Theorem climserle 11114

Description: The partial sums of a converging infinite series with nonnegative terms are bounded by its limit. (Contributed by NM, 27-Dec-2005.) (Revised by Mario Carneiro, 9-Feb-2014.)

Hypotheses

Ref	Expression
clim2iser.1	|- Z = ( ZZ>= ` M )
climserle.2	|- ( ph -> N e. Z )
climserle.3	|- ( ph -> seq M ( + , F ) ~~> A )
climserle.4	|- ( ( ph /\ k e. Z ) -> ( F ` k ) e. RR )
climserle.5	|- ( ( ph /\ k e. Z ) -> 0 <_ ( F ` k ) )

Assertion

Ref	Expression
climserle	|- ( ph -> ( seq M ( + , F ) ` N ) <_ A )

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

Proof of Theorem climserle

Dummy variables j v are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		clim2iser.1	. 2 |- Z = ( ZZ>= ` M )
2		climserle.2	. 2 |- ( ph -> N e. Z )
3		climserle.3	. 2 |- ( ph -> seq M ( + , F ) ~~> A )
4	2, 1	eleqtrdi 2232	. . . . 5 |- ( ph -> N e. ( ZZ>= ` M ) )
5		eluzel2 9331	. . . . 5 |- ( N e. ( ZZ>= ` M ) -> M e. ZZ )
6	4, 5	syl 14	. . . 4 |- ( ph -> M e. ZZ )
7		climserle.4	. . . 4 |- ( ( ph /\ k e. Z ) -> ( F ` k ) e. RR )
8	1, 6, 7	serfre 10248	. . 3 |- ( ph -> seq M ( + , F ) : Z --> RR )
9	8	ffvelrnda 5555	. 2 |- ( ( ph /\ j e. Z ) -> ( seq M ( + , F ) ` j ) e. RR )
10	1	peano2uzs 9379	. . . . 5 |- ( j e. Z -> ( j + 1 ) e. Z )
11		fveq2 5421	. . . . . . . . 9 |- ( k = ( j + 1 ) -> ( F ` k ) = ( F ` ( j + 1 ) ) )
12	11	breq2d 3941	. . . . . . . 8 |- ( k = ( j + 1 ) -> ( 0 <_ ( F ` k ) <-> 0 <_ ( F ` ( j + 1 ) ) ) )
13	12	imbi2d 229	. . . . . . 7 |- ( k = ( j + 1 ) -> ( ( ph -> 0 <_ ( F ` k ) ) <-> ( ph -> 0 <_ ( F ` ( j + 1 ) ) ) ) )
14		climserle.5	. . . . . . . 8 |- ( ( ph /\ k e. Z ) -> 0 <_ ( F ` k ) )
15	14	expcom 115	. . . . . . 7 |- ( k e. Z -> ( ph -> 0 <_ ( F ` k ) ) )
16	13, 15	vtoclga 2752	. . . . . 6 |- ( ( j + 1 ) e. Z -> ( ph -> 0 <_ ( F ` ( j + 1 ) ) ) )
17	16	impcom 124	. . . . 5 |- ( ( ph /\ ( j + 1 ) e. Z ) -> 0 <_ ( F ` ( j + 1 ) ) )
18	10, 17	sylan2 284	. . . 4 |- ( ( ph /\ j e. Z ) -> 0 <_ ( F ` ( j + 1 ) ) )
19	11	eleq1d 2208	. . . . . . . . 9 |- ( k = ( j + 1 ) -> ( ( F ` k ) e. RR <-> ( F ` ( j + 1 ) ) e. RR ) )
20	19	imbi2d 229	. . . . . . . 8 |- ( k = ( j + 1 ) -> ( ( ph -> ( F ` k ) e. RR ) <-> ( ph -> ( F ` ( j + 1 ) ) e. RR ) ) )
21	7	expcom 115	. . . . . . . 8 |- ( k e. Z -> ( ph -> ( F ` k ) e. RR ) )
22	20, 21	vtoclga 2752	. . . . . . 7 |- ( ( j + 1 ) e. Z -> ( ph -> ( F ` ( j + 1 ) ) e. RR ) )
23	22	impcom 124	. . . . . 6 |- ( ( ph /\ ( j + 1 ) e. Z ) -> ( F ` ( j + 1 ) ) e. RR )
24	10, 23	sylan2 284	. . . . 5 |- ( ( ph /\ j e. Z ) -> ( F ` ( j + 1 ) ) e. RR )
25	9, 24	addge01d 8295	. . . 4 |- ( ( ph /\ j e. Z ) -> ( 0 <_ ( F ` ( j + 1 ) ) <-> ( seq M ( + , F ) ` j ) <_ ( ( seq M ( + , F ) ` j ) + ( F ` ( j + 1 ) ) ) ) )
26	18, 25	mpbid 146	. . 3 |- ( ( ph /\ j e. Z ) -> ( seq M ( + , F ) ` j ) <_ ( ( seq M ( + , F ) ` j ) + ( F ` ( j + 1 ) ) ) )
27	1	eleq2i 2206	. . . . . 6 |- ( j e. Z <-> j e. ( ZZ>= ` M ) )
28	27	biimpi 119	. . . . 5 |- ( j e. Z -> j e. ( ZZ>= ` M ) )
29	28	adantl 275	. . . 4 |- ( ( ph /\ j e. Z ) -> j e. ( ZZ>= ` M ) )
30		simpll 518	. . . . 5 |- ( ( ( ph /\ j e. Z ) /\ k e. ( ZZ>= ` M ) ) -> ph )
31	1	eleq2i 2206	. . . . . . 7 |- ( k e. Z <-> k e. ( ZZ>= ` M ) )
32	31	biimpri 132	. . . . . 6 |- ( k e. ( ZZ>= ` M ) -> k e. Z )
33	32	adantl 275	. . . . 5 |- ( ( ( ph /\ j e. Z ) /\ k e. ( ZZ>= ` M ) ) -> k e. Z )
34	30, 33, 7	syl2anc 408	. . . 4 |- ( ( ( ph /\ j e. Z ) /\ k e. ( ZZ>= ` M ) ) -> ( F ` k ) e. RR )
35		readdcl 7746	. . . . 5 |- ( ( k e. RR /\ v e. RR ) -> ( k + v ) e. RR )
36	35	adantl 275	. . . 4 |- ( ( ( ph /\ j e. Z ) /\ ( k e. RR /\ v e. RR ) ) -> ( k + v ) e. RR )
37	29, 34, 36	seq3p1 10235	. . 3 |- ( ( ph /\ j e. Z ) -> ( seq M ( + , F ) ` ( j + 1 ) ) = ( ( seq M ( + , F ) ` j ) + ( F ` ( j + 1 ) ) ) )
38	26, 37	breqtrrd 3956	. 2 |- ( ( ph /\ j e. Z ) -> ( seq M ( + , F ) ` j ) <_ ( seq M ( + , F ) ` ( j + 1 ) ) )
39	1, 2, 3, 9, 38	climub 11113	1 |- ( ph -> ( seq M ( + , F ) ` N ) <_ A )

Syntax hints:   -> wi 4   /\ wa 103   = wceq 1331   e. wcel 1480   class class class wbr 3929   ` cfv 5123  (class class class)co 5774   RRcr 7619   0cc0 7620   1c1 7621   + caddc 7623   <_ cle 7801   ZZcz 9054   ZZ>=cuz 9326   seqcseq 10218   ~~> cli 11047

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-coll 4043  ax-sep 4046  ax-nul 4054  ax-pow 4098  ax-pr 4131  ax-un 4355  ax-setind 4452  ax-iinf 4502  ax-cnex 7711  ax-resscn 7712  ax-1cn 7713  ax-1re 7714  ax-icn 7715  ax-addcl 7716  ax-addrcl 7717  ax-mulcl 7718  ax-mulrcl 7719  ax-addcom 7720  ax-mulcom 7721  ax-addass 7722  ax-mulass 7723  ax-distr 7724  ax-i2m1 7725  ax-0lt1 7726  ax-1rid 7727  ax-0id 7728  ax-rnegex 7729  ax-precex 7730  ax-cnre 7731  ax-pre-ltirr 7732  ax-pre-ltwlin 7733  ax-pre-lttrn 7734  ax-pre-apti 7735  ax-pre-ltadd 7736  ax-pre-mulgt0 7737  ax-pre-mulext 7738  ax-arch 7739  ax-caucvg 7740

This theorem depends on definitions:  df-bi 116  df-dc 820  df-3or 963  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ne 2309  df-nel 2404  df-ral 2421  df-rex 2422  df-reu 2423  df-rmo 2424  df-rab 2425  df-v 2688  df-sbc 2910  df-csb 3004  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-nul 3364  df-if 3475  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-int 3772  df-iun 3815  df-br 3930  df-opab 3990  df-mpt 3991  df-tr 4027  df-id 4215  df-po 4218  df-iso 4219  df-iord 4288  df-on 4290  df-ilim 4291  df-suc 4293  df-iom 4505  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-rn 4550  df-res 4551  df-ima 4552  df-iota 5088  df-fun 5125  df-fn 5126  df-f 5127  df-f1 5128  df-fo 5129  df-f1o 5130  df-fv 5131  df-riota 5730  df-ov 5777  df-oprab 5778  df-mpo 5779  df-1st 6038  df-2nd 6039  df-recs 6202  df-frec 6288  df-pnf 7802  df-mnf 7803  df-xr 7804  df-ltxr 7805  df-le 7806  df-sub 7935  df-neg 7936  df-reap 8337  df-ap 8344  df-div 8433  df-inn 8721  df-2 8779  df-3 8780  df-4 8781  df-n0 8978  df-z 9055  df-uz 9327  df-rp 9442  df-fz 9791  df-seqfrec 10219  df-exp 10293  df-cj 10614  df-re 10615  df-im 10616  df-rsqrt 10770  df-abs 10771  df-clim 11048

This theorem is referenced by:  isumrpcl  11263  ege2le3  11377