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Theorem lmmcvg 19171
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 19170 . . . . 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 202 . . . 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 971 . . 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 4180 . . . . . 6  |-  ( x  =  R  ->  (
( ( F `  k ) D P )  <  x  <->  ( ( F `  k ) D P )  <  R
) )
11103anbi3d 1260 . . . . 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 2714 . . . 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 3012 . . 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 58 . 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 10463 . . . . . 6  |-  ( ( j  e.  Z  /\  k  e.  ( ZZ>= `  j ) )  -> 
k  e.  Z )
16 3simpc 956 . . . . . . 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 2474 . . . . . . . 8  |-  ( (
ph  /\  k  e.  Z )  ->  (
( F `  k
)  e.  X  <->  A  e.  X ) )
1917oveq1d 6059 . . . . . . . . 9  |-  ( (
ph  /\  k  e.  Z )  ->  (
( F `  k
) D P )  =  ( A D P ) )
2019breq1d 4186 . . . . . . . 8  |-  ( (
ph  /\  k  e.  Z )  ->  (
( ( F `  k ) D P )  <  R  <->  ( A D P )  <  R
) )
2118, 20anbi12d 692 . . . . . . 7  |-  ( (
ph  /\  k  e.  Z )  ->  (
( ( F `  k )  e.  X  /\  ( ( F `  k ) D P )  <  R )  <-> 
( A  e.  X  /\  ( A D P )  <  R ) ) )
2216, 21syl5ib 211 . . . . . 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 461 . . . . 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 630 . . . 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 2748 . . 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 2782 . 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 359    /\ w3a 936    = wceq 1649    e. wcel 1721   A.wral 2670   E.wrex 2671   class class class wbr 4176   dom cdm 4841   ` cfv 5417  (class class class)co 6044    ^pm cpm 6982   CCcc 8948    < clt 9080   ZZcz 10242   ZZ>=cuz 10448   RR+crp 10572   * Metcxmt 16645   MetOpencmopn 16650   ~~> tclm 17248
This theorem is referenced by:  bfplem2  26426
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2389  ax-sep 4294  ax-nul 4302  ax-pow 4341  ax-pr 4367  ax-un 4664  ax-cnex 9006  ax-resscn 9007  ax-1cn 9008  ax-icn 9009  ax-addcl 9010  ax-addrcl 9011  ax-mulcl 9012  ax-mulrcl 9013  ax-mulcom 9014  ax-addass 9015  ax-mulass 9016  ax-distr 9017  ax-i2m1 9018  ax-1ne0 9019  ax-1rid 9020  ax-rnegex 9021  ax-rrecex 9022  ax-cnre 9023  ax-pre-lttri 9024  ax-pre-lttrn 9025  ax-pre-ltadd 9026  ax-pre-mulgt0 9027  ax-pre-sup 9028
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2262  df-mo 2263  df-clab 2395  df-cleq 2401  df-clel 2404  df-nfc 2533  df-ne 2573  df-nel 2574  df-ral 2675  df-rex 2676  df-reu 2677  df-rmo 2678  df-rab 2679  df-v 2922  df-sbc 3126  df-csb 3216  df-dif 3287  df-un 3289  df-in 3291  df-ss 3298  df-pss 3300  df-nul 3593  df-if 3704  df-pw 3765  df-sn 3784  df-pr 3785  df-tp 3786  df-op 3787  df-uni 3980  df-iun 4059  df-br 4177  df-opab 4231  df-mpt 4232  df-tr 4267  df-eprel 4458  df-id 4462  df-po 4467  df-so 4468  df-fr 4505  df-we 4507  df-ord 4548  df-on 4549  df-lim 4550  df-suc 4551  df-om 4809  df-xp 4847  df-rel 4848  df-cnv 4849  df-co 4850  df-dm 4851  df-rn 4852  df-res 4853  df-ima 4854  df-iota 5381  df-fun 5419  df-fn 5420  df-f 5421  df-f1 5422  df-fo 5423  df-f1o 5424  df-fv 5425  df-ov 6047  df-oprab 6048  df-mpt2 6049  df-1st 6312  df-2nd 6313  df-riota 6512  df-recs 6596  df-rdg 6631  df-er 6868  df-map 6983  df-pm 6984  df-en 7073  df-dom 7074  df-sdom 7075  df-sup 7408  df-pnf 9082  df-mnf 9083  df-xr 9084  df-ltxr 9085  df-le 9086  df-sub 9253  df-neg 9254  df-div 9638  df-nn 9961  df-2 10018  df-n0 10182  df-z 10243  df-uz 10449  df-q 10535  df-rp 10573  df-xneg 10670  df-xadd 10671  df-xmul 10672  df-topgen 13626  df-psmet 16653  df-xmet 16654  df-bl 16656  df-mopn 16657  df-top 16922  df-bases 16924  df-topon 16925  df-lm 17251
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