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Theorem lmmcvg 22173
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 22172 . . . . 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 213 . . . 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 1019 . . 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 4370 . . . . . 6  |-  ( x  =  R  ->  (
( ( F `  k ) D P )  <  x  <->  ( ( F `  k ) D P )  <  R
) )
11103anbi3d 1341 . . . . 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 2886 . . . 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 3121 . . 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 62 . 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 11127 . . . . . 6  |-  ( ( j  e.  Z  /\  k  e.  ( ZZ>= `  j ) )  -> 
k  e.  Z )
16 3simpc 1004 . . . . . . 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 2490 . . . . . . . 8  |-  ( (
ph  /\  k  e.  Z )  ->  (
( F `  k
)  e.  X  <->  A  e.  X ) )
1917oveq1d 6264 . . . . . . . . 9  |-  ( (
ph  /\  k  e.  Z )  ->  (
( F `  k
) D P )  =  ( A D P ) )
2019breq1d 4376 . . . . . . . 8  |-  ( (
ph  /\  k  e.  Z )  ->  (
( ( F `  k ) D P )  <  R  <->  ( A D P )  <  R
) )
2118, 20anbi12d 715 . . . . . . 7  |-  ( (
ph  /\  k  e.  Z )  ->  (
( ( F `  k )  e.  X  /\  ( ( F `  k ) D P )  <  R )  <-> 
( A  e.  X  /\  ( A D P )  <  R ) ) )
2216, 21syl5ib 222 . . . . . 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 476 . . . . 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 652 . . . 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 2773 . . 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 2839 . 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 370    /\ w3a 982    = wceq 1437    e. wcel 1872   A.wral 2714   E.wrex 2715   class class class wbr 4366   dom cdm 4796   ` cfv 5544  (class class class)co 6249    ^pm cpm 7428   CCcc 9488    < clt 9626   ZZcz 10888   ZZ>=cuz 11110   RR+crp 11253   *Metcxmt 18898   MetOpencmopn 18903   ~~> tclm 20184
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-8 1874  ax-9 1876  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2063  ax-ext 2408  ax-sep 4489  ax-nul 4498  ax-pow 4545  ax-pr 4603  ax-un 6541  ax-cnex 9546  ax-resscn 9547  ax-1cn 9548  ax-icn 9549  ax-addcl 9550  ax-addrcl 9551  ax-mulcl 9552  ax-mulrcl 9553  ax-mulcom 9554  ax-addass 9555  ax-mulass 9556  ax-distr 9557  ax-i2m1 9558  ax-1ne0 9559  ax-1rid 9560  ax-rnegex 9561  ax-rrecex 9562  ax-cnre 9563  ax-pre-lttri 9564  ax-pre-lttrn 9565  ax-pre-ltadd 9566  ax-pre-mulgt0 9567  ax-pre-sup 9568
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 1658  df-nf 1662  df-sb 1791  df-eu 2280  df-mo 2281  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2558  df-ne 2601  df-nel 2602  df-ral 2719  df-rex 2720  df-reu 2721  df-rmo 2722  df-rab 2723  df-v 3024  df-sbc 3243  df-csb 3339  df-dif 3382  df-un 3384  df-in 3386  df-ss 3393  df-pss 3395  df-nul 3705  df-if 3855  df-pw 3926  df-sn 3942  df-pr 3944  df-tp 3946  df-op 3948  df-uni 4163  df-iun 4244  df-br 4367  df-opab 4426  df-mpt 4427  df-tr 4462  df-eprel 4707  df-id 4711  df-po 4717  df-so 4718  df-fr 4755  df-we 4757  df-xp 4802  df-rel 4803  df-cnv 4804  df-co 4805  df-dm 4806  df-rn 4807  df-res 4808  df-ima 4809  df-pred 5342  df-ord 5388  df-on 5389  df-lim 5390  df-suc 5391  df-iota 5508  df-fun 5546  df-fn 5547  df-f 5548  df-f1 5549  df-fo 5550  df-f1o 5551  df-fv 5552  df-riota 6211  df-ov 6252  df-oprab 6253  df-mpt2 6254  df-om 6651  df-1st 6751  df-2nd 6752  df-wrecs 6983  df-recs 7045  df-rdg 7083  df-er 7318  df-map 7429  df-pm 7430  df-en 7525  df-dom 7526  df-sdom 7527  df-sup 7909  df-inf 7910  df-pnf 9628  df-mnf 9629  df-xr 9630  df-ltxr 9631  df-le 9632  df-sub 9813  df-neg 9814  df-div 10221  df-nn 10561  df-2 10619  df-n0 10821  df-z 10889  df-uz 11111  df-q 11216  df-rp 11254  df-xneg 11360  df-xadd 11361  df-xmul 11362  df-topgen 15285  df-psmet 18905  df-xmet 18906  df-bl 18908  df-mopn 18909  df-top 19863  df-bases 19864  df-topon 19865  df-lm 20187
This theorem is referenced by:  bfplem2  32062
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