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Theorem dvfsumrlimge0 26264

Description: Lemma for dvfsumrlim 26265. Satisfy the assumption of dvfsumlem4 26263. (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 13464	. . . . . 6 ⊢ (𝑇(,)+∞) ⊆ ℝ
3	1, 2	eqsstri 3980	. . . . 5 ⊢ 𝑆 ⊆ ℝ
4		simprl 783	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝐷 ≤ 𝑥)) → 𝑥 ∈ 𝑆)
5	3, 4	sselid 3932	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝐷 ≤ 𝑥)) → 𝑥 ∈ ℝ)
6	5	rexrd 11287	. . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝐷 ≤ 𝑥)) → 𝑥 ∈ ℝ^*)
7	5	renepnfd 11288	. . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝐷 ≤ 𝑥)) → 𝑥 ≠ +∞)
8		icopnfsup 13930	. . 3 ⊢ ((𝑥 ∈ ℝ^* ∧ 𝑥 ≠ +∞) → sup((𝑥[,)+∞), ℝ^*, < ) = +∞)
9	6, 7, 8	syl2anc 596	. 2 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝐷 ≤ 𝑥)) → sup((𝑥[,)+∞), ℝ^*, < ) = +∞)
10		dvfsum.t	. . . . . 6 ⊢ (𝜑 → 𝑇 ∈ ℝ)
11	10	rexrd 11287	. . . . 5 ⊢ (𝜑 → 𝑇 ∈ ℝ^*)
12	4, 1	eleqtrdi 2872	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝐷 ≤ 𝑥)) → 𝑥 ∈ (𝑇(,)+∞))
13	11	adantr 486	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝐷 ≤ 𝑥)) → 𝑇 ∈ ℝ^*)
14		elioopnf 13500	. . . . . . . 8 ⊢ (𝑇 ∈ ℝ^* → (𝑥 ∈ (𝑇(,)+∞) ↔ (𝑥 ∈ ℝ ∧ 𝑇 < 𝑥)))
15	13, 14	syl 18	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝐷 ≤ 𝑥)) → (𝑥 ∈ (𝑇(,)+∞) ↔ (𝑥 ∈ ℝ ∧ 𝑇 < 𝑥)))
16	12, 15	mpbid 235	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝐷 ≤ 𝑥)) → (𝑥 ∈ ℝ ∧ 𝑇 < 𝑥))
17	16	simprd 501	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝐷 ≤ 𝑥)) → 𝑇 < 𝑥)
18		df-ioo 13406	. . . . . 6 ⊢ (,) = (𝑢 ∈ ℝ^, 𝑣 ∈ ℝ^ ↦ {𝑤 ∈ ℝ^* ∣ (𝑢 < 𝑤 ∧ 𝑤 < 𝑣)})
19		df-ico 13408	. . . . . 6 ⊢ [,) = (𝑢 ∈ ℝ^, 𝑣 ∈ ℝ^ ↦ {𝑤 ∈ ℝ^* ∣ (𝑢 ≤ 𝑤 ∧ 𝑤 < 𝑣)})
20		xrltletr 13212	. . . . . 6 ⊢ ((𝑇 ∈ ℝ^* ∧ 𝑥 ∈ ℝ^* ∧ 𝑧 ∈ ℝ^*) → ((𝑇 < 𝑥 ∧ 𝑥 ≤ 𝑧) → 𝑇 < 𝑧))
21	18, 19, 20	ixxss1 13420	. . . . 5 ⊢ ((𝑇 ∈ ℝ^* ∧ 𝑇 < 𝑥) → (𝑥[,)+∞) ⊆ (𝑇(,)+∞))
22	11, 17, 21	syl2an2r 698	. . . 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 5213	. . . 4 ⊢ (𝑥 ∈ 𝑆 ↦ 𝐵) = (𝑘 ∈ 𝑆 ↦ 𝐶)
26		dvfsumrlim.k	. . . . 5 ⊢ (𝜑 → (𝑥 ∈ 𝑆 ↦ 𝐵) ⇝_𝑟 0)
27	26	adantr 486	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝐷 ≤ 𝑥)) → (𝑥 ∈ 𝑆 ↦ 𝐵) ⇝_𝑟 0)
28	25, 27	eqbrtrrid 5145	. . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝐷 ≤ 𝑥)) → (𝑘 ∈ 𝑆 ↦ 𝐶) ⇝_𝑟 0)
29	23, 28	rlimres2 15652	. 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 26258	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆) → 𝐵 ∈ ℝ)
35	34	adantrr 730	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝐷 ≤ 𝑥)) → 𝐵 ∈ ℝ)
36	35	recnd 11265	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝐷 ≤ 𝑥)) → 𝐵 ∈ ℂ)
37		rlimconst 15635	. . . 4 ⊢ ((𝑆 ⊆ ℝ ∧ 𝐵 ∈ ℂ) → (𝑘 ∈ 𝑆 ↦ 𝐵) ⇝_𝑟 𝐵)
38	30, 36, 37	syl2an2r 698	. . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝐷 ≤ 𝑥)) → (𝑘 ∈ 𝑆 ↦ 𝐵) ⇝_𝑟 𝐵)
39	23, 38	rlimres2 15652	. 2 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝐷 ≤ 𝑥)) → (𝑘 ∈ (𝑥[,)+∞) ↦ 𝐵) ⇝_𝑟 𝐵)
40	34	ralrimiva 3156	. . . 4 ⊢ (𝜑 → ∀𝑥 ∈ 𝑆 𝐵 ∈ ℝ)
41	40	adantr 486	. . 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 2847	. . . 4 ⊢ (𝑥 = 𝑘 → (𝐵 ∈ ℝ ↔ 𝐶 ∈ ℝ))
44	43	rspccva 3578	. . 3 ⊢ ((∀𝑥 ∈ 𝑆 𝐵 ∈ ℝ ∧ 𝑘 ∈ 𝑆) → 𝐶 ∈ ℝ)
45	41, 42, 44	syl2an2r 698	. 2 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝐷 ≤ 𝑥)) ∧ 𝑘 ∈ (𝑥[,)+∞)) → 𝐶 ∈ ℝ)
46	35	adantr 486	. 2 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝐷 ≤ 𝑥)) ∧ 𝑘 ∈ (𝑥[,)+∞)) → 𝐵 ∈ ℝ)
47		simpll 779	. . 3 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝐷 ≤ 𝑥)) ∧ 𝑘 ∈ (𝑥[,)+∞)) → 𝜑)
48		simplrl 789	. . 3 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝐷 ≤ 𝑥)) ∧ 𝑘 ∈ (𝑥[,)+∞)) → 𝑥 ∈ 𝑆)
49		simplrr 790	. . 3 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝐷 ≤ 𝑥)) ∧ 𝑘 ∈ (𝑥[,)+∞)) → 𝐷 ≤ 𝑥)
50		elicopnf 13502	. . . . 5 ⊢ (𝑥 ∈ ℝ → (𝑘 ∈ (𝑥[,)+∞) ↔ (𝑘 ∈ ℝ ∧ 𝑥 ≤ 𝑘)))
51	5, 50	syl 18	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝐷 ≤ 𝑥)) → (𝑘 ∈ (𝑥[,)+∞) ↔ (𝑘 ∈ ℝ ∧ 𝑥 ≤ 𝑘)))
52	51	simplbda 505	. . 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 1406	. 2 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝐷 ≤ 𝑥)) ∧ 𝑘 ∈ (𝑥[,)+∞)) → 𝐶 ≤ 𝐵)
55	9, 29, 39, 45, 46, 54	rlimle 15739	1 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝐷 ≤ 𝑥)) → 0 ≤ 𝐵)

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ↔ wb 209 ∧ wa 401 ∧ w3a 1103 = wceq 1570 ∈ wcel 2145 ≠ wne 2957 ∀wral 3078 ⊆ wss 3902 class class class wbr 5107 ↦ cmpt 5190 ‘cfv 6537 (class class class)co 7417 supcsup 9414 ℂcc 11126 ℝcr 11127 0cc0 11128 1c1 11129 + caddc 11131 +∞cpnf 11268 ℝ^*cxr 11270 < clt 11271 ≤ cle 11272 − cmin 11469 ℤcz 12619 ℤ_≥cuz 12891 (,)cioo 13402 [,)cico 13404 ...cfz 13565 ⌊cfl 13855 ⇝_𝑟 crli 15576 Σcsu 15777 D cdv 26097

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1943 ax-6 2000 ax-7 2041 ax-8 2147 ax-9 2155 ax-10 2178 ax-11 2194 ax-12 2215 ax-ext 2734 ax-rep 5236 ax-sep 5255 ax-nul 5267 ax-pow 5334 ax-pr 5402 ax-un 7740 ax-cnex 11184 ax-resscn 11185 ax-1cn 11186 ax-icn 11187 ax-addcl 11188 ax-addrcl 11189 ax-mulcl 11190 ax-mulrcl 11191 ax-mulcom 11192 ax-addass 11193 ax-mulass 11194 ax-distr 11195 ax-i2m1 11196 ax-1ne0 11197 ax-1rid 11198 ax-rnegex 11199 ax-rrecex 11200 ax-cnre 11201 ax-pre-lttri 11202 ax-pre-lttrn 11203 ax-pre-ltadd 11204 ax-pre-mulgt0 11205 ax-pre-sup 11206

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2100 df-mo 2566 df-eu 2596 df-clab 2741 df-cleq 2754 df-clel 2837 df-nfc 2911 df-ne 2958 df-nel 3064 df-ral 3079 df-rex 3089 df-rmo 3367 df-reu 3368 df-rab 3415 df-v 3455 df-sbc 3743 df-csb 3851 df-dif 3905 df-un 3907 df-in 3909 df-ss 3919 df-pss 3922 df-nul 4283 df-if 4486 df-pw 4562 df-sn 4588 df-pr 4590 df-tp 4592 df-op 4594 df-uni 4871 df-int 4911 df-iun 4956 df-iin 4957 df-br 5108 df-opab 5172 df-mpt 5191 df-tr 5217 df-id 5554 df-eprel 5559 df-po 5567 df-so 5568 df-fr 5612 df-we 5614 df-xp 5665 df-rel 5666 df-cnv 5667 df-co 5668 df-dm 5669 df-rn 5670 df-res 5671 df-ima 5672 df-pred 6303 df-ord 6364 df-on 6365 df-lim 6366 df-suc 6367 df-iota 6493 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-riota 7374 df-ov 7420 df-oprab 7421 df-mpo 7422 df-om 7867 df-1st 7990 df-2nd 7991 df-frecs 8284 df-wrecs 8315 df-recs 8364 df-rdg 8403 df-1o 8459 df-er 8700 df-map 8832 df-pm 8833 df-en 8957 df-dom 8958 df-sdom 8959 df-fin 8960 df-fi 9385 df-sup 9416 df-inf 9417 df-pnf 11273 df-mnf 11274 df-xr 11275 df-ltxr 11276 df-le 11277 df-sub 11471 df-neg 11472 df-div 11900 df-nn 12262 df-2 12331 df-3 12332 df-4 12333 df-5 12334 df-6 12335 df-7 12336 df-8 12337 df-9 12338 df-n0 12533 df-z 12620 df-dec 12741 df-uz 12892 df-q 13002 df-rp 13047 df-xneg 13167 df-xadd 13168 df-xmul 13169 df-ioo 13406 df-ico 13408 df-icc 13409 df-fz 13566 df-seq 14070 df-exp 14130 df-cj 15190 df-re 15191 df-im 15192 df-sqrt 15326 df-abs 15327 df-rlim 15580 df-struct 17245 df-slot 17280 df-ndx 17292 df-base 17308 df-plusg 17361 df-mulr 17362 df-starv 17363 df-tset 17367 df-ple 17368 df-ds 17370 df-unif 17371 df-rest 17513 df-topn 17514 df-topgen 17534 df-psmet 21583 df-xmet 21584 df-met 21585 df-bl 21586 df-mopn 21587 df-fbas 21588 df-fg 21589 df-cnfld 21592 df-top 23125 df-topon 23142 df-topsp 23164 df-bases 23177 df-cld 23250 df-ntr 23251 df-cls 23252 df-nei 23329 df-lp 23367 df-perf 23368 df-cn 23458 df-cnp 23459 df-haus 23546 df-fil 24078 df-fm 24170 df-flim 24171 df-flf 24172 df-xms 24552 df-ms 24553 df-cncf 25112 df-limc 26100 df-dv 26101

This theorem is used by: dvfsumrlim 26265 dvfsumrlim2 26266

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