dvfsumrlim3 - Metamath Proof Explorer

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Theorem dvfsumrlim3 26025

Description: Conjoin the statements of dvfsumrlim 26023 and dvfsumrlim2 26024. (This is useful as a target for lemmas, because the hypotheses to this theorem are complex, and we don't want to repeat ourselves.) (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)
dvfsumrlim3.1	⊢ (𝑥 = 𝑋 → 𝐵 = 𝐸)

**Assertion**
Ref	Expression
dvfsumrlim3	⊢ (𝜑 → (𝐺:𝑆⟶ℝ ∧ 𝐺 ∈ dom ⇝_𝑟 ∧ ((𝐺 ⇝_𝑟 𝐿 ∧ 𝑋 ∈ 𝑆 ∧ 𝐷 ≤ 𝑋) → (abs‘((𝐺‘𝑋) − 𝐿)) ≤ 𝐸)))

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

**Proof of Theorem dvfsumrlim3**
Step	Hyp	Ref	Expression
1		dvfsum.s	. . 3 ⊢ 𝑆 = (𝑇(,)+∞)
2		dvfsum.z	. . 3 ⊢ 𝑍 = (ℤ_≥‘𝑀)
3		dvfsum.m	. . 3 ⊢ (𝜑 → 𝑀 ∈ ℤ)
4		dvfsum.d	. . 3 ⊢ (𝜑 → 𝐷 ∈ ℝ)
5		dvfsum.md	. . 3 ⊢ (𝜑 → 𝑀 ≤ (𝐷 + 1))
6		dvfsum.t	. . 3 ⊢ (𝜑 → 𝑇 ∈ ℝ)
7		dvfsum.a	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆) → 𝐴 ∈ ℝ)
8		dvfsum.b1	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆) → 𝐵 ∈ 𝑉)
9		dvfsum.b2	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑍) → 𝐵 ∈ ℝ)
10		dvfsum.b3	. . 3 ⊢ (𝜑 → (ℝ D (𝑥 ∈ 𝑆 ↦ 𝐴)) = (𝑥 ∈ 𝑆 ↦ 𝐵))
11		dvfsum.c	. . 3 ⊢ (𝑥 = 𝑘 → 𝐵 = 𝐶)
12		dvfsumrlim.g	. . 3 ⊢ 𝐺 = (𝑥 ∈ 𝑆 ↦ (Σ𝑘 ∈ (𝑀...(⌊‘𝑥))𝐶 − 𝐴))
13	1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12	dvfsumrlimf 26017	. 2 ⊢ (𝜑 → 𝐺:𝑆⟶ℝ)
14		dvfsumrlim.l	. . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑘 ∈ 𝑆) ∧ (𝐷 ≤ 𝑥 ∧ 𝑥 ≤ 𝑘)) → 𝐶 ≤ 𝐵)
15		dvfsumrlim.k	. . 3 ⊢ (𝜑 → (𝑥 ∈ 𝑆 ↦ 𝐵) ⇝_𝑟 0)
16	1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 14, 12, 15	dvfsumrlim 26023	. 2 ⊢ (𝜑 → 𝐺 ∈ dom ⇝_𝑟 )
17	3	adantr 481	. . . . . . 7 ⊢ ((𝜑 ∧ (𝐷 ≤ 𝑋 ∧ 𝑋 ∈ 𝑆)) → 𝑀 ∈ ℤ)
18	4	adantr 481	. . . . . . 7 ⊢ ((𝜑 ∧ (𝐷 ≤ 𝑋 ∧ 𝑋 ∈ 𝑆)) → 𝐷 ∈ ℝ)
19	5	adantr 481	. . . . . . 7 ⊢ ((𝜑 ∧ (𝐷 ≤ 𝑋 ∧ 𝑋 ∈ 𝑆)) → 𝑀 ≤ (𝐷 + 1))
20	6	adantr 481	. . . . . . 7 ⊢ ((𝜑 ∧ (𝐷 ≤ 𝑋 ∧ 𝑋 ∈ 𝑆)) → 𝑇 ∈ ℝ)
21	7	adantlr 721	. . . . . . 7 ⊢ (((𝜑 ∧ (𝐷 ≤ 𝑋 ∧ 𝑋 ∈ 𝑆)) ∧ 𝑥 ∈ 𝑆) → 𝐴 ∈ ℝ)
22	8	adantlr 721	. . . . . . 7 ⊢ (((𝜑 ∧ (𝐷 ≤ 𝑋 ∧ 𝑋 ∈ 𝑆)) ∧ 𝑥 ∈ 𝑆) → 𝐵 ∈ 𝑉)
23	9	adantlr 721	. . . . . . 7 ⊢ (((𝜑 ∧ (𝐷 ≤ 𝑋 ∧ 𝑋 ∈ 𝑆)) ∧ 𝑥 ∈ 𝑍) → 𝐵 ∈ ℝ)
24	10	adantr 481	. . . . . . 7 ⊢ ((𝜑 ∧ (𝐷 ≤ 𝑋 ∧ 𝑋 ∈ 𝑆)) → (ℝ D (𝑥 ∈ 𝑆 ↦ 𝐴)) = (𝑥 ∈ 𝑆 ↦ 𝐵))
25	14	3adant1r 1184	. . . . . . 7 ⊢ (((𝜑 ∧ (𝐷 ≤ 𝑋 ∧ 𝑋 ∈ 𝑆)) ∧ (𝑥 ∈ 𝑆 ∧ 𝑘 ∈ 𝑆) ∧ (𝐷 ≤ 𝑥 ∧ 𝑥 ≤ 𝑘)) → 𝐶 ≤ 𝐵)
26	15	adantr 481	. . . . . . 7 ⊢ ((𝜑 ∧ (𝐷 ≤ 𝑋 ∧ 𝑋 ∈ 𝑆)) → (𝑥 ∈ 𝑆 ↦ 𝐵) ⇝_𝑟 0)
27		simprr 778	. . . . . . 7 ⊢ ((𝜑 ∧ (𝐷 ≤ 𝑋 ∧ 𝑋 ∈ 𝑆)) → 𝑋 ∈ 𝑆)
28		simprl 776	. . . . . . 7 ⊢ ((𝜑 ∧ (𝐷 ≤ 𝑋 ∧ 𝑋 ∈ 𝑆)) → 𝐷 ≤ 𝑋)
29	1, 2, 17, 18, 19, 20, 21, 22, 23, 24, 11, 25, 12, 26, 27, 28	dvfsumrlim2 26024	. . . . . 6 ⊢ (((𝜑 ∧ (𝐷 ≤ 𝑋 ∧ 𝑋 ∈ 𝑆)) ∧ 𝐺 ⇝_𝑟 𝐿) → (abs‘((𝐺‘𝑋) − 𝐿)) ≤ ⦋𝑋 / 𝑥⦌𝐵)
30	27	adantr 481	. . . . . . 7 ⊢ (((𝜑 ∧ (𝐷 ≤ 𝑋 ∧ 𝑋 ∈ 𝑆)) ∧ 𝐺 ⇝_𝑟 𝐿) → 𝑋 ∈ 𝑆)
31		nfcvd 2903	. . . . . . . 8 ⊢ (𝑋 ∈ 𝑆 → Ⅎ𝑥𝐸)
32		dvfsumrlim3.1	. . . . . . . 8 ⊢ (𝑥 = 𝑋 → 𝐵 = 𝐸)
33	31, 32	csbiegf 3871	. . . . . . 7 ⊢ (𝑋 ∈ 𝑆 → ⦋𝑋 / 𝑥⦌𝐵 = 𝐸)
34	30, 33	syl 17	. . . . . 6 ⊢ (((𝜑 ∧ (𝐷 ≤ 𝑋 ∧ 𝑋 ∈ 𝑆)) ∧ 𝐺 ⇝_𝑟 𝐿) → ⦋𝑋 / 𝑥⦌𝐵 = 𝐸)
35	29, 34	breqtrd 5105	. . . . 5 ⊢ (((𝜑 ∧ (𝐷 ≤ 𝑋 ∧ 𝑋 ∈ 𝑆)) ∧ 𝐺 ⇝_𝑟 𝐿) → (abs‘((𝐺‘𝑋) − 𝐿)) ≤ 𝐸)
36	35	exp42 436	. . . 4 ⊢ (𝜑 → (𝐷 ≤ 𝑋 → (𝑋 ∈ 𝑆 → (𝐺 ⇝_𝑟 𝐿 → (abs‘((𝐺‘𝑋) − 𝐿)) ≤ 𝐸))))
37	36	com24 95	. . 3 ⊢ (𝜑 → (𝐺 ⇝_𝑟 𝐿 → (𝑋 ∈ 𝑆 → (𝐷 ≤ 𝑋 → (abs‘((𝐺‘𝑋) − 𝐿)) ≤ 𝐸))))
38	37	3impd 1355	. 2 ⊢ (𝜑 → ((𝐺 ⇝_𝑟 𝐿 ∧ 𝑋 ∈ 𝑆 ∧ 𝐷 ≤ 𝑋) → (abs‘((𝐺‘𝑋) − 𝐿)) ≤ 𝐸))
39	13, 16, 38	3jca 1134	1 ⊢ (𝜑 → (𝐺:𝑆⟶ℝ ∧ 𝐺 ∈ dom ⇝_𝑟 ∧ ((𝐺 ⇝_𝑟 𝐿 ∧ 𝑋 ∈ 𝑆 ∧ 𝐷 ≤ 𝑋) → (abs‘((𝐺‘𝑋) − 𝐿)) ≤ 𝐸)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 396 ∧ w3a 1092 = wceq 1547 ∈ wcel 2119 ⦋csb 3838 class class class wbr 5079 ↦ cmpt 5160 dom cdm 5625 ⟶wf 6488 ‘cfv 6492 (class class class)co 7363 ℝcr 11035 0cc0 11036 1c1 11037 + caddc 11039 +∞cpnf 11174 ≤ cle 11178 − cmin 11375 ℤcz 12522 ℤ_≥cuz 12786 (,)cioo 13296 ...cfz 13459 ⌊cfl 13747 abscabs 15194 ⇝_𝑟 crli 15445 Σcsu 15646 D cdv 25855

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1974 ax-7 2015 ax-8 2121 ax-9 2129 ax-10 2152 ax-11 2168 ax-12 2189 ax-ext 2712 ax-rep 5206 ax-sep 5225 ax-nul 5235 ax-pow 5301 ax-pr 5369 ax-un 7685 ax-inf2 9560 ax-cnex 11092 ax-resscn 11093 ax-1cn 11094 ax-icn 11095 ax-addcl 11096 ax-addrcl 11097 ax-mulcl 11098 ax-mulrcl 11099 ax-mulcom 11100 ax-addass 11101 ax-mulass 11102 ax-distr 11103 ax-i2m1 11104 ax-1ne0 11105 ax-1rid 11106 ax-rnegex 11107 ax-rrecex 11108 ax-cnre 11109 ax-pre-lttri 11110 ax-pre-lttrn 11111 ax-pre-ltadd 11112 ax-pre-mulgt0 11113 ax-pre-sup 11114 ax-addf 11115

This theorem depends on definitions: df-bi 208 df-an 397 df-or 854 df-3or 1093 df-3an 1094 df-tru 1550 df-fal 1560 df-ex 1787 df-nf 1791 df-sb 2074 df-mo 2543 df-eu 2573 df-clab 2719 df-cleq 2732 df-clel 2815 df-nfc 2889 df-ne 2936 df-nel 3040 df-ral 3055 df-rex 3065 df-rmo 3345 df-reu 3346 df-rab 3393 df-v 3434 df-sbc 3731 df-csb 3839 df-dif 3893 df-un 3895 df-in 3897 df-ss 3907 df-pss 3910 df-nul 4269 df-if 4462 df-pw 4538 df-sn 4563 df-pr 4565 df-tp 4567 df-op 4569 df-uni 4846 df-int 4885 df-iun 4930 df-iin 4931 df-br 5080 df-opab 5142 df-mpt 5161 df-tr 5187 df-id 5520 df-eprel 5525 df-po 5533 df-so 5534 df-fr 5578 df-se 5579 df-we 5580 df-xp 5631 df-rel 5632 df-cnv 5633 df-co 5634 df-dm 5635 df-rn 5636 df-res 5637 df-ima 5638 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-isom 6501 df-riota 7320 df-ov 7366 df-oprab 7367 df-mpo 7368 df-of 7627 df-om 7814 df-1st 7938 df-2nd 7939 df-supp 8108 df-frecs 8228 df-wrecs 8259 df-recs 8308 df-rdg 8346 df-1o 8402 df-2o 8403 df-er 8640 df-map 8772 df-pm 8773 df-ixp 8843 df-en 8891 df-dom 8892 df-sdom 8893 df-fin 8894 df-fsupp 9272 df-fi 9321 df-sup 9352 df-inf 9353 df-oi 9422 df-card 9861 df-pnf 11179 df-mnf 11180 df-xr 11181 df-ltxr 11182 df-le 11183 df-sub 11377 df-neg 11378 df-div 11806 df-nn 12173 df-2 12242 df-3 12243 df-4 12244 df-5 12245 df-6 12246 df-7 12247 df-8 12248 df-9 12249 df-n0 12436 df-z 12523 df-dec 12643 df-uz 12787 df-q 12897 df-rp 12941 df-xneg 13061 df-xadd 13062 df-xmul 13063 df-ioo 13300 df-ico 13302 df-icc 13303 df-fz 13460 df-fzo 13607 df-fl 13749 df-seq 13962 df-exp 14022 df-hash 14291 df-cj 15059 df-re 15060 df-im 15061 df-sqrt 15195 df-abs 15196 df-limsup 15431 df-clim 15448 df-rlim 15449 df-sum 15647 df-struct 17115 df-sets 17132 df-slot 17150 df-ndx 17162 df-base 17178 df-ress 17199 df-plusg 17231 df-mulr 17232 df-starv 17233 df-sca 17234 df-vsca 17235 df-ip 17236 df-tset 17237 df-ple 17238 df-ds 17240 df-unif 17241 df-hom 17242 df-cco 17243 df-rest 17383 df-topn 17384 df-0g 17402 df-gsum 17403 df-topgen 17404 df-pt 17405 df-prds 17408 df-xrs 17464 df-qtop 17469 df-imas 17470 df-xps 17472 df-mre 17546 df-mrc 17547 df-acs 17549 df-mgm 18606 df-sgrp 18685 df-mnd 18701 df-submnd 18750 df-mulg 19042 df-cntz 19290 df-cmn 19755 df-psmet 21346 df-xmet 21347 df-met 21348 df-bl 21349 df-mopn 21350 df-fbas 21351 df-fg 21352 df-cnfld 21355 df-top 22884 df-topon 22901 df-topsp 22923 df-bases 22936 df-cld 23009 df-ntr 23010 df-cls 23011 df-nei 23088 df-lp 23126 df-perf 23127 df-cn 23217 df-cnp 23218 df-haus 23305 df-cmp 23377 df-tx 23552 df-hmeo 23745 df-fil 23836 df-fm 23928 df-flim 23929 df-flf 23930 df-xms 24310 df-ms 24311 df-tms 24312 df-cncf 24870 df-limc 25858 df-dv 25859

This theorem is referenced by: divsqrtsumlem 26968 logdivsum 27521

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