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Theorem climfsum 12601
Description: Limit of a finite sum of converging sequences. Note that  F ( k ) is a collection of functions with implicit parameter  k, each of which converges to  B ( k ) as  n  ~~>  +oo. (Contributed by Mario Carneiro, 22-Jul-2014.) (Proof shortened by Mario Carneiro, 22-May-2016.)
Hypotheses
Ref Expression
climfsum.1  |-  Z  =  ( ZZ>= `  M )
climfsum.2  |-  ( ph  ->  M  e.  ZZ )
climfsum.3  |-  ( ph  ->  A  e.  Fin )
climfsum.5  |-  ( (
ph  /\  k  e.  A )  ->  F  ~~>  B )
climfsum.6  |-  ( ph  ->  H  e.  W )
climfsum.7  |-  ( (
ph  /\  ( k  e.  A  /\  n  e.  Z ) )  -> 
( F `  n
)  e.  CC )
climfsum.8  |-  ( (
ph  /\  n  e.  Z )  ->  ( H `  n )  =  sum_ k  e.  A  ( F `  n ) )
Assertion
Ref Expression
climfsum  |-  ( ph  ->  H  ~~>  sum_ k  e.  A  B )
Distinct variable groups:    k, n, A    n, H    ph, k, n   
k, Z, n    B, n    n, F    n, M
Allowed substitution hints:    B( k)    F( k)    H( k)    M( k)    W( k, n)

Proof of Theorem climfsum
StepHypRef Expression
1 climfsum.8 . . . 4  |-  ( (
ph  /\  n  e.  Z )  ->  ( H `  n )  =  sum_ k  e.  A  ( F `  n ) )
21mpteq2dva 4297 . . 3  |-  ( ph  ->  ( n  e.  Z  |->  ( H `  n
) )  =  ( n  e.  Z  |->  sum_ k  e.  A  ( F `  n ) ) )
3 climfsum.1 . . . . . . . 8  |-  Z  =  ( ZZ>= `  M )
4 uzssz 10507 . . . . . . . 8  |-  ( ZZ>= `  M )  C_  ZZ
53, 4eqsstri 3380 . . . . . . 7  |-  Z  C_  ZZ
6 zssre 10291 . . . . . . 7  |-  ZZ  C_  RR
75, 6sstri 3359 . . . . . 6  |-  Z  C_  RR
87a1i 11 . . . . 5  |-  ( ph  ->  Z  C_  RR )
9 climfsum.3 . . . . 5  |-  ( ph  ->  A  e.  Fin )
10 fvex 5744 . . . . . 6  |-  ( F `
 n )  e. 
_V
1110a1i 11 . . . . 5  |-  ( (
ph  /\  ( n  e.  Z  /\  k  e.  A ) )  -> 
( F `  n
)  e.  _V )
12 climfsum.5 . . . . . . 7  |-  ( (
ph  /\  k  e.  A )  ->  F  ~~>  B )
13 climfsum.2 . . . . . . . . 9  |-  ( ph  ->  M  e.  ZZ )
1413adantr 453 . . . . . . . 8  |-  ( (
ph  /\  k  e.  A )  ->  M  e.  ZZ )
15 climrel 12288 . . . . . . . . . 10  |-  Rel  ~~>
1615brrelexi 4920 . . . . . . . . 9  |-  ( F  ~~>  B  ->  F  e.  _V )
1712, 16syl 16 . . . . . . . 8  |-  ( (
ph  /\  k  e.  A )  ->  F  e.  _V )
18 eqid 2438 . . . . . . . . 9  |-  ( n  e.  Z  |->  ( F `
 n ) )  =  ( n  e.  Z  |->  ( F `  n ) )
193, 18climmpt 12367 . . . . . . . 8  |-  ( ( M  e.  ZZ  /\  F  e.  _V )  ->  ( F  ~~>  B  <->  ( n  e.  Z  |->  ( F `
 n ) )  ~~>  B ) )
2014, 17, 19syl2anc 644 . . . . . . 7  |-  ( (
ph  /\  k  e.  A )  ->  ( F 
~~>  B  <->  ( n  e.  Z  |->  ( F `  n ) )  ~~>  B ) )
2112, 20mpbid 203 . . . . . 6  |-  ( (
ph  /\  k  e.  A )  ->  (
n  e.  Z  |->  ( F `  n ) )  ~~>  B )
22 climfsum.7 . . . . . . . . 9  |-  ( (
ph  /\  ( k  e.  A  /\  n  e.  Z ) )  -> 
( F `  n
)  e.  CC )
2322anassrs 631 . . . . . . . 8  |-  ( ( ( ph  /\  k  e.  A )  /\  n  e.  Z )  ->  ( F `  n )  e.  CC )
2423, 18fmptd 5895 . . . . . . 7  |-  ( (
ph  /\  k  e.  A )  ->  (
n  e.  Z  |->  ( F `  n ) ) : Z --> CC )
253, 14, 24rlimclim 12342 . . . . . 6  |-  ( (
ph  /\  k  e.  A )  ->  (
( n  e.  Z  |->  ( F `  n
) )  ~~> r  B  <->  ( n  e.  Z  |->  ( F `  n ) )  ~~>  B ) )
2621, 25mpbird 225 . . . . 5  |-  ( (
ph  /\  k  e.  A )  ->  (
n  e.  Z  |->  ( F `  n ) )  ~~> r  B )
278, 9, 11, 26fsumrlim 12592 . . . 4  |-  ( ph  ->  ( n  e.  Z  |-> 
sum_ k  e.  A  ( F `  n ) )  ~~> r  sum_ k  e.  A  B )
289adantr 453 . . . . . . 7  |-  ( (
ph  /\  n  e.  Z )  ->  A  e.  Fin )
2922anass1rs 784 . . . . . . 7  |-  ( ( ( ph  /\  n  e.  Z )  /\  k  e.  A )  ->  ( F `  n )  e.  CC )
3028, 29fsumcl 12529 . . . . . 6  |-  ( (
ph  /\  n  e.  Z )  ->  sum_ k  e.  A  ( F `  n )  e.  CC )
31 eqid 2438 . . . . . 6  |-  ( n  e.  Z  |->  sum_ k  e.  A  ( F `  n ) )  =  ( n  e.  Z  |-> 
sum_ k  e.  A  ( F `  n ) )
3230, 31fmptd 5895 . . . . 5  |-  ( ph  ->  ( n  e.  Z  |-> 
sum_ k  e.  A  ( F `  n ) ) : Z --> CC )
333, 13, 32rlimclim 12342 . . . 4  |-  ( ph  ->  ( ( n  e.  Z  |->  sum_ k  e.  A  ( F `  n ) )  ~~> r  sum_ k  e.  A  B  <->  ( n  e.  Z  |->  sum_ k  e.  A  ( F `  n ) )  ~~>  sum_ k  e.  A  B )
)
3427, 33mpbid 203 . . 3  |-  ( ph  ->  ( n  e.  Z  |-> 
sum_ k  e.  A  ( F `  n ) )  ~~>  sum_ k  e.  A  B )
352, 34eqbrtrd 4234 . 2  |-  ( ph  ->  ( n  e.  Z  |->  ( H `  n
) )  ~~>  sum_ k  e.  A  B )
36 climfsum.6 . . 3  |-  ( ph  ->  H  e.  W )
37 eqid 2438 . . . 4  |-  ( n  e.  Z  |->  ( H `
 n ) )  =  ( n  e.  Z  |->  ( H `  n ) )
383, 37climmpt 12367 . . 3  |-  ( ( M  e.  ZZ  /\  H  e.  W )  ->  ( H  ~~>  sum_ k  e.  A  B  <->  ( n  e.  Z  |->  ( H `
 n ) )  ~~> 
sum_ k  e.  A  B ) )
3913, 36, 38syl2anc 644 . 2  |-  ( ph  ->  ( H  ~~>  sum_ k  e.  A  B  <->  ( n  e.  Z  |->  ( H `
 n ) )  ~~> 
sum_ k  e.  A  B ) )
4035, 39mpbird 225 1  |-  ( ph  ->  H  ~~>  sum_ k  e.  A  B )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 178    /\ wa 360    = wceq 1653    e. wcel 1726   _Vcvv 2958    C_ wss 3322   class class class wbr 4214    e. cmpt 4268   ` cfv 5456   Fincfn 7111   CCcc 8990   RRcr 8991   ZZcz 10284   ZZ>=cuz 10490    ~~> cli 12280    ~~> r crli 12281   sum_csu 12481
This theorem is referenced by:  itg1climres  19608  plyeq0lem  20131
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 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-rep 4322  ax-sep 4332  ax-nul 4340  ax-pow 4379  ax-pr 4405  ax-un 4703  ax-inf2 7598  ax-cnex 9048  ax-resscn 9049  ax-1cn 9050  ax-icn 9051  ax-addcl 9052  ax-addrcl 9053  ax-mulcl 9054  ax-mulrcl 9055  ax-mulcom 9056  ax-addass 9057  ax-mulass 9058  ax-distr 9059  ax-i2m1 9060  ax-1ne0 9061  ax-1rid 9062  ax-rnegex 9063  ax-rrecex 9064  ax-cnre 9065  ax-pre-lttri 9066  ax-pre-lttrn 9067  ax-pre-ltadd 9068  ax-pre-mulgt0 9069  ax-pre-sup 9070  ax-addf 9071
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-int 4053  df-iun 4097  df-br 4215  df-opab 4269  df-mpt 4270  df-tr 4305  df-eprel 4496  df-id 4500  df-po 4505  df-so 4506  df-fr 4543  df-se 4544  df-we 4545  df-ord 4586  df-on 4587  df-lim 4588  df-suc 4589  df-om 4848  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-res 4892  df-ima 4893  df-iota 5420  df-fun 5458  df-fn 5459  df-f 5460  df-f1 5461  df-fo 5462  df-f1o 5463  df-fv 5464  df-isom 5465  df-ov 6086  df-oprab 6087  df-mpt2 6088  df-1st 6351  df-2nd 6352  df-riota 6551  df-recs 6635  df-rdg 6670  df-1o 6726  df-oadd 6730  df-er 6907  df-pm 7023  df-en 7112  df-dom 7113  df-sdom 7114  df-fin 7115  df-sup 7448  df-oi 7481  df-card 7828  df-pnf 9124  df-mnf 9125  df-xr 9126  df-ltxr 9127  df-le 9128  df-sub 9295  df-neg 9296  df-div 9680  df-nn 10003  df-2 10060  df-3 10061  df-n0 10224  df-z 10285  df-uz 10491  df-rp 10615  df-fz 11046  df-fzo 11138  df-fl 11204  df-seq 11326  df-exp 11385  df-hash 11621  df-cj 11906  df-re 11907  df-im 11908  df-sqr 12042  df-abs 12043  df-clim 12284  df-rlim 12285  df-sum 12482
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