Theorem climserle 11006

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	syl6eleq 2207	. . . . 5 |- ( ph -> N e. ( ZZ>= ` M ) )
5		eluzel2 9233	. . . . 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 10141	. . 3 |- ( ph -> seq M ( + , F ) : Z --> RR )
9	8	ffvelrnda 5509	. 2 |- ( ( ph /\ j e. Z ) -> ( seq M ( + , F ) ` j ) e. RR )
10	1	peano2uzs 9281	. . . . 5 |- ( j e. Z -> ( j + 1 ) e. Z )
11		fveq2 5375	. . . . . . . . 9 |- ( k = ( j + 1 ) -> ( F ` k ) = ( F ` ( j + 1 ) ) )
12	11	breq2d 3907	. . . . . . . 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 2723	. . . . . 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 282	. . . 4 |- ( ( ph /\ j e. Z ) -> 0 <_ ( F ` ( j + 1 ) ) )
19	11	eleq1d 2183	. . . . . . . . 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 2723	. . . . . . 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 282	. . . . 5 |- ( ( ph /\ j e. Z ) -> ( F ` ( j + 1 ) ) e. RR )
25	9, 24	addge01d 8213	. . . 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 2181	. . . . . 6 |- ( j e. Z <-> j e. ( ZZ>= ` M ) )
28	27	biimpi 119	. . . . 5 |- ( j e. Z -> j e. ( ZZ>= ` M ) )
29	28	adantl 273	. . . 4 |- ( ( ph /\ j e. Z ) -> j e. ( ZZ>= ` M ) )
30		simpll 501	. . . . 5 |- ( ( ( ph /\ j e. Z ) /\ k e. ( ZZ>= ` M ) ) -> ph )
31	1	eleq2i 2181	. . . . . . 7 |- ( k e. Z <-> k e. ( ZZ>= ` M ) )
32	31	biimpri 132	. . . . . 6 |- ( k e. ( ZZ>= ` M ) -> k e. Z )
33	32	adantl 273	. . . . 5 |- ( ( ( ph /\ j e. Z ) /\ k e. ( ZZ>= ` M ) ) -> k e. Z )
34	30, 33, 7	syl2anc 406	. . . 4 |- ( ( ( ph /\ j e. Z ) /\ k e. ( ZZ>= ` M ) ) -> ( F ` k ) e. RR )
35		readdcl 7670	. . . . 5 |- ( ( k e. RR /\ v e. RR ) -> ( k + v ) e. RR )
36	35	adantl 273	. . . 4 |- ( ( ( ph /\ j e. Z ) /\ ( k e. RR /\ v e. RR ) ) -> ( k + v ) e. RR )
37	29, 34, 36	seq3p1 10128	. . 3 |- ( ( ph /\ j e. Z ) -> ( seq M ( + , F ) ` ( j + 1 ) ) = ( ( seq M ( + , F ) ` j ) + ( F ` ( j + 1 ) ) ) )
38	26, 37	breqtrrd 3921	. 2 |- ( ( ph /\ j e. Z ) -> ( seq M ( + , F ) ` j ) <_ ( seq M ( + , F ) ` ( j + 1 ) ) )
39	1, 2, 3, 9, 38	climub 11005	1 |- ( ph -> ( seq M ( + , F ) ` N ) <_ A )

Syntax hints:   -> wi 4   /\ wa 103   = wceq 1314   e. wcel 1463   class class class wbr 3895   ` cfv 5081  (class class class)co 5728   RRcr 7546   0cc0 7547   1c1 7548   + caddc 7550   <_ cle 7725   ZZcz 8958   ZZ>=cuz 9228   seqcseq 10111   ~~> cli 10939

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 586  ax-in2 587  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-13 1474  ax-14 1475  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-coll 4003  ax-sep 4006  ax-nul 4014  ax-pow 4058  ax-pr 4091  ax-un 4315  ax-setind 4412  ax-iinf 4462  ax-cnex 7636  ax-resscn 7637  ax-1cn 7638  ax-1re 7639  ax-icn 7640  ax-addcl 7641  ax-addrcl 7642  ax-mulcl 7643  ax-mulrcl 7644  ax-addcom 7645  ax-mulcom 7646  ax-addass 7647  ax-mulass 7648  ax-distr 7649  ax-i2m1 7650  ax-0lt1 7651  ax-1rid 7652  ax-0id 7653  ax-rnegex 7654  ax-precex 7655  ax-cnre 7656  ax-pre-ltirr 7657  ax-pre-ltwlin 7658  ax-pre-lttrn 7659  ax-pre-apti 7660  ax-pre-ltadd 7661  ax-pre-mulgt0 7662  ax-pre-mulext 7663  ax-arch 7664  ax-caucvg 7665

This theorem depends on definitions:  df-bi 116  df-dc 803  df-3or 946  df-3an 947  df-tru 1317  df-fal 1320  df-nf 1420  df-sb 1719  df-eu 1978  df-mo 1979  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2244  df-ne 2283  df-nel 2378  df-ral 2395  df-rex 2396  df-reu 2397  df-rmo 2398  df-rab 2399  df-v 2659  df-sbc 2879  df-csb 2972  df-dif 3039  df-un 3041  df-in 3043  df-ss 3050  df-nul 3330  df-if 3441  df-pw 3478  df-sn 3499  df-pr 3500  df-op 3502  df-uni 3703  df-int 3738  df-iun 3781  df-br 3896  df-opab 3950  df-mpt 3951  df-tr 3987  df-id 4175  df-po 4178  df-iso 4179  df-iord 4248  df-on 4250  df-ilim 4251  df-suc 4253  df-iom 4465  df-xp 4505  df-rel 4506  df-cnv 4507  df-co 4508  df-dm 4509  df-rn 4510  df-res 4511  df-ima 4512  df-iota 5046  df-fun 5083  df-fn 5084  df-f 5085  df-f1 5086  df-fo 5087  df-f1o 5088  df-fv 5089  df-riota 5684  df-ov 5731  df-oprab 5732  df-mpo 5733  df-1st 5992  df-2nd 5993  df-recs 6156  df-frec 6242  df-pnf 7726  df-mnf 7727  df-xr 7728  df-ltxr 7729  df-le 7730  df-sub 7858  df-neg 7859  df-reap 8255  df-ap 8262  df-div 8346  df-inn 8631  df-2 8689  df-3 8690  df-4 8691  df-n0 8882  df-z 8959  df-uz 9229  df-rp 9344  df-fz 9684  df-seqfrec 10112  df-exp 10186  df-cj 10507  df-re 10508  df-im 10509  df-rsqrt 10662  df-abs 10663  df-clim 10940

This theorem is referenced by:  isumrpcl  11155  ege2le3  11228