Theorem fisumcvg 10730

Description: The sequence of partial sums of a finite sum converges to the whole sum. (Contributed by Mario Carneiro, 20-Apr-2014.) Use fsum3cvg 10731 instead. (New usage is discouraged.)

Hypotheses

Ref	Expression
isummo.1	|- F = ( k e. ZZ |-> if ( k e. A , B , 0 ) )
isummo.2	|- ( ( ph /\ k e. A ) -> B e. CC )
isummo.dc	|- ( ( ph /\ k e. ( ZZ>= ` M ) ) -> DECID k e. A )
isumrb.3	|- ( ph -> N e. ( ZZ>= ` M ) )
fisumcvg.4	|- ( ph -> A C_ ( M ... N ) )

Assertion

Ref	Expression
fisumcvg	|- ( ph -> seq M ( + , F , CC ) ~~> ( seq M ( + , F , CC ) ` N ) )

Distinct variable groups:   A, k   k, N   ph, k   k, M   k, F

Allowed substitution hint:   B( k)

Proof of Theorem fisumcvg

Dummy variables n z m are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqid 2088	. 2 |- ( ZZ>= ` N ) = ( ZZ>= ` N )
2		isumrb.3	. . 3 |- ( ph -> N e. ( ZZ>= ` M ) )
3		eluzelz 8997	. . 3 |- ( N e. ( ZZ>= ` M ) -> N e. ZZ )
4	2, 3	syl 14	. 2 |- ( ph -> N e. ZZ )
5		iseqex 9821	. . 3 |- seq M ( + , F , CC ) e. _V
6	5	a1i 9	. 2 |- ( ph -> seq M ( + , F , CC ) e. _V )
7		eqid 2088	. . . 4 |- ( ZZ>= ` M ) = ( ZZ>= ` M )
8		eluzel2 8993	. . . . 5 |- ( N e. ( ZZ>= ` M ) -> M e. ZZ )
9	2, 8	syl 14	. . . 4 |- ( ph -> M e. ZZ )
10		eluzelz 8997	. . . . . . 7 |- ( k e. ( ZZ>= ` M ) -> k e. ZZ )
11	10	adantl 271	. . . . . 6 |- ( ( ph /\ k e. ( ZZ>= ` M ) ) -> k e. ZZ )
12		iftrue 3394	. . . . . . . . . . 11 |- ( k e. A -> if ( k e. A , B , 0 ) = B )
13	12	adantl 271	. . . . . . . . . 10 |- ( ( ph /\ k e. A ) -> if ( k e. A , B , 0 ) = B )
14		isummo.2	. . . . . . . . . 10 |- ( ( ph /\ k e. A ) -> B e. CC )
15	13, 14	eqeltrd 2164	. . . . . . . . 9 |- ( ( ph /\ k e. A ) -> if ( k e. A , B , 0 ) e. CC )
16	15	ex 113	. . . . . . . 8 |- ( ph -> ( k e. A -> if ( k e. A , B , 0 ) e. CC ) )
17	16	adantr 270	. . . . . . 7 |- ( ( ph /\ k e. ( ZZ>= ` M ) ) -> ( k e. A -> if ( k e. A , B , 0 ) e. CC ) )
18		iffalse 3397	. . . . . . . . 9 |- ( -. k e. A -> if ( k e. A , B , 0 ) = 0 )
19		0cn 7459	. . . . . . . . 9 |- 0 e. CC
20	18, 19	syl6eqel 2178	. . . . . . . 8 |- ( -. k e. A -> if ( k e. A , B , 0 ) e. CC )
21	20	a1i 9	. . . . . . 7 |- ( ( ph /\ k e. ( ZZ>= ` M ) ) -> ( -. k e. A -> if ( k e. A , B , 0 ) e. CC ) )
22		isummo.dc	. . . . . . . 8 |- ( ( ph /\ k e. ( ZZ>= ` M ) ) -> DECID k e. A )
23		exmiddc 782	. . . . . . . 8 |- (DECID k e. A -> ( k e. A \/ -. k e. A ) )
24	22, 23	syl 14	. . . . . . 7 |- ( ( ph /\ k e. ( ZZ>= ` M ) ) -> ( k e. A \/ -. k e. A ) )
25	17, 21, 24	mpjaod 673	. . . . . 6 |- ( ( ph /\ k e. ( ZZ>= ` M ) ) -> if ( k e. A , B , 0 ) e. CC )
26		isummo.1	. . . . . . 7 |- F = ( k e. ZZ |-> if ( k e. A , B , 0 ) )
27	26	fvmpt2 5370	. . . . . 6 |- ( ( k e. ZZ /\ if ( k e. A , B , 0 ) e. CC ) -> ( F ` k ) = if ( k e. A , B , 0 ) )
28	11, 25, 27	syl2anc 403	. . . . 5 |- ( ( ph /\ k e. ( ZZ>= ` M ) ) -> ( F ` k ) = if ( k e. A , B , 0 ) )
29	28, 25	eqeltrd 2164	. . . 4 |- ( ( ph /\ k e. ( ZZ>= ` M ) ) -> ( F ` k ) e. CC )
30	7, 9, 29	iserf 9868	. . 3 |- ( ph -> seq M ( + , F , CC ) : ( ZZ>= ` M ) --> CC )
31	30, 2	ffvelrnd 5419	. 2 |- ( ph -> ( seq M ( + , F , CC ) ` N ) e. CC )
32		addid1 7599	. . . . 5 |- ( m e. CC -> ( m + 0 ) = m )
33	32	adantl 271	. . . 4 |- ( ( ( ph /\ n e. ( ZZ>= ` N ) ) /\ m e. CC ) -> ( m + 0 ) = m )
34	2	adantr 270	. . . 4 |- ( ( ph /\ n e. ( ZZ>= ` N ) ) -> N e. ( ZZ>= ` M ) )
35		simpr 108	. . . 4 |- ( ( ph /\ n e. ( ZZ>= ` N ) ) -> n e. ( ZZ>= ` N ) )
36	31	adantr 270	. . . 4 |- ( ( ph /\ n e. ( ZZ>= ` N ) ) -> ( seq M ( + , F , CC ) ` N ) e. CC )
37		elfzuz 9405	. . . . . 6 |- ( m e. ( ( N + 1 ) ... n ) -> m e. ( ZZ>= ` ( N + 1 ) ) )
38		eluzelz 8997	. . . . . . . . 9 |- ( m e. ( ZZ>= ` ( N + 1 ) ) -> m e. ZZ )
39	38	adantl 271	. . . . . . . 8 |- ( ( ph /\ m e. ( ZZ>= ` ( N + 1 ) ) ) -> m e. ZZ )
40		fisumcvg.4	. . . . . . . . . . . 12 |- ( ph -> A C_ ( M ... N ) )
41	40	sseld 3022	. . . . . . . . . . 11 |- ( ph -> ( m e. A -> m e. ( M ... N ) ) )
42		fznuz 9483	. . . . . . . . . . 11 |- ( m e. ( M ... N ) -> -. m e. ( ZZ>= ` ( N + 1 ) ) )
43	41, 42	syl6 33	. . . . . . . . . 10 |- ( ph -> ( m e. A -> -. m e. ( ZZ>= ` ( N + 1 ) ) ) )
44	43	con2d 589	. . . . . . . . 9 |- ( ph -> ( m e. ( ZZ>= ` ( N + 1 ) ) -> -. m e. A ) )
45	44	imp 122	. . . . . . . 8 |- ( ( ph /\ m e. ( ZZ>= ` ( N + 1 ) ) ) -> -. m e. A )
46	39, 45	eldifd 3007	. . . . . . 7 |- ( ( ph /\ m e. ( ZZ>= ` ( N + 1 ) ) ) -> m e. ( ZZ \ A ) )
47		fveq2 5289	. . . . . . . . 9 |- ( k = m -> ( F ` k ) = ( F ` m ) )
48	47	eqeq1d 2096	. . . . . . . 8 |- ( k = m -> ( ( F ` k ) = 0 <-> ( F ` m ) = 0 ) )
49		eldifi 3120	. . . . . . . . . 10 |- ( k e. ( ZZ \ A ) -> k e. ZZ )
50		eldifn 3121	. . . . . . . . . . . 12 |- ( k e. ( ZZ \ A ) -> -. k e. A )
51	50, 18	syl 14	. . . . . . . . . . 11 |- ( k e. ( ZZ \ A ) -> if ( k e. A , B , 0 ) = 0 )
52	51, 19	syl6eqel 2178	. . . . . . . . . 10 |- ( k e. ( ZZ \ A ) -> if ( k e. A , B , 0 ) e. CC )
53	49, 52, 27	syl2anc 403	. . . . . . . . 9 |- ( k e. ( ZZ \ A ) -> ( F ` k ) = if ( k e. A , B , 0 ) )
54	53, 51	eqtrd 2120	. . . . . . . 8 |- ( k e. ( ZZ \ A ) -> ( F ` k ) = 0 )
55	48, 54	vtoclga 2685	. . . . . . 7 |- ( m e. ( ZZ \ A ) -> ( F ` m ) = 0 )
56	46, 55	syl 14	. . . . . 6 |- ( ( ph /\ m e. ( ZZ>= ` ( N + 1 ) ) ) -> ( F ` m ) = 0 )
57	37, 56	sylan2 280	. . . . 5 |- ( ( ph /\ m e. ( ( N + 1 ) ... n ) ) -> ( F ` m ) = 0 )
58	57	adantlr 461	. . . 4 |- ( ( ( ph /\ n e. ( ZZ>= ` N ) ) /\ m e. ( ( N + 1 ) ... n ) ) -> ( F ` m ) = 0 )
59	47	eleq1d 2156	. . . . 5 |- ( k = m -> ( ( F ` k ) e. CC <-> ( F ` m ) e. CC ) )
60	29	ralrimiva 2446	. . . . . 6 |- ( ph -> A. k e. ( ZZ>= ` M ) ( F ` k ) e. CC )
61	60	ad2antrr 472	. . . . 5 |- ( ( ( ph /\ n e. ( ZZ>= ` N ) ) /\ m e. ( ZZ>= ` M ) ) -> A. k e. ( ZZ>= ` M ) ( F ` k ) e. CC )
62		simpr 108	. . . . 5 |- ( ( ( ph /\ n e. ( ZZ>= ` N ) ) /\ m e. ( ZZ>= ` M ) ) -> m e. ( ZZ>= ` M ) )
63	59, 61, 62	rspcdva 2727	. . . 4 |- ( ( ( ph /\ n e. ( ZZ>= ` N ) ) /\ m e. ( ZZ>= ` M ) ) -> ( F ` m ) e. CC )
64		addcl 7446	. . . . 5 |- ( ( m e. CC /\ z e. CC ) -> ( m + z ) e. CC )
65	64	adantl 271	. . . 4 |- ( ( ( ph /\ n e. ( ZZ>= ` N ) ) /\ ( m e. CC /\ z e. CC ) ) -> ( m + z ) e. CC )
66	33, 34, 35, 36, 58, 63, 65	iseqid2 9906	. . 3 |- ( ( ph /\ n e. ( ZZ>= ` N ) ) -> ( seq M ( + , F , CC ) ` N ) = ( seq M ( + , F , CC ) ` n ) )
67	66	eqcomd 2093	. 2 |- ( ( ph /\ n e. ( ZZ>= ` N ) ) -> ( seq M ( + , F , CC ) ` n ) = ( seq M ( + , F , CC ) ` N ) )
68	1, 4, 6, 31, 67	climconst 10642	1 |- ( ph -> seq M ( + , F , CC ) ~~> ( seq M ( + , F , CC ) ` N ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 102   \/ wo 664  DECID wdc 780   = wceq 1289   e. wcel 1438   A.wral 2359   _Vcvv 2619   \ cdif 2994   C_ wss 2997   ifcif 3389   class class class wbr 3837   |-> cmpt 3891   ` cfv 5002  (class class class)co 5634   CCcc 7327   0cc0 7329   1c1 7330   + caddc 7332   ZZcz 8720   ZZ>=cuz 8988   ...cfz 9393   seqcseq4 9816   ~~> cli 10630

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 579  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-13 1449  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-coll 3946  ax-sep 3949  ax-nul 3957  ax-pow 4001  ax-pr 4027  ax-un 4251  ax-setind 4343  ax-iinf 4393  ax-cnex 7415  ax-resscn 7416  ax-1cn 7417  ax-1re 7418  ax-icn 7419  ax-addcl 7420  ax-addrcl 7421  ax-mulcl 7422  ax-mulrcl 7423  ax-addcom 7424  ax-mulcom 7425  ax-addass 7426  ax-mulass 7427  ax-distr 7428  ax-i2m1 7429  ax-0lt1 7430  ax-1rid 7431  ax-0id 7432  ax-rnegex 7433  ax-precex 7434  ax-cnre 7435  ax-pre-ltirr 7436  ax-pre-ltwlin 7437  ax-pre-lttrn 7438  ax-pre-apti 7439  ax-pre-ltadd 7440  ax-pre-mulgt0 7441  ax-pre-mulext 7442

This theorem depends on definitions:  df-bi 115  df-dc 781  df-3or 925  df-3an 926  df-tru 1292  df-fal 1295  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ne 2256  df-nel 2351  df-ral 2364  df-rex 2365  df-reu 2366  df-rmo 2367  df-rab 2368  df-v 2621  df-sbc 2839  df-csb 2932  df-dif 2999  df-un 3001  df-in 3003  df-ss 3010  df-nul 3285  df-if 3390  df-pw 3427  df-sn 3447  df-pr 3448  df-op 3450  df-uni 3649  df-int 3684  df-iun 3727  df-br 3838  df-opab 3892  df-mpt 3893  df-tr 3929  df-id 4111  df-po 4114  df-iso 4115  df-iord 4184  df-on 4186  df-ilim 4187  df-suc 4189  df-iom 4396  df-xp 4434  df-rel 4435  df-cnv 4436  df-co 4437  df-dm 4438  df-rn 4439  df-res 4440  df-ima 4441  df-iota 4967  df-fun 5004  df-fn 5005  df-f 5006  df-f1 5007  df-fo 5008  df-f1o 5009  df-fv 5010  df-riota 5590  df-ov 5637  df-oprab 5638  df-mpt2 5639  df-1st 5893  df-2nd 5894  df-recs 6052  df-frec 6138  df-pnf 7503  df-mnf 7504  df-xr 7505  df-ltxr 7506  df-le 7507  df-sub 7634  df-neg 7635  df-reap 8028  df-ap 8035  df-div 8114  df-inn 8395  df-2 8452  df-n0 8644  df-z 8721  df-uz 8989  df-rp 9104  df-fz 9394  df-iseq 9818  df-seq3 9819  df-exp 9920  df-cj 10241  df-rsqrt 10396  df-abs 10397  df-clim 10631

This theorem is referenced by:  isummolem2a  10735  fisumcvg2  10750