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Theorem isumadd 12506
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 fveq2 5687 . . . . . 6  |-  ( m  =  k  ->  ( F `  m )  =  ( F `  k ) )
4 fveq2 5687 . . . . . 6  |-  ( m  =  k  ->  ( G `  m )  =  ( G `  k ) )
53, 4oveq12d 6058 . . . . 5  |-  ( m  =  k  ->  (
( F `  m
)  +  ( G `
 m ) )  =  ( ( F `
 k )  +  ( G `  k
) ) )
6 eqid 2404 . . . . 5  |-  ( m  e.  Z  |->  ( ( F `  m )  +  ( G `  m ) ) )  =  ( m  e.  Z  |->  ( ( F `
 m )  +  ( G `  m
) ) )
7 ovex 6065 . . . . 5  |-  ( ( F `  k )  +  ( G `  k ) )  e. 
_V
85, 6, 7fvmpt 5765 . . . 4  |-  ( k  e.  Z  ->  (
( m  e.  Z  |->  ( ( F `  m )  +  ( G `  m ) ) ) `  k
)  =  ( ( F `  k )  +  ( G `  k ) ) )
98adantl 453 . . 3  |-  ( (
ph  /\  k  e.  Z )  ->  (
( m  e.  Z  |->  ( ( F `  m )  +  ( G `  m ) ) ) `  k
)  =  ( ( F `  k )  +  ( G `  k ) ) )
10 isumadd.3 . . . 4  |-  ( (
ph  /\  k  e.  Z )  ->  ( F `  k )  =  A )
11 isumadd.5 . . . 4  |-  ( (
ph  /\  k  e.  Z )  ->  ( G `  k )  =  B )
1210, 11oveq12d 6058 . . 3  |-  ( (
ph  /\  k  e.  Z )  ->  (
( F `  k
)  +  ( G `
 k ) )  =  ( A  +  B ) )
139, 12eqtrd 2436 . 2  |-  ( (
ph  /\  k  e.  Z )  ->  (
( m  e.  Z  |->  ( ( F `  m )  +  ( G `  m ) ) ) `  k
)  =  ( A  +  B ) )
14 isumadd.4 . . 3  |-  ( (
ph  /\  k  e.  Z )  ->  A  e.  CC )
15 isumadd.6 . . 3  |-  ( (
ph  /\  k  e.  Z )  ->  B  e.  CC )
1614, 15addcld 9063 . 2  |-  ( (
ph  /\  k  e.  Z )  ->  ( A  +  B )  e.  CC )
17 isumadd.7 . . . 4  |-  ( ph  ->  seq  M (  +  ,  F )  e. 
dom 
~~>  )
181, 2, 10, 14, 17isumclim2 12497 . . 3  |-  ( ph  ->  seq  M (  +  ,  F )  ~~>  sum_ k  e.  Z  A )
19 seqex 11280 . . . 4  |-  seq  M
(  +  ,  ( m  e.  Z  |->  ( ( F `  m
)  +  ( G `
 m ) ) ) )  e.  _V
2019a1i 11 . . 3  |-  ( ph  ->  seq  M (  +  ,  ( m  e.  Z  |->  ( ( F `
 m )  +  ( G `  m
) ) ) )  e.  _V )
21 isumadd.8 . . . 4  |-  ( ph  ->  seq  M (  +  ,  G )  e. 
dom 
~~>  )
221, 2, 11, 15, 21isumclim2 12497 . . 3  |-  ( ph  ->  seq  M (  +  ,  G )  ~~>  sum_ k  e.  Z  B )
2310, 14eqeltrd 2478 . . . . 5  |-  ( (
ph  /\  k  e.  Z )  ->  ( F `  k )  e.  CC )
241, 2, 23serf 11306 . . . 4  |-  ( ph  ->  seq  M (  +  ,  F ) : Z --> CC )
2524ffvelrnda 5829 . . 3  |-  ( (
ph  /\  j  e.  Z )  ->  (  seq  M (  +  ,  F ) `  j
)  e.  CC )
2611, 15eqeltrd 2478 . . . . 5  |-  ( (
ph  /\  k  e.  Z )  ->  ( G `  k )  e.  CC )
271, 2, 26serf 11306 . . . 4  |-  ( ph  ->  seq  M (  +  ,  G ) : Z --> CC )
2827ffvelrnda 5829 . . 3  |-  ( (
ph  /\  j  e.  Z )  ->  (  seq  M (  +  ,  G ) `  j
)  e.  CC )
29 simpr 448 . . . . 5  |-  ( (
ph  /\  j  e.  Z )  ->  j  e.  Z )
3029, 1syl6eleq 2494 . . . 4  |-  ( (
ph  /\  j  e.  Z )  ->  j  e.  ( ZZ>= `  M )
)
31 simpll 731 . . . . 5  |-  ( ( ( ph  /\  j  e.  Z )  /\  k  e.  ( M ... j
) )  ->  ph )
32 elfzuz 11011 . . . . . . 7  |-  ( k  e.  ( M ... j )  ->  k  e.  ( ZZ>= `  M )
)
3332, 1syl6eleqr 2495 . . . . . 6  |-  ( k  e.  ( M ... j )  ->  k  e.  Z )
3433adantl 453 . . . . 5  |-  ( ( ( ph  /\  j  e.  Z )  /\  k  e.  ( M ... j
) )  ->  k  e.  Z )
3531, 34, 23syl2anc 643 . . . 4  |-  ( ( ( ph  /\  j  e.  Z )  /\  k  e.  ( M ... j
) )  ->  ( F `  k )  e.  CC )
3631, 34, 26syl2anc 643 . . . 4  |-  ( ( ( ph  /\  j  e.  Z )  /\  k  e.  ( M ... j
) )  ->  ( G `  k )  e.  CC )
3734, 8syl 16 . . . 4  |-  ( ( ( ph  /\  j  e.  Z )  /\  k  e.  ( M ... j
) )  ->  (
( m  e.  Z  |->  ( ( F `  m )  +  ( G `  m ) ) ) `  k
)  =  ( ( F `  k )  +  ( G `  k ) ) )
3830, 35, 36, 37seradd 11320 . . 3  |-  ( (
ph  /\  j  e.  Z )  ->  (  seq  M (  +  , 
( m  e.  Z  |->  ( ( F `  m )  +  ( G `  m ) ) ) ) `  j )  =  ( (  seq  M (  +  ,  F ) `
 j )  +  (  seq  M (  +  ,  G ) `
 j ) ) )
391, 2, 18, 20, 22, 25, 28, 38climadd 12380 . 2  |-  ( ph  ->  seq  M (  +  ,  ( m  e.  Z  |->  ( ( F `
 m )  +  ( G `  m
) ) ) )  ~~>  ( sum_ k  e.  Z  A  +  sum_ k  e.  Z  B ) )
401, 2, 13, 16, 39isumclim 12496 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 359    = wceq 1649    e. wcel 1721   _Vcvv 2916    e. cmpt 4226   dom cdm 4837   ` cfv 5413  (class class class)co 6040   CCcc 8944    + caddc 8949   ZZcz 10238   ZZ>=cuz 10444   ...cfz 10999    seq cseq 11278    ~~> cli 12233   sum_csu 12434
This theorem is referenced by:  sumsplit  12507
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660  ax-inf2 7552  ax-cnex 9002  ax-resscn 9003  ax-1cn 9004  ax-icn 9005  ax-addcl 9006  ax-addrcl 9007  ax-mulcl 9008  ax-mulrcl 9009  ax-mulcom 9010  ax-addass 9011  ax-mulass 9012  ax-distr 9013  ax-i2m1 9014  ax-1ne0 9015  ax-1rid 9016  ax-rnegex 9017  ax-rrecex 9018  ax-cnre 9019  ax-pre-lttri 9020  ax-pre-lttrn 9021  ax-pre-ltadd 9022  ax-pre-mulgt0 9023  ax-pre-sup 9024
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-reu 2673  df-rmo 2674  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-pss 3296  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-tp 3782  df-op 3783  df-uni 3976  df-int 4011  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-tr 4263  df-eprel 4454  df-id 4458  df-po 4463  df-so 4464  df-fr 4501  df-se 4502  df-we 4503  df-ord 4544  df-on 4545  df-lim 4546  df-suc 4547  df-om 4805  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-isom 5422  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-1st 6308  df-2nd 6309  df-riota 6508  df-recs 6592  df-rdg 6627  df-1o 6683  df-oadd 6687  df-er 6864  df-en 7069  df-dom 7070  df-sdom 7071  df-fin 7072  df-sup 7404  df-oi 7435  df-card 7782  df-pnf 9078  df-mnf 9079  df-xr 9080  df-ltxr 9081  df-le 9082  df-sub 9249  df-neg 9250  df-div 9634  df-nn 9957  df-2 10014  df-3 10015  df-n0 10178  df-z 10239  df-uz 10445  df-rp 10569  df-fz 11000  df-fzo 11091  df-seq 11279  df-exp 11338  df-hash 11574  df-cj 11859  df-re 11860  df-im 11861  df-sqr 11995  df-abs 11996  df-clim 12237  df-sum 12435
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