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Theorem dvfsumrlimge0 25993

Description: Lemma for dvfsumrlim 25994. Satisfy the assumption of dvfsumlem4 25992. (Contributed by Mario Carneiro, 18-May-2016.)

**Hypotheses**
Ref	Expression
dvfsum.s	⊢ 𝑆 = (𝑇(,)+∞)
dvfsum.z	⊢ 𝑍 = (ℤ_≥‘𝑀)
dvfsum.m	⊢ (𝜑 → 𝑀 ∈ ℤ)
dvfsum.d	⊢ (𝜑 → 𝐷 ∈ ℝ)
dvfsum.md	⊢ (𝜑 → 𝑀 ≤ (𝐷 + 1))
dvfsum.t	⊢ (𝜑 → 𝑇 ∈ ℝ)
dvfsum.a	⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆) → 𝐴 ∈ ℝ)
dvfsum.b1	⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆) → 𝐵 ∈ 𝑉)
dvfsum.b2	⊢ ((𝜑 ∧ 𝑥 ∈ 𝑍) → 𝐵 ∈ ℝ)
dvfsum.b3	⊢ (𝜑 → (ℝ D (𝑥 ∈ 𝑆 ↦ 𝐴)) = (𝑥 ∈ 𝑆 ↦ 𝐵))
dvfsum.c	⊢ (𝑥 = 𝑘 → 𝐵 = 𝐶)
dvfsumrlim.l	⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑘 ∈ 𝑆) ∧ (𝐷 ≤ 𝑥 ∧ 𝑥 ≤ 𝑘)) → 𝐶 ≤ 𝐵)
dvfsumrlim.g	⊢ 𝐺 = (𝑥 ∈ 𝑆 ↦ (Σ𝑘 ∈ (𝑀...(⌊‘𝑥))𝐶 − 𝐴))
dvfsumrlim.k	⊢ (𝜑 → (𝑥 ∈ 𝑆 ↦ 𝐵) ⇝_𝑟 0)

**Assertion**
Ref	Expression
dvfsumrlimge0	⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝐷 ≤ 𝑥)) → 0 ≤ 𝐵)

Distinct variable groups: 𝐵,𝑘 𝑥,𝐶 𝑥,𝑘,𝐷 𝜑,𝑘,𝑥 𝑆,𝑘,𝑥 𝑘,𝑀,𝑥 𝑥,𝑇 𝑥,𝑍 Allowed substitution hints: 𝐴(𝑥,𝑘) 𝐵(𝑥) 𝐶(𝑘) 𝑇(𝑘) 𝐺(𝑥,𝑘) 𝑉(𝑥,𝑘) 𝑍(𝑘)

Proof of Theorem dvfsumrlimge0 Dummy variables 𝑧 𝑢 𝑣 𝑤 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		dvfsum.s	. . . . . 6 ⊢ 𝑆 = (𝑇(,)+∞)
2		ioossre 13323	. . . . . 6 ⊢ (𝑇(,)+∞) ⊆ ℝ
3	1, 2	eqsstri 3980	. . . . 5 ⊢ 𝑆 ⊆ ℝ
4		simprl 770	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝐷 ≤ 𝑥)) → 𝑥 ∈ 𝑆)
5	3, 4	sselid 3931	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝐷 ≤ 𝑥)) → 𝑥 ∈ ℝ)
6	5	rexrd 11182	. . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝐷 ≤ 𝑥)) → 𝑥 ∈ ℝ^*)
7	5	renepnfd 11183	. . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝐷 ≤ 𝑥)) → 𝑥 ≠ +∞)
8		icopnfsup 13785	. . 3 ⊢ ((𝑥 ∈ ℝ^* ∧ 𝑥 ≠ +∞) → sup((𝑥[,)+∞), ℝ^*, < ) = +∞)
9	6, 7, 8	syl2anc 584	. 2 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝐷 ≤ 𝑥)) → sup((𝑥[,)+∞), ℝ^*, < ) = +∞)
10		dvfsum.t	. . . . . 6 ⊢ (𝜑 → 𝑇 ∈ ℝ)
11	10	rexrd 11182	. . . . 5 ⊢ (𝜑 → 𝑇 ∈ ℝ^*)
12	4, 1	eleqtrdi 2846	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝐷 ≤ 𝑥)) → 𝑥 ∈ (𝑇(,)+∞))
13	11	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝐷 ≤ 𝑥)) → 𝑇 ∈ ℝ^*)
14		elioopnf 13359	. . . . . . . 8 ⊢ (𝑇 ∈ ℝ^* → (𝑥 ∈ (𝑇(,)+∞) ↔ (𝑥 ∈ ℝ ∧ 𝑇 < 𝑥)))
15	13, 14	syl 17	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝐷 ≤ 𝑥)) → (𝑥 ∈ (𝑇(,)+∞) ↔ (𝑥 ∈ ℝ ∧ 𝑇 < 𝑥)))
16	12, 15	mpbid 232	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝐷 ≤ 𝑥)) → (𝑥 ∈ ℝ ∧ 𝑇 < 𝑥))
17	16	simprd 495	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝐷 ≤ 𝑥)) → 𝑇 < 𝑥)
18		df-ioo 13265	. . . . . 6 ⊢ (,) = (𝑢 ∈ ℝ^, 𝑣 ∈ ℝ^ ↦ {𝑤 ∈ ℝ^* ∣ (𝑢 < 𝑤 ∧ 𝑤 < 𝑣)})
19		df-ico 13267	. . . . . 6 ⊢ [,) = (𝑢 ∈ ℝ^, 𝑣 ∈ ℝ^ ↦ {𝑤 ∈ ℝ^* ∣ (𝑢 ≤ 𝑤 ∧ 𝑤 < 𝑣)})
20		xrltletr 13071	. . . . . 6 ⊢ ((𝑇 ∈ ℝ^* ∧ 𝑥 ∈ ℝ^* ∧ 𝑧 ∈ ℝ^*) → ((𝑇 < 𝑥 ∧ 𝑥 ≤ 𝑧) → 𝑇 < 𝑧))
21	18, 19, 20	ixxss1 13279	. . . . 5 ⊢ ((𝑇 ∈ ℝ^* ∧ 𝑇 < 𝑥) → (𝑥[,)+∞) ⊆ (𝑇(,)+∞))
22	11, 17, 21	syl2an2r 685	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝐷 ≤ 𝑥)) → (𝑥[,)+∞) ⊆ (𝑇(,)+∞))
23	22, 1	sseqtrrdi 3975	. . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝐷 ≤ 𝑥)) → (𝑥[,)+∞) ⊆ 𝑆)
24		dvfsum.c	. . . . 5 ⊢ (𝑥 = 𝑘 → 𝐵 = 𝐶)
25	24	cbvmptv 5202	. . . 4 ⊢ (𝑥 ∈ 𝑆 ↦ 𝐵) = (𝑘 ∈ 𝑆 ↦ 𝐶)
26		dvfsumrlim.k	. . . . 5 ⊢ (𝜑 → (𝑥 ∈ 𝑆 ↦ 𝐵) ⇝_𝑟 0)
27	26	adantr 480	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝐷 ≤ 𝑥)) → (𝑥 ∈ 𝑆 ↦ 𝐵) ⇝_𝑟 0)
28	25, 27	eqbrtrrid 5134	. . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝐷 ≤ 𝑥)) → (𝑘 ∈ 𝑆 ↦ 𝐶) ⇝_𝑟 0)
29	23, 28	rlimres2 15484	. 2 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝐷 ≤ 𝑥)) → (𝑘 ∈ (𝑥[,)+∞) ↦ 𝐶) ⇝_𝑟 0)
30	3	a1i 11	. . . 4 ⊢ (𝜑 → 𝑆 ⊆ ℝ)
31		dvfsum.a	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆) → 𝐴 ∈ ℝ)
32		dvfsum.b1	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆) → 𝐵 ∈ 𝑉)
33		dvfsum.b3	. . . . . . 7 ⊢ (𝜑 → (ℝ D (𝑥 ∈ 𝑆 ↦ 𝐴)) = (𝑥 ∈ 𝑆 ↦ 𝐵))
34	30, 31, 32, 33	dvmptrecl 25986	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆) → 𝐵 ∈ ℝ)
35	34	adantrr 717	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝐷 ≤ 𝑥)) → 𝐵 ∈ ℝ)
36	35	recnd 11160	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝐷 ≤ 𝑥)) → 𝐵 ∈ ℂ)
37		rlimconst 15467	. . . 4 ⊢ ((𝑆 ⊆ ℝ ∧ 𝐵 ∈ ℂ) → (𝑘 ∈ 𝑆 ↦ 𝐵) ⇝_𝑟 𝐵)
38	30, 36, 37	syl2an2r 685	. . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝐷 ≤ 𝑥)) → (𝑘 ∈ 𝑆 ↦ 𝐵) ⇝_𝑟 𝐵)
39	23, 38	rlimres2 15484	. 2 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝐷 ≤ 𝑥)) → (𝑘 ∈ (𝑥[,)+∞) ↦ 𝐵) ⇝_𝑟 𝐵)
40	34	ralrimiva 3128	. . . 4 ⊢ (𝜑 → ∀𝑥 ∈ 𝑆 𝐵 ∈ ℝ)
41	40	adantr 480	. . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝐷 ≤ 𝑥)) → ∀𝑥 ∈ 𝑆 𝐵 ∈ ℝ)
42	23	sselda 3933	. . 3 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝐷 ≤ 𝑥)) ∧ 𝑘 ∈ (𝑥[,)+∞)) → 𝑘 ∈ 𝑆)
43	24	eleq1d 2821	. . . 4 ⊢ (𝑥 = 𝑘 → (𝐵 ∈ ℝ ↔ 𝐶 ∈ ℝ))
44	43	rspccva 3575	. . 3 ⊢ ((∀𝑥 ∈ 𝑆 𝐵 ∈ ℝ ∧ 𝑘 ∈ 𝑆) → 𝐶 ∈ ℝ)
45	41, 42, 44	syl2an2r 685	. 2 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝐷 ≤ 𝑥)) ∧ 𝑘 ∈ (𝑥[,)+∞)) → 𝐶 ∈ ℝ)
46	35	adantr 480	. 2 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝐷 ≤ 𝑥)) ∧ 𝑘 ∈ (𝑥[,)+∞)) → 𝐵 ∈ ℝ)
47		simpll 766	. . 3 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝐷 ≤ 𝑥)) ∧ 𝑘 ∈ (𝑥[,)+∞)) → 𝜑)
48		simplrl 776	. . 3 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝐷 ≤ 𝑥)) ∧ 𝑘 ∈ (𝑥[,)+∞)) → 𝑥 ∈ 𝑆)
49		simplrr 777	. . 3 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝐷 ≤ 𝑥)) ∧ 𝑘 ∈ (𝑥[,)+∞)) → 𝐷 ≤ 𝑥)
50		elicopnf 13361	. . . . 5 ⊢ (𝑥 ∈ ℝ → (𝑘 ∈ (𝑥[,)+∞) ↔ (𝑘 ∈ ℝ ∧ 𝑥 ≤ 𝑘)))
51	5, 50	syl 17	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝐷 ≤ 𝑥)) → (𝑘 ∈ (𝑥[,)+∞) ↔ (𝑘 ∈ ℝ ∧ 𝑥 ≤ 𝑘)))
52	51	simplbda 499	. . 3 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝐷 ≤ 𝑥)) ∧ 𝑘 ∈ (𝑥[,)+∞)) → 𝑥 ≤ 𝑘)
53		dvfsumrlim.l	. . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑘 ∈ 𝑆) ∧ (𝐷 ≤ 𝑥 ∧ 𝑥 ≤ 𝑘)) → 𝐶 ≤ 𝐵)
54	47, 48, 42, 49, 52, 53	syl122anc 1381	. 2 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝐷 ≤ 𝑥)) ∧ 𝑘 ∈ (𝑥[,)+∞)) → 𝐶 ≤ 𝐵)
55	9, 29, 39, 45, 46, 54	rlimle 15571	1 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝐷 ≤ 𝑥)) → 0 ≤ 𝐵)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∧ w3a 1086 = wceq 1541 ∈ wcel 2113 ≠ wne 2932 ∀wral 3051 ⊆ wss 3901 class class class wbr 5098 ↦ cmpt 5179 ‘cfv 6492 (class class class)co 7358 supcsup 9343 ℂcc 11024 ℝcr 11025 0cc0 11026 1c1 11027 + caddc 11029 +∞cpnf 11163 ℝ^*cxr 11165 < clt 11166 ≤ cle 11167 − cmin 11364 ℤcz 12488 ℤ_≥cuz 12751 (,)cioo 13261 [,)cico 13263 ...cfz 13423 ⌊cfl 13710 ⇝_𝑟 crli 15408 Σcsu 15609 D cdv 25820

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1968 ax-7 2009 ax-8 2115 ax-9 2123 ax-10 2146 ax-11 2162 ax-12 2184 ax-ext 2708 ax-rep 5224 ax-sep 5241 ax-nul 5251 ax-pow 5310 ax-pr 5377 ax-un 7680 ax-cnex 11082 ax-resscn 11083 ax-1cn 11084 ax-icn 11085 ax-addcl 11086 ax-addrcl 11087 ax-mulcl 11088 ax-mulrcl 11089 ax-mulcom 11090 ax-addass 11091 ax-mulass 11092 ax-distr 11093 ax-i2m1 11094 ax-1ne0 11095 ax-1rid 11096 ax-rnegex 11097 ax-rrecex 11098 ax-cnre 11099 ax-pre-lttri 11100 ax-pre-lttrn 11101 ax-pre-ltadd 11102 ax-pre-mulgt0 11103 ax-pre-sup 11104

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2068 df-mo 2539 df-eu 2569 df-clab 2715 df-cleq 2728 df-clel 2811 df-nfc 2885 df-ne 2933 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3350 df-reu 3351 df-rab 3400 df-v 3442 df-sbc 3741 df-csb 3850 df-dif 3904 df-un 3906 df-in 3908 df-ss 3918 df-pss 3921 df-nul 4286 df-if 4480 df-pw 4556 df-sn 4581 df-pr 4583 df-tp 4585 df-op 4587 df-uni 4864 df-int 4903 df-iun 4948 df-iin 4949 df-br 5099 df-opab 5161 df-mpt 5180 df-tr 5206 df-id 5519 df-eprel 5524 df-po 5532 df-so 5533 df-fr 5577 df-we 5579 df-xp 5630 df-rel 5631 df-cnv 5632 df-co 5633 df-dm 5634 df-rn 5635 df-res 5636 df-ima 5637 df-pred 6259 df-ord 6320 df-on 6321 df-lim 6322 df-suc 6323 df-iota 6448 df-fun 6494 df-fn 6495 df-f 6496 df-f1 6497 df-fo 6498 df-f1o 6499 df-fv 6500 df-riota 7315 df-ov 7361 df-oprab 7362 df-mpo 7363 df-om 7809 df-1st 7933 df-2nd 7934 df-frecs 8223 df-wrecs 8254 df-recs 8303 df-rdg 8341 df-1o 8397 df-er 8635 df-map 8765 df-pm 8766 df-en 8884 df-dom 8885 df-sdom 8886 df-fin 8887 df-fi 9314 df-sup 9345 df-inf 9346 df-pnf 11168 df-mnf 11169 df-xr 11170 df-ltxr 11171 df-le 11172 df-sub 11366 df-neg 11367 df-div 11795 df-nn 12146 df-2 12208 df-3 12209 df-4 12210 df-5 12211 df-6 12212 df-7 12213 df-8 12214 df-9 12215 df-n0 12402 df-z 12489 df-dec 12608 df-uz 12752 df-q 12862 df-rp 12906 df-xneg 13026 df-xadd 13027 df-xmul 13028 df-ioo 13265 df-ico 13267 df-icc 13268 df-fz 13424 df-seq 13925 df-exp 13985 df-cj 15022 df-re 15023 df-im 15024 df-sqrt 15158 df-abs 15159 df-rlim 15412 df-struct 17074 df-slot 17109 df-ndx 17121 df-base 17137 df-plusg 17190 df-mulr 17191 df-starv 17192 df-tset 17196 df-ple 17197 df-ds 17199 df-unif 17200 df-rest 17342 df-topn 17343 df-topgen 17363 df-psmet 21301 df-xmet 21302 df-met 21303 df-bl 21304 df-mopn 21305 df-fbas 21306 df-fg 21307 df-cnfld 21310 df-top 22838 df-topon 22855 df-topsp 22877 df-bases 22890 df-cld 22963 df-ntr 22964 df-cls 22965 df-nei 23042 df-lp 23080 df-perf 23081 df-cn 23171 df-cnp 23172 df-haus 23259 df-fil 23790 df-fm 23882 df-flim 23883 df-flf 23884 df-xms 24264 df-ms 24265 df-cncf 24827 df-limc 25823 df-dv 25824

This theorem is referenced by: dvfsumrlim 25994 dvfsumrlim2 25995

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