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Theorem isumadd 12544
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 5721 . . . . . 6  |-  ( m  =  k  ->  ( F `  m )  =  ( F `  k ) )
4 fveq2 5721 . . . . . 6  |-  ( m  =  k  ->  ( G `  m )  =  ( G `  k ) )
53, 4oveq12d 6092 . . . . 5  |-  ( m  =  k  ->  (
( F `  m
)  +  ( G `
 m ) )  =  ( ( F `
 k )  +  ( G `  k
) ) )
6 eqid 2436 . . . . 5  |-  ( m  e.  Z  |->  ( ( F `  m )  +  ( G `  m ) ) )  =  ( m  e.  Z  |->  ( ( F `
 m )  +  ( G `  m
) ) )
7 ovex 6099 . . . . 5  |-  ( ( F `  k )  +  ( G `  k ) )  e. 
_V
85, 6, 7fvmpt 5799 . . . 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 6092 . . 3  |-  ( (
ph  /\  k  e.  Z )  ->  (
( F `  k
)  +  ( G `
 k ) )  =  ( A  +  B ) )
139, 12eqtrd 2468 . 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 9100 . 2  |-  ( (
ph  /\  k  e.  Z )  ->  ( A  +  B )  e.  CC )
17 isumadd.7 . . . 4  |-  ( ph  ->  seq  M (  +  ,  F )  e. 
dom 
~~>  )
181, 2, 10, 14, 17isumclim2 12535 . . 3  |-  ( ph  ->  seq  M (  +  ,  F )  ~~>  sum_ k  e.  Z  A )
19 seqex 11318 . . . 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 12535 . . 3  |-  ( ph  ->  seq  M (  +  ,  G )  ~~>  sum_ k  e.  Z  B )
2310, 14eqeltrd 2510 . . . . 5  |-  ( (
ph  /\  k  e.  Z )  ->  ( F `  k )  e.  CC )
241, 2, 23serf 11344 . . . 4  |-  ( ph  ->  seq  M (  +  ,  F ) : Z --> CC )
2524ffvelrnda 5863 . . 3  |-  ( (
ph  /\  j  e.  Z )  ->  (  seq  M (  +  ,  F ) `  j
)  e.  CC )
2611, 15eqeltrd 2510 . . . . 5  |-  ( (
ph  /\  k  e.  Z )  ->  ( G `  k )  e.  CC )
271, 2, 26serf 11344 . . . 4  |-  ( ph  ->  seq  M (  +  ,  G ) : Z --> CC )
2827ffvelrnda 5863 . . 3  |-  ( (
ph  /\  j  e.  Z )  ->  (  seq  M (  +  ,  G ) `  j
)  e.  CC )
29 simpr 448 . . . . 5  |-  ( (
ph  /\  j  e.  Z )  ->  j  e.  Z )
3029, 1syl6eleq 2526 . . . 4  |-  ( (
ph  /\  j  e.  Z )  ->  j  e.  ( ZZ>= `  M )
)
31 simpll 731 . . . . 5  |-  ( ( ( ph  /\  j  e.  Z )  /\  k  e.  ( M ... j
) )  ->  ph )
32 elfzuz 11048 . . . . . . 7  |-  ( k  e.  ( M ... j )  ->  k  e.  ( ZZ>= `  M )
)
3332, 1syl6eleqr 2527 . . . . . 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 11358 . . 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 12418 . 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 12534 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 1652    e. wcel 1725   _Vcvv 2949    e. cmpt 4259   dom cdm 4871   ` cfv 5447  (class class class)co 6074   CCcc 8981    + caddc 8986   ZZcz 10275   ZZ>=cuz 10481   ...cfz 11036    seq cseq 11316    ~~> cli 12271   sum_csu 12472
This theorem is referenced by:  sumsplit  12545
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-rep 4313  ax-sep 4323  ax-nul 4331  ax-pow 4370  ax-pr 4396  ax-un 4694  ax-inf2 7589  ax-cnex 9039  ax-resscn 9040  ax-1cn 9041  ax-icn 9042  ax-addcl 9043  ax-addrcl 9044  ax-mulcl 9045  ax-mulrcl 9046  ax-mulcom 9047  ax-addass 9048  ax-mulass 9049  ax-distr 9050  ax-i2m1 9051  ax-1ne0 9052  ax-1rid 9053  ax-rnegex 9054  ax-rrecex 9055  ax-cnre 9056  ax-pre-lttri 9057  ax-pre-lttrn 9058  ax-pre-ltadd 9059  ax-pre-mulgt0 9060  ax-pre-sup 9061
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-nel 2602  df-ral 2703  df-rex 2704  df-reu 2705  df-rmo 2706  df-rab 2707  df-v 2951  df-sbc 3155  df-csb 3245  df-dif 3316  df-un 3318  df-in 3320  df-ss 3327  df-pss 3329  df-nul 3622  df-if 3733  df-pw 3794  df-sn 3813  df-pr 3814  df-tp 3815  df-op 3816  df-uni 4009  df-int 4044  df-iun 4088  df-br 4206  df-opab 4260  df-mpt 4261  df-tr 4296  df-eprel 4487  df-id 4491  df-po 4496  df-so 4497  df-fr 4534  df-se 4535  df-we 4536  df-ord 4577  df-on 4578  df-lim 4579  df-suc 4580  df-om 4839  df-xp 4877  df-rel 4878  df-cnv 4879  df-co 4880  df-dm 4881  df-rn 4882  df-res 4883  df-ima 4884  df-iota 5411  df-fun 5449  df-fn 5450  df-f 5451  df-f1 5452  df-fo 5453  df-f1o 5454  df-fv 5455  df-isom 5456  df-ov 6077  df-oprab 6078  df-mpt2 6079  df-1st 6342  df-2nd 6343  df-riota 6542  df-recs 6626  df-rdg 6661  df-1o 6717  df-oadd 6721  df-er 6898  df-en 7103  df-dom 7104  df-sdom 7105  df-fin 7106  df-sup 7439  df-oi 7472  df-card 7819  df-pnf 9115  df-mnf 9116  df-xr 9117  df-ltxr 9118  df-le 9119  df-sub 9286  df-neg 9287  df-div 9671  df-nn 9994  df-2 10051  df-3 10052  df-n0 10215  df-z 10276  df-uz 10482  df-rp 10606  df-fz 11037  df-fzo 11129  df-seq 11317  df-exp 11376  df-hash 11612  df-cj 11897  df-re 11898  df-im 11899  df-sqr 12033  df-abs 12034  df-clim 12275  df-sum 12473
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