Theorem isumltss 13424

Description: A partial sum of a series with positive terms is less than the infinite sum. (Contributed by Jeff Madsen, 2-Sep-2009.) (Proof shortened by Mario Carneiro, 12-Mar-2015.)

Hypotheses

Ref	Expression
isumltss.1	|- Z = ( ZZ>= ` M )
isumltss.2	|- ( ph -> M e. ZZ )
isumltss.3	|- ( ph -> A e. Fin )
isumltss.4	|- ( ph -> A C_ Z )
isumltss.5	|- ( ( ph /\ k e. Z ) -> ( F ` k ) = B )
isumltss.6	|- ( ( ph /\ k e. Z ) -> B e. RR+ )
isumltss.7	|- ( ph -> seq M ( + , F ) e. dom ~~> )

Assertion

Ref	Expression
isumltss	|- ( ph -> sum_ k e. A B < sum_ k e. Z B )

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

Allowed substitution hint:   B( k)

Proof of Theorem isumltss

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		isumltss.2	. . . . 5 |- ( ph -> M e. ZZ )
2		isumltss.1	. . . . . 6 |- Z = ( ZZ>= ` M )
3	2	uzinf 11900	. . . . 5 |- ( M e. ZZ -> -. Z e. Fin )
4	1, 3	syl 16	. . . 4 |- ( ph -> -. Z e. Fin )
5		ssdif0 3840	. . . . 5 |- ( Z C_ A <-> ( Z \ A ) = (/) )
6		isumltss.4	. . . . . 6 |- ( ph -> A C_ Z )
7		eqss 3474	. . . . . . 7 |- ( A = Z <-> ( A C_ Z /\ Z C_ A ) )
8		isumltss.3	. . . . . . . 8 |- ( ph -> A e. Fin )
9		eleq1 2524	. . . . . . . 8 |- ( A = Z -> ( A e. Fin <-> Z e. Fin ) )
10	8, 9	syl5ibcom 220	. . . . . . 7 |- ( ph -> ( A = Z -> Z e. Fin ) )
11	7, 10	syl5bir 218	. . . . . 6 |- ( ph -> ( ( A C_ Z /\ Z C_ A ) -> Z e. Fin ) )
12	6, 11	mpand 675	. . . . 5 |- ( ph -> ( Z C_ A -> Z e. Fin ) )
13	5, 12	syl5bir 218	. . . 4 |- ( ph -> ( ( Z \ A ) = (/) -> Z e. Fin ) )
14	4, 13	mtod 177	. . 3 |- ( ph -> -. ( Z \ A ) = (/) )
15		neq0 3750	. . 3 |- ( -. ( Z \ A ) = (/) <-> E. x x e. ( Z \ A ) )
16	14, 15	sylib 196	. 2 |- ( ph -> E. x x e. ( Z \ A ) )
17	8	adantr 465	. . . 4 |- ( ( ph /\ x e. ( Z \ A ) ) -> A e. Fin )
18	6	adantr 465	. . . . . 6 |- ( ( ph /\ x e. ( Z \ A ) ) -> A C_ Z )
19	18	sselda 3459	. . . . 5 |- ( ( ( ph /\ x e. ( Z \ A ) ) /\ k e. A ) -> k e. Z )
20		isumltss.6	. . . . . . 7 |- ( ( ph /\ k e. Z ) -> B e. RR+ )
21	20	adantlr 714	. . . . . 6 |- ( ( ( ph /\ x e. ( Z \ A ) ) /\ k e. Z ) -> B e. RR+ )
22	21	rpred 11133	. . . . 5 |- ( ( ( ph /\ x e. ( Z \ A ) ) /\ k e. Z ) -> B e. RR )
23	19, 22	syldan 470	. . . 4 |- ( ( ( ph /\ x e. ( Z \ A ) ) /\ k e. A ) -> B e. RR )
24	17, 23	fsumrecl 13324	. . 3 |- ( ( ph /\ x e. ( Z \ A ) ) -> sum_ k e. A B e. RR )
25		snfi 7495	. . . . 5 |- { x } e. Fin
26		unfi 7685	. . . . 5 |- ( ( A e. Fin /\ { x } e. Fin ) -> ( A u. { x } ) e. Fin )
27	17, 25, 26	sylancl 662	. . . 4 |- ( ( ph /\ x e. ( Z \ A ) ) -> ( A u. { x } ) e. Fin )
28		eldifi 3581	. . . . . . . . 9 |- ( x e. ( Z \ A ) -> x e. Z )
29	28	snssd 4121	. . . . . . . 8 |- ( x e. ( Z \ A ) -> { x } C_ Z )
30	6, 29	anim12i 566	. . . . . . 7 |- ( ( ph /\ x e. ( Z \ A ) ) -> ( A C_ Z /\ { x } C_ Z ) )
31		unss 3633	. . . . . . 7 |- ( ( A C_ Z /\ { x } C_ Z ) <-> ( A u. { x } ) C_ Z )
32	30, 31	sylib 196	. . . . . 6 |- ( ( ph /\ x e. ( Z \ A ) ) -> ( A u. { x } ) C_ Z )
33	32	sselda 3459	. . . . 5 |- ( ( ( ph /\ x e. ( Z \ A ) ) /\ k e. ( A u. { x } ) ) -> k e. Z )
34	33, 22	syldan 470	. . . 4 |- ( ( ( ph /\ x e. ( Z \ A ) ) /\ k e. ( A u. { x } ) ) -> B e. RR )
35	27, 34	fsumrecl 13324	. . 3 |- ( ( ph /\ x e. ( Z \ A ) ) -> sum_ k e. ( A u. { x } ) B e. RR )
36	1	adantr 465	. . . 4 |- ( ( ph /\ x e. ( Z \ A ) ) -> M e. ZZ )
37		isumltss.5	. . . . 5 |- ( ( ph /\ k e. Z ) -> ( F ` k ) = B )
38	37	adantlr 714	. . . 4 |- ( ( ( ph /\ x e. ( Z \ A ) ) /\ k e. Z ) -> ( F ` k ) = B )
39		isumltss.7	. . . . 5 |- ( ph -> seq M ( + , F ) e. dom ~~> )
40	39	adantr 465	. . . 4 |- ( ( ph /\ x e. ( Z \ A ) ) -> seq M ( + , F ) e. dom ~~> )
41	2, 36, 38, 22, 40	isumrecl 13345	. . 3 |- ( ( ph /\ x e. ( Z \ A ) ) -> sum_ k e. Z B e. RR )
42	25	a1i 11	. . . . . 6 |- ( ( ph /\ x e. ( Z \ A ) ) -> { x } e. Fin )
43		vex 3075	. . . . . . . 8 |- x e. _V
44	43	snnz 4096	. . . . . . 7 |- { x } =/= (/)
45	44	a1i 11	. . . . . 6 |- ( ( ph /\ x e. ( Z \ A ) ) -> { x } =/= (/) )
46	29	adantl 466	. . . . . . . 8 |- ( ( ph /\ x e. ( Z \ A ) ) -> { x } C_ Z )
47	46	sselda 3459	. . . . . . 7 |- ( ( ( ph /\ x e. ( Z \ A ) ) /\ k e. { x } ) -> k e. Z )
48	47, 21	syldan 470	. . . . . 6 |- ( ( ( ph /\ x e. ( Z \ A ) ) /\ k e. { x } ) -> B e. RR+ )
49	42, 45, 48	fsumrpcl 13327	. . . . 5 |- ( ( ph /\ x e. ( Z \ A ) ) -> sum_ k e. { x } B e. RR+ )
50	24, 49	ltaddrpd 11162	. . . 4 |- ( ( ph /\ x e. ( Z \ A ) ) -> sum_ k e. A B < ( sum_ k e. A B + sum_ k e. { x } B ) )
51		eldifn 3582	. . . . . . 7 |- ( x e. ( Z \ A ) -> -. x e. A )
52	51	adantl 466	. . . . . 6 |- ( ( ph /\ x e. ( Z \ A ) ) -> -. x e. A )
53		disjsn 4039	. . . . . 6 |- ( ( A i^i { x } ) = (/) <-> -. x e. A )
54	52, 53	sylibr 212	. . . . 5 |- ( ( ph /\ x e. ( Z \ A ) ) -> ( A i^i { x } ) = (/) )
55		eqidd 2453	. . . . 5 |- ( ( ph /\ x e. ( Z \ A ) ) -> ( A u. { x } ) = ( A u. { x } ) )
56	21	rpcnd 11135	. . . . . 6 |- ( ( ( ph /\ x e. ( Z \ A ) ) /\ k e. Z ) -> B e. CC )
57	33, 56	syldan 470	. . . . 5 |- ( ( ( ph /\ x e. ( Z \ A ) ) /\ k e. ( A u. { x } ) ) -> B e. CC )
58	54, 55, 27, 57	fsumsplit 13329	. . . 4 |- ( ( ph /\ x e. ( Z \ A ) ) -> sum_ k e. ( A u. { x } ) B = ( sum_ k e. A B + sum_ k e. { x } B ) )
59	50, 58	breqtrrd 4421	. . 3 |- ( ( ph /\ x e. ( Z \ A ) ) -> sum_ k e. A B < sum_ k e. ( A u. { x } ) B )
60	21	rpge0d 11137	. . . 4 |- ( ( ( ph /\ x e. ( Z \ A ) ) /\ k e. Z ) -> 0 <_ B )
61	2, 36, 27, 32, 38, 22, 60, 40	isumless 13421	. . 3 |- ( ( ph /\ x e. ( Z \ A ) ) -> sum_ k e. ( A u. { x } ) B <_ sum_ k e. Z B )
62	24, 35, 41, 59, 61	ltletrd 9637	. 2 |- ( ( ph /\ x e. ( Z \ A ) ) -> sum_ k e. A B < sum_ k e. Z B )
63	16, 62	exlimddv 1693	1 |- ( ph -> sum_ k e. A B < sum_ k e. Z B )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 369   = wceq 1370   E.wex 1587   e. wcel 1758   =/= wne 2645   \ cdif 3428   u. cun 3429   i^i cin 3430   C_ wss 3431   (/)c0 3740   {csn 3980   class class class wbr 4395   dom cdm 4943   ` cfv 5521  (class class class)co 6195   Fincfn 7415   CCcc 9386   RRcr 9387   + caddc 9391   < clt 9524   ZZcz 10752   ZZ>=cuz 10967   RR+crp 11097   seqcseq 11918   ~~> cli 13075   sum_csu 13276

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1954  ax-ext 2431  ax-rep 4506  ax-sep 4516  ax-nul 4524  ax-pow 4573  ax-pr 4634  ax-un 6477  ax-inf2 7953  ax-cnex 9444  ax-resscn 9445  ax-1cn 9446  ax-icn 9447  ax-addcl 9448  ax-addrcl 9449  ax-mulcl 9450  ax-mulrcl 9451  ax-mulcom 9452  ax-addass 9453  ax-mulass 9454  ax-distr 9455  ax-i2m1 9456  ax-1ne0 9457  ax-1rid 9458  ax-rnegex 9459  ax-rrecex 9460  ax-cnre 9461  ax-pre-lttri 9462  ax-pre-lttrn 9463  ax-pre-ltadd 9464  ax-pre-mulgt0 9465  ax-pre-sup 9466

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-fal 1376  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2265  df-mo 2266  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2602  df-ne 2647  df-nel 2648  df-ral 2801  df-rex 2802  df-reu 2803  df-rmo 2804  df-rab 2805  df-v 3074  df-sbc 3289  df-csb 3391  df-dif 3434  df-un 3436  df-in 3438  df-ss 3445  df-pss 3447  df-nul 3741  df-if 3895  df-pw 3965  df-sn 3981  df-pr 3983  df-tp 3985  df-op 3987  df-uni 4195  df-int 4232  df-iun 4276  df-br 4396  df-opab 4454  df-mpt 4455  df-tr 4489  df-eprel 4735  df-id 4739  df-po 4744  df-so 4745  df-fr 4782  df-se 4783  df-we 4784  df-ord 4825  df-on 4826  df-lim 4827  df-suc 4828  df-xp 4949  df-rel 4950  df-cnv 4951  df-co 4952  df-dm 4953  df-rn 4954  df-res 4955  df-ima 4956  df-iota 5484  df-fun 5523  df-fn 5524  df-f 5525  df-f1 5526  df-fo 5527  df-f1o 5528  df-fv 5529  df-isom 5530  df-riota 6156  df-ov 6198  df-oprab 6199  df-mpt2 6200  df-om 6582  df-1st 6682  df-2nd 6683  df-recs 6937  df-rdg 6971  df-1o 7025  df-oadd 7029  df-er 7206  df-pm 7322  df-en 7416  df-dom 7417  df-sdom 7418  df-fin 7419  df-sup 7797  df-oi 7830  df-card 8215  df-pnf 9526  df-mnf 9527  df-xr 9528  df-ltxr 9529  df-le 9530  df-sub 9703  df-neg 9704  df-div 10100  df-nn 10429  df-2 10486  df-3 10487  df-n0 10686  df-z 10753  df-uz 10968  df-rp 11098  df-fz 11550  df-fzo 11661  df-fl 11754  df-seq 11919  df-exp 11978  df-hash 12216  df-cj 12701  df-re 12702  df-im 12703  df-sqr 12837  df-abs 12838  df-clim 13079  df-rlim 13080  df-sum 13277

This theorem is referenced by: (None)