Theorem isumadd 14498

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		fveq2 6191	. . . . . 6 |- ( m = k -> ( F ` m ) = ( F ` k ) )
4		fveq2 6191	. . . . . 6 |- ( m = k -> ( G ` m ) = ( G ` k ) )
5	3, 4	oveq12d 6668	. . . . 5 |- ( m = k -> ( ( F ` m ) + ( G ` m ) ) = ( ( F ` k ) + ( G ` k ) ) )
6		eqid 2622	. . . . 5 |- ( m e. Z |-> ( ( F ` m ) + ( G ` m ) ) ) = ( m e. Z |-> ( ( F ` m ) + ( G ` m ) ) )
7		ovex 6678	. . . . 5 |- ( ( F ` k ) + ( G ` k ) ) e. _V
8	5, 6, 7	fvmpt 6282	. . . 4 |- ( k e. Z -> ( ( m e. Z |-> ( ( F ` m ) + ( G ` m ) ) ) ` k ) = ( ( F ` k ) + ( G ` k ) ) )
9	8	adantl 482	. . 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 )
12	10, 11	oveq12d 6668	. . 3 |- ( ( ph /\ k e. Z ) -> ( ( F ` k ) + ( G ` k ) ) = ( A + B ) )
13	9, 12	eqtrd 2656	. 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 )
16	14, 15	addcld 10059	. 2 |- ( ( ph /\ k e. Z ) -> ( A + B ) e. CC )
17		isumadd.7	. . . 4 |- ( ph -> seq M ( + , F ) e. dom ~~> )
18	1, 2, 10, 14, 17	isumclim2 14489	. . 3 |- ( ph -> seq M ( + , F ) ~~> sum_ k e. Z A )
19		seqex 12803	. . . 4 |- seq M ( + , ( m e. Z |-> ( ( F ` m ) + ( G ` m ) ) ) ) e. _V
20	19	a1i 11	. . 3 |- ( ph -> seq M ( + , ( m e. Z |-> ( ( F ` m ) + ( G ` m ) ) ) ) e. _V )
21		isumadd.8	. . . 4 |- ( ph -> seq M ( + , G ) e. dom ~~> )
22	1, 2, 11, 15, 21	isumclim2 14489	. . 3 |- ( ph -> seq M ( + , G ) ~~> sum_ k e. Z B )
23	10, 14	eqeltrd 2701	. . . . 5 |- ( ( ph /\ k e. Z ) -> ( F ` k ) e. CC )
24	1, 2, 23	serf 12829	. . . 4 |- ( ph -> seq M ( + , F ) : Z --> CC )
25	24	ffvelrnda 6359	. . 3 |- ( ( ph /\ j e. Z ) -> ( seq M ( + , F ) ` j ) e. CC )
26	11, 15	eqeltrd 2701	. . . . 5 |- ( ( ph /\ k e. Z ) -> ( G ` k ) e. CC )
27	1, 2, 26	serf 12829	. . . 4 |- ( ph -> seq M ( + , G ) : Z --> CC )
28	27	ffvelrnda 6359	. . 3 |- ( ( ph /\ j e. Z ) -> ( seq M ( + , G ) ` j ) e. CC )
29		simpr 477	. . . . 5 |- ( ( ph /\ j e. Z ) -> j e. Z )
30	29, 1	syl6eleq 2711	. . . 4 |- ( ( ph /\ j e. Z ) -> j e. ( ZZ>= ` M ) )
31		simpll 790	. . . . 5 |- ( ( ( ph /\ j e. Z ) /\ k e. ( M ... j ) ) -> ph )
32		elfzuz 12338	. . . . . . 7 |- ( k e. ( M ... j ) -> k e. ( ZZ>= ` M ) )
33	32, 1	syl6eleqr 2712	. . . . . 6 |- ( k e. ( M ... j ) -> k e. Z )
34	33	adantl 482	. . . . 5 |- ( ( ( ph /\ j e. Z ) /\ k e. ( M ... j ) ) -> k e. Z )
35	31, 34, 23	syl2anc 693	. . . 4 |- ( ( ( ph /\ j e. Z ) /\ k e. ( M ... j ) ) -> ( F ` k ) e. CC )
36	31, 34, 26	syl2anc 693	. . . 4 |- ( ( ( ph /\ j e. Z ) /\ k e. ( M ... j ) ) -> ( G ` k ) e. CC )
37	34, 8	syl 17	. . . 4 |- ( ( ( ph /\ j e. Z ) /\ k e. ( M ... j ) ) -> ( ( m e. Z |-> ( ( F ` m ) + ( G ` m ) ) ) ` k ) = ( ( F ` k ) + ( G ` k ) ) )
38	30, 35, 36, 37	seradd 12843	. . 3 |- ( ( ph /\ j e. Z ) -> ( seq M ( + , ( m e. Z |-> ( ( F ` m ) + ( G ` m ) ) ) ) ` j ) = ( ( seq M ( + , F ) ` j ) + ( seq M ( + , G ) ` j ) ) )
39	1, 2, 18, 20, 22, 25, 28, 38	climadd 14362	. 2 |- ( ph -> seq M ( + , ( m e. Z |-> ( ( F ` m ) + ( G ` m ) ) ) ) ~~> ( sum_ k e. Z A + sum_ k e. Z B ) )
40	1, 2, 13, 16, 39	isumclim 14488	1 |- ( ph -> sum_ k e. Z ( A + B ) = ( sum_ k e. Z A + sum_ k e. Z B ) )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   _Vcvv 3200   |-> cmpt 4729   dom cdm 5114   ` cfv 5888  (class class class)co 6650   CCcc 9934   + caddc 9939   ZZcz 11377   ZZ>=cuz 11687   ...cfz 12326   seqcseq 12801   ~~> cli 14215   sum_csu 14416

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-inf2 8538  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013  ax-pre-sup 10014

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-se 5074  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-isom 5897  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-oadd 7564  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-sup 8348  df-oi 8415  df-card 8765  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-div 10685  df-nn 11021  df-2 11079  df-3 11080  df-n0 11293  df-z 11378  df-uz 11688  df-rp 11833  df-fz 12327  df-fzo 12466  df-seq 12802  df-exp 12861  df-hash 13118  df-cj 13839  df-re 13840  df-im 13841  df-sqrt 13975  df-abs 13976  df-clim 14219  df-sum 14417

This theorem is referenced by:  sumsplit  14499  binomcxplemnotnn0  38555