Theorem isumadd 11742

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 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 )
6	4, 5	eqeltrd 2282	. . . . 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 2282	. . . . 5 |- ( ( ph /\ k e. Z ) -> ( G ` k ) e. CC )
10	6, 9	addcld 8092	. . . 4 |- ( ( ph /\ k e. Z ) -> ( ( F ` k ) + ( G ` k ) ) e. CC )
11		fveq2 5576	. . . . . 6 |- ( m = k -> ( F ` m ) = ( F ` k ) )
12		fveq2 5576	. . . . . 6 |- ( m = k -> ( G ` m ) = ( G ` k ) )
13	11, 12	oveq12d 5962	. . . . 5 |- ( m = k -> ( ( F ` m ) + ( G ` m ) ) = ( ( F ` k ) + ( G ` k ) ) )
14		eqid 2205	. . . . 5 |- ( m e. Z |-> ( ( F ` m ) + ( G ` m ) ) ) = ( m e. Z |-> ( ( F ` m ) + ( G ` m ) ) )
15	13, 14	fvmptg 5655	. . . 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 411	. . 3 |- ( ( ph /\ k e. Z ) -> ( ( m e. Z |-> ( ( F ` m ) + ( G ` m ) ) ) ` k ) = ( ( F ` k ) + ( G ` k ) ) )
17	4, 7	oveq12d 5962	. . 3 |- ( ( ph /\ k e. Z ) -> ( ( F ` k ) + ( G ` k ) ) = ( A + B ) )
18	16, 17	eqtrd 2238	. 2 |- ( ( ph /\ k e. Z ) -> ( ( m e. Z |-> ( ( F ` m ) + ( G ` m ) ) ) ` k ) = ( A + B ) )
19	5, 8	addcld 8092	. 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 11733	. . 3 |- ( ph -> seq M ( + , F ) ~~> sum_ k e. Z A )
22		seqex 10594	. . . 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 11733	. . 3 |- ( ph -> seq M ( + , G ) ~~> sum_ k e. Z B )
26	1, 2, 6	serf 10628	. . . 4 |- ( ph -> seq M ( + , F ) : Z --> CC )
27	26	ffvelcdmda 5715	. . 3 |- ( ( ph /\ j e. Z ) -> ( seq M ( + , F ) ` j ) e. CC )
28	1, 2, 9	serf 10628	. . . 4 |- ( ph -> seq M ( + , G ) : Z --> CC )
29	28	ffvelcdmda 5715	. . 3 |- ( ( ph /\ j e. Z ) -> ( seq M ( + , G ) ` j ) e. CC )
30		simpr 110	. . . . 5 |- ( ( ph /\ j e. Z ) -> j e. Z )
31	30, 1	eleqtrdi 2298	. . . 4 |- ( ( ph /\ j e. Z ) -> j e. ( ZZ>= ` M ) )
32		simpll 527	. . . . 5 |- ( ( ( ph /\ j e. Z ) /\ k e. ( ZZ>= ` M ) ) -> ph )
33	1	eleq2i 2272	. . . . . . 7 |- ( k e. Z <-> k e. ( ZZ>= ` M ) )
34	33	biimpri 133	. . . . . 6 |- ( k e. ( ZZ>= ` M ) -> k e. Z )
35	34	adantl 277	. . . . 5 |- ( ( ( ph /\ j e. Z ) /\ k e. ( ZZ>= ` M ) ) -> k e. Z )
36	32, 35, 6	syl2anc 411	. . . 4 |- ( ( ( ph /\ j e. Z ) /\ k e. ( ZZ>= ` M ) ) -> ( F ` k ) e. CC )
37	32, 35, 9	syl2anc 411	. . . 4 |- ( ( ( ph /\ j e. Z ) /\ k e. ( ZZ>= ` M ) ) -> ( G ` k ) e. CC )
38	32, 35, 10	syl2anc 411	. . . . 5 |- ( ( ( ph /\ j e. Z ) /\ k e. ( ZZ>= ` M ) ) -> ( ( F ` k ) + ( G ` k ) ) e. CC )
39	35, 38, 15	syl2anc 411	. . . 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 10667	. . 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 11637	. 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 11732	1 |- ( ph -> sum_ k e. Z ( A + B ) = ( sum_ k e. Z A + sum_ k e. Z B ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1373   e. wcel 2176   _Vcvv 2772   |-> cmpt 4105   dom cdm 4675   ` cfv 5271  (class class class)co 5944   CCcc 7923   + caddc 7928   ZZcz 9372   ZZ>=cuz 9648   seqcseq 10592   ~~> cli 11589   sum_csu 11664

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 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-13 2178  ax-14 2179  ax-ext 2187  ax-coll 4159  ax-sep 4162  ax-nul 4170  ax-pow 4218  ax-pr 4253  ax-un 4480  ax-setind 4585  ax-iinf 4636  ax-cnex 8016  ax-resscn 8017  ax-1cn 8018  ax-1re 8019  ax-icn 8020  ax-addcl 8021  ax-addrcl 8022  ax-mulcl 8023  ax-mulrcl 8024  ax-addcom 8025  ax-mulcom 8026  ax-addass 8027  ax-mulass 8028  ax-distr 8029  ax-i2m1 8030  ax-0lt1 8031  ax-1rid 8032  ax-0id 8033  ax-rnegex 8034  ax-precex 8035  ax-cnre 8036  ax-pre-ltirr 8037  ax-pre-ltwlin 8038  ax-pre-lttrn 8039  ax-pre-apti 8040  ax-pre-ltadd 8041  ax-pre-mulgt0 8042  ax-pre-mulext 8043  ax-arch 8044  ax-caucvg 8045

This theorem depends on definitions:  df-bi 117  df-dc 837  df-3or 982  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1484  df-sb 1786  df-eu 2057  df-mo 2058  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ne 2377  df-nel 2472  df-ral 2489  df-rex 2490  df-reu 2491  df-rmo 2492  df-rab 2493  df-v 2774  df-sbc 2999  df-csb 3094  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3461  df-if 3572  df-pw 3618  df-sn 3639  df-pr 3640  df-op 3642  df-uni 3851  df-int 3886  df-iun 3929  df-br 4045  df-opab 4106  df-mpt 4107  df-tr 4143  df-id 4340  df-po 4343  df-iso 4344  df-iord 4413  df-on 4415  df-ilim 4416  df-suc 4418  df-iom 4639  df-xp 4681  df-rel 4682  df-cnv 4683  df-co 4684  df-dm 4685  df-rn 4686  df-res 4687  df-ima 4688  df-iota 5232  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-riota 5899  df-ov 5947  df-oprab 5948  df-mpo 5949  df-1st 6226  df-2nd 6227  df-recs 6391  df-irdg 6456  df-frec 6477  df-1o 6502  df-oadd 6506  df-er 6620  df-en 6828  df-dom 6829  df-fin 6830  df-pnf 8109  df-mnf 8110  df-xr 8111  df-ltxr 8112  df-le 8113  df-sub 8245  df-neg 8246  df-reap 8648  df-ap 8655  df-div 8746  df-inn 9037  df-2 9095  df-3 9096  df-4 9097  df-n0 9296  df-z 9373  df-uz 9649  df-q 9741  df-rp 9776  df-fz 10131  df-fzo 10265  df-seqfrec 10593  df-exp 10684  df-ihash 10921  df-cj 11153  df-re 11154  df-im 11155  df-rsqrt 11309  df-abs 11310  df-clim 11590  df-sumdc 11665

This theorem is referenced by:  sumsplitdc  11743