Theorem isumltss 13616

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 12039	. . . . 5 |- ( M e. ZZ -> -. Z e. Fin )
4	1, 3	syl 16	. . . 4 |- ( ph -> -. Z e. Fin )
5		ssdif0 3885	. . . . 5 |- ( Z C_ A <-> ( Z \ A ) = (/) )
6		isumltss.4	. . . . . 6 |- ( ph -> A C_ Z )
7		eqss 3519	. . . . . . 7 |- ( A = Z <-> ( A C_ Z /\ Z C_ A ) )
8		isumltss.3	. . . . . . . 8 |- ( ph -> A e. Fin )
9		eleq1 2539	. . . . . . . 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 3795	. . 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 3504	. . . . 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 11252	. . . . 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 13512	. . 3 |- ( ( ph /\ x e. ( Z \ A ) ) -> sum_ k e. A B e. RR )
25		snfi 7593	. . . . 5 |- { x } e. Fin
26		unfi 7783	. . . . 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 3626	. . . . . . . . 9 |- ( x e. ( Z \ A ) -> x e. Z )
29	28	snssd 4172	. . . . . . . 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 3678	. . . . . . 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 3504	. . . . 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 13512	. . 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 13536	. . 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 3116	. . . . . . . 8 |- x e. _V
44	43	snnz 4145	. . . . . . 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 3504	. . . . . . 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 13515	. . . . 5 |- ( ( ph /\ x e. ( Z \ A ) ) -> sum_ k e. { x } B e. RR+ )
50	24, 49	ltaddrpd 11281	. . . 4 |- ( ( ph /\ x e. ( Z \ A ) ) -> sum_ k e. A B < ( sum_ k e. A B + sum_ k e. { x } B ) )
51		eldifn 3627	. . . . . . 7 |- ( x e. ( Z \ A ) -> -. x e. A )
52	51	adantl 466	. . . . . 6 |- ( ( ph /\ x e. ( Z \ A ) ) -> -. x e. A )
53		disjsn 4088	. . . . . 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 2468	. . . . 5 |- ( ( ph /\ x e. ( Z \ A ) ) -> ( A u. { x } ) = ( A u. { x } ) )
56	21	rpcnd 11254	. . . . . 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 13518	. . . 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 4473	. . 3 |- ( ( ph /\ x e. ( Z \ A ) ) -> sum_ k e. A B < sum_ k e. ( A u. { x } ) B )
60	21	rpge0d 11256	. . . 4 |- ( ( ( ph /\ x e. ( Z \ A ) ) /\ k e. Z ) -> 0 <_ B )
61	2, 36, 27, 32, 38, 22, 60, 40	isumless 13613	. . 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 9737	. 2 |- ( ( ph /\ x e. ( Z \ A ) ) -> sum_ k e. A B < sum_ k e. Z B )
63	16, 62	exlimddv 1702	1 |- ( ph -> sum_ k e. A B < sum_ k e. Z B )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 369   = wceq 1379   E.wex 1596   e. wcel 1767   =/= wne 2662   \ cdif 3473   u. cun 3474   i^i cin 3475   C_ wss 3476   (/)c0 3785   {csn 4027   class class class wbr 4447   dom cdm 4999   ` cfv 5586  (class class class)co 6282   Fincfn 7513   CCcc 9486   RRcr 9487   + caddc 9491   < clt 9624   ZZcz 10860   ZZ>=cuz 11078   RR+crp 11216   seqcseq 12070   ~~> cli 13263   sum_csu 13464

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6574  ax-inf2 8054  ax-cnex 9544  ax-resscn 9545  ax-1cn 9546  ax-icn 9547  ax-addcl 9548  ax-addrcl 9549  ax-mulcl 9550  ax-mulrcl 9551  ax-mulcom 9552  ax-addass 9553  ax-mulass 9554  ax-distr 9555  ax-i2m1 9556  ax-1ne0 9557  ax-1rid 9558  ax-rnegex 9559  ax-rrecex 9560  ax-cnre 9561  ax-pre-lttri 9562  ax-pre-lttrn 9563  ax-pre-ltadd 9564  ax-pre-mulgt0 9565  ax-pre-sup 9566

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-fal 1385  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-int 4283  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-se 4839  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fun 5588  df-fn 5589  df-f 5590  df-f1 5591  df-fo 5592  df-f1o 5593  df-fv 5594  df-isom 5595  df-riota 6243  df-ov 6285  df-oprab 6286  df-mpt2 6287  df-om 6679  df-1st 6781  df-2nd 6782  df-recs 7039  df-rdg 7073  df-1o 7127  df-oadd 7131  df-er 7308  df-pm 7420  df-en 7514  df-dom 7515  df-sdom 7516  df-fin 7517  df-sup 7897  df-oi 7931  df-card 8316  df-pnf 9626  df-mnf 9627  df-xr 9628  df-ltxr 9629  df-le 9630  df-sub 9803  df-neg 9804  df-div 10203  df-nn 10533  df-2 10590  df-3 10591  df-n0 10792  df-z 10861  df-uz 11079  df-rp 11217  df-fz 11669  df-fzo 11789  df-fl 11893  df-seq 12071  df-exp 12130  df-hash 12368  df-cj 12889  df-re 12890  df-im 12891  df-sqrt 13025  df-abs 13026  df-clim 13267  df-rlim 13268  df-sum 13465

This theorem is referenced by: (None)