Theorem isumltss 13307

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 11784	. . . . 5 |- ( M e. ZZ -> -. Z e. Fin )
4	1, 3	syl 16	. . . 4 |- ( ph -> -. Z e. Fin )
5		ssdif0 3734	. . . . 5 |- ( Z C_ A <-> ( Z \ A ) = (/) )
6		isumltss.4	. . . . . 6 |- ( ph -> A C_ Z )
7		eqss 3368	. . . . . . 7 |- ( A = Z <-> ( A C_ Z /\ Z C_ A ) )
8		isumltss.3	. . . . . . . 8 |- ( ph -> A e. Fin )
9		eleq1 2501	. . . . . . . 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 670	. . . . 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 3644	. . 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 462	. . . 4 |- ( ( ph /\ x e. ( Z \ A ) ) -> A e. Fin )
18	6	adantr 462	. . . . . 6 |- ( ( ph /\ x e. ( Z \ A ) ) -> A C_ Z )
19	18	sselda 3353	. . . . 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 709	. . . . . 6 |- ( ( ( ph /\ x e. ( Z \ A ) ) /\ k e. Z ) -> B e. RR+ )
22	21	rpred 11023	. . . . 5 |- ( ( ( ph /\ x e. ( Z \ A ) ) /\ k e. Z ) -> B e. RR )
23	19, 22	syldan 467	. . . 4 |- ( ( ( ph /\ x e. ( Z \ A ) ) /\ k e. A ) -> B e. RR )
24	17, 23	fsumrecl 13207	. . 3 |- ( ( ph /\ x e. ( Z \ A ) ) -> sum_ k e. A B e. RR )
25		snfi 7386	. . . . 5 |- { x } e. Fin
26		unfi 7575	. . . . 5 |- ( ( A e. Fin /\ { x } e. Fin ) -> ( A u. { x } ) e. Fin )
27	17, 25, 26	sylancl 657	. . . 4 |- ( ( ph /\ x e. ( Z \ A ) ) -> ( A u. { x } ) e. Fin )
28		eldifi 3475	. . . . . . . . 9 |- ( x e. ( Z \ A ) -> x e. Z )
29	28	snssd 4015	. . . . . . . 8 |- ( x e. ( Z \ A ) -> { x } C_ Z )
30	6, 29	anim12i 563	. . . . . . 7 |- ( ( ph /\ x e. ( Z \ A ) ) -> ( A C_ Z /\ { x } C_ Z ) )
31		unss 3527	. . . . . . 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 3353	. . . . 5 |- ( ( ( ph /\ x e. ( Z \ A ) ) /\ k e. ( A u. { x } ) ) -> k e. Z )
34	33, 22	syldan 467	. . . 4 |- ( ( ( ph /\ x e. ( Z \ A ) ) /\ k e. ( A u. { x } ) ) -> B e. RR )
35	27, 34	fsumrecl 13207	. . 3 |- ( ( ph /\ x e. ( Z \ A ) ) -> sum_ k e. ( A u. { x } ) B e. RR )
36	1	adantr 462	. . . 4 |- ( ( ph /\ x e. ( Z \ A ) ) -> M e. ZZ )
37		isumltss.5	. . . . 5 |- ( ( ph /\ k e. Z ) -> ( F ` k ) = B )
38	37	adantlr 709	. . . 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 462	. . . 4 |- ( ( ph /\ x e. ( Z \ A ) ) -> seq M ( + , F ) e. dom ~~> )
41	2, 36, 38, 22, 40	isumrecl 13228	. . 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 2973	. . . . . . . 8 |- x e. _V
44	43	snnz 3990	. . . . . . 7 |- { x } =/= (/)
45	44	a1i 11	. . . . . 6 |- ( ( ph /\ x e. ( Z \ A ) ) -> { x } =/= (/) )
46	29	adantl 463	. . . . . . . 8 |- ( ( ph /\ x e. ( Z \ A ) ) -> { x } C_ Z )
47	46	sselda 3353	. . . . . . 7 |- ( ( ( ph /\ x e. ( Z \ A ) ) /\ k e. { x } ) -> k e. Z )
48	47, 21	syldan 467	. . . . . 6 |- ( ( ( ph /\ x e. ( Z \ A ) ) /\ k e. { x } ) -> B e. RR+ )
49	42, 45, 48	fsumrpcl 13210	. . . . 5 |- ( ( ph /\ x e. ( Z \ A ) ) -> sum_ k e. { x } B e. RR+ )
50	24, 49	ltaddrpd 11052	. . . 4 |- ( ( ph /\ x e. ( Z \ A ) ) -> sum_ k e. A B < ( sum_ k e. A B + sum_ k e. { x } B ) )
51		eldifn 3476	. . . . . . 7 |- ( x e. ( Z \ A ) -> -. x e. A )
52	51	adantl 463	. . . . . 6 |- ( ( ph /\ x e. ( Z \ A ) ) -> -. x e. A )
53		disjsn 3933	. . . . . 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 2442	. . . . 5 |- ( ( ph /\ x e. ( Z \ A ) ) -> ( A u. { x } ) = ( A u. { x } ) )
56	21	rpcnd 11025	. . . . . 6 |- ( ( ( ph /\ x e. ( Z \ A ) ) /\ k e. Z ) -> B e. CC )
57	33, 56	syldan 467	. . . . 5 |- ( ( ( ph /\ x e. ( Z \ A ) ) /\ k e. ( A u. { x } ) ) -> B e. CC )
58	54, 55, 27, 57	fsumsplit 13212	. . . 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 4315	. . 3 |- ( ( ph /\ x e. ( Z \ A ) ) -> sum_ k e. A B < sum_ k e. ( A u. { x } ) B )
60	21	rpge0d 11027	. . . 4 |- ( ( ( ph /\ x e. ( Z \ A ) ) /\ k e. Z ) -> 0 <_ B )
61	2, 36, 27, 32, 38, 22, 60, 40	isumless 13304	. . 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 9527	. 2 |- ( ( ph /\ x e. ( Z \ A ) ) -> sum_ k e. A B < sum_ k e. Z B )
63	16, 62	exlimddv 1697	1 |- ( ph -> sum_ k e. A B < sum_ k e. Z B )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 369   = wceq 1364   E.wex 1591   e. wcel 1761   =/= wne 2604   \ cdif 3322   u. cun 3323   i^i cin 3324   C_ wss 3325   (/)c0 3634   {csn 3874   class class class wbr 4289   dom cdm 4836   ` cfv 5415  (class class class)co 6090   Fincfn 7306   CCcc 9276   RRcr 9277   + caddc 9281   < clt 9414   ZZcz 10642   ZZ>=cuz 10857   RR+crp 10987   seqcseq 11802   ~~> cli 12958   sum_csu 13159

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1713  ax-7 1733  ax-8 1763  ax-9 1765  ax-10 1780  ax-11 1785  ax-12 1797  ax-13 1948  ax-ext 2422  ax-rep 4400  ax-sep 4410  ax-nul 4418  ax-pow 4467  ax-pr 4528  ax-un 6371  ax-inf2 7843  ax-cnex 9334  ax-resscn 9335  ax-1cn 9336  ax-icn 9337  ax-addcl 9338  ax-addrcl 9339  ax-mulcl 9340  ax-mulrcl 9341  ax-mulcom 9342  ax-addass 9343  ax-mulass 9344  ax-distr 9345  ax-i2m1 9346  ax-1ne0 9347  ax-1rid 9348  ax-rnegex 9349  ax-rrecex 9350  ax-cnre 9351  ax-pre-lttri 9352  ax-pre-lttrn 9353  ax-pre-ltadd 9354  ax-pre-mulgt0 9355  ax-pre-sup 9356

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 961  df-3an 962  df-tru 1367  df-fal 1370  df-ex 1592  df-nf 1595  df-sb 1706  df-eu 2261  df-mo 2262  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-nel 2607  df-ral 2718  df-rex 2719  df-reu 2720  df-rmo 2721  df-rab 2722  df-v 2972  df-sbc 3184  df-csb 3286  df-dif 3328  df-un 3330  df-in 3332  df-ss 3339  df-pss 3341  df-nul 3635  df-if 3789  df-pw 3859  df-sn 3875  df-pr 3877  df-tp 3879  df-op 3881  df-uni 4089  df-int 4126  df-iun 4170  df-br 4290  df-opab 4348  df-mpt 4349  df-tr 4383  df-eprel 4628  df-id 4632  df-po 4637  df-so 4638  df-fr 4675  df-se 4676  df-we 4677  df-ord 4718  df-on 4719  df-lim 4720  df-suc 4721  df-xp 4842  df-rel 4843  df-cnv 4844  df-co 4845  df-dm 4846  df-rn 4847  df-res 4848  df-ima 4849  df-iota 5378  df-fun 5417  df-fn 5418  df-f 5419  df-f1 5420  df-fo 5421  df-f1o 5422  df-fv 5423  df-isom 5424  df-riota 6049  df-ov 6093  df-oprab 6094  df-mpt2 6095  df-om 6476  df-1st 6576  df-2nd 6577  df-recs 6828  df-rdg 6862  df-1o 6916  df-oadd 6920  df-er 7097  df-pm 7213  df-en 7307  df-dom 7308  df-sdom 7309  df-fin 7310  df-sup 7687  df-oi 7720  df-card 8105  df-pnf 9416  df-mnf 9417  df-xr 9418  df-ltxr 9419  df-le 9420  df-sub 9593  df-neg 9594  df-div 9990  df-nn 10319  df-2 10376  df-3 10377  df-n0 10576  df-z 10643  df-uz 10858  df-rp 10988  df-fz 11434  df-fzo 11545  df-fl 11638  df-seq 11803  df-exp 11862  df-hash 12100  df-cj 12584  df-re 12585  df-im 12586  df-sqr 12720  df-abs 12721  df-clim 12962  df-rlim 12963  df-sum 13160

This theorem is referenced by: (None)