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Theorem dvfsumrlim2 22184
 Description: Compare a finite sum to an integral (the integral here is given as a function with a known derivative). The statement here says that if is a decreasing function with antiderivative converging to zero, then the difference between and converges to a constant limit value, with the remainder term bounded by . (Contributed by Mario Carneiro, 18-May-2016.)
Hypotheses
Ref Expression
dvfsum.s
dvfsum.z
dvfsum.m
dvfsum.d
dvfsum.md
dvfsum.t
dvfsum.a
dvfsum.b1
dvfsum.b2
dvfsum.b3
dvfsum.c
dvfsumrlim.l
dvfsumrlim.g
dvfsumrlim.k
dvfsumrlim2.1
dvfsumrlim2.2
Assertion
Ref Expression
dvfsumrlim2
Distinct variable groups:   ,   ,   ,,   ,,   ,,   ,,   ,   ,   ,,
Allowed substitution hints:   (,)   ()   ()   ()   (,)   (,)   (,)   ()

Proof of Theorem dvfsumrlim2
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 dvfsum.s . . . . . . 7
2 ioossre 11585 . . . . . . 7
31, 2eqsstri 3534 . . . . . 6
4 dvfsumrlim2.1 . . . . . 6
53, 4sseldi 3502 . . . . 5
65rexrd 9642 . . . 4
75renepnfd 9643 . . . 4
8 icopnfsup 11959 . . . 4
96, 7, 8syl2anc 661 . . 3
11 dvfsum.z . . . . . . . 8
12 dvfsum.m . . . . . . . 8
13 dvfsum.d . . . . . . . 8
14 dvfsum.md . . . . . . . 8
15 dvfsum.t . . . . . . . 8
16 dvfsum.a . . . . . . . 8
17 dvfsum.b1 . . . . . . . 8
18 dvfsum.b2 . . . . . . . 8
19 dvfsum.b3 . . . . . . . 8
20 dvfsum.c . . . . . . . 8
21 dvfsumrlim.g . . . . . . . 8
221, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21dvfsumrlimf 22177 . . . . . . 7
2322ad2antrr 725 . . . . . 6
244ad2antrr 725 . . . . . 6
2523, 24ffvelrnd 6021 . . . . 5
2625recnd 9621 . . . 4
2715rexrd 9642 . . . . . . . . . 10
284, 1syl6eleq 2565 . . . . . . . . . . . 12
29 elioopnf 11617 . . . . . . . . . . . . 13
3027, 29syl 16 . . . . . . . . . . . 12
3128, 30mpbid 210 . . . . . . . . . . 11
3231simprd 463 . . . . . . . . . 10
33 df-ioo 11532 . . . . . . . . . . 11
34 df-ico 11534 . . . . . . . . . . 11
35 xrltletr 11359 . . . . . . . . . . 11
3633, 34, 35ixxss1 11546 . . . . . . . . . 10
3727, 32, 36syl2anc 661 . . . . . . . . 9
3837, 1syl6sseqr 3551 . . . . . . . 8
3938adantr 465 . . . . . . 7
4039sselda 3504 . . . . . 6
4123, 40ffvelrnd 6021 . . . . 5
4241recnd 9621 . . . 4
4326, 42subcld 9929 . . 3
44 pnfxr 11320 . . . . . . 7
45 icossre 11604 . . . . . . 7
465, 44, 45sylancl 662 . . . . . 6
4746adantr 465 . . . . 5
48 rlimf 13286 . . . . . . . 8
4948adantl 466 . . . . . . 7
50 ovex 6308 . . . . . . . . 9
5150, 21dmmpti 5709 . . . . . . . 8
5251feq2i 5723 . . . . . . 7
5349, 52sylib 196 . . . . . 6
544adantr 465 . . . . . 6
5553, 54ffvelrnd 6021 . . . . 5
56 rlimconst 13329 . . . . 5
5747, 55, 56syl2anc 661 . . . 4
5853feqmptd 5919 . . . . . 6
59 simpr 461 . . . . . 6
6058, 59eqbrtrrd 4469 . . . . 5
6139, 60rlimres2 13346 . . . 4
6226, 42, 57, 61rlimsub 13428 . . 3
6343, 62rlimabs 13393 . 2
643a1i 11 . . . . . . . 8
6564, 16, 17, 19dvmptrecl 22176 . . . . . . 7
6665ralrimiva 2878 . . . . . 6
67 nfcsb1v 3451 . . . . . . . 8
6867nfel1 2645 . . . . . . 7
69 csbeq1a 3444 . . . . . . . 8
7069eleq1d 2536 . . . . . . 7
7168, 70rspc 3208 . . . . . 6
724, 66, 71sylc 60 . . . . 5
7372recnd 9621 . . . 4
74 rlimconst 13329 . . . 4
7546, 73, 74syl2anc 661 . . 3
7743abscld 13229 . 2
7926, 42abssubd 13246 . . 3
8012adantr 465 . . . . 5
8113adantr 465 . . . . 5
8214adantr 465 . . . . 5
8315adantr 465 . . . . 5
8416adantlr 714 . . . . 5
8517adantlr 714 . . . . 5
8618adantlr 714 . . . . 5
8719adantr 465 . . . . 5
8844a1i 11 . . . . 5
89 3simpa 993 . . . . . . 7
90 dvfsumrlim.l . . . . . . 7
9189, 90syl3an3 1263 . . . . . 6
92913adant1r 1221 . . . . 5
93 dvfsumrlim.k . . . . . . . 8
941, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 90, 21, 93dvfsumrlimge0 22182 . . . . . . 7
95943adantr3 1157 . . . . . 6
9695adantlr 714 . . . . 5
974adantr 465 . . . . 5
9838sselda 3504 . . . . 5
99 dvfsumrlim2.2 . . . . . 6
10099adantr 465 . . . . 5
101 elicopnf 11619 . . . . . . 7
1025, 101syl 16 . . . . . 6
103102simplbda 624 . . . . 5
104102simprbda 623 . . . . . . 7
105104rexrd 9642 . . . . . 6
106 pnfge 11338 . . . . . 6
107105, 106syl 16 . . . . 5
1081, 11, 80, 81, 82, 83, 84, 85, 86, 87, 20, 88, 92, 21, 96, 97, 98, 100, 103, 107dvfsumlem4 22181 . . . 4