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Theorem lmmcvg 20771
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 20770 . . . . 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 1002 . . 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 4295 . . . . . 6  |-  ( x  =  R  ->  (
( ( F `  k ) D P )  <  x  <->  ( ( F `  k ) D P )  <  R
) )
11103anbi3d 1295 . . . . 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 2758 . . . 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 3068 . . 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 10877 . . . . . 6  |-  ( ( j  e.  Z  /\  k  e.  ( ZZ>= `  j ) )  -> 
k  e.  Z )
16 3simpc 987 . . . . . . 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 2508 . . . . . . . 8  |-  ( (
ph  /\  k  e.  Z )  ->  (
( F `  k
)  e.  X  <->  A  e.  X ) )
1917oveq1d 6105 . . . . . . . . 9  |-  ( (
ph  /\  k  e.  Z )  ->  (
( F `  k
) D P )  =  ( A D P ) )
2019breq1d 4301 . . . . . . . 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 2793 . . 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 2827 . 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 965    = wceq 1369    e. wcel 1756   A.wral 2714   E.wrex 2715   class class class wbr 4291   dom cdm 4839   ` cfv 5417  (class class class)co 6090    ^pm cpm 7214   CCcc 9279    < clt 9417   ZZcz 10645   ZZ>=cuz 10860   RR+crp 10990   *Metcxmt 17800   MetOpencmopn 17805   ~~> tclm 18829
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-sep 4412  ax-nul 4420  ax-pow 4469  ax-pr 4530  ax-un 6371  ax-cnex 9337  ax-resscn 9338  ax-1cn 9339  ax-icn 9340  ax-addcl 9341  ax-addrcl 9342  ax-mulcl 9343  ax-mulrcl 9344  ax-mulcom 9345  ax-addass 9346  ax-mulass 9347  ax-distr 9348  ax-i2m1 9349  ax-1ne0 9350  ax-1rid 9351  ax-rnegex 9352  ax-rrecex 9353  ax-cnre 9354  ax-pre-lttri 9355  ax-pre-lttrn 9356  ax-pre-ltadd 9357  ax-pre-mulgt0 9358  ax-pre-sup 9359
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 2429  df-cleq 2435  df-clel 2438  df-nfc 2567  df-ne 2607  df-nel 2608  df-ral 2719  df-rex 2720  df-reu 2721  df-rmo 2722  df-rab 2723  df-v 2973  df-sbc 3186  df-csb 3288  df-dif 3330  df-un 3332  df-in 3334  df-ss 3341  df-pss 3343  df-nul 3637  df-if 3791  df-pw 3861  df-sn 3877  df-pr 3879  df-tp 3881  df-op 3883  df-uni 4091  df-iun 4172  df-br 4292  df-opab 4350  df-mpt 4351  df-tr 4385  df-eprel 4631  df-id 4635  df-po 4640  df-so 4641  df-fr 4678  df-we 4680  df-ord 4721  df-on 4722  df-lim 4723  df-suc 4724  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-iota 5380  df-fun 5419  df-fn 5420  df-f 5421  df-f1 5422  df-fo 5423  df-f1o 5424  df-fv 5425  df-riota 6051  df-ov 6093  df-oprab 6094  df-mpt2 6095  df-om 6476  df-1st 6576  df-2nd 6577  df-recs 6831  df-rdg 6865  df-er 7100  df-map 7215  df-pm 7216  df-en 7310  df-dom 7311  df-sdom 7312  df-sup 7690  df-pnf 9419  df-mnf 9420  df-xr 9421  df-ltxr 9422  df-le 9423  df-sub 9596  df-neg 9597  df-div 9993  df-nn 10322  df-2 10379  df-n0 10579  df-z 10646  df-uz 10861  df-q 10953  df-rp 10991  df-xneg 11088  df-xadd 11089  df-xmul 11090  df-topgen 14381  df-psmet 17808  df-xmet 17809  df-bl 17811  df-mopn 17812  df-top 18502  df-bases 18504  df-topon 18505  df-lm 18832
This theorem is referenced by:  bfplem2  28720
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