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Theorem climfsum 13823
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 4453 . . 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 11129 . . . . . . . 8  |-  ( ZZ>= `  M )  C_  ZZ
53, 4eqsstri 3437 . . . . . . 7  |-  Z  C_  ZZ
6 zssre 10895 . . . . . . 7  |-  ZZ  C_  RR
75, 6sstri 3416 . . . . . 6  |-  Z  C_  RR
87a1i 11 . . . . 5  |-  ( ph  ->  Z  C_  RR )
9 climfsum.3 . . . . 5  |-  ( ph  ->  A  e.  Fin )
10 fvex 5835 . . . . . 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 466 . . . . . . . 8  |-  ( (
ph  /\  k  e.  A )  ->  M  e.  ZZ )
15 climrel 13499 . . . . . . . . . 10  |-  Rel  ~~>
1615brrelexi 4837 . . . . . . . . 9  |-  ( F  ~~>  B  ->  F  e.  _V )
1712, 16syl 17 . . . . . . . 8  |-  ( (
ph  /\  k  e.  A )  ->  F  e.  _V )
18 eqid 2428 . . . . . . . . 9  |-  ( n  e.  Z  |->  ( F `
 n ) )  =  ( n  e.  Z  |->  ( F `  n ) )
193, 18climmpt 13578 . . . . . . . 8  |-  ( ( M  e.  ZZ  /\  F  e.  _V )  ->  ( F  ~~>  B  <->  ( n  e.  Z  |->  ( F `
 n ) )  ~~>  B ) )
2014, 17, 19syl2anc 665 . . . . . . 7  |-  ( (
ph  /\  k  e.  A )  ->  ( F 
~~>  B  <->  ( n  e.  Z  |->  ( F `  n ) )  ~~>  B ) )
2112, 20mpbid 213 . . . . . 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 652 . . . . . . . 8  |-  ( ( ( ph  /\  k  e.  A )  /\  n  e.  Z )  ->  ( F `  n )  e.  CC )
2423, 18fmptd 6005 . . . . . . 7  |-  ( (
ph  /\  k  e.  A )  ->  (
n  e.  Z  |->  ( F `  n ) ) : Z --> CC )
253, 14, 24rlimclim 13553 . . . . . 6  |-  ( (
ph  /\  k  e.  A )  ->  (
( n  e.  Z  |->  ( F `  n
) )  ~~> r  B  <->  ( n  e.  Z  |->  ( F `  n ) )  ~~>  B ) )
2621, 25mpbird 235 . . . . 5  |-  ( (
ph  /\  k  e.  A )  ->  (
n  e.  Z  |->  ( F `  n ) )  ~~> r  B )
278, 9, 11, 26fsumrlim 13814 . . . 4  |-  ( ph  ->  ( n  e.  Z  |-> 
sum_ k  e.  A  ( F `  n ) )  ~~> r  sum_ k  e.  A  B )
289adantr 466 . . . . . . 7  |-  ( (
ph  /\  n  e.  Z )  ->  A  e.  Fin )
2922anass1rs 814 . . . . . . 7  |-  ( ( ( ph  /\  n  e.  Z )  /\  k  e.  A )  ->  ( F `  n )  e.  CC )
3028, 29fsumcl 13742 . . . . . 6  |-  ( (
ph  /\  n  e.  Z )  ->  sum_ k  e.  A  ( F `  n )  e.  CC )
31 eqid 2428 . . . . . 6  |-  ( n  e.  Z  |->  sum_ k  e.  A  ( F `  n ) )  =  ( n  e.  Z  |-> 
sum_ k  e.  A  ( F `  n ) )
3230, 31fmptd 6005 . . . . 5  |-  ( ph  ->  ( n  e.  Z  |-> 
sum_ k  e.  A  ( F `  n ) ) : Z --> CC )
333, 13, 32rlimclim 13553 . . . 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 213 . . 3  |-  ( ph  ->  ( n  e.  Z  |-> 
sum_ k  e.  A  ( F `  n ) )  ~~>  sum_ k  e.  A  B )
352, 34eqbrtrd 4387 . 2  |-  ( ph  ->  ( n  e.  Z  |->  ( H `  n
) )  ~~>  sum_ k  e.  A  B )
36 climfsum.6 . . 3  |-  ( ph  ->  H  e.  W )
37 eqid 2428 . . . 4  |-  ( n  e.  Z  |->  ( H `
 n ) )  =  ( n  e.  Z  |->  ( H `  n ) )
383, 37climmpt 13578 . . 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 665 . 2  |-  ( ph  ->  ( H  ~~>  sum_ k  e.  A  B  <->  ( n  e.  Z  |->  ( H `
 n ) )  ~~> 
sum_ k  e.  A  B ) )
4035, 39mpbird 235 1  |-  ( ph  ->  H  ~~>  sum_ k  e.  A  B )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 187    /\ wa 370    = wceq 1437    e. wcel 1872   _Vcvv 3022    C_ wss 3379   class class class wbr 4366    |-> cmpt 4425   ` cfv 5544   Fincfn 7524   CCcc 9488   RRcr 9489   ZZcz 10888   ZZ>=cuz 11110    ~~> cli 13491    ~~> r crli 13492   sum_csu 13695
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-rep 4479  ax-sep 4489  ax-nul 4498  ax-pow 4545  ax-pr 4603  ax-un 6541  ax-inf2 8099  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  ax-addf 9569
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-fal 1443  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-int 4199  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-se 4756  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-isom 5553  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-1o 7137  df-oadd 7141  df-er 7318  df-pm 7430  df-en 7525  df-dom 7526  df-sdom 7527  df-fin 7528  df-sup 7909  df-inf 7910  df-oi 7978  df-card 8325  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-3 10620  df-n0 10821  df-z 10889  df-uz 11111  df-rp 11254  df-fz 11736  df-fzo 11867  df-fl 11978  df-seq 12164  df-exp 12223  df-hash 12466  df-cj 13106  df-re 13107  df-im 13108  df-sqrt 13242  df-abs 13243  df-clim 13495  df-rlim 13496  df-sum 13696
This theorem is referenced by:  itg1climres  22614  plyeq0lem  23106
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