Theorem isumadd 11394

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 2247	. . . . 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 2247	. . . . 5 |- ( ( ph /\ k e. Z ) -> ( G ` k ) e. CC )
10	6, 9	addcld 7939	. . . 4 |- ( ( ph /\ k e. Z ) -> ( ( F ` k ) + ( G ` k ) ) e. CC )
11		fveq2 5496	. . . . . 6 |- ( m = k -> ( F ` m ) = ( F ` k ) )
12		fveq2 5496	. . . . . 6 |- ( m = k -> ( G ` m ) = ( G ` k ) )
13	11, 12	oveq12d 5871	. . . . 5 |- ( m = k -> ( ( F ` m ) + ( G ` m ) ) = ( ( F ` k ) + ( G ` k ) ) )
14		eqid 2170	. . . . 5 |- ( m e. Z |-> ( ( F ` m ) + ( G ` m ) ) ) = ( m e. Z |-> ( ( F ` m ) + ( G ` m ) ) )
15	13, 14	fvmptg 5572	. . . 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 409	. . 3 |- ( ( ph /\ k e. Z ) -> ( ( m e. Z |-> ( ( F ` m ) + ( G ` m ) ) ) ` k ) = ( ( F ` k ) + ( G ` k ) ) )
17	4, 7	oveq12d 5871	. . 3 |- ( ( ph /\ k e. Z ) -> ( ( F ` k ) + ( G ` k ) ) = ( A + B ) )
18	16, 17	eqtrd 2203	. 2 |- ( ( ph /\ k e. Z ) -> ( ( m e. Z |-> ( ( F ` m ) + ( G ` m ) ) ) ` k ) = ( A + B ) )
19	5, 8	addcld 7939	. 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 11385	. . 3 |- ( ph -> seq M ( + , F ) ~~> sum_ k e. Z A )
22		seqex 10403	. . . 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 11385	. . 3 |- ( ph -> seq M ( + , G ) ~~> sum_ k e. Z B )
26	1, 2, 6	serf 10430	. . . 4 |- ( ph -> seq M ( + , F ) : Z --> CC )
27	26	ffvelrnda 5631	. . 3 |- ( ( ph /\ j e. Z ) -> ( seq M ( + , F ) ` j ) e. CC )
28	1, 2, 9	serf 10430	. . . 4 |- ( ph -> seq M ( + , G ) : Z --> CC )
29	28	ffvelrnda 5631	. . 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 2263	. . . 4 |- ( ( ph /\ j e. Z ) -> j e. ( ZZ>= ` M ) )
32		simpll 524	. . . . 5 |- ( ( ( ph /\ j e. Z ) /\ k e. ( ZZ>= ` M ) ) -> ph )
33	1	eleq2i 2237	. . . . . . 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 409	. . . 4 |- ( ( ( ph /\ j e. Z ) /\ k e. ( ZZ>= ` M ) ) -> ( F ` k ) e. CC )
37	32, 35, 9	syl2anc 409	. . . 4 |- ( ( ( ph /\ j e. Z ) /\ k e. ( ZZ>= ` M ) ) -> ( G ` k ) e. CC )
38	32, 35, 10	syl2anc 409	. . . . 5 |- ( ( ( ph /\ j e. Z ) /\ k e. ( ZZ>= ` M ) ) -> ( ( F ` k ) + ( G ` k ) ) e. CC )
39	35, 38, 15	syl2anc 409	. . . 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 10461	. . 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 11289	. 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 11384	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 1348   e. wcel 2141   _Vcvv 2730   |-> cmpt 4050   dom cdm 4611   ` cfv 5198  (class class class)co 5853   CCcc 7772   + caddc 7777   ZZcz 9212   ZZ>=cuz 9487   seqcseq 10401   ~~> cli 11241   sum_csu 11316

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 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-coll 4104  ax-sep 4107  ax-nul 4115  ax-pow 4160  ax-pr 4194  ax-un 4418  ax-setind 4521  ax-iinf 4572  ax-cnex 7865  ax-resscn 7866  ax-1cn 7867  ax-1re 7868  ax-icn 7869  ax-addcl 7870  ax-addrcl 7871  ax-mulcl 7872  ax-mulrcl 7873  ax-addcom 7874  ax-mulcom 7875  ax-addass 7876  ax-mulass 7877  ax-distr 7878  ax-i2m1 7879  ax-0lt1 7880  ax-1rid 7881  ax-0id 7882  ax-rnegex 7883  ax-precex 7884  ax-cnre 7885  ax-pre-ltirr 7886  ax-pre-ltwlin 7887  ax-pre-lttrn 7888  ax-pre-apti 7889  ax-pre-ltadd 7890  ax-pre-mulgt0 7891  ax-pre-mulext 7892  ax-arch 7893  ax-caucvg 7894

This theorem depends on definitions:  df-bi 116  df-dc 830  df-3or 974  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-nel 2436  df-ral 2453  df-rex 2454  df-reu 2455  df-rmo 2456  df-rab 2457  df-v 2732  df-sbc 2956  df-csb 3050  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-nul 3415  df-if 3527  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-int 3832  df-iun 3875  df-br 3990  df-opab 4051  df-mpt 4052  df-tr 4088  df-id 4278  df-po 4281  df-iso 4282  df-iord 4351  df-on 4353  df-ilim 4354  df-suc 4356  df-iom 4575  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-rn 4622  df-res 4623  df-ima 4624  df-iota 5160  df-fun 5200  df-fn 5201  df-f 5202  df-f1 5203  df-fo 5204  df-f1o 5205  df-fv 5206  df-isom 5207  df-riota 5809  df-ov 5856  df-oprab 5857  df-mpo 5858  df-1st 6119  df-2nd 6120  df-recs 6284  df-irdg 6349  df-frec 6370  df-1o 6395  df-oadd 6399  df-er 6513  df-en 6719  df-dom 6720  df-fin 6721  df-pnf 7956  df-mnf 7957  df-xr 7958  df-ltxr 7959  df-le 7960  df-sub 8092  df-neg 8093  df-reap 8494  df-ap 8501  df-div 8590  df-inn 8879  df-2 8937  df-3 8938  df-4 8939  df-n0 9136  df-z 9213  df-uz 9488  df-q 9579  df-rp 9611  df-fz 9966  df-fzo 10099  df-seqfrec 10402  df-exp 10476  df-ihash 10710  df-cj 10806  df-re 10807  df-im 10808  df-rsqrt 10962  df-abs 10963  df-clim 11242  df-sumdc 11317

This theorem is referenced by:  sumsplitdc  11395