Theorem isumadd 11827

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 2283	. . . . 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 2283	. . . . 5 |- ( ( ph /\ k e. Z ) -> ( G ` k ) e. CC )
10	6, 9	addcld 8122	. . . 4 |- ( ( ph /\ k e. Z ) -> ( ( F ` k ) + ( G ` k ) ) e. CC )
11		fveq2 5594	. . . . . 6 |- ( m = k -> ( F ` m ) = ( F ` k ) )
12		fveq2 5594	. . . . . 6 |- ( m = k -> ( G ` m ) = ( G ` k ) )
13	11, 12	oveq12d 5980	. . . . 5 |- ( m = k -> ( ( F ` m ) + ( G ` m ) ) = ( ( F ` k ) + ( G ` k ) ) )
14		eqid 2206	. . . . 5 |- ( m e. Z |-> ( ( F ` m ) + ( G ` m ) ) ) = ( m e. Z |-> ( ( F ` m ) + ( G ` m ) ) )
15	13, 14	fvmptg 5673	. . . 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 5980	. . 3 |- ( ( ph /\ k e. Z ) -> ( ( F ` k ) + ( G ` k ) ) = ( A + B ) )
18	16, 17	eqtrd 2239	. 2 |- ( ( ph /\ k e. Z ) -> ( ( m e. Z |-> ( ( F ` m ) + ( G ` m ) ) ) ` k ) = ( A + B ) )
19	5, 8	addcld 8122	. 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 11818	. . 3 |- ( ph -> seq M ( + , F ) ~~> sum_ k e. Z A )
22		seqex 10626	. . . 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 11818	. . 3 |- ( ph -> seq M ( + , G ) ~~> sum_ k e. Z B )
26	1, 2, 6	serf 10660	. . . 4 |- ( ph -> seq M ( + , F ) : Z --> CC )
27	26	ffvelcdmda 5733	. . 3 |- ( ( ph /\ j e. Z ) -> ( seq M ( + , F ) ` j ) e. CC )
28	1, 2, 9	serf 10660	. . . 4 |- ( ph -> seq M ( + , G ) : Z --> CC )
29	28	ffvelcdmda 5733	. . 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 2299	. . . 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 2273	. . . . . . 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 10699	. . 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 11722	. 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 11817	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 2177   _Vcvv 2773   |-> cmpt 4116   dom cdm 4688   ` cfv 5285  (class class class)co 5962   CCcc 7953   + caddc 7958   ZZcz 9402   ZZ>=cuz 9678   seqcseq 10624   ~~> cli 11674   sum_csu 11749

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 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2179  ax-14 2180  ax-ext 2188  ax-coll 4170  ax-sep 4173  ax-nul 4181  ax-pow 4229  ax-pr 4264  ax-un 4493  ax-setind 4598  ax-iinf 4649  ax-cnex 8046  ax-resscn 8047  ax-1cn 8048  ax-1re 8049  ax-icn 8050  ax-addcl 8051  ax-addrcl 8052  ax-mulcl 8053  ax-mulrcl 8054  ax-addcom 8055  ax-mulcom 8056  ax-addass 8057  ax-mulass 8058  ax-distr 8059  ax-i2m1 8060  ax-0lt1 8061  ax-1rid 8062  ax-0id 8063  ax-rnegex 8064  ax-precex 8065  ax-cnre 8066  ax-pre-ltirr 8067  ax-pre-ltwlin 8068  ax-pre-lttrn 8069  ax-pre-apti 8070  ax-pre-ltadd 8071  ax-pre-mulgt0 8072  ax-pre-mulext 8073  ax-arch 8074  ax-caucvg 8075

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 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-ne 2378  df-nel 2473  df-ral 2490  df-rex 2491  df-reu 2492  df-rmo 2493  df-rab 2494  df-v 2775  df-sbc 3003  df-csb 3098  df-dif 3172  df-un 3174  df-in 3176  df-ss 3183  df-nul 3465  df-if 3576  df-pw 3623  df-sn 3644  df-pr 3645  df-op 3647  df-uni 3860  df-int 3895  df-iun 3938  df-br 4055  df-opab 4117  df-mpt 4118  df-tr 4154  df-id 4353  df-po 4356  df-iso 4357  df-iord 4426  df-on 4428  df-ilim 4429  df-suc 4431  df-iom 4652  df-xp 4694  df-rel 4695  df-cnv 4696  df-co 4697  df-dm 4698  df-rn 4699  df-res 4700  df-ima 4701  df-iota 5246  df-fun 5287  df-fn 5288  df-f 5289  df-f1 5290  df-fo 5291  df-f1o 5292  df-fv 5293  df-isom 5294  df-riota 5917  df-ov 5965  df-oprab 5966  df-mpo 5967  df-1st 6244  df-2nd 6245  df-recs 6409  df-irdg 6474  df-frec 6495  df-1o 6520  df-oadd 6524  df-er 6638  df-en 6846  df-dom 6847  df-fin 6848  df-pnf 8139  df-mnf 8140  df-xr 8141  df-ltxr 8142  df-le 8143  df-sub 8275  df-neg 8276  df-reap 8678  df-ap 8685  df-div 8776  df-inn 9067  df-2 9125  df-3 9126  df-4 9127  df-n0 9326  df-z 9403  df-uz 9679  df-q 9771  df-rp 9806  df-fz 10161  df-fzo 10295  df-seqfrec 10625  df-exp 10716  df-ihash 10953  df-cj 11238  df-re 11239  df-im 11240  df-rsqrt 11394  df-abs 11395  df-clim 11675  df-sumdc 11750

This theorem is referenced by:  sumsplitdc  11828