Theorem isumltss 13672

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 12079	. . . . 5 |- ( M e. ZZ -> -. Z e. Fin )
4	1, 3	syl 16	. . . 4 |- ( ph -> -. Z e. Fin )
5		ssdif0 3888	. . . . 5 |- ( Z C_ A <-> ( Z \ A ) = (/) )
6		isumltss.4	. . . . . 6 |- ( ph -> A C_ Z )
7		eqss 3514	. . . . . . 7 |- ( A = Z <-> ( A C_ Z /\ Z C_ A ) )
8		isumltss.3	. . . . . . . 8 |- ( ph -> A e. Fin )
9		eleq1 2529	. . . . . . . 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 3804	. . 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 3499	. . . . 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 11281	. . . . 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 13568	. . 3 |- ( ( ph /\ x e. ( Z \ A ) ) -> sum_ k e. A B e. RR )
25		snfi 7615	. . . . 5 |- { x } e. Fin
26		unfi 7805	. . . . 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 3622	. . . . . . . . 9 |- ( x e. ( Z \ A ) -> x e. Z )
29	28	snssd 4177	. . . . . . . 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 3674	. . . . . . 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 3499	. . . . 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 13568	. . 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 13592	. . 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 3112	. . . . . . . 8 |- x e. _V
44	43	snnz 4150	. . . . . . 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 3499	. . . . . . 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 13571	. . . . 5 |- ( ( ph /\ x e. ( Z \ A ) ) -> sum_ k e. { x } B e. RR+ )
50	24, 49	ltaddrpd 11310	. . . 4 |- ( ( ph /\ x e. ( Z \ A ) ) -> sum_ k e. A B < ( sum_ k e. A B + sum_ k e. { x } B ) )
51		eldifn 3623	. . . . . . 7 |- ( x e. ( Z \ A ) -> -. x e. A )
52	51	adantl 466	. . . . . 6 |- ( ( ph /\ x e. ( Z \ A ) ) -> -. x e. A )
53		disjsn 4092	. . . . . 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 2458	. . . . 5 |- ( ( ph /\ x e. ( Z \ A ) ) -> ( A u. { x } ) = ( A u. { x } ) )
56	21	rpcnd 11283	. . . . . 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 13574	. . . 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 4482	. . 3 |- ( ( ph /\ x e. ( Z \ A ) ) -> sum_ k e. A B < sum_ k e. ( A u. { x } ) B )
60	21	rpge0d 11285	. . . 4 |- ( ( ( ph /\ x e. ( Z \ A ) ) /\ k e. Z ) -> 0 <_ B )
61	2, 36, 27, 32, 38, 22, 60, 40	isumless 13669	. . 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 9759	. 2 |- ( ( ph /\ x e. ( Z \ A ) ) -> sum_ k e. A B < sum_ k e. Z B )
63	16, 62	exlimddv 1727	1 |- ( ph -> sum_ k e. A B < sum_ k e. Z B )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 369   = wceq 1395   E.wex 1613   e. wcel 1819   =/= wne 2652   \ cdif 3468   u. cun 3469   i^i cin 3470   C_ wss 3471   (/)c0 3793   {csn 4032   class class class wbr 4456   dom cdm 5008   ` cfv 5594  (class class class)co 6296   Fincfn 7535   CCcc 9507   RRcr 9508   + caddc 9512   < clt 9645   ZZcz 10885   ZZ>=cuz 11106   RR+crp 11245   seqcseq 12110   ~~> cli 13319   sum_csu 13520

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-rep 4568  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591  ax-inf2 8075  ax-cnex 9565  ax-resscn 9566  ax-1cn 9567  ax-icn 9568  ax-addcl 9569  ax-addrcl 9570  ax-mulcl 9571  ax-mulrcl 9572  ax-mulcom 9573  ax-addass 9574  ax-mulass 9575  ax-distr 9576  ax-i2m1 9577  ax-1ne0 9578  ax-1rid 9579  ax-rnegex 9580  ax-rrecex 9581  ax-cnre 9582  ax-pre-lttri 9583  ax-pre-lttrn 9584  ax-pre-ltadd 9585  ax-pre-mulgt0 9586  ax-pre-sup 9587

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-fal 1401  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-nel 2655  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-tp 4037  df-op 4039  df-uni 4252  df-int 4289  df-iun 4334  df-br 4457  df-opab 4516  df-mpt 4517  df-tr 4551  df-eprel 4800  df-id 4804  df-po 4809  df-so 4810  df-fr 4847  df-se 4848  df-we 4849  df-ord 4890  df-on 4891  df-lim 4892  df-suc 4893  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-isom 5603  df-riota 6258  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-om 6700  df-1st 6799  df-2nd 6800  df-recs 7060  df-rdg 7094  df-1o 7148  df-oadd 7152  df-er 7329  df-pm 7441  df-en 7536  df-dom 7537  df-sdom 7538  df-fin 7539  df-sup 7919  df-oi 7953  df-card 8337  df-pnf 9647  df-mnf 9648  df-xr 9649  df-ltxr 9650  df-le 9651  df-sub 9826  df-neg 9827  df-div 10228  df-nn 10557  df-2 10615  df-3 10616  df-n0 10817  df-z 10886  df-uz 11107  df-rp 11246  df-fz 11698  df-fzo 11822  df-fl 11932  df-seq 12111  df-exp 12170  df-hash 12409  df-cj 12944  df-re 12945  df-im 12946  df-sqrt 13080  df-abs 13081  df-clim 13323  df-rlim 13324  df-sum 13521

This theorem is referenced by: (None)