . . . 6  |-  ( ( ( ph  /\  (
j  e.  Z  /\  ( F  |`  ( ZZ>= `  j ) ) : ( ZZ>= `  j ) --> X ) )  /\  k  e.  ( ZZ>= `  j ) )  -> 
( ( F  |`  ( ZZ>= `  j )
) `  k )  e.  X )
2320, 22eqeltrrd 2518 . . . . 5  |-  ( ( ( ph  /\  (
j  e.  Z  /\  ( F  |`  ( ZZ>= `  j ) ) : ( ZZ>= `  j ) --> X ) )  /\  k  e.  ( ZZ>= `  j ) )  -> 
( F `  k
)  e.  X )
241uztrn2 10897 . . . . . . 7  |-  ( ( j  e.  Z  /\  k  e.  ( ZZ>= `  j ) )  -> 
k  e.  Z )
2514, 24sylan 471 . . . . . 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 714 . . . . . 6  |-  ( ( ( ph  /\  (
j  e.  Z  /\  ( F  |`  ( ZZ>= `  j ) ) : ( ZZ>= `  j ) --> X ) )  /\  k  e.  Z )  ->  ( Q D ( F `  k ) )  <_  R )
2825, 27syldan 470 . . . . 5  |-  ( ( ( ph  /\  (
j  e.  Z  /\  ( F  |`  ( ZZ>= `  j ) ) : ( ZZ>= `  j ) --> X ) )  /\  k  e.  ( ZZ>= `  j ) )  -> 
( Q D ( F `  k ) )  <_  R )
29 oveq2 6118 . . . . . . 7  |-  ( x  =  ( F `  k )  ->  ( Q D x )  =  ( Q D ( F `  k ) ) )
3029breq1d 4321 . . . . . 6  |-  ( x  =  ( F `  k )  ->  (
( Q D x )  <_  R  <->  ( Q D ( F `  k ) )  <_  R ) )
3130elrab 3136 . . . . 5  |-  ( ( F `  k )  e.  { x  e.  X  |  ( Q D x )  <_  R }  <->  ( ( F `
 k )  e.  X  /\  ( Q D ( F `  k ) )  <_  R ) )
3223, 28, 31sylanbrc 664 . . . 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 2443 . . . . . . 7  |-  { x  e.  X  |  ( Q D x )  <_  R }  =  {
x  e.  X  | 
( Q D x )  <_  R }
363, 35blcld 20099 . . . . . 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 1218 . . . . 5  |-  ( ph  ->  { x  e.  X  |  ( Q D x )  <_  R }  e.  ( Clsd `  J ) )
3837adantr 465 . . . 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 18926 . . 3  |-  ( (
ph  /\  ( j  e.  Z  /\  ( F  |`  ( ZZ>= `  j
) ) : (
ZZ>= `  j ) --> X ) )  ->  P  e.  { x  e.  X  |  ( Q D x )  <_  R } )
4011, 39rexlimddv 2864 . 2  |-  ( ph  ->  P  e.  { x  e.  X  |  ( Q D x )  <_  R } )
41 oveq2 6118 . . . . 5  |-  ( x  =  P  ->  ( Q D x )  =  ( Q D P ) )
4241breq1d 4321 . . . 4  |-  ( x  =  P  ->  (
( Q D x )  <_  R  <->  ( Q D P )  <_  R
) )
4342elrab 3136 . . 3  |-  ( P  e.  { x  e.  X  |  ( Q D x )  <_  R }  <->  ( P  e.  X  /\  ( Q D P )  <_  R ) )
4443simprbi 464 . 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 setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1369    e. wcel 1756   {crab 2738   class class class wbr 4311   dom cdm 4859    |` cres 4861   Rel wrel 4864   -->wf 5433   ` cfv 5437  (class class class)co 6110   RR*cxr 9436    <_ cle 9438   ZZcz 10665   ZZ>=cuz 10880   *Metcxmt 17820   MetOpencmopn 17825  TopOnctopon 18518   Clsdccld 18639   ~~> tclm 18849
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-rep 4422  ax-sep 4432  ax-nul 4440  ax-pow 4489  ax-pr 4550  ax-un 6391  ax-cnex 9357  ax-resscn 9358  ax-1cn 9359  ax-icn 9360  ax-addcl 9361  ax-addrcl 9362  ax-mulcl 9363  ax-mulrcl 9364  ax-mulcom 9365  ax-addass 9366  ax-mulass 9367  ax-distr 9368  ax-i2m1 9369  ax-1ne0 9370  ax-1rid 9371  ax-rnegex 9372  ax-rrecex 9373  ax-cnre 9374  ax-pre-lttri 9375  ax-pre-lttrn 9376  ax-pre-ltadd 9377  ax-pre-mulgt0 9378  ax-pre-sup 9379
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 2577  df-ne 2622  df-nel 2623  df-ral 2739  df-rex 2740  df-reu 2741  df-rmo 2742  df-rab 2743  df-v 2993  df-sbc 3206  df-csb 3308  df-dif 3350  df-un 3352  df-in 3354  df-ss 3361  df-pss 3363  df-nul 3657  df-if 3811  df-pw 3881  df-sn 3897  df-pr 3899  df-tp 3901  df-op 3903  df-uni 4111  df-int 4148  df-iun 4192  df-iin 4193  df-br 4312  df-opab 4370  df-mpt 4371  df-tr 4405  df-eprel 4651  df-id 4655  df-po 4660  df-so 4661  df-fr 4698  df-we 4700  df-ord 4741  df-on 4742  df-lim 4743  df-suc 4744  df-xp 4865  df-rel 4866  df-cnv 4867  df-co 4868  df-dm 4869  df-rn 4870  df-res 4871  df-ima 4872  df-iota 5400  df-fun 5439  df-fn 5440  df-f 5441  df-f1 5442  df-fo 5443  df-f1o 5444  df-fv 5445  df-riota 6071  df-ov 6113  df-oprab 6114  df-mpt2 6115  df-om 6496  df-1st 6596  df-2nd 6597  df-recs 6851  df-rdg 6885  df-er 7120  df-map 7235  df-pm 7236  df-en 7330  df-dom 7331  df-sdom 7332  df-sup 7710  df-pnf 9439  df-mnf 9440  df-xr 9441  df-ltxr 9442  df-le 9443  df-sub 9616  df-neg 9617  df-div 10013  df-nn 10342  df-2 10399  df-n0 10599  df-z 10666  df-uz 10881  df-q 10973  df-rp 11011  df-xneg 11108  df-xadd 11109  df-xmul 11110  df-topgen 14401  df-psmet 17828  df-xmet 17829  df-bl 17831  df-mopn 17832  df-top 18522  df-bases 18524  df-topon 18525  df-cld 18642  df-ntr 18643  df-cls 18644  df-lm 18852
This theorem is referenced by:  nvlmle  24106  minvecolem4  24300
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