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Theorem climfsum 13279
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 4375 . . 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 10876 . . . . . . . 8  |-  ( ZZ>= `  M )  C_  ZZ
53, 4eqsstri 3383 . . . . . . 7  |-  Z  C_  ZZ
6 zssre 10649 . . . . . . 7  |-  ZZ  C_  RR
75, 6sstri 3362 . . . . . 6  |-  Z  C_  RR
87a1i 11 . . . . 5  |-  ( ph  ->  Z  C_  RR )
9 climfsum.3 . . . . 5  |-  ( ph  ->  A  e.  Fin )
10 fvex 5698 . . . . . 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 462 . . . . . . . 8  |-  ( (
ph  /\  k  e.  A )  ->  M  e.  ZZ )
15 climrel 12966 . . . . . . . . . 10  |-  Rel  ~~>
1615brrelexi 4875 . . . . . . . . 9  |-  ( F  ~~>  B  ->  F  e.  _V )
1712, 16syl 16 . . . . . . . 8  |-  ( (
ph  /\  k  e.  A )  ->  F  e.  _V )
18 eqid 2441 . . . . . . . . 9  |-  ( n  e.  Z  |->  ( F `
 n ) )  =  ( n  e.  Z  |->  ( F `  n ) )
193, 18climmpt 13045 . . . . . . . 8  |-  ( ( M  e.  ZZ  /\  F  e.  _V )  ->  ( F  ~~>  B  <->  ( n  e.  Z  |->  ( F `
 n ) )  ~~>  B ) )
2014, 17, 19syl2anc 656 . . . . . . 7  |-  ( (
ph  /\  k  e.  A )  ->  ( F 
~~>  B  <->  ( n  e.  Z  |->  ( F `  n ) )  ~~>  B ) )
2112, 20mpbid 210 . . . . . 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 643 . . . . . . . 8  |-  ( ( ( ph  /\  k  e.  A )  /\  n  e.  Z )  ->  ( F `  n )  e.  CC )
2423, 18fmptd 5864 . . . . . . 7  |-  ( (
ph  /\  k  e.  A )  ->  (
n  e.  Z  |->  ( F `  n ) ) : Z --> CC )
253, 14, 24rlimclim 13020 . . . . . 6  |-  ( (
ph  /\  k  e.  A )  ->  (
( n  e.  Z  |->  ( F `  n
) )  ~~> r  B  <->  ( n  e.  Z  |->  ( F `  n ) )  ~~>  B ) )
2621, 25mpbird 232 . . . . 5  |-  ( (
ph  /\  k  e.  A )  ->  (
n  e.  Z  |->  ( F `  n ) )  ~~> r  B )
278, 9, 11, 26fsumrlim 13270 . . . 4  |-  ( ph  ->  ( n  e.  Z  |-> 
sum_ k  e.  A  ( F `  n ) )  ~~> r  sum_ k  e.  A  B )
289adantr 462 . . . . . . 7  |-  ( (
ph  /\  n  e.  Z )  ->  A  e.  Fin )
2922anass1rs 800 . . . . . . 7  |-  ( ( ( ph  /\  n  e.  Z )  /\  k  e.  A )  ->  ( F `  n )  e.  CC )
3028, 29fsumcl 13206 . . . . . 6  |-  ( (
ph  /\  n  e.  Z )  ->  sum_ k  e.  A  ( F `  n )  e.  CC )
31 eqid 2441 . . . . . 6  |-  ( n  e.  Z  |->  sum_ k  e.  A  ( F `  n ) )  =  ( n  e.  Z  |-> 
sum_ k  e.  A  ( F `  n ) )
3230, 31fmptd 5864 . . . . 5  |-  ( ph  ->  ( n  e.  Z  |-> 
sum_ k  e.  A  ( F `  n ) ) : Z --> CC )
333, 13, 32rlimclim 13020 . . . 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 210 . . 3  |-  ( ph  ->  ( n  e.  Z  |-> 
sum_ k  e.  A  ( F `  n ) )  ~~>  sum_ k  e.  A  B )
352, 34eqbrtrd 4309 . 2  |-  ( ph  ->  ( n  e.  Z  |->  ( H `  n
) )  ~~>  sum_ k  e.  A  B )
36 climfsum.6 . . 3  |-  ( ph  ->  H  e.  W )
37 eqid 2441 . . . 4  |-  ( n  e.  Z  |->  ( H `
 n ) )  =  ( n  e.  Z  |->  ( H `  n ) )
383, 37climmpt 13045 . . 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 656 . 2  |-  ( ph  ->  ( H  ~~>  sum_ k  e.  A  B  <->  ( n  e.  Z  |->  ( H `
 n ) )  ~~> 
sum_ k  e.  A  B ) )
4035, 39mpbird 232 1  |-  ( ph  ->  H  ~~>  sum_ k  e.  A  B )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1364    e. wcel 1761   _Vcvv 2970    C_ wss 3325   class class class wbr 4289    e. cmpt 4347   ` cfv 5415   Fincfn 7306   CCcc 9276   RRcr 9277   ZZcz 10642   ZZ>=cuz 10857    ~~> cli 12958    ~~> r crli 12959   sum_csu 13159
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1713  ax-7 1733  ax-8 1763  ax-9 1765  ax-10 1780  ax-11 1785  ax-12 1797  ax-13 1948  ax-ext 2422  ax-rep 4400  ax-sep 4410  ax-nul 4418  ax-pow 4467  ax-pr 4528  ax-un 6371  ax-inf2 7843  ax-cnex 9334  ax-resscn 9335  ax-1cn 9336  ax-icn 9337  ax-addcl 9338  ax-addrcl 9339  ax-mulcl 9340  ax-mulrcl 9341  ax-mulcom 9342  ax-addass 9343  ax-mulass 9344  ax-distr 9345  ax-i2m1 9346  ax-1ne0 9347  ax-1rid 9348  ax-rnegex 9349  ax-rrecex 9350  ax-cnre 9351  ax-pre-lttri 9352  ax-pre-lttrn 9353  ax-pre-ltadd 9354  ax-pre-mulgt0 9355  ax-pre-sup 9356  ax-addf 9357
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 961  df-3an 962  df-tru 1367  df-fal 1370  df-ex 1592  df-nf 1595  df-sb 1706  df-eu 2261  df-mo 2262  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-nel 2607  df-ral 2718  df-rex 2719  df-reu 2720  df-rmo 2721  df-rab 2722  df-v 2972  df-sbc 3184  df-csb 3286  df-dif 3328  df-un 3330  df-in 3332  df-ss 3339  df-pss 3341  df-nul 3635  df-if 3789  df-pw 3859  df-sn 3875  df-pr 3877  df-tp 3879  df-op 3881  df-uni 4089  df-int 4126  df-iun 4170  df-br 4290  df-opab 4348  df-mpt 4349  df-tr 4383  df-eprel 4628  df-id 4632  df-po 4637  df-so 4638  df-fr 4675  df-se 4676  df-we 4677  df-ord 4718  df-on 4719  df-lim 4720  df-suc 4721  df-xp 4842  df-rel 4843  df-cnv 4844  df-co 4845  df-dm 4846  df-rn 4847  df-res 4848  df-ima 4849  df-iota 5378  df-fun 5417  df-fn 5418  df-f 5419  df-f1 5420  df-fo 5421  df-f1o 5422  df-fv 5423  df-isom 5424  df-riota 6049  df-ov 6093  df-oprab 6094  df-mpt2 6095  df-om 6476  df-1st 6576  df-2nd 6577  df-recs 6828  df-rdg 6862  df-1o 6916  df-oadd 6920  df-er 7097  df-pm 7213  df-en 7307  df-dom 7308  df-sdom 7309  df-fin 7310  df-sup 7687  df-oi 7720  df-card 8105  df-pnf 9416  df-mnf 9417  df-xr 9418  df-ltxr 9419  df-le 9420  df-sub 9593  df-neg 9594  df-div 9990  df-nn 10319  df-2 10376  df-3 10377  df-n0 10576  df-z 10643  df-uz 10858  df-rp 10988  df-fz 11434  df-fzo 11545  df-fl 11638  df-seq 11803  df-exp 11862  df-hash 12100  df-cj 12584  df-re 12585  df-im 12586  df-sqr 12720  df-abs 12721  df-clim 12962  df-rlim 12963  df-sum 13160
This theorem is referenced by:  itg1climres  21151  plyeq0lem  21637
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