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Theorem dvfsumrlimge0 25965

Description: Lemma for dvfsumrlim 25966. Satisfy the assumption of dvfsumlem4 25964. (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 13307	. . . . . 6 ⊢ (𝑇(,)+∞) ⊆ ℝ
3	1, 2	eqsstri 3981	. . . . 5 ⊢ 𝑆 ⊆ ℝ
4		simprl 770	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝐷 ≤ 𝑥)) → 𝑥 ∈ 𝑆)
5	3, 4	sselid 3932	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝐷 ≤ 𝑥)) → 𝑥 ∈ ℝ)
6	5	rexrd 11162	. . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝐷 ≤ 𝑥)) → 𝑥 ∈ ℝ^*)
7	5	renepnfd 11163	. . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝐷 ≤ 𝑥)) → 𝑥 ≠ +∞)
8		icopnfsup 13769	. . 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 11162	. . . . 5 ⊢ (𝜑 → 𝑇 ∈ ℝ^*)
12	4, 1	eleqtrdi 2841	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝐷 ≤ 𝑥)) → 𝑥 ∈ (𝑇(,)+∞))
13	11	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝐷 ≤ 𝑥)) → 𝑇 ∈ ℝ^*)
14		elioopnf 13343	. . . . . . . 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 13249	. . . . . 6 ⊢ (,) = (𝑢 ∈ ℝ^, 𝑣 ∈ ℝ^ ↦ {𝑤 ∈ ℝ^* ∣ (𝑢 < 𝑤 ∧ 𝑤 < 𝑣)})
19		df-ico 13251	. . . . . 6 ⊢ [,) = (𝑢 ∈ ℝ^, 𝑣 ∈ ℝ^ ↦ {𝑤 ∈ ℝ^* ∣ (𝑢 ≤ 𝑤 ∧ 𝑤 < 𝑣)})
20		xrltletr 13056	. . . . . 6 ⊢ ((𝑇 ∈ ℝ^* ∧ 𝑥 ∈ ℝ^* ∧ 𝑧 ∈ ℝ^*) → ((𝑇 < 𝑥 ∧ 𝑥 ≤ 𝑧) → 𝑇 < 𝑧))
21	18, 19, 20	ixxss1 13263	. . . . 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 3976	. . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝐷 ≤ 𝑥)) → (𝑥[,)+∞) ⊆ 𝑆)
24		dvfsum.c	. . . . 5 ⊢ (𝑥 = 𝑘 → 𝐵 = 𝐶)
25	24	cbvmptv 5195	. . . 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 5127	. . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝐷 ≤ 𝑥)) → (𝑘 ∈ 𝑆 ↦ 𝐶) ⇝_𝑟 0)
29	23, 28	rlimres2 15468	. 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 25958	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆) → 𝐵 ∈ ℝ)
35	34	adantrr 717	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝐷 ≤ 𝑥)) → 𝐵 ∈ ℝ)
36	35	recnd 11140	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝐷 ≤ 𝑥)) → 𝐵 ∈ ℂ)
37		rlimconst 15451	. . . 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 15468	. 2 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝐷 ≤ 𝑥)) → (𝑘 ∈ (𝑥[,)+∞) ↦ 𝐵) ⇝_𝑟 𝐵)
40	34	ralrimiva 3124	. . . 4 ⊢ (𝜑 → ∀𝑥 ∈ 𝑆 𝐵 ∈ ℝ)
41	40	adantr 480	. . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝐷 ≤ 𝑥)) → ∀𝑥 ∈ 𝑆 𝐵 ∈ ℝ)
42	23	sselda 3934	. . 3 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝐷 ≤ 𝑥)) ∧ 𝑘 ∈ (𝑥[,)+∞)) → 𝑘 ∈ 𝑆)
43	24	eleq1d 2816	. . . 4 ⊢ (𝑥 = 𝑘 → (𝐵 ∈ ℝ ↔ 𝐶 ∈ ℝ))
44	43	rspccva 3576	. . 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 13345	. . . . 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 15555	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 2111 ≠ wne 2928 ∀wral 3047 ⊆ wss 3902 class class class wbr 5091 ↦ cmpt 5172 ‘cfv 6481 (class class class)co 7346 supcsup 9324 ℂcc 11004 ℝcr 11005 0cc0 11006 1c1 11007 + caddc 11009 +∞cpnf 11143 ℝ^*cxr 11145 < clt 11146 ≤ cle 11147 − cmin 11344 ℤcz 12468 ℤ_≥cuz 12732 (,)cioo 13245 [,)cico 13247 ...cfz 13407 ⌊cfl 13694 ⇝_𝑟 crli 15392 Σcsu 15593 D cdv 25792

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 2113 ax-9 2121 ax-10 2144 ax-11 2160 ax-12 2180 ax-ext 2703 ax-rep 5217 ax-sep 5234 ax-nul 5244 ax-pow 5303 ax-pr 5370 ax-un 7668 ax-cnex 11062 ax-resscn 11063 ax-1cn 11064 ax-icn 11065 ax-addcl 11066 ax-addrcl 11067 ax-mulcl 11068 ax-mulrcl 11069 ax-mulcom 11070 ax-addass 11071 ax-mulass 11072 ax-distr 11073 ax-i2m1 11074 ax-1ne0 11075 ax-1rid 11076 ax-rnegex 11077 ax-rrecex 11078 ax-cnre 11079 ax-pre-lttri 11080 ax-pre-lttrn 11081 ax-pre-ltadd 11082 ax-pre-mulgt0 11083 ax-pre-sup 11084

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 2535 df-eu 2564 df-clab 2710 df-cleq 2723 df-clel 2806 df-nfc 2881 df-ne 2929 df-nel 3033 df-ral 3048 df-rex 3057 df-rmo 3346 df-reu 3347 df-rab 3396 df-v 3438 df-sbc 3742 df-csb 3851 df-dif 3905 df-un 3907 df-in 3909 df-ss 3919 df-pss 3922 df-nul 4284 df-if 4476 df-pw 4552 df-sn 4577 df-pr 4579 df-tp 4581 df-op 4583 df-uni 4860 df-int 4898 df-iun 4943 df-iin 4944 df-br 5092 df-opab 5154 df-mpt 5173 df-tr 5199 df-id 5511 df-eprel 5516 df-po 5524 df-so 5525 df-fr 5569 df-we 5571 df-xp 5622 df-rel 5623 df-cnv 5624 df-co 5625 df-dm 5626 df-rn 5627 df-res 5628 df-ima 5629 df-pred 6248 df-ord 6309 df-on 6310 df-lim 6311 df-suc 6312 df-iota 6437 df-fun 6483 df-fn 6484 df-f 6485 df-f1 6486 df-fo 6487 df-f1o 6488 df-fv 6489 df-riota 7303 df-ov 7349 df-oprab 7350 df-mpo 7351 df-om 7797 df-1st 7921 df-2nd 7922 df-frecs 8211 df-wrecs 8242 df-recs 8291 df-rdg 8329 df-1o 8385 df-er 8622 df-map 8752 df-pm 8753 df-en 8870 df-dom 8871 df-sdom 8872 df-fin 8873 df-fi 9295 df-sup 9326 df-inf 9327 df-pnf 11148 df-mnf 11149 df-xr 11150 df-ltxr 11151 df-le 11152 df-sub 11346 df-neg 11347 df-div 11775 df-nn 12126 df-2 12188 df-3 12189 df-4 12190 df-5 12191 df-6 12192 df-7 12193 df-8 12194 df-9 12195 df-n0 12382 df-z 12469 df-dec 12589 df-uz 12733 df-q 12847 df-rp 12891 df-xneg 13011 df-xadd 13012 df-xmul 13013 df-ioo 13249 df-ico 13251 df-icc 13252 df-fz 13408 df-seq 13909 df-exp 13969 df-cj 15006 df-re 15007 df-im 15008 df-sqrt 15142 df-abs 15143 df-rlim 15396 df-struct 17058 df-slot 17093 df-ndx 17105 df-base 17121 df-plusg 17174 df-mulr 17175 df-starv 17176 df-tset 17180 df-ple 17181 df-ds 17183 df-unif 17184 df-rest 17326 df-topn 17327 df-topgen 17347 df-psmet 21284 df-xmet 21285 df-met 21286 df-bl 21287 df-mopn 21288 df-fbas 21289 df-fg 21290 df-cnfld 21293 df-top 22810 df-topon 22827 df-topsp 22849 df-bases 22862 df-cld 22935 df-ntr 22936 df-cls 22937 df-nei 23014 df-lp 23052 df-perf 23053 df-cn 23143 df-cnp 23144 df-haus 23231 df-fil 23762 df-fm 23854 df-flim 23855 df-flf 23856 df-xms 24236 df-ms 24237 df-cncf 24799 df-limc 25795 df-dv 25796

This theorem is referenced by: dvfsumrlim 25966 dvfsumrlim2 25967

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