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Theorem climfsum 13614
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 4539 . . 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 11113 . . . . . . . 8  |-  ( ZZ>= `  M )  C_  ZZ
53, 4eqsstri 3539 . . . . . . 7  |-  Z  C_  ZZ
6 zssre 10883 . . . . . . 7  |-  ZZ  C_  RR
75, 6sstri 3518 . . . . . 6  |-  Z  C_  RR
87a1i 11 . . . . 5  |-  ( ph  ->  Z  C_  RR )
9 climfsum.3 . . . . 5  |-  ( ph  ->  A  e.  Fin )
10 fvex 5882 . . . . . 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 465 . . . . . . . 8  |-  ( (
ph  /\  k  e.  A )  ->  M  e.  ZZ )
15 climrel 13295 . . . . . . . . . 10  |-  Rel  ~~>
1615brrelexi 5046 . . . . . . . . 9  |-  ( F  ~~>  B  ->  F  e.  _V )
1712, 16syl 16 . . . . . . . 8  |-  ( (
ph  /\  k  e.  A )  ->  F  e.  _V )
18 eqid 2467 . . . . . . . . 9  |-  ( n  e.  Z  |->  ( F `
 n ) )  =  ( n  e.  Z  |->  ( F `  n ) )
193, 18climmpt 13374 . . . . . . . 8  |-  ( ( M  e.  ZZ  /\  F  e.  _V )  ->  ( F  ~~>  B  <->  ( n  e.  Z  |->  ( F `
 n ) )  ~~>  B ) )
2014, 17, 19syl2anc 661 . . . . . . 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 648 . . . . . . . 8  |-  ( ( ( ph  /\  k  e.  A )  /\  n  e.  Z )  ->  ( F `  n )  e.  CC )
2423, 18fmptd 6056 . . . . . . 7  |-  ( (
ph  /\  k  e.  A )  ->  (
n  e.  Z  |->  ( F `  n ) ) : Z --> CC )
253, 14, 24rlimclim 13349 . . . . . 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 13605 . . . 4  |-  ( ph  ->  ( n  e.  Z  |-> 
sum_ k  e.  A  ( F `  n ) )  ~~> r  sum_ k  e.  A  B )
289adantr 465 . . . . . . 7  |-  ( (
ph  /\  n  e.  Z )  ->  A  e.  Fin )
2922anass1rs 805 . . . . . . 7  |-  ( ( ( ph  /\  n  e.  Z )  /\  k  e.  A )  ->  ( F `  n )  e.  CC )
3028, 29fsumcl 13535 . . . . . 6  |-  ( (
ph  /\  n  e.  Z )  ->  sum_ k  e.  A  ( F `  n )  e.  CC )
31 eqid 2467 . . . . . 6  |-  ( n  e.  Z  |->  sum_ k  e.  A  ( F `  n ) )  =  ( n  e.  Z  |-> 
sum_ k  e.  A  ( F `  n ) )
3230, 31fmptd 6056 . . . . 5  |-  ( ph  ->  ( n  e.  Z  |-> 
sum_ k  e.  A  ( F `  n ) ) : Z --> CC )
333, 13, 32rlimclim 13349 . . . 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 4473 . 2  |-  ( ph  ->  ( n  e.  Z  |->  ( H `  n
) )  ~~>  sum_ k  e.  A  B )
36 climfsum.6 . . 3  |-  ( ph  ->  H  e.  W )
37 eqid 2467 . . . 4  |-  ( n  e.  Z  |->  ( H `
 n ) )  =  ( n  e.  Z  |->  ( H `  n ) )
383, 37climmpt 13374 . . 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 661 . 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 1379    e. wcel 1767   _Vcvv 3118    C_ wss 3481   class class class wbr 4453    |-> cmpt 4511   ` cfv 5594   Fincfn 7528   CCcc 9502   RRcr 9503   ZZcz 10876   ZZ>=cuz 11094    ~~> cli 13287    ~~> r crli 13288   sum_csu 13488
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4564  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-un 6587  ax-inf2 8070  ax-cnex 9560  ax-resscn 9561  ax-1cn 9562  ax-icn 9563  ax-addcl 9564  ax-addrcl 9565  ax-mulcl 9566  ax-mulrcl 9567  ax-mulcom 9568  ax-addass 9569  ax-mulass 9570  ax-distr 9571  ax-i2m1 9572  ax-1ne0 9573  ax-1rid 9574  ax-rnegex 9575  ax-rrecex 9576  ax-cnre 9577  ax-pre-lttri 9578  ax-pre-lttrn 9579  ax-pre-ltadd 9580  ax-pre-mulgt0 9581  ax-pre-sup 9582  ax-addf 9583
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-fal 1385  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2822  df-rex 2823  df-reu 2824  df-rmo 2825  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-pss 3497  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-tp 4038  df-op 4040  df-uni 4252  df-int 4289  df-iun 4333  df-br 4454  df-opab 4512  df-mpt 4513  df-tr 4547  df-eprel 4797  df-id 4801  df-po 4806  df-so 4807  df-fr 4844  df-se 4845  df-we 4846  df-ord 4887  df-on 4888  df-lim 4889  df-suc 4890  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-isom 5603  df-riota 6256  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-om 6696  df-1st 6795  df-2nd 6796  df-recs 7054  df-rdg 7088  df-1o 7142  df-oadd 7146  df-er 7323  df-pm 7435  df-en 7529  df-dom 7530  df-sdom 7531  df-fin 7532  df-sup 7913  df-oi 7947  df-card 8332  df-pnf 9642  df-mnf 9643  df-xr 9644  df-ltxr 9645  df-le 9646  df-sub 9819  df-neg 9820  df-div 10219  df-nn 10549  df-2 10606  df-3 10607  df-n0 10808  df-z 10877  df-uz 11095  df-rp 11233  df-fz 11685  df-fzo 11805  df-fl 11909  df-seq 12088  df-exp 12147  df-hash 12386  df-cj 12912  df-re 12913  df-im 12914  df-sqrt 13048  df-abs 13049  df-clim 13291  df-rlim 13292  df-sum 13489
This theorem is referenced by:  itg1climres  21989  plyeq0lem  22475
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