Theorem climserle 11926

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 2324	. . . . 5 |- ( ph -> N e. ( ZZ>= ` M ) )
5		eluzel2 9763	. . . . 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 10750	. . 3 |- ( ph -> seq M ( + , F ) : Z --> RR )
9	8	ffvelcdmda 5783	. 2 |- ( ( ph /\ j e. Z ) -> ( seq M ( + , F ) ` j ) e. RR )
10	1	peano2uzs 9821	. . . . 5 |- ( j e. Z -> ( j + 1 ) e. Z )
11		fveq2 5640	. . . . . . . . 9 |- ( k = ( j + 1 ) -> ( F ` k ) = ( F ` ( j + 1 ) ) )
12	11	breq2d 4100	. . . . . . . 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 2870	. . . . . 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 2300	. . . . . . . . 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 2870	. . . . . . 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 8716	. . . 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 2298	. . . . . 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 527	. . . . 5 |- ( ( ( ph /\ j e. Z ) /\ k e. ( ZZ>= ` M ) ) -> ph )
31	1	eleq2i 2298	. . . . . . 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 411	. . . 4 |- ( ( ( ph /\ j e. Z ) /\ k e. ( ZZ>= ` M ) ) -> ( F ` k ) e. RR )
35		readdcl 8161	. . . . 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 10731	. . 3 |- ( ( ph /\ j e. Z ) -> ( seq M ( + , F ) ` ( j + 1 ) ) = ( ( seq M ( + , F ) ` j ) + ( F ` ( j + 1 ) ) ) )
38	26, 37	breqtrrd 4116	. 2 |- ( ( ph /\ j e. Z ) -> ( seq M ( + , F ) ` j ) <_ ( seq M ( + , F ) ` ( j + 1 ) ) )
39	1, 2, 3, 9, 38	climub 11925	1 |- ( ph -> ( seq M ( + , F ) ` N ) <_ A )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1397   e. wcel 2202   class class class wbr 4088   ` cfv 5326  (class class class)co 6021   RRcr 8034   0cc0 8035   1c1 8036   + caddc 8038   <_ cle 8218   ZZcz 9482   ZZ>=cuz 9758   seqcseq 10713   ~~> cli 11859

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 619  ax-in2 620  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-coll 4204  ax-sep 4207  ax-nul 4215  ax-pow 4264  ax-pr 4299  ax-un 4530  ax-setind 4635  ax-iinf 4686  ax-cnex 8126  ax-resscn 8127  ax-1cn 8128  ax-1re 8129  ax-icn 8130  ax-addcl 8131  ax-addrcl 8132  ax-mulcl 8133  ax-mulrcl 8134  ax-addcom 8135  ax-mulcom 8136  ax-addass 8137  ax-mulass 8138  ax-distr 8139  ax-i2m1 8140  ax-0lt1 8141  ax-1rid 8142  ax-0id 8143  ax-rnegex 8144  ax-precex 8145  ax-cnre 8146  ax-pre-ltirr 8147  ax-pre-ltwlin 8148  ax-pre-lttrn 8149  ax-pre-apti 8150  ax-pre-ltadd 8151  ax-pre-mulgt0 8152  ax-pre-mulext 8153  ax-arch 8154  ax-caucvg 8155

This theorem depends on definitions:  df-bi 117  df-dc 842  df-3or 1005  df-3an 1006  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ne 2403  df-nel 2498  df-ral 2515  df-rex 2516  df-reu 2517  df-rmo 2518  df-rab 2519  df-v 2804  df-sbc 3032  df-csb 3128  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-nul 3495  df-if 3606  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-int 3929  df-iun 3972  df-br 4089  df-opab 4151  df-mpt 4152  df-tr 4188  df-id 4390  df-po 4393  df-iso 4394  df-iord 4463  df-on 4465  df-ilim 4466  df-suc 4468  df-iom 4689  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-res 4737  df-ima 4738  df-iota 5286  df-fun 5328  df-fn 5329  df-f 5330  df-f1 5331  df-fo 5332  df-f1o 5333  df-fv 5334  df-riota 5974  df-ov 6024  df-oprab 6025  df-mpo 6026  df-1st 6306  df-2nd 6307  df-recs 6474  df-frec 6560  df-pnf 8219  df-mnf 8220  df-xr 8221  df-ltxr 8222  df-le 8223  df-sub 8355  df-neg 8356  df-reap 8758  df-ap 8765  df-div 8856  df-inn 9147  df-2 9205  df-3 9206  df-4 9207  df-n0 9406  df-z 9483  df-uz 9759  df-rp 9892  df-fz 10247  df-seqfrec 10714  df-exp 10805  df-cj 11423  df-re 11424  df-im 11425  df-rsqrt 11579  df-abs 11580  df-clim 11860

This theorem is referenced by:  isumrpcl  12076  ege2le3  12253