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Theorem isumadd 10886
Description: Addition of infinite sums. (Contributed by Mario Carneiro, 18-Aug-2013.) (Revised by Mario Carneiro, 23-Apr-2014.)
Hypotheses
Ref Expression
isumadd.1  |-  Z  =  ( ZZ>= `  M )
isumadd.2  |-  ( ph  ->  M  e.  ZZ )
isumadd.3  |-  ( (
ph  /\  k  e.  Z )  ->  ( F `  k )  =  A )
isumadd.4  |-  ( (
ph  /\  k  e.  Z )  ->  A  e.  CC )
isumadd.5  |-  ( (
ph  /\  k  e.  Z )  ->  ( G `  k )  =  B )
isumadd.6  |-  ( (
ph  /\  k  e.  Z )  ->  B  e.  CC )
isumadd.7  |-  ( ph  ->  seq M (  +  ,  F )  e. 
dom 
~~>  )
isumadd.8  |-  ( ph  ->  seq M (  +  ,  G )  e. 
dom 
~~>  )
Assertion
Ref Expression
isumadd  |-  ( ph  -> 
sum_ k  e.  Z  ( A  +  B
)  =  ( sum_ k  e.  Z  A  +  sum_ k  e.  Z  B ) )
Distinct variable groups:    k, F    k, G    k, M    ph, k    k, Z
Allowed substitution hints:    A( k)    B( k)

Proof of Theorem isumadd
Dummy variables  j  m are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 isumadd.1 . 2  |-  Z  =  ( ZZ>= `  M )
2 isumadd.2 . 2  |-  ( ph  ->  M  e.  ZZ )
3 simpr 109 . . . 4  |-  ( (
ph  /\  k  e.  Z )  ->  k  e.  Z )
4 isumadd.3 . . . . . 6  |-  ( (
ph  /\  k  e.  Z )  ->  ( F `  k )  =  A )
5 isumadd.4 . . . . . 6  |-  ( (
ph  /\  k  e.  Z )  ->  A  e.  CC )
64, 5eqeltrd 2165 . . . . 5  |-  ( (
ph  /\  k  e.  Z )  ->  ( F `  k )  e.  CC )
7 isumadd.5 . . . . . 6  |-  ( (
ph  /\  k  e.  Z )  ->  ( G `  k )  =  B )
8 isumadd.6 . . . . . 6  |-  ( (
ph  /\  k  e.  Z )  ->  B  e.  CC )
97, 8eqeltrd 2165 . . . . 5  |-  ( (
ph  /\  k  e.  Z )  ->  ( G `  k )  e.  CC )
106, 9addcld 7568 . . . 4  |-  ( (
ph  /\  k  e.  Z )  ->  (
( F `  k
)  +  ( G `
 k ) )  e.  CC )
11 fveq2 5318 . . . . . 6  |-  ( m  =  k  ->  ( F `  m )  =  ( F `  k ) )
12 fveq2 5318 . . . . . 6  |-  ( m  =  k  ->  ( G `  m )  =  ( G `  k ) )
1311, 12oveq12d 5684 . . . . 5  |-  ( m  =  k  ->  (
( F `  m
)  +  ( G `
 m ) )  =  ( ( F `
 k )  +  ( G `  k
) ) )
14 eqid 2089 . . . . 5  |-  ( m  e.  Z  |->  ( ( F `  m )  +  ( G `  m ) ) )  =  ( m  e.  Z  |->  ( ( F `
 m )  +  ( G `  m
) ) )
1513, 14fvmptg 5393 . . . 4  |-  ( ( k  e.  Z  /\  ( ( F `  k )  +  ( G `  k ) )  e.  CC )  ->  ( ( m  e.  Z  |->  ( ( F `  m )  +  ( G `  m ) ) ) `
 k )  =  ( ( F `  k )  +  ( G `  k ) ) )
163, 10, 15syl2anc 404 . . 3  |-  ( (
ph  /\  k  e.  Z )  ->  (
( m  e.  Z  |->  ( ( F `  m )  +  ( G `  m ) ) ) `  k
)  =  ( ( F `  k )  +  ( G `  k ) ) )
174, 7oveq12d 5684 . . 3  |-  ( (
ph  /\  k  e.  Z )  ->  (
( F `  k
)  +  ( G `
 k ) )  =  ( A  +  B ) )
1816, 17eqtrd 2121 . 2  |-  ( (
ph  /\  k  e.  Z )  ->  (
( m  e.  Z  |->  ( ( F `  m )  +  ( G `  m ) ) ) `  k
)  =  ( A  +  B ) )
195, 8addcld 7568 . 2  |-  ( (
ph  /\  k  e.  Z )  ->  ( A  +  B )  e.  CC )
20 isumadd.7 . . . 4  |-  ( ph  ->  seq M (  +  ,  F )  e. 
dom 
~~>  )
211, 2, 4, 5, 20isumclim2 10877 . . 3  |-  ( ph  ->  seq M (  +  ,  F )  ~~>  sum_ k  e.  Z  A )
22 seqex 9918 . . . 4  |-  seq M
(  +  ,  ( m  e.  Z  |->  ( ( F `  m
)  +  ( G `
 m ) ) ) )  e.  _V
2322a1i 9 . . 3  |-  ( ph  ->  seq M (  +  ,  ( m  e.  Z  |->  ( ( F `
 m )  +  ( G `  m
) ) ) )  e.  _V )
24 isumadd.8 . . . 4  |-  ( ph  ->  seq M (  +  ,  G )  e. 
dom 
~~>  )
251, 2, 7, 8, 24isumclim2 10877 . . 3  |-  ( ph  ->  seq M (  +  ,  G )  ~~>  sum_ k  e.  Z  B )
261, 2, 6serf 9961 . . . 4  |-  ( ph  ->  seq M (  +  ,  F ) : Z --> CC )
2726ffvelrnda 5448 . . 3  |-  ( (
ph  /\  j  e.  Z )  ->  (  seq M (  +  ,  F ) `  j
)  e.  CC )
281, 2, 9serf 9961 . . . 4  |-  ( ph  ->  seq M (  +  ,  G ) : Z --> CC )
2928ffvelrnda 5448 . . 3  |-  ( (
ph  /\  j  e.  Z )  ->  (  seq M (  +  ,  G ) `  j
)  e.  CC )
30 simpr 109 . . . . 5  |-  ( (
ph  /\  j  e.  Z )  ->  j  e.  Z )
3130, 1syl6eleq 2181 . . . 4  |-  ( (
ph  /\  j  e.  Z )  ->  j  e.  ( ZZ>= `  M )
)
32 simpll 497 . . . . 5  |-  ( ( ( ph  /\  j  e.  Z )  /\  k  e.  ( ZZ>= `  M )
)  ->  ph )
331eleq2i 2155 . . . . . . 7  |-  ( k  e.  Z  <->  k  e.  ( ZZ>= `  M )
)
3433biimpri 132 . . . . . 6  |-  ( k  e.  ( ZZ>= `  M
)  ->  k  e.  Z )
3534adantl 272 . . . . 5  |-  ( ( ( ph  /\  j  e.  Z )  /\  k  e.  ( ZZ>= `  M )
)  ->  k  e.  Z )
3632, 35, 6syl2anc 404 . . . 4  |-  ( ( ( ph  /\  j  e.  Z )  /\  k  e.  ( ZZ>= `  M )
)  ->  ( F `  k )  e.  CC )
3732, 35, 9syl2anc 404 . . . 4  |-  ( ( ( ph  /\  j  e.  Z )  /\  k  e.  ( ZZ>= `  M )
)  ->  ( G `  k )  e.  CC )
3832, 35, 10syl2anc 404 . . . . 5  |-  ( ( ( ph  /\  j  e.  Z )  /\  k  e.  ( ZZ>= `  M )
)  ->  ( ( F `  k )  +  ( G `  k ) )  e.  CC )
3935, 38, 15syl2anc 404 . . . 4  |-  ( ( ( ph  /\  j  e.  Z )  /\  k  e.  ( ZZ>= `  M )
)  ->  ( (
m  e.  Z  |->  ( ( F `  m
)  +  ( G `
 m ) ) ) `  k )  =  ( ( F `
 k )  +  ( G `  k
) ) )
4031, 36, 37, 39ser3add 9996 . . 3  |-  ( (
ph  /\  j  e.  Z )  ->  (  seq M (  +  , 
( m  e.  Z  |->  ( ( F `  m )  +  ( G `  m ) ) ) ) `  j )  =  ( (  seq M (  +  ,  F ) `
 j )  +  (  seq M (  +  ,  G ) `
 j ) ) )
411, 2, 21, 23, 25, 27, 29, 40climadd 10775 . 2  |-  ( ph  ->  seq M (  +  ,  ( m  e.  Z  |->  ( ( F `
 m )  +  ( G `  m
) ) ) )  ~~>  ( sum_ k  e.  Z  A  +  sum_ k  e.  Z  B ) )
421, 2, 18, 19, 41isumclim 10876 1  |-  ( ph  -> 
sum_ k  e.  Z  ( A  +  B
)  =  ( sum_ k  e.  Z  A  +  sum_ k  e.  Z  B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    = wceq 1290    e. wcel 1439   _Vcvv 2620    |-> cmpt 3905   dom cdm 4452   ` cfv 5028  (class class class)co 5666   CCcc 7409    + caddc 7414   ZZcz 8811   ZZ>=cuz 9080    seqcseq 9913    ~~> cli 10727   sum_csu 10803
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 580  ax-in2 581  ax-io 666  ax-5 1382  ax-7 1383  ax-gen 1384  ax-ie1 1428  ax-ie2 1429  ax-8 1441  ax-10 1442  ax-11 1443  ax-i12 1444  ax-bndl 1445  ax-4 1446  ax-13 1450  ax-14 1451  ax-17 1465  ax-i9 1469  ax-ial 1473  ax-i5r 1474  ax-ext 2071  ax-coll 3960  ax-sep 3963  ax-nul 3971  ax-pow 4015  ax-pr 4045  ax-un 4269  ax-setind 4366  ax-iinf 4416  ax-cnex 7497  ax-resscn 7498  ax-1cn 7499  ax-1re 7500  ax-icn 7501  ax-addcl 7502  ax-addrcl 7503  ax-mulcl 7504  ax-mulrcl 7505  ax-addcom 7506  ax-mulcom 7507  ax-addass 7508  ax-mulass 7509  ax-distr 7510  ax-i2m1 7511  ax-0lt1 7512  ax-1rid 7513  ax-0id 7514  ax-rnegex 7515  ax-precex 7516  ax-cnre 7517  ax-pre-ltirr 7518  ax-pre-ltwlin 7519  ax-pre-lttrn 7520  ax-pre-apti 7521  ax-pre-ltadd 7522  ax-pre-mulgt0 7523  ax-pre-mulext 7524  ax-arch 7525  ax-caucvg 7526
This theorem depends on definitions:  df-bi 116  df-dc 782  df-3or 926  df-3an 927  df-tru 1293  df-fal 1296  df-nf 1396  df-sb 1694  df-eu 1952  df-mo 1953  df-clab 2076  df-cleq 2082  df-clel 2085  df-nfc 2218  df-ne 2257  df-nel 2352  df-ral 2365  df-rex 2366  df-reu 2367  df-rmo 2368  df-rab 2369  df-v 2622  df-sbc 2842  df-csb 2935  df-dif 3002  df-un 3004  df-in 3006  df-ss 3013  df-nul 3288  df-if 3398  df-pw 3435  df-sn 3456  df-pr 3457  df-op 3459  df-uni 3660  df-int 3695  df-iun 3738  df-br 3852  df-opab 3906  df-mpt 3907  df-tr 3943  df-id 4129  df-po 4132  df-iso 4133  df-iord 4202  df-on 4204  df-ilim 4205  df-suc 4207  df-iom 4419  df-xp 4458  df-rel 4459  df-cnv 4460  df-co 4461  df-dm 4462  df-rn 4463  df-res 4464  df-ima 4465  df-iota 4993  df-fun 5030  df-fn 5031  df-f 5032  df-f1 5033  df-fo 5034  df-f1o 5035  df-fv 5036  df-isom 5037  df-riota 5622  df-ov 5669  df-oprab 5670  df-mpt2 5671  df-1st 5925  df-2nd 5926  df-recs 6084  df-irdg 6149  df-frec 6170  df-1o 6195  df-oadd 6199  df-er 6306  df-en 6512  df-dom 6513  df-fin 6514  df-pnf 7585  df-mnf 7586  df-xr 7587  df-ltxr 7588  df-le 7589  df-sub 7716  df-neg 7717  df-reap 8113  df-ap 8120  df-div 8201  df-inn 8484  df-2 8542  df-3 8543  df-4 8544  df-n0 8735  df-z 8812  df-uz 9081  df-q 9166  df-rp 9196  df-fz 9486  df-fzo 9615  df-iseq 9914  df-seq3 9915  df-exp 10016  df-ihash 10245  df-cj 10337  df-re 10338  df-im 10339  df-rsqrt 10492  df-abs 10493  df-clim 10728  df-isum 10804
This theorem is referenced by:  sumsplitdc  10887
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