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Theorem isumadd 12246
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 5541 . . . . . 6  |-  ( m  =  k  ->  ( F `  m )  =  ( F `  k ) )
4 fveq2 5541 . . . . . 6  |-  ( m  =  k  ->  ( G `  m )  =  ( G `  k ) )
53, 4oveq12d 5892 . . . . 5  |-  ( m  =  k  ->  (
( F `  m
)  +  ( G `
 m ) )  =  ( ( F `
 k )  +  ( G `  k
) ) )
6 eqid 2296 . . . . 5  |-  ( m  e.  Z  |->  ( ( F `  m )  +  ( G `  m ) ) )  =  ( m  e.  Z  |->  ( ( F `
 m )  +  ( G `  m
) ) )
7 ovex 5899 . . . . 5  |-  ( ( F `  k )  +  ( G `  k ) )  e. 
_V
85, 6, 7fvmpt 5618 . . . 4  |-  ( k  e.  Z  ->  (
( m  e.  Z  |->  ( ( F `  m )  +  ( G `  m ) ) ) `  k
)  =  ( ( F `  k )  +  ( G `  k ) ) )
98adantl 452 . . 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 5892 . . 3  |-  ( (
ph  /\  k  e.  Z )  ->  (
( F `  k
)  +  ( G `
 k ) )  =  ( A  +  B ) )
139, 12eqtrd 2328 . 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 8870 . 2  |-  ( (
ph  /\  k  e.  Z )  ->  ( A  +  B )  e.  CC )
17 isumadd.7 . . . 4  |-  ( ph  ->  seq  M (  +  ,  F )  e. 
dom 
~~>  )
181, 2, 10, 14, 17isumclim2 12237 . . 3  |-  ( ph  ->  seq  M (  +  ,  F )  ~~>  sum_ k  e.  Z  A )
19 seqex 11064 . . . 4  |-  seq  M
(  +  ,  ( m  e.  Z  |->  ( ( F `  m
)  +  ( G `
 m ) ) ) )  e.  _V
2019a1i 10 . . 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 12237 . . 3  |-  ( ph  ->  seq  M (  +  ,  G )  ~~>  sum_ k  e.  Z  B )
2310, 14eqeltrd 2370 . . . . 5  |-  ( (
ph  /\  k  e.  Z )  ->  ( F `  k )  e.  CC )
241, 2, 23serf 11090 . . . 4  |-  ( ph  ->  seq  M (  +  ,  F ) : Z --> CC )
25 ffvelrn 5679 . . . 4  |-  ( (  seq  M (  +  ,  F ) : Z --> CC  /\  j  e.  Z )  ->  (  seq  M (  +  ,  F ) `  j
)  e.  CC )
2624, 25sylan 457 . . 3  |-  ( (
ph  /\  j  e.  Z )  ->  (  seq  M (  +  ,  F ) `  j
)  e.  CC )
2711, 15eqeltrd 2370 . . . . 5  |-  ( (
ph  /\  k  e.  Z )  ->  ( G `  k )  e.  CC )
281, 2, 27serf 11090 . . . 4  |-  ( ph  ->  seq  M (  +  ,  G ) : Z --> CC )
29 ffvelrn 5679 . . . 4  |-  ( (  seq  M (  +  ,  G ) : Z --> CC  /\  j  e.  Z )  ->  (  seq  M (  +  ,  G ) `  j
)  e.  CC )
3028, 29sylan 457 . . 3  |-  ( (
ph  /\  j  e.  Z )  ->  (  seq  M (  +  ,  G ) `  j
)  e.  CC )
31 simpr 447 . . . . 5  |-  ( (
ph  /\  j  e.  Z )  ->  j  e.  Z )
3231, 1syl6eleq 2386 . . . 4  |-  ( (
ph  /\  j  e.  Z )  ->  j  e.  ( ZZ>= `  M )
)
33 simpll 730 . . . . 5  |-  ( ( ( ph  /\  j  e.  Z )  /\  k  e.  ( M ... j
) )  ->  ph )
34 elfzuz 10810 . . . . . . 7  |-  ( k  e.  ( M ... j )  ->  k  e.  ( ZZ>= `  M )
)
3534, 1syl6eleqr 2387 . . . . . 6  |-  ( k  e.  ( M ... j )  ->  k  e.  Z )
3635adantl 452 . . . . 5  |-  ( ( ( ph  /\  j  e.  Z )  /\  k  e.  ( M ... j
) )  ->  k  e.  Z )
3733, 36, 23syl2anc 642 . . . 4  |-  ( ( ( ph  /\  j  e.  Z )  /\  k  e.  ( M ... j
) )  ->  ( F `  k )  e.  CC )
3833, 36, 27syl2anc 642 . . . 4  |-  ( ( ( ph  /\  j  e.  Z )  /\  k  e.  ( M ... j
) )  ->  ( G `  k )  e.  CC )
3936, 8syl 15 . . . 4  |-  ( ( ( ph  /\  j  e.  Z )  /\  k  e.  ( M ... j
) )  ->  (
( m  e.  Z  |->  ( ( F `  m )  +  ( G `  m ) ) ) `  k
)  =  ( ( F `  k )  +  ( G `  k ) ) )
4032, 37, 38, 39seradd 11104 . . 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, 18, 20, 22, 26, 30, 40climadd 12121 . 2  |-  ( ph  ->  seq  M (  +  ,  ( m  e.  Z  |->  ( ( F `
 m )  +  ( G `  m
) ) ) )  ~~>  ( sum_ k  e.  Z  A  +  sum_ k  e.  Z  B ) )
421, 2, 13, 16, 41isumclim 12236 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 358    = wceq 1632    e. wcel 1696   _Vcvv 2801    e. cmpt 4093   dom cdm 4705   -->wf 5267   ` cfv 5271  (class class class)co 5874   CCcc 8751    + caddc 8756   ZZcz 10040   ZZ>=cuz 10246   ...cfz 10798    seq cseq 11062    ~~> cli 11974   sum_csu 12174
This theorem is referenced by:  sumsplit  12247
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
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-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-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-sum 12175
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