Theorem climserle 12092

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 2331	. . . . 5 |- ( ph -> N e. ( ZZ>= ` M ) )
5		eluzel2 9908	. . . . 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 10902	. . 3 |- ( ph -> seq M ( + , F ) : Z --> RR )
9	8	ffvelcdmda 5837	. 2 |- ( ( ph /\ j e. Z ) -> ( seq M ( + , F ) ` j ) e. RR )
10	1	peano2uzs 9966	. . . . 5 |- ( j e. Z -> ( j + 1 ) e. Z )
11		fveq2 5693	. . . . . . . . 9 |- ( k = ( j + 1 ) -> ( F ` k ) = ( F ` ( j + 1 ) ) )
12	11	breq2d 4140	. . . . . . . 8 |- ( k = ( j + 1 ) -> ( 0 <_ ( F ` k ) <-> 0 <_ ( F ` ( j + 1 ) ) ) )
13	12	imbi2d 230	. . . . . . 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 116	. . . . . . 7 |- ( k e. Z -> ( ph -> 0 <_ ( F ` k ) ) )
16	13, 15	vtoclga 2889	. . . . . 6 |- ( ( j + 1 ) e. Z -> ( ph -> 0 <_ ( F ` ( j + 1 ) ) ) )
17	16	impcom 125	. . . . 5 |- ( ( ph /\ ( j + 1 ) e. Z ) -> 0 <_ ( F ` ( j + 1 ) ) )
18	10, 17	sylan2 286	. . . 4 |- ( ( ph /\ j e. Z ) -> 0 <_ ( F ` ( j + 1 ) ) )
19	11	eleq1d 2307	. . . . . . . . 9 |- ( k = ( j + 1 ) -> ( ( F ` k ) e. RR <-> ( F ` ( j + 1 ) ) e. RR ) )
20	19	imbi2d 230	. . . . . . . 8 |- ( k = ( j + 1 ) -> ( ( ph -> ( F ` k ) e. RR ) <-> ( ph -> ( F ` ( j + 1 ) ) e. RR ) ) )
21	7	expcom 116	. . . . . . . 8 |- ( k e. Z -> ( ph -> ( F ` k ) e. RR ) )
22	20, 21	vtoclga 2889	. . . . . . 7 |- ( ( j + 1 ) e. Z -> ( ph -> ( F ` ( j + 1 ) ) e. RR ) )
23	22	impcom 125	. . . . . 6 |- ( ( ph /\ ( j + 1 ) e. Z ) -> ( F ` ( j + 1 ) ) e. RR )
24	10, 23	sylan2 286	. . . . 5 |- ( ( ph /\ j e. Z ) -> ( F ` ( j + 1 ) ) e. RR )
25	9, 24	addge01d 8854	. . . 4 |- ( ( ph /\ j e. Z ) -> ( 0 <_ ( F ` ( j + 1 ) ) <-> ( seq M ( + , F ) ` j ) <_ ( ( seq M ( + , F ) ` j ) + ( F ` ( j + 1 ) ) ) ) )
26	18, 25	mpbid 147	. . 3 |- ( ( ph /\ j e. Z ) -> ( seq M ( + , F ) ` j ) <_ ( ( seq M ( + , F ) ` j ) + ( F ` ( j + 1 ) ) ) )
27	1	eleq2i 2305	. . . . . 6 |- ( j e. Z <-> j e. ( ZZ>= ` M ) )
28	27	biimpi 120	. . . . 5 |- ( j e. Z -> j e. ( ZZ>= ` M ) )
29	28	adantl 277	. . . 4 |- ( ( ph /\ j e. Z ) -> j e. ( ZZ>= ` M ) )
30		simpll 531	. . . . 5 |- ( ( ( ph /\ j e. Z ) /\ k e. ( ZZ>= ` M ) ) -> ph )
31	1	eleq2i 2305	. . . . . . 7 |- ( k e. Z <-> k e. ( ZZ>= ` M ) )
32	31	biimpri 133	. . . . . 6 |- ( k e. ( ZZ>= ` M ) -> k e. Z )
33	32	adantl 277	. . . . 5 |- ( ( ( ph /\ j e. Z ) /\ k e. ( ZZ>= ` M ) ) -> k e. Z )
34	30, 33, 7	syl2anc 415	. . . 4 |- ( ( ( ph /\ j e. Z ) /\ k e. ( ZZ>= ` M ) ) -> ( F ` k ) e. RR )
35		readdcl 8298	. . . . 5 |- ( ( k e. RR /\ v e. RR ) -> ( k + v ) e. RR )
36	35	adantl 277	. . . 4 |- ( ( ( ph /\ j e. Z ) /\ ( k e. RR /\ v e. RR ) ) -> ( k + v ) e. RR )
37	29, 34, 36	seq3p1 10883	. . 3 |- ( ( ph /\ j e. Z ) -> ( seq M ( + , F ) ` ( j + 1 ) ) = ( ( seq M ( + , F ) ` j ) + ( F ` ( j + 1 ) ) ) )
38	26, 37	breqtrrd 4156	. 2 |- ( ( ph /\ j e. Z ) -> ( seq M ( + , F ) ` j ) <_ ( seq M ( + , F ) ` ( j + 1 ) ) )
39	1, 2, 3, 9, 38	climub 12091	1 |- ( ph -> ( seq M ( + , F ) ` N ) <_ A )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1402   e. wcel 2209   class class class wbr 4128   ` cfv 5375  (class class class)co 6078   RRcr 8171   0cc0 8172   1c1 8173   + caddc 8175   <_ cle 8354   ZZcz 9626   ZZ>=cuz 9903   seqcseq 10865   ~~> cli 12025

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4244  ax-sep 4247  ax-nul 4257  ax-pow 4309  ax-pr 4344  ax-un 4576  ax-setind 4682  ax-iinf 4733  ax-cnex 8263  ax-resscn 8264  ax-1cn 8265  ax-1re 8266  ax-icn 8267  ax-addcl 8268  ax-addrcl 8269  ax-mulcl 8270  ax-mulrcl 8271  ax-addcom 8272  ax-mulcom 8273  ax-addass 8274  ax-mulass 8275  ax-distr 8276  ax-i2m1 8277  ax-0lt1 8278  ax-1rid 8279  ax-0id 8280  ax-rnegex 8281  ax-precex 8282  ax-cnre 8283  ax-pre-ltirr 8284  ax-pre-ltwlin 8285  ax-pre-lttrn 8286  ax-pre-apti 8287  ax-pre-ltadd 8288  ax-pre-mulgt0 8289  ax-pre-mulext 8290  ax-arch 8291  ax-caucvg 8292

This theorem depends on definitions:  df-bi 117  df-dc 847  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-reu 2535  df-rmo 2536  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-if 3639  df-pw 3690  df-sn 3714  df-pr 3715  df-op 3717  df-uni 3934  df-int 3969  df-iun 4012  df-br 4129  df-opab 4191  df-mpt 4192  df-tr 4228  df-id 4436  df-po 4439  df-iso 4440  df-iord 4509  df-on 4511  df-ilim 4512  df-suc 4514  df-iom 4736  df-xp 4778  df-rel 4779  df-cnv 4780  df-co 4781  df-dm 4782  df-rn 4783  df-res 4784  df-ima 4785  df-iota 5335  df-fun 5377  df-fn 5378  df-f 5379  df-f1 5380  df-fo 5381  df-f1o 5382  df-fv 5383  df-riota 6031  df-ov 6081  df-oprab 6082  df-mpo 6083  df-1st 6367  df-2nd 6368  df-recs 6569  df-frec 6655  df-pnf 8355  df-mnf 8356  df-xr 8357  df-ltxr 8358  df-le 8359  df-sub 8492  df-neg 8493  df-reap 8896  df-ap 8903  df-div 8996  df-inn 9287  df-2 9345  df-3 9346  df-4 9347  df-n0 9546  df-z 9627  df-uz 9904  df-rp 10037  df-fz 10394  df-seqfrec 10866  df-exp 10957  df-cj 11588  df-re 11589  df-im 11590  df-rsqrt 11745  df-abs 11746  df-clim 12026

This theorem is referenced by:  isumrpcl  12242  ege2le3  12419