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Theorem climfsum 12294
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 4122 . . 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 10263 . . . . . . . 8  |-  ( ZZ>= `  M )  C_  ZZ
53, 4eqsstri 3221 . . . . . . 7  |-  Z  C_  ZZ
6 zssre 10047 . . . . . . 7  |-  ZZ  C_  RR
75, 6sstri 3201 . . . . . 6  |-  Z  C_  RR
87a1i 10 . . . . 5  |-  ( ph  ->  Z  C_  RR )
9 climfsum.3 . . . . 5  |-  ( ph  ->  A  e.  Fin )
10 fvex 5555 . . . . . 6  |-  ( F `
 n )  e. 
_V
1110a1i 10 . . . . 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 451 . . . . . . . 8  |-  ( (
ph  /\  k  e.  A )  ->  M  e.  ZZ )
15 climrel 11982 . . . . . . . . . 10  |-  Rel  ~~>
1615brrelexi 4745 . . . . . . . . 9  |-  ( F  ~~>  B  ->  F  e.  _V )
1712, 16syl 15 . . . . . . . 8  |-  ( (
ph  /\  k  e.  A )  ->  F  e.  _V )
18 eqid 2296 . . . . . . . . 9  |-  ( n  e.  Z  |->  ( F `
 n ) )  =  ( n  e.  Z  |->  ( F `  n ) )
193, 18climmpt 12061 . . . . . . . 8  |-  ( ( M  e.  ZZ  /\  F  e.  _V )  ->  ( F  ~~>  B  <->  ( n  e.  Z  |->  ( F `
 n ) )  ~~>  B ) )
2014, 17, 19syl2anc 642 . . . . . . 7  |-  ( (
ph  /\  k  e.  A )  ->  ( F 
~~>  B  <->  ( n  e.  Z  |->  ( F `  n ) )  ~~>  B ) )
2112, 20mpbid 201 . . . . . 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 629 . . . . . . . 8  |-  ( ( ( ph  /\  k  e.  A )  /\  n  e.  Z )  ->  ( F `  n )  e.  CC )
2423, 18fmptd 5700 . . . . . . 7  |-  ( (
ph  /\  k  e.  A )  ->  (
n  e.  Z  |->  ( F `  n ) ) : Z --> CC )
253, 14, 24rlimclim 12036 . . . . . 6  |-  ( (
ph  /\  k  e.  A )  ->  (
( n  e.  Z  |->  ( F `  n
) )  ~~> r  B  <->  ( n  e.  Z  |->  ( F `  n ) )  ~~>  B ) )
2621, 25mpbird 223 . . . . 5  |-  ( (
ph  /\  k  e.  A )  ->  (
n  e.  Z  |->  ( F `  n ) )  ~~> r  B )
278, 9, 11, 26fsumrlim 12285 . . . 4  |-  ( ph  ->  ( n  e.  Z  |-> 
sum_ k  e.  A  ( F `  n ) )  ~~> r  sum_ k  e.  A  B )
289adantr 451 . . . . . . 7  |-  ( (
ph  /\  n  e.  Z )  ->  A  e.  Fin )
2922anass1rs 782 . . . . . . 7  |-  ( ( ( ph  /\  n  e.  Z )  /\  k  e.  A )  ->  ( F `  n )  e.  CC )
3028, 29fsumcl 12222 . . . . . 6  |-  ( (
ph  /\  n  e.  Z )  ->  sum_ k  e.  A  ( F `  n )  e.  CC )
31 eqid 2296 . . . . . 6  |-  ( n  e.  Z  |->  sum_ k  e.  A  ( F `  n ) )  =  ( n  e.  Z  |-> 
sum_ k  e.  A  ( F `  n ) )
3230, 31fmptd 5700 . . . . 5  |-  ( ph  ->  ( n  e.  Z  |-> 
sum_ k  e.  A  ( F `  n ) ) : Z --> CC )
333, 13, 32rlimclim 12036 . . . 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 201 . . 3  |-  ( ph  ->  ( n  e.  Z  |-> 
sum_ k  e.  A  ( F `  n ) )  ~~>  sum_ k  e.  A  B )
352, 34eqbrtrd 4059 . 2  |-  ( ph  ->  ( n  e.  Z  |->  ( H `  n
) )  ~~>  sum_ k  e.  A  B )
36 climfsum.6 . . 3  |-  ( ph  ->  H  e.  W )
37 eqid 2296 . . . 4  |-  ( n  e.  Z  |->  ( H `
 n ) )  =  ( n  e.  Z  |->  ( H `  n ) )
383, 37climmpt 12061 . . 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 642 . 2  |-  ( ph  ->  ( H  ~~>  sum_ k  e.  A  B  <->  ( n  e.  Z  |->  ( H `
 n ) )  ~~> 
sum_ k  e.  A  B ) )
4035, 39mpbird 223 1  |-  ( ph  ->  H  ~~>  sum_ k  e.  A  B )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1632    e. wcel 1696   _Vcvv 2801    C_ wss 3165   class class class wbr 4039    e. cmpt 4093   ` cfv 5271   Fincfn 6879   CCcc 8751   RRcr 8752   ZZcz 10040   ZZ>=cuz 10246    ~~> cli 11974    ~~> r crli 11975   sum_csu 12174
This theorem is referenced by:  itg1climres  19085  plyeq0lem  19608
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-inf2 7358  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830  ax-pre-sup 8831  ax-addf 8832
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-int 3879  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-se 4369  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-isom 5280  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-riota 6320  df-recs 6404  df-rdg 6439  df-1o 6495  df-oadd 6499  df-er 6676  df-pm 6791  df-en 6880  df-dom 6881  df-sdom 6882  df-fin 6883  df-sup 7210  df-oi 7241  df-card 7588  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-div 9440  df-nn 9763  df-2 9820  df-3 9821  df-n0 9982  df-z 10041  df-uz 10247  df-rp 10371  df-fz 10799  df-fzo 10887  df-fl 10941  df-seq 11063  df-exp 11121  df-hash 11354  df-cj 11600  df-re 11601  df-im 11602  df-sqr 11736  df-abs 11737  df-clim 11978  df-rlim 11979  df-sum 12175
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