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Theorem lmmcvg 21566
Description: Convergence property of a converging sequence. (Contributed by NM, 1-Jun-2007.) (Revised by Mario Carneiro, 1-May-2014.)
Hypotheses
Ref Expression
lmmbr.2  |-  J  =  ( MetOpen `  D )
lmmbr.3  |-  ( ph  ->  D  e.  ( *Met `  X ) )
lmmbr3.5  |-  Z  =  ( ZZ>= `  M )
lmmbr3.6  |-  ( ph  ->  M  e.  ZZ )
lmmbrf.7  |-  ( (
ph  /\  k  e.  Z )  ->  ( F `  k )  =  A )
lmmcvg.8  |-  ( ph  ->  F ( ~~> t `  J ) P )
lmmcvg.9  |-  ( ph  ->  R  e.  RR+ )
Assertion
Ref Expression
lmmcvg  |-  ( ph  ->  E. j  e.  Z  A. k  e.  ( ZZ>=
`  j ) ( A  e.  X  /\  ( A D P )  <  R ) )
Distinct variable groups:    j, k, D    j, F, k    P, j, k    j, X, k   
j, M    ph, j, k    R, j, k    j, Z, k
Allowed substitution hints:    A( j, k)    J( j, k)    M( k)

Proof of Theorem lmmcvg
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 lmmcvg.9 . . 3  |-  ( ph  ->  R  e.  RR+ )
2 lmmcvg.8 . . . . 5  |-  ( ph  ->  F ( ~~> t `  J ) P )
3 lmmbr.2 . . . . . 6  |-  J  =  ( MetOpen `  D )
4 lmmbr.3 . . . . . 6  |-  ( ph  ->  D  e.  ( *Met `  X ) )
5 lmmbr3.5 . . . . . 6  |-  Z  =  ( ZZ>= `  M )
6 lmmbr3.6 . . . . . 6  |-  ( ph  ->  M  e.  ZZ )
73, 4, 5, 6lmmbr3 21565 . . . . 5  |-  ( ph  ->  ( F ( ~~> t `  J ) P  <->  ( F  e.  ( X  ^pm  CC )  /\  P  e.  X  /\  A. x  e.  RR+  E. j  e.  Z  A. k  e.  ( ZZ>= `  j ) ( k  e.  dom  F  /\  ( F `  k )  e.  X  /\  (
( F `  k
) D P )  <  x ) ) ) )
82, 7mpbid 210 . . . 4  |-  ( ph  ->  ( F  e.  ( X  ^pm  CC )  /\  P  e.  X  /\  A. x  e.  RR+  E. j  e.  Z  A. k  e.  ( ZZ>= `  j ) ( k  e.  dom  F  /\  ( F `  k )  e.  X  /\  (
( F `  k
) D P )  <  x ) ) )
98simp3d 1010 . . 3  |-  ( ph  ->  A. x  e.  RR+  E. j  e.  Z  A. k  e.  ( ZZ>= `  j ) ( k  e.  dom  F  /\  ( F `  k )  e.  X  /\  (
( F `  k
) D P )  <  x ) )
10 breq2 4457 . . . . . 6  |-  ( x  =  R  ->  (
( ( F `  k ) D P )  <  x  <->  ( ( F `  k ) D P )  <  R
) )
11103anbi3d 1305 . . . . 5  |-  ( x  =  R  ->  (
( k  e.  dom  F  /\  ( F `  k )  e.  X  /\  ( ( F `  k ) D P )  <  x )  <-> 
( k  e.  dom  F  /\  ( F `  k )  e.  X  /\  ( ( F `  k ) D P )  <  R ) ) )
1211rexralbidv 2986 . . . 4  |-  ( x  =  R  ->  ( E. j  e.  Z  A. k  e.  ( ZZ>=
`  j ) ( k  e.  dom  F  /\  ( F `  k
)  e.  X  /\  ( ( F `  k ) D P )  <  x )  <->  E. j  e.  Z  A. k  e.  ( ZZ>=
`  j ) ( k  e.  dom  F  /\  ( F `  k
)  e.  X  /\  ( ( F `  k ) D P )  <  R ) ) )
1312rspcv 3215 . . 3  |-  ( R  e.  RR+  ->  ( A. x  e.  RR+  E. j  e.  Z  A. k  e.  ( ZZ>= `  j )
( k  e.  dom  F  /\  ( F `  k )  e.  X  /\  ( ( F `  k ) D P )  <  x )  ->  E. j  e.  Z  A. k  e.  ( ZZ>=
`  j ) ( k  e.  dom  F  /\  ( F `  k
)  e.  X  /\  ( ( F `  k ) D P )  <  R ) ) )
141, 9, 13sylc 60 . 2  |-  ( ph  ->  E. j  e.  Z  A. k  e.  ( ZZ>=
`  j ) ( k  e.  dom  F  /\  ( F `  k
)  e.  X  /\  ( ( F `  k ) D P )  <  R ) )
155uztrn2 11111 . . . . . 6  |-  ( ( j  e.  Z  /\  k  e.  ( ZZ>= `  j ) )  -> 
k  e.  Z )
16 3simpc 995 . . . . . . 7  |-  ( ( k  e.  dom  F  /\  ( F `  k
)  e.  X  /\  ( ( F `  k ) D P )  <  R )  ->  ( ( F `
 k )  e.  X  /\  ( ( F `  k ) D P )  < 
R ) )
17 lmmbrf.7 . . . . . . . . 9  |-  ( (
ph  /\  k  e.  Z )  ->  ( F `  k )  =  A )
1817eleq1d 2536 . . . . . . . 8  |-  ( (
ph  /\  k  e.  Z )  ->  (
( F `  k
)  e.  X  <->  A  e.  X ) )
1917oveq1d 6310 . . . . . . . . 9  |-  ( (
ph  /\  k  e.  Z )  ->  (
( F `  k
) D P )  =  ( A D P ) )
2019breq1d 4463 . . . . . . . 8  |-  ( (
ph  /\  k  e.  Z )  ->  (
( ( F `  k ) D P )  <  R  <->  ( A D P )  <  R
) )
2118, 20anbi12d 710 . . . . . . 7  |-  ( (
ph  /\  k  e.  Z )  ->  (
( ( F `  k )  e.  X  /\  ( ( F `  k ) D P )  <  R )  <-> 
( A  e.  X  /\  ( A D P )  <  R ) ) )
2216, 21syl5ib 219 . . . . . 6  |-  ( (
ph  /\  k  e.  Z )  ->  (
( k  e.  dom  F  /\  ( F `  k )  e.  X  /\  ( ( F `  k ) D P )  <  R )  ->  ( A  e.  X  /\  ( A D P )  < 
R ) ) )
2315, 22sylan2 474 . . . . 5  |-  ( (
ph  /\  ( j  e.  Z  /\  k  e.  ( ZZ>= `  j )
) )  ->  (
( k  e.  dom  F  /\  ( F `  k )  e.  X  /\  ( ( F `  k ) D P )  <  R )  ->  ( A  e.  X  /\  ( A D P )  < 
R ) ) )
2423anassrs 648 . . . 4  |-  ( ( ( ph  /\  j  e.  Z )  /\  k  e.  ( ZZ>= `  j )
)  ->  ( (
k  e.  dom  F  /\  ( F `  k
)  e.  X  /\  ( ( F `  k ) D P )  <  R )  ->  ( A  e.  X  /\  ( A D P )  < 
R ) ) )
2524ralimdva 2875 . . 3  |-  ( (
ph  /\  j  e.  Z )  ->  ( A. k  e.  ( ZZ>=
`  j ) ( k  e.  dom  F  /\  ( F `  k
)  e.  X  /\  ( ( F `  k ) D P )  <  R )  ->  A. k  e.  (
ZZ>= `  j ) ( A  e.  X  /\  ( A D P )  <  R ) ) )
2625reximdva 2942 . 2  |-  ( ph  ->  ( E. j  e.  Z  A. k  e.  ( ZZ>= `  j )
( k  e.  dom  F  /\  ( F `  k )  e.  X  /\  ( ( F `  k ) D P )  <  R )  ->  E. j  e.  Z  A. k  e.  ( ZZ>=
`  j ) ( A  e.  X  /\  ( A D P )  <  R ) ) )
2714, 26mpd 15 1  |-  ( ph  ->  E. j  e.  Z  A. k  e.  ( ZZ>=
`  j ) ( A  e.  X  /\  ( A D P )  <  R ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 973    = wceq 1379    e. wcel 1767   A.wral 2817   E.wrex 2818   class class class wbr 4453   dom cdm 5005   ` cfv 5594  (class class class)co 6295    ^pm cpm 7433   CCcc 9502    < clt 9640   ZZcz 10876   ZZ>=cuz 11094   RR+crp 11232   *Metcxmt 18271   MetOpencmopn 18276   ~~> tclm 19593
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-un 6587  ax-cnex 9560  ax-resscn 9561  ax-1cn 9562  ax-icn 9563  ax-addcl 9564  ax-addrcl 9565  ax-mulcl 9566  ax-mulrcl 9567  ax-mulcom 9568  ax-addass 9569  ax-mulass 9570  ax-distr 9571  ax-i2m1 9572  ax-1ne0 9573  ax-1rid 9574  ax-rnegex 9575  ax-rrecex 9576  ax-cnre 9577  ax-pre-lttri 9578  ax-pre-lttrn 9579  ax-pre-ltadd 9580  ax-pre-mulgt0 9581  ax-pre-sup 9582
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2822  df-rex 2823  df-reu 2824  df-rmo 2825  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-pss 3497  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-tp 4038  df-op 4040  df-uni 4252  df-iun 4333  df-br 4454  df-opab 4512  df-mpt 4513  df-tr 4547  df-eprel 4797  df-id 4801  df-po 4806  df-so 4807  df-fr 4844  df-we 4846  df-ord 4887  df-on 4888  df-lim 4889  df-suc 4890  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-riota 6256  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-om 6696  df-1st 6795  df-2nd 6796  df-recs 7054  df-rdg 7088  df-er 7323  df-map 7434  df-pm 7435  df-en 7529  df-dom 7530  df-sdom 7531  df-sup 7913  df-pnf 9642  df-mnf 9643  df-xr 9644  df-ltxr 9645  df-le 9646  df-sub 9819  df-neg 9820  df-div 10219  df-nn 10549  df-2 10606  df-n0 10808  df-z 10877  df-uz 11095  df-q 11195  df-rp 11233  df-xneg 11330  df-xadd 11331  df-xmul 11332  df-topgen 14715  df-psmet 18279  df-xmet 18280  df-bl 18282  df-mopn 18283  df-top 19266  df-bases 19268  df-topon 19269  df-lm 19596
This theorem is referenced by:  bfplem2  30253
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