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Theorem isumadd 11453
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 110 . . . 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 2264 . . . . 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 2264 . . . . 5  |-  ( (
ph  /\  k  e.  Z )  ->  ( G `  k )  e.  CC )
106, 9addcld 7991 . . . 4  |-  ( (
ph  /\  k  e.  Z )  ->  (
( F `  k
)  +  ( G `
 k ) )  e.  CC )
11 fveq2 5527 . . . . . 6  |-  ( m  =  k  ->  ( F `  m )  =  ( F `  k ) )
12 fveq2 5527 . . . . . 6  |-  ( m  =  k  ->  ( G `  m )  =  ( G `  k ) )
1311, 12oveq12d 5906 . . . . 5  |-  ( m  =  k  ->  (
( F `  m
)  +  ( G `
 m ) )  =  ( ( F `
 k )  +  ( G `  k
) ) )
14 eqid 2187 . . . . 5  |-  ( m  e.  Z  |->  ( ( F `  m )  +  ( G `  m ) ) )  =  ( m  e.  Z  |->  ( ( F `
 m )  +  ( G `  m
) ) )
1513, 14fvmptg 5605 . . . 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 411 . . 3  |-  ( (
ph  /\  k  e.  Z )  ->  (
( m  e.  Z  |->  ( ( F `  m )  +  ( G `  m ) ) ) `  k
)  =  ( ( F `  k )  +  ( G `  k ) ) )
174, 7oveq12d 5906 . . 3  |-  ( (
ph  /\  k  e.  Z )  ->  (
( F `  k
)  +  ( G `
 k ) )  =  ( A  +  B ) )
1816, 17eqtrd 2220 . 2  |-  ( (
ph  /\  k  e.  Z )  ->  (
( m  e.  Z  |->  ( ( F `  m )  +  ( G `  m ) ) ) `  k
)  =  ( A  +  B ) )
195, 8addcld 7991 . 2  |-  ( (
ph  /\  k  e.  Z )  ->  ( A  +  B )  e.  CC )
20 isumadd.7 . . . 4  |-  ( ph  ->  seq M (  +  ,  F )  e. 
dom 
~~>  )
211, 2, 4, 5, 20isumclim2 11444 . . 3  |-  ( ph  ->  seq M (  +  ,  F )  ~~>  sum_ k  e.  Z  A )
22 seqex 10461 . . . 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 11444 . . 3  |-  ( ph  ->  seq M (  +  ,  G )  ~~>  sum_ k  e.  Z  B )
261, 2, 6serf 10488 . . . 4  |-  ( ph  ->  seq M (  +  ,  F ) : Z --> CC )
2726ffvelcdmda 5664 . . 3  |-  ( (
ph  /\  j  e.  Z )  ->  (  seq M (  +  ,  F ) `  j
)  e.  CC )
281, 2, 9serf 10488 . . . 4  |-  ( ph  ->  seq M (  +  ,  G ) : Z --> CC )
2928ffvelcdmda 5664 . . 3  |-  ( (
ph  /\  j  e.  Z )  ->  (  seq M (  +  ,  G ) `  j
)  e.  CC )
30 simpr 110 . . . . 5  |-  ( (
ph  /\  j  e.  Z )  ->  j  e.  Z )
3130, 1eleqtrdi 2280 . . . 4  |-  ( (
ph  /\  j  e.  Z )  ->  j  e.  ( ZZ>= `  M )
)
32 simpll 527 . . . . 5  |-  ( ( ( ph  /\  j  e.  Z )  /\  k  e.  ( ZZ>= `  M )
)  ->  ph )
331eleq2i 2254 . . . . . . 7  |-  ( k  e.  Z  <->  k  e.  ( ZZ>= `  M )
)
3433biimpri 133 . . . . . 6  |-  ( k  e.  ( ZZ>= `  M
)  ->  k  e.  Z )
3534adantl 277 . . . . 5  |-  ( ( ( ph  /\  j  e.  Z )  /\  k  e.  ( ZZ>= `  M )
)  ->  k  e.  Z )
3632, 35, 6syl2anc 411 . . . 4  |-  ( ( ( ph  /\  j  e.  Z )  /\  k  e.  ( ZZ>= `  M )
)  ->  ( F `  k )  e.  CC )
3732, 35, 9syl2anc 411 . . . 4  |-  ( ( ( ph  /\  j  e.  Z )  /\  k  e.  ( ZZ>= `  M )
)  ->  ( G `  k )  e.  CC )
3832, 35, 10syl2anc 411 . . . . 5  |-  ( ( ( ph  /\  j  e.  Z )  /\  k  e.  ( ZZ>= `  M )
)  ->  ( ( F `  k )  +  ( G `  k ) )  e.  CC )
3935, 38, 15syl2anc 411 . . . 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 10519 . . 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 11348 . 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 11443 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 104    = wceq 1363    e. wcel 2158   _Vcvv 2749    |-> cmpt 4076   dom cdm 4638   ` cfv 5228  (class class class)co 5888   CCcc 7823    + caddc 7828   ZZcz 9267   ZZ>=cuz 9542    seqcseq 10459    ~~> cli 11300   sum_csu 11375
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-13 2160  ax-14 2161  ax-ext 2169  ax-coll 4130  ax-sep 4133  ax-nul 4141  ax-pow 4186  ax-pr 4221  ax-un 4445  ax-setind 4548  ax-iinf 4599  ax-cnex 7916  ax-resscn 7917  ax-1cn 7918  ax-1re 7919  ax-icn 7920  ax-addcl 7921  ax-addrcl 7922  ax-mulcl 7923  ax-mulrcl 7924  ax-addcom 7925  ax-mulcom 7926  ax-addass 7927  ax-mulass 7928  ax-distr 7929  ax-i2m1 7930  ax-0lt1 7931  ax-1rid 7932  ax-0id 7933  ax-rnegex 7934  ax-precex 7935  ax-cnre 7936  ax-pre-ltirr 7937  ax-pre-ltwlin 7938  ax-pre-lttrn 7939  ax-pre-apti 7940  ax-pre-ltadd 7941  ax-pre-mulgt0 7942  ax-pre-mulext 7943  ax-arch 7944  ax-caucvg 7945
This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 980  df-3an 981  df-tru 1366  df-fal 1369  df-nf 1471  df-sb 1773  df-eu 2039  df-mo 2040  df-clab 2174  df-cleq 2180  df-clel 2183  df-nfc 2318  df-ne 2358  df-nel 2453  df-ral 2470  df-rex 2471  df-reu 2472  df-rmo 2473  df-rab 2474  df-v 2751  df-sbc 2975  df-csb 3070  df-dif 3143  df-un 3145  df-in 3147  df-ss 3154  df-nul 3435  df-if 3547  df-pw 3589  df-sn 3610  df-pr 3611  df-op 3613  df-uni 3822  df-int 3857  df-iun 3900  df-br 4016  df-opab 4077  df-mpt 4078  df-tr 4114  df-id 4305  df-po 4308  df-iso 4309  df-iord 4378  df-on 4380  df-ilim 4381  df-suc 4383  df-iom 4602  df-xp 4644  df-rel 4645  df-cnv 4646  df-co 4647  df-dm 4648  df-rn 4649  df-res 4650  df-ima 4651  df-iota 5190  df-fun 5230  df-fn 5231  df-f 5232  df-f1 5233  df-fo 5234  df-f1o 5235  df-fv 5236  df-isom 5237  df-riota 5844  df-ov 5891  df-oprab 5892  df-mpo 5893  df-1st 6155  df-2nd 6156  df-recs 6320  df-irdg 6385  df-frec 6406  df-1o 6431  df-oadd 6435  df-er 6549  df-en 6755  df-dom 6756  df-fin 6757  df-pnf 8008  df-mnf 8009  df-xr 8010  df-ltxr 8011  df-le 8012  df-sub 8144  df-neg 8145  df-reap 8546  df-ap 8553  df-div 8644  df-inn 8934  df-2 8992  df-3 8993  df-4 8994  df-n0 9191  df-z 9268  df-uz 9543  df-q 9634  df-rp 9668  df-fz 10023  df-fzo 10157  df-seqfrec 10460  df-exp 10534  df-ihash 10770  df-cj 10865  df-re 10866  df-im 10867  df-rsqrt 11021  df-abs 11022  df-clim 11301  df-sumdc 11376
This theorem is referenced by:  sumsplitdc  11454
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