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Theorem lmle 22164
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 21385 . . . . 5  |-  ( D  e.  ( *Met `  X )  ->  J  e.  (TopOn `  X )
)
52, 4syl 17 . . . 4  |-  ( ph  ->  J  e.  (TopOn `  X ) )
6 lmle.6 . . . 4  |-  ( ph  ->  M  e.  ZZ )
7 lmrel 20177 . . . . 5  |-  Rel  ( ~~> t `  J )
8 lmle.7 . . . . 5  |-  ( ph  ->  F ( ~~> t `  J ) P )
9 releldm 5087 . . . . 5  |-  ( ( Rel  ( ~~> t `  J )  /\  F
( ~~> t `  J
) P )  ->  F  e.  dom  ( ~~> t `  J ) )
107, 8, 9sylancr 667 . . . 4  |-  ( ph  ->  F  e.  dom  ( ~~> t `  J )
)
111, 5, 6, 10lmff 20248 . . 3  |-  ( ph  ->  E. j  e.  Z  ( F  |`  ( ZZ>= `  j ) ) : ( ZZ>= `  j ) --> X )
12 eqid 2429 . . . 4  |-  ( ZZ>= `  j )  =  (
ZZ>= `  j )
135adantr 466 . . . 4  |-  ( (
ph  /\  ( j  e.  Z  /\  ( F  |`  ( ZZ>= `  j
) ) : (
ZZ>= `  j ) --> X ) )  ->  J  e.  (TopOn `  X )
)
14 simprl 762 . . . . . 6  |-  ( (
ph  /\  ( j  e.  Z  /\  ( F  |`  ( ZZ>= `  j
) ) : (
ZZ>= `  j ) --> X ) )  ->  j  e.  Z )
1514, 1syl6eleq 2527 . . . . 5  |-  ( (
ph  /\  ( j  e.  Z  /\  ( F  |`  ( ZZ>= `  j
) ) : (
ZZ>= `  j ) --> X ) )  ->  j  e.  ( ZZ>= `  M )
)
16 eluzelz 11168 . . . . 5  |-  ( j  e.  ( ZZ>= `  M
)  ->  j  e.  ZZ )
1715, 16syl 17 . . . 4  |-  ( (
ph  /\  ( j  e.  Z  /\  ( F  |`  ( ZZ>= `  j
) ) : (
ZZ>= `  j ) --> X ) )  ->  j  e.  ZZ )
188adantr 466 . . . 4  |-  ( (
ph  /\  ( j  e.  Z  /\  ( F  |`  ( ZZ>= `  j
) ) : (
ZZ>= `  j ) --> X ) )  ->  F
( ~~> t `  J
) P )
19 fvres 5895 . . . . . . 7  |-  ( k  e.  ( ZZ>= `  j
)  ->  ( ( F  |`  ( ZZ>= `  j
) ) `  k
)  =  ( F `
 k ) )
2019adantl 467 . . . . . 6  |-  ( ( ( ph  /\  (
j  e.  Z  /\  ( F  |`  ( ZZ>= `  j ) ) : ( ZZ>= `  j ) --> X ) )  /\  k  e.  ( ZZ>= `  j ) )  -> 
( ( F  |`  ( ZZ>= `  j )
) `  k )  =  ( F `  k ) )
21 simprr 764 . . . . . . 7  |-  ( (
ph  /\  ( j  e.  Z  /\  ( F  |`  ( ZZ>= `  j
) ) : (
ZZ>= `  j ) --> X ) )  ->  ( F  |`  ( ZZ>= `  j
) ) : (
ZZ>= `  j ) --> X )
2221ffvelrnda 6037 . . . . . 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 11176 . . . . . . 7  |-  ( ( j  e.  Z  /\  k  e.  ( ZZ>= `  j ) )  -> 
k  e.  Z )
2514, 24sylan 473 . . . . . 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 719 . . . . . 6  |-  ( ( ( ph  /\  (
j  e.  Z  /\  ( F  |`  ( ZZ>= `  j ) ) : ( ZZ>= `  j ) --> X ) )  /\  k  e.  Z )  ->  ( Q D ( F `  k ) )  <_  R )
2825, 27syldan 472 . . . . 5  |-  ( ( ( ph  /\  (
j  e.  Z  /\  ( F  |`  ( ZZ>= `  j ) ) : ( ZZ>= `  j ) --> X ) )  /\  k  e.  ( ZZ>= `  j ) )  -> 
( Q D ( F `  k ) )  <_  R )
29 oveq2 6313 . . . . . . 7  |-  ( x  =  ( F `  k )  ->  ( Q D x )  =  ( Q D ( F `  k ) ) )
3029breq1d 4436 . . . . . 6  |-  ( x  =  ( F `  k )  ->  (
( Q D x )  <_  R  <->  ( Q D ( F `  k ) )  <_  R ) )
3130elrab 3235 . . . . 5  |-  ( ( F `  k )  e.  { x  e.  X  |  ( Q D x )  <_  R }  <->  ( ( F `
 k )  e.  X  /\  ( Q D ( F `  k ) )  <_  R ) )
3223, 28, 31sylanbrc 668 . . . 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 2429 . . . . . . 7  |-  { x  e.  X  |  ( Q D x )  <_  R }  =  {
x  e.  X  | 
( Q D x )  <_  R }
363, 35blcld 21451 . . . . . 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 1264 . . . . 5  |-  ( ph  ->  { x  e.  X  |  ( Q D x )  <_  R }  e.  ( Clsd `  J ) )
3837adantr 466 . . . 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 20250 . . 3  |-  ( (
ph  /\  ( j  e.  Z  /\  ( F  |`  ( ZZ>= `  j
) ) : (
ZZ>= `  j ) --> X ) )  ->  P  e.  { x  e.  X  |  ( Q D x )  <_  R } )
4011, 39rexlimddv 2928 . 2  |-  ( ph  ->  P  e.  { x  e.  X  |  ( Q D x )  <_  R } )
41 oveq2 6313 . . . . 5  |-  ( x  =  P  ->  ( Q D x )  =  ( Q D P ) )
4241breq1d 4436 . . . 4  |-  ( x  =  P  ->  (
( Q D x )  <_  R  <->  ( Q D P )  <_  R
) )
4342elrab 3235 . . 3  |-  ( P  e.  { x  e.  X  |  ( Q D x )  <_  R }  <->  ( P  e.  X  /\  ( Q D P )  <_  R ) )
4443simprbi 465 . 2  |-  ( P  e.  { x  e.  X  |  ( Q D x )  <_  R }  ->  ( Q D P )  <_  R )
4540, 44syl 17 1  |-  ( ph  ->  ( Q D P )  <_  R )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 370    = wceq 1437    e. wcel 1870   {crab 2786   class class class wbr 4426   dom cdm 4854    |` cres 4856   Rel wrel 4859   -->wf 5597   ` cfv 5601  (class class class)co 6305   RR*cxr 9673    <_ cle 9675   ZZcz 10937   ZZ>=cuz 11159   *Metcxmt 18890   MetOpencmopn 18895  TopOnctopon 19849   Clsdccld 19962   ~~> tclm 20173
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-8 1872  ax-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-rep 4538  ax-sep 4548  ax-nul 4556  ax-pow 4603  ax-pr 4661  ax-un 6597  ax-cnex 9594  ax-resscn 9595  ax-1cn 9596  ax-icn 9597  ax-addcl 9598  ax-addrcl 9599  ax-mulcl 9600  ax-mulrcl 9601  ax-mulcom 9602  ax-addass 9603  ax-mulass 9604  ax-distr 9605  ax-i2m1 9606  ax-1ne0 9607  ax-1rid 9608  ax-rnegex 9609  ax-rrecex 9610  ax-cnre 9611  ax-pre-lttri 9612  ax-pre-lttrn 9613  ax-pre-ltadd 9614  ax-pre-mulgt0 9615  ax-pre-sup 9616
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1790  df-eu 2270  df-mo 2271  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-nel 2628  df-ral 2787  df-rex 2788  df-reu 2789  df-rmo 2790  df-rab 2791  df-v 3089  df-sbc 3306  df-csb 3402  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-pss 3458  df-nul 3768  df-if 3916  df-pw 3987  df-sn 4003  df-pr 4005  df-tp 4007  df-op 4009  df-uni 4223  df-int 4259  df-iun 4304  df-iin 4305  df-br 4427  df-opab 4485  df-mpt 4486  df-tr 4521  df-eprel 4765  df-id 4769  df-po 4775  df-so 4776  df-fr 4813  df-we 4815  df-xp 4860  df-rel 4861  df-cnv 4862  df-co 4863  df-dm 4864  df-rn 4865  df-res 4866  df-ima 4867  df-pred 5399  df-ord 5445  df-on 5446  df-lim 5447  df-suc 5448  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-riota 6267  df-ov 6308  df-oprab 6309  df-mpt2 6310  df-om 6707  df-1st 6807  df-2nd 6808  df-wrecs 7036  df-recs 7098  df-rdg 7136  df-er 7371  df-map 7482  df-pm 7483  df-en 7578  df-dom 7579  df-sdom 7580  df-sup 7962  df-pnf 9676  df-mnf 9677  df-xr 9678  df-ltxr 9679  df-le 9680  df-sub 9861  df-neg 9862  df-div 10269  df-nn 10610  df-2 10668  df-n0 10870  df-z 10938  df-uz 11160  df-q 11265  df-rp 11303  df-xneg 11409  df-xadd 11410  df-xmul 11411  df-topgen 15301  df-psmet 18897  df-xmet 18898  df-bl 18900  df-mopn 18901  df-top 19852  df-bases 19853  df-topon 19854  df-cld 19965  df-ntr 19966  df-cls 19967  df-lm 20176
This theorem is referenced by:  nvlmle  26173  minvecolem4  26367
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