Theorem isumadd 11200

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.

Step	Hyp	Ref	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 )
6	4, 5	eqeltrd 2216	. . . . 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 )
9	7, 8	eqeltrd 2216	. . . . 5 |- ( ( ph /\ k e. Z ) -> ( G ` k ) e. CC )
10	6, 9	addcld 7785	. . . 4 |- ( ( ph /\ k e. Z ) -> ( ( F ` k ) + ( G ` k ) ) e. CC )
11		fveq2 5421	. . . . . 6 |- ( m = k -> ( F ` m ) = ( F ` k ) )
12		fveq2 5421	. . . . . 6 |- ( m = k -> ( G ` m ) = ( G ` k ) )
13	11, 12	oveq12d 5792	. . . . 5 |- ( m = k -> ( ( F ` m ) + ( G ` m ) ) = ( ( F ` k ) + ( G ` k ) ) )
14		eqid 2139	. . . . 5 |- ( m e. Z |-> ( ( F ` m ) + ( G ` m ) ) ) = ( m e. Z |-> ( ( F ` m ) + ( G ` m ) ) )
15	13, 14	fvmptg 5497	. . . 4 |- ( ( k e. Z /\ ( ( F ` k ) + ( G ` k ) ) e. CC ) -> ( ( m e. Z |-> ( ( F ` m ) + ( G ` m ) ) ) ` k ) = ( ( F ` k ) + ( G ` k ) ) )
16	3, 10, 15	syl2anc 408	. . 3 |- ( ( ph /\ k e. Z ) -> ( ( m e. Z |-> ( ( F ` m ) + ( G ` m ) ) ) ` k ) = ( ( F ` k ) + ( G ` k ) ) )
17	4, 7	oveq12d 5792	. . 3 |- ( ( ph /\ k e. Z ) -> ( ( F ` k ) + ( G ` k ) ) = ( A + B ) )
18	16, 17	eqtrd 2172	. 2 |- ( ( ph /\ k e. Z ) -> ( ( m e. Z |-> ( ( F ` m ) + ( G ` m ) ) ) ` k ) = ( A + B ) )
19	5, 8	addcld 7785	. 2 |- ( ( ph /\ k e. Z ) -> ( A + B ) e. CC )
20		isumadd.7	. . . 4 |- ( ph -> seq M ( + , F ) e. dom ~~> )
21	1, 2, 4, 5, 20	isumclim2 11191	. . 3 |- ( ph -> seq M ( + , F ) ~~> sum_ k e. Z A )
22		seqex 10220	. . . 4 |- seq M ( + , ( m e. Z |-> ( ( F ` m ) + ( G ` m ) ) ) ) e. _V
23	22	a1i 9	. . 3 |- ( ph -> seq M ( + , ( m e. Z |-> ( ( F ` m ) + ( G ` m ) ) ) ) e. _V )
24		isumadd.8	. . . 4 |- ( ph -> seq M ( + , G ) e. dom ~~> )
25	1, 2, 7, 8, 24	isumclim2 11191	. . 3 |- ( ph -> seq M ( + , G ) ~~> sum_ k e. Z B )
26	1, 2, 6	serf 10247	. . . 4 |- ( ph -> seq M ( + , F ) : Z --> CC )
27	26	ffvelrnda 5555	. . 3 |- ( ( ph /\ j e. Z ) -> ( seq M ( + , F ) ` j ) e. CC )
28	1, 2, 9	serf 10247	. . . 4 |- ( ph -> seq M ( + , G ) : Z --> CC )
29	28	ffvelrnda 5555	. . 3 |- ( ( ph /\ j e. Z ) -> ( seq M ( + , G ) ` j ) e. CC )
30		simpr 109	. . . . 5 |- ( ( ph /\ j e. Z ) -> j e. Z )
31	30, 1	eleqtrdi 2232	. . . 4 |- ( ( ph /\ j e. Z ) -> j e. ( ZZ>= ` M ) )
32		simpll 518	. . . . 5 |- ( ( ( ph /\ j e. Z ) /\ k e. ( ZZ>= ` M ) ) -> ph )
33	1	eleq2i 2206	. . . . . . 7 |- ( k e. Z <-> k e. ( ZZ>= ` M ) )
34	33	biimpri 132	. . . . . 6 |- ( k e. ( ZZ>= ` M ) -> k e. Z )
35	34	adantl 275	. . . . 5 |- ( ( ( ph /\ j e. Z ) /\ k e. ( ZZ>= ` M ) ) -> k e. Z )
36	32, 35, 6	syl2anc 408	. . . 4 |- ( ( ( ph /\ j e. Z ) /\ k e. ( ZZ>= ` M ) ) -> ( F ` k ) e. CC )
37	32, 35, 9	syl2anc 408	. . . 4 |- ( ( ( ph /\ j e. Z ) /\ k e. ( ZZ>= ` M ) ) -> ( G ` k ) e. CC )
38	32, 35, 10	syl2anc 408	. . . . 5 |- ( ( ( ph /\ j e. Z ) /\ k e. ( ZZ>= ` M ) ) -> ( ( F ` k ) + ( G ` k ) ) e. CC )
39	35, 38, 15	syl2anc 408	. . . 4 |- ( ( ( ph /\ j e. Z ) /\ k e. ( ZZ>= ` M ) ) -> ( ( m e. Z |-> ( ( F ` m ) + ( G ` m ) ) ) ` k ) = ( ( F ` k ) + ( G ` k ) ) )
40	31, 36, 37, 39	ser3add 10278	. . 3 |- ( ( ph /\ j e. Z ) -> ( seq M ( + , ( m e. Z |-> ( ( F ` m ) + ( G ` m ) ) ) ) ` j ) = ( ( seq M ( + , F ) ` j ) + ( seq M ( + , G ) ` j ) ) )
41	1, 2, 21, 23, 25, 27, 29, 40	climadd 11095	. 2 |- ( ph -> seq M ( + , ( m e. Z |-> ( ( F ` m ) + ( G ` m ) ) ) ) ~~> ( sum_ k e. Z A + sum_ k e. Z B ) )
42	1, 2, 18, 19, 41	isumclim 11190	1 |- ( ph -> sum_ k e. Z ( A + B ) = ( sum_ k e. Z A + sum_ k e. Z B ) )

Syntax hints:   -> wi 4   /\ wa 103   = wceq 1331   e. wcel 1480   _Vcvv 2686   |-> cmpt 3989   dom cdm 4539   ` cfv 5123  (class class class)co 5774   CCcc 7618   + caddc 7623   ZZcz 9054   ZZ>=cuz 9326   seqcseq 10218   ~~> cli 11047   sum_csu 11122

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-coll 4043  ax-sep 4046  ax-nul 4054  ax-pow 4098  ax-pr 4131  ax-un 4355  ax-setind 4452  ax-iinf 4502  ax-cnex 7711  ax-resscn 7712  ax-1cn 7713  ax-1re 7714  ax-icn 7715  ax-addcl 7716  ax-addrcl 7717  ax-mulcl 7718  ax-mulrcl 7719  ax-addcom 7720  ax-mulcom 7721  ax-addass 7722  ax-mulass 7723  ax-distr 7724  ax-i2m1 7725  ax-0lt1 7726  ax-1rid 7727  ax-0id 7728  ax-rnegex 7729  ax-precex 7730  ax-cnre 7731  ax-pre-ltirr 7732  ax-pre-ltwlin 7733  ax-pre-lttrn 7734  ax-pre-apti 7735  ax-pre-ltadd 7736  ax-pre-mulgt0 7737  ax-pre-mulext 7738  ax-arch 7739  ax-caucvg 7740

This theorem depends on definitions:  df-bi 116  df-dc 820  df-3or 963  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ne 2309  df-nel 2404  df-ral 2421  df-rex 2422  df-reu 2423  df-rmo 2424  df-rab 2425  df-v 2688  df-sbc 2910  df-csb 3004  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-nul 3364  df-if 3475  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-int 3772  df-iun 3815  df-br 3930  df-opab 3990  df-mpt 3991  df-tr 4027  df-id 4215  df-po 4218  df-iso 4219  df-iord 4288  df-on 4290  df-ilim 4291  df-suc 4293  df-iom 4505  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-rn 4550  df-res 4551  df-ima 4552  df-iota 5088  df-fun 5125  df-fn 5126  df-f 5127  df-f1 5128  df-fo 5129  df-f1o 5130  df-fv 5131  df-isom 5132  df-riota 5730  df-ov 5777  df-oprab 5778  df-mpo 5779  df-1st 6038  df-2nd 6039  df-recs 6202  df-irdg 6267  df-frec 6288  df-1o 6313  df-oadd 6317  df-er 6429  df-en 6635  df-dom 6636  df-fin 6637  df-pnf 7802  df-mnf 7803  df-xr 7804  df-ltxr 7805  df-le 7806  df-sub 7935  df-neg 7936  df-reap 8337  df-ap 8344  df-div 8433  df-inn 8721  df-2 8779  df-3 8780  df-4 8781  df-n0 8978  df-z 9055  df-uz 9327  df-q 9412  df-rp 9442  df-fz 9791  df-fzo 9920  df-seqfrec 10219  df-exp 10293  df-ihash 10522  df-cj 10614  df-re 10615  df-im 10616  df-rsqrt 10770  df-abs 10771  df-clim 11048  df-sumdc 11123

This theorem is referenced by:  sumsplitdc  11201