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Theorem lmmcvg 19219
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 19218 . . . . 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 203 . . . 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 972 . . 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 4219 . . . . . 6  |-  ( x  =  R  ->  (
( ( F `  k ) D P )  <  x  <->  ( ( F `  k ) D P )  <  R
) )
11103anbi3d 1261 . . . . 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 2751 . . . 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 3050 . . 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 59 . 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 10508 . . . . . 6  |-  ( ( j  e.  Z  /\  k  e.  ( ZZ>= `  j ) )  -> 
k  e.  Z )
16 3simpc 957 . . . . . . 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 2504 . . . . . . . 8  |-  ( (
ph  /\  k  e.  Z )  ->  (
( F `  k
)  e.  X  <->  A  e.  X ) )
1917oveq1d 6099 . . . . . . . . 9  |-  ( (
ph  /\  k  e.  Z )  ->  (
( F `  k
) D P )  =  ( A D P ) )
2019breq1d 4225 . . . . . . . 8  |-  ( (
ph  /\  k  e.  Z )  ->  (
( ( F `  k ) D P )  <  R  <->  ( A D P )  <  R
) )
2118, 20anbi12d 693 . . . . . . 7  |-  ( (
ph  /\  k  e.  Z )  ->  (
( ( F `  k )  e.  X  /\  ( ( F `  k ) D P )  <  R )  <-> 
( A  e.  X  /\  ( A D P )  <  R ) ) )
2216, 21syl5ib 212 . . . . . 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 462 . . . . 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 631 . . . 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 2786 . . 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 2820 . 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 set class
Syntax hints:    -> wi 4    /\ wa 360    /\ w3a 937    = wceq 1653    e. wcel 1726   A.wral 2707   E.wrex 2708   class class class wbr 4215   dom cdm 4881   ` cfv 5457  (class class class)co 6084    ^pm cpm 7022   CCcc 8993    < clt 9125   ZZcz 10287   ZZ>=cuz 10493   RR+crp 10617   * Metcxmt 16691   MetOpencmopn 16696   ~~> tclm 17295
This theorem is referenced by:  bfplem2  26546
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 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-sep 4333  ax-nul 4341  ax-pow 4380  ax-pr 4406  ax-un 4704  ax-cnex 9051  ax-resscn 9052  ax-1cn 9053  ax-icn 9054  ax-addcl 9055  ax-addrcl 9056  ax-mulcl 9057  ax-mulrcl 9058  ax-mulcom 9059  ax-addass 9060  ax-mulass 9061  ax-distr 9062  ax-i2m1 9063  ax-1ne0 9064  ax-1rid 9065  ax-rnegex 9066  ax-rrecex 9067  ax-cnre 9068  ax-pre-lttri 9069  ax-pre-lttrn 9070  ax-pre-ltadd 9071  ax-pre-mulgt0 9072  ax-pre-sup 9073
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-iun 4097  df-br 4216  df-opab 4270  df-mpt 4271  df-tr 4306  df-eprel 4497  df-id 4501  df-po 4506  df-so 4507  df-fr 4544  df-we 4546  df-ord 4587  df-on 4588  df-lim 4589  df-suc 4590  df-om 4849  df-xp 4887  df-rel 4888  df-cnv 4889  df-co 4890  df-dm 4891  df-rn 4892  df-res 4893  df-ima 4894  df-iota 5421  df-fun 5459  df-fn 5460  df-f 5461  df-f1 5462  df-fo 5463  df-f1o 5464  df-fv 5465  df-ov 6087  df-oprab 6088  df-mpt2 6089  df-1st 6352  df-2nd 6353  df-riota 6552  df-recs 6636  df-rdg 6671  df-er 6908  df-map 7023  df-pm 7024  df-en 7113  df-dom 7114  df-sdom 7115  df-sup 7449  df-pnf 9127  df-mnf 9128  df-xr 9129  df-ltxr 9130  df-le 9131  df-sub 9298  df-neg 9299  df-div 9683  df-nn 10006  df-2 10063  df-n0 10227  df-z 10288  df-uz 10494  df-q 10580  df-rp 10618  df-xneg 10715  df-xadd 10716  df-xmul 10717  df-topgen 13672  df-psmet 16699  df-xmet 16700  df-bl 16702  df-mopn 16703  df-top 16968  df-bases 16970  df-topon 16971  df-lm 17298
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