Theorem isumltss 13955

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 12211	. . . . 5 |- ( M e. ZZ -> -. Z e. Fin )
4	1, 3	syl 17	. . . 4 |- ( ph -> -. Z e. Fin )
5		ssdif0 3835	. . . . 5 |- ( Z C_ A <-> ( Z \ A ) = (/) )
6		isumltss.4	. . . . . 6 |- ( ph -> A C_ Z )
7		eqss 3459	. . . . . . 7 |- ( A = Z <-> ( A C_ Z /\ Z C_ A ) )
8		isumltss.3	. . . . . . . 8 |- ( ph -> A e. Fin )
9		eleq1 2528	. . . . . . . 8 |- ( A = Z -> ( A e. Fin <-> Z e. Fin ) )
10	8, 9	syl5ibcom 228	. . . . . . 7 |- ( ph -> ( A = Z -> Z e. Fin ) )
11	7, 10	syl5bir 226	. . . . . 6 |- ( ph -> ( ( A C_ Z /\ Z C_ A ) -> Z e. Fin ) )
12	6, 11	mpand 686	. . . . 5 |- ( ph -> ( Z C_ A -> Z e. Fin ) )
13	5, 12	syl5bir 226	. . . 4 |- ( ph -> ( ( Z \ A ) = (/) -> Z e. Fin ) )
14	4, 13	mtod 182	. . 3 |- ( ph -> -. ( Z \ A ) = (/) )
15		neq0 3754	. . 3 |- ( -. ( Z \ A ) = (/) <-> E. x x e. ( Z \ A ) )
16	14, 15	sylib 201	. 2 |- ( ph -> E. x x e. ( Z \ A ) )
17	8	adantr 471	. . . 4 |- ( ( ph /\ x e. ( Z \ A ) ) -> A e. Fin )
18	6	adantr 471	. . . . . 6 |- ( ( ph /\ x e. ( Z \ A ) ) -> A C_ Z )
19	18	sselda 3444	. . . . 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 726	. . . . . 6 |- ( ( ( ph /\ x e. ( Z \ A ) ) /\ k e. Z ) -> B e. RR+ )
22	21	rpred 11370	. . . . 5 |- ( ( ( ph /\ x e. ( Z \ A ) ) /\ k e. Z ) -> B e. RR )
23	19, 22	syldan 477	. . . 4 |- ( ( ( ph /\ x e. ( Z \ A ) ) /\ k e. A ) -> B e. RR )
24	17, 23	fsumrecl 13849	. . 3 |- ( ( ph /\ x e. ( Z \ A ) ) -> sum_ k e. A B e. RR )
25		snfi 7676	. . . . 5 |- { x } e. Fin
26		unfi 7864	. . . . 5 |- ( ( A e. Fin /\ { x } e. Fin ) -> ( A u. { x } ) e. Fin )
27	17, 25, 26	sylancl 673	. . . 4 |- ( ( ph /\ x e. ( Z \ A ) ) -> ( A u. { x } ) e. Fin )
28		eldifi 3567	. . . . . . . . 9 |- ( x e. ( Z \ A ) -> x e. Z )
29	28	snssd 4130	. . . . . . . 8 |- ( x e. ( Z \ A ) -> { x } C_ Z )
30	6, 29	anim12i 574	. . . . . . 7 |- ( ( ph /\ x e. ( Z \ A ) ) -> ( A C_ Z /\ { x } C_ Z ) )
31		unss 3620	. . . . . . 7 |- ( ( A C_ Z /\ { x } C_ Z ) <-> ( A u. { x } ) C_ Z )
32	30, 31	sylib 201	. . . . . 6 |- ( ( ph /\ x e. ( Z \ A ) ) -> ( A u. { x } ) C_ Z )
33	32	sselda 3444	. . . . 5 |- ( ( ( ph /\ x e. ( Z \ A ) ) /\ k e. ( A u. { x } ) ) -> k e. Z )
34	33, 22	syldan 477	. . . 4 |- ( ( ( ph /\ x e. ( Z \ A ) ) /\ k e. ( A u. { x } ) ) -> B e. RR )
35	27, 34	fsumrecl 13849	. . 3 |- ( ( ph /\ x e. ( Z \ A ) ) -> sum_ k e. ( A u. { x } ) B e. RR )
36	1	adantr 471	. . . 4 |- ( ( ph /\ x e. ( Z \ A ) ) -> M e. ZZ )
37		isumltss.5	. . . . 5 |- ( ( ph /\ k e. Z ) -> ( F ` k ) = B )
38	37	adantlr 726	. . . 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 471	. . . 4 |- ( ( ph /\ x e. ( Z \ A ) ) -> seq M ( + , F ) e. dom ~~> )
41	2, 36, 38, 22, 40	isumrecl 13875	. . 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 3060	. . . . . . . 8 |- x e. _V
44	43	snnz 4103	. . . . . . 7 |- { x } =/= (/)
45	44	a1i 11	. . . . . 6 |- ( ( ph /\ x e. ( Z \ A ) ) -> { x } =/= (/) )
46	29	adantl 472	. . . . . . . 8 |- ( ( ph /\ x e. ( Z \ A ) ) -> { x } C_ Z )
47	46	sselda 3444	. . . . . . 7 |- ( ( ( ph /\ x e. ( Z \ A ) ) /\ k e. { x } ) -> k e. Z )
48	47, 21	syldan 477	. . . . . 6 |- ( ( ( ph /\ x e. ( Z \ A ) ) /\ k e. { x } ) -> B e. RR+ )
49	42, 45, 48	fsumrpcl 13852	. . . . 5 |- ( ( ph /\ x e. ( Z \ A ) ) -> sum_ k e. { x } B e. RR+ )
50	24, 49	ltaddrpd 11400	. . . 4 |- ( ( ph /\ x e. ( Z \ A ) ) -> sum_ k e. A B < ( sum_ k e. A B + sum_ k e. { x } B ) )
51		eldifn 3568	. . . . . . 7 |- ( x e. ( Z \ A ) -> -. x e. A )
52	51	adantl 472	. . . . . 6 |- ( ( ph /\ x e. ( Z \ A ) ) -> -. x e. A )
53		disjsn 4044	. . . . . 6 |- ( ( A i^i { x } ) = (/) <-> -. x e. A )
54	52, 53	sylibr 217	. . . . 5 |- ( ( ph /\ x e. ( Z \ A ) ) -> ( A i^i { x } ) = (/) )
55		eqidd 2463	. . . . 5 |- ( ( ph /\ x e. ( Z \ A ) ) -> ( A u. { x } ) = ( A u. { x } ) )
56	21	rpcnd 11372	. . . . . 6 |- ( ( ( ph /\ x e. ( Z \ A ) ) /\ k e. Z ) -> B e. CC )
57	33, 56	syldan 477	. . . . 5 |- ( ( ( ph /\ x e. ( Z \ A ) ) /\ k e. ( A u. { x } ) ) -> B e. CC )
58	54, 55, 27, 57	fsumsplit 13855	. . . 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 4443	. . 3 |- ( ( ph /\ x e. ( Z \ A ) ) -> sum_ k e. A B < sum_ k e. ( A u. { x } ) B )
60	21	rpge0d 11374	. . . 4 |- ( ( ( ph /\ x e. ( Z \ A ) ) /\ k e. Z ) -> 0 <_ B )
61	2, 36, 27, 32, 38, 22, 60, 40	isumless 13952	. . 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 9821	. 2 |- ( ( ph /\ x e. ( Z \ A ) ) -> sum_ k e. A B < sum_ k e. Z B )
63	16, 62	exlimddv 1792	1 |- ( ph -> sum_ k e. A B < sum_ k e. Z B )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 375   = wceq 1455   E.wex 1674   e. wcel 1898   =/= wne 2633   \ cdif 3413   u. cun 3414   i^i cin 3415   C_ wss 3416   (/)c0 3743   {csn 3980   class class class wbr 4416   dom cdm 4853   ` cfv 5601  (class class class)co 6315   Fincfn 7595   CCcc 9563   RRcr 9564   + caddc 9568   < clt 9701   ZZcz 10966   ZZ>=cuz 11188   RR+crp 11331   seqcseq 12245   ~~> cli 13597   sum_csu 13801

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1680  ax-4 1693  ax-5 1769  ax-6 1816  ax-7 1862  ax-8 1900  ax-9 1907  ax-10 1926  ax-11 1931  ax-12 1944  ax-13 2102  ax-ext 2442  ax-rep 4529  ax-sep 4539  ax-nul 4548  ax-pow 4595  ax-pr 4653  ax-un 6610  ax-inf2 8172  ax-cnex 9621  ax-resscn 9622  ax-1cn 9623  ax-icn 9624  ax-addcl 9625  ax-addrcl 9626  ax-mulcl 9627  ax-mulrcl 9628  ax-mulcom 9629  ax-addass 9630  ax-mulass 9631  ax-distr 9632  ax-i2m1 9633  ax-1ne0 9634  ax-1rid 9635  ax-rnegex 9636  ax-rrecex 9637  ax-cnre 9638  ax-pre-lttri 9639  ax-pre-lttrn 9640  ax-pre-ltadd 9641  ax-pre-mulgt0 9642  ax-pre-sup 9643

This theorem depends on definitions:  df-bi 190  df-or 376  df-an 377  df-3or 992  df-3an 993  df-tru 1458  df-fal 1461  df-ex 1675  df-nf 1679  df-sb 1809  df-eu 2314  df-mo 2315  df-clab 2449  df-cleq 2455  df-clel 2458  df-nfc 2592  df-ne 2635  df-nel 2636  df-ral 2754  df-rex 2755  df-reu 2756  df-rmo 2757  df-rab 2758  df-v 3059  df-sbc 3280  df-csb 3376  df-dif 3419  df-un 3421  df-in 3423  df-ss 3430  df-pss 3432  df-nul 3744  df-if 3894  df-pw 3965  df-sn 3981  df-pr 3983  df-tp 3985  df-op 3987  df-uni 4213  df-int 4249  df-iun 4294  df-br 4417  df-opab 4476  df-mpt 4477  df-tr 4512  df-eprel 4764  df-id 4768  df-po 4774  df-so 4775  df-fr 4812  df-se 4813  df-we 4814  df-xp 4859  df-rel 4860  df-cnv 4861  df-co 4862  df-dm 4863  df-rn 4864  df-res 4865  df-ima 4866  df-pred 5399  df-ord 5445  df-on 5446  df-lim 5447  df-suc 5448  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-isom 5610  df-riota 6277  df-ov 6318  df-oprab 6319  df-mpt2 6320  df-om 6720  df-1st 6820  df-2nd 6821  df-wrecs 7054  df-recs 7116  df-rdg 7154  df-1o 7208  df-oadd 7212  df-er 7389  df-pm 7501  df-en 7596  df-dom 7597  df-sdom 7598  df-fin 7599  df-sup 7982  df-inf 7983  df-oi 8051  df-card 8399  df-pnf 9703  df-mnf 9704  df-xr 9705  df-ltxr 9706  df-le 9707  df-sub 9888  df-neg 9889  df-div 10298  df-nn 10638  df-2 10696  df-3 10697  df-n0 10899  df-z 10967  df-uz 11189  df-rp 11332  df-fz 11814  df-fzo 11947  df-fl 12060  df-seq 12246  df-exp 12305  df-hash 12548  df-cj 13211  df-re 13212  df-im 13213  df-sqrt 13347  df-abs 13348  df-clim 13601  df-rlim 13602  df-sum 13802

This theorem is referenced by: (None)