knoppndvlem22 - Mathbox for Asger C. Ipsen

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Theorem knoppndvlem22 36507

Description: Lemma for knoppndv 36508. (Contributed by Asger C. Ipsen, 19-Aug-2021.)

**Hypotheses**
Ref	Expression
knoppndvlem22.t	⊢ 𝑇 = (𝑥 ∈ ℝ ↦ (abs‘((⌊‘(𝑥 + (1 / 2))) − 𝑥)))
knoppndvlem22.f	⊢ 𝐹 = (𝑦 ∈ ℝ ↦ (𝑛 ∈ ℕ₀ ↦ ((𝐶↑𝑛) · (𝑇‘(((2 · 𝑁)↑𝑛) · 𝑦)))))
knoppndvlem22.w	⊢ 𝑊 = (𝑤 ∈ ℝ ↦ Σ𝑖 ∈ ℕ₀ ((𝐹‘𝑤)‘𝑖))
knoppndvlem22.c	⊢ (𝜑 → 𝐶 ∈ (-1(,)1))
knoppndvlem22.d	⊢ (𝜑 → 𝐷 ∈ ℝ⁺)
knoppndvlem22.e	⊢ (𝜑 → 𝐸 ∈ ℝ⁺)
knoppndvlem22.h	⊢ (𝜑 → 𝐻 ∈ ℝ)
knoppndvlem22.n	⊢ (𝜑 → 𝑁 ∈ ℕ)
knoppndvlem22.1	⊢ (𝜑 → 1 < (𝑁 · (abs‘𝐶)))

**Assertion**
Ref	Expression
knoppndvlem22	⊢ (𝜑 → ∃𝑎 ∈ ℝ ∃𝑏 ∈ ℝ ((𝑎 ≤ 𝐻 ∧ 𝐻 ≤ 𝑏) ∧ ((𝑏 − 𝑎) < 𝐷 ∧ 𝑎 ≠ 𝑏) ∧ 𝐸 ≤ ((abs‘((𝑊‘𝑏) − (𝑊‘𝑎))) / (𝑏 − 𝑎))))

Distinct variable groups: 𝐶,𝑖,𝑛,𝑤,𝑦 𝐷,𝑎,𝑏 𝐷,𝑖,𝑛,𝑤,𝑦 𝐸,𝑎,𝑏 𝑖,𝐸,𝑛,𝑤,𝑦 𝑖,𝐹,𝑤 𝐻,𝑎,𝑏 𝑁,𝑎,𝑏 𝑖,𝑁,𝑛,𝑤,𝑦 𝑥,𝑁,𝑖,𝑤 𝑇,𝑛,𝑦 𝑊,𝑎,𝑏 𝜑,𝑖,𝑛,𝑤,𝑦 Allowed substitution hints: 𝜑(𝑥,𝑎,𝑏) 𝐶(𝑥,𝑎,𝑏) 𝐷(𝑥) 𝑇(𝑥,𝑤,𝑖,𝑎,𝑏) 𝐸(𝑥) 𝐹(𝑥,𝑦,𝑛,𝑎,𝑏) 𝐻(𝑥,𝑦,𝑤,𝑖,𝑛) 𝑊(𝑥,𝑦,𝑤,𝑖,𝑛)

Proof of Theorem knoppndvlem22 Dummy variable 𝑗 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		knoppndvlem22.c	. . 3 ⊢ (𝜑 → 𝐶 ∈ (-1(,)1))
2		knoppndvlem22.n	. . 3 ⊢ (𝜑 → 𝑁 ∈ ℕ)
3		knoppndvlem22.d	. . 3 ⊢ (𝜑 → 𝐷 ∈ ℝ⁺)
4		knoppndvlem22.e	. . 3 ⊢ (𝜑 → 𝐸 ∈ ℝ⁺)
5		knoppndvlem22.1	. . . 4 ⊢ (𝜑 → 1 < (𝑁 · (abs‘𝐶)))
6	1, 2, 5	knoppndvlem20 36505	. . 3 ⊢ (𝜑 → (1 − (1 / (((2 · 𝑁) · (abs‘𝐶)) − 1))) ∈ ℝ⁺)
7	1, 2, 3, 4, 6, 5	knoppndvlem18 36503	. 2 ⊢ (𝜑 → ∃𝑗 ∈ ℕ₀ ((((2 · 𝑁)↑-𝑗) / 2) < 𝐷 ∧ 𝐸 ≤ ((((2 · 𝑁) · (abs‘𝐶))↑𝑗) · (1 − (1 / (((2 · 𝑁) · (abs‘𝐶)) − 1))))))
8		knoppndvlem22.t	. . 3 ⊢ 𝑇 = (𝑥 ∈ ℝ ↦ (abs‘((⌊‘(𝑥 + (1 / 2))) − 𝑥)))
9		knoppndvlem22.f	. . 3 ⊢ 𝐹 = (𝑦 ∈ ℝ ↦ (𝑛 ∈ ℕ₀ ↦ ((𝐶↑𝑛) · (𝑇‘(((2 · 𝑁)↑𝑛) · 𝑦)))))
10		knoppndvlem22.w	. . 3 ⊢ 𝑊 = (𝑤 ∈ ℝ ↦ Σ𝑖 ∈ ℕ₀ ((𝐹‘𝑤)‘𝑖))
11		eqid 2729	. . 3 ⊢ (1 − (1 / (((2 · 𝑁) · (abs‘𝐶)) − 1))) = (1 − (1 / (((2 · 𝑁) · (abs‘𝐶)) − 1)))
12	1	adantr 480	. . 3 ⊢ ((𝜑 ∧ (𝑗 ∈ ℕ₀ ∧ ((((2 · 𝑁)↑-𝑗) / 2) < 𝐷 ∧ 𝐸 ≤ ((((2 · 𝑁) · (abs‘𝐶))↑𝑗) · (1 − (1 / (((2 · 𝑁) · (abs‘𝐶)) − 1))))))) → 𝐶 ∈ (-1(,)1))
13	3	adantr 480	. . 3 ⊢ ((𝜑 ∧ (𝑗 ∈ ℕ₀ ∧ ((((2 · 𝑁)↑-𝑗) / 2) < 𝐷 ∧ 𝐸 ≤ ((((2 · 𝑁) · (abs‘𝐶))↑𝑗) · (1 − (1 / (((2 · 𝑁) · (abs‘𝐶)) − 1))))))) → 𝐷 ∈ ℝ⁺)
14	4	adantr 480	. . 3 ⊢ ((𝜑 ∧ (𝑗 ∈ ℕ₀ ∧ ((((2 · 𝑁)↑-𝑗) / 2) < 𝐷 ∧ 𝐸 ≤ ((((2 · 𝑁) · (abs‘𝐶))↑𝑗) · (1 − (1 / (((2 · 𝑁) · (abs‘𝐶)) − 1))))))) → 𝐸 ∈ ℝ⁺)
15		knoppndvlem22.h	. . . 4 ⊢ (𝜑 → 𝐻 ∈ ℝ)
16	15	adantr 480	. . 3 ⊢ ((𝜑 ∧ (𝑗 ∈ ℕ₀ ∧ ((((2 · 𝑁)↑-𝑗) / 2) < 𝐷 ∧ 𝐸 ≤ ((((2 · 𝑁) · (abs‘𝐶))↑𝑗) · (1 − (1 / (((2 · 𝑁) · (abs‘𝐶)) − 1))))))) → 𝐻 ∈ ℝ)
17		simprl 770	. . 3 ⊢ ((𝜑 ∧ (𝑗 ∈ ℕ₀ ∧ ((((2 · 𝑁)↑-𝑗) / 2) < 𝐷 ∧ 𝐸 ≤ ((((2 · 𝑁) · (abs‘𝐶))↑𝑗) · (1 − (1 / (((2 · 𝑁) · (abs‘𝐶)) − 1))))))) → 𝑗 ∈ ℕ₀)
18	2	adantr 480	. . 3 ⊢ ((𝜑 ∧ (𝑗 ∈ ℕ₀ ∧ ((((2 · 𝑁)↑-𝑗) / 2) < 𝐷 ∧ 𝐸 ≤ ((((2 · 𝑁) · (abs‘𝐶))↑𝑗) · (1 − (1 / (((2 · 𝑁) · (abs‘𝐶)) − 1))))))) → 𝑁 ∈ ℕ)
19	5	adantr 480	. . 3 ⊢ ((𝜑 ∧ (𝑗 ∈ ℕ₀ ∧ ((((2 · 𝑁)↑-𝑗) / 2) < 𝐷 ∧ 𝐸 ≤ ((((2 · 𝑁) · (abs‘𝐶))↑𝑗) · (1 − (1 / (((2 · 𝑁) · (abs‘𝐶)) − 1))))))) → 1 < (𝑁 · (abs‘𝐶)))
20		simprrl 780	. . 3 ⊢ ((𝜑 ∧ (𝑗 ∈ ℕ₀ ∧ ((((2 · 𝑁)↑-𝑗) / 2) < 𝐷 ∧ 𝐸 ≤ ((((2 · 𝑁) · (abs‘𝐶))↑𝑗) · (1 − (1 / (((2 · 𝑁) · (abs‘𝐶)) − 1))))))) → (((2 · 𝑁)↑-𝑗) / 2) < 𝐷)
21		simprrr 781	. . 3 ⊢ ((𝜑 ∧ (𝑗 ∈ ℕ₀ ∧ ((((2 · 𝑁)↑-𝑗) / 2) < 𝐷 ∧ 𝐸 ≤ ((((2 · 𝑁) · (abs‘𝐶))↑𝑗) · (1 − (1 / (((2 · 𝑁) · (abs‘𝐶)) − 1))))))) → 𝐸 ≤ ((((2 · 𝑁) · (abs‘𝐶))↑𝑗) · (1 − (1 / (((2 · 𝑁) · (abs‘𝐶)) − 1)))))
22	8, 9, 10, 11, 12, 13, 14, 16, 17, 18, 19, 20, 21	knoppndvlem21 36506	. 2 ⊢ ((𝜑 ∧ (𝑗 ∈ ℕ₀ ∧ ((((2 · 𝑁)↑-𝑗) / 2) < 𝐷 ∧ 𝐸 ≤ ((((2 · 𝑁) · (abs‘𝐶))↑𝑗) · (1 − (1 / (((2 · 𝑁) · (abs‘𝐶)) − 1))))))) → ∃𝑎 ∈ ℝ ∃𝑏 ∈ ℝ ((𝑎 ≤ 𝐻 ∧ 𝐻 ≤ 𝑏) ∧ ((𝑏 − 𝑎) < 𝐷 ∧ 𝑎 ≠ 𝑏) ∧ 𝐸 ≤ ((abs‘((𝑊‘𝑏) − (𝑊‘𝑎))) / (𝑏 − 𝑎))))
23	7, 22	rexlimddv 3136	1 ⊢ (𝜑 → ∃𝑎 ∈ ℝ ∃𝑏 ∈ ℝ ((𝑎 ≤ 𝐻 ∧ 𝐻 ≤ 𝑏) ∧ ((𝑏 − 𝑎) < 𝐷 ∧ 𝑎 ≠ 𝑏) ∧ 𝐸 ≤ ((abs‘((𝑊‘𝑏) − (𝑊‘𝑎))) / (𝑏 − 𝑎))))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 ∧ w3a 1086 = wceq 1540 ∈ wcel 2109 ≠ wne 2925 ∃wrex 3053 class class class wbr 5092 ↦ cmpt 5173 ‘cfv 6482 (class class class)co 7349 ℝcr 11008 1c1 11010 + caddc 11012 · cmul 11014 < clt 11149 ≤ cle 11150 − cmin 11347 -cneg 11348 / cdiv 11777 ℕcn 12128 2c2 12183 ℕ₀cn0 12384 ℝ⁺crp 12893 (,)cioo 13248 ⌊cfl 13694 ↑cexp 13968 abscabs 15141 Σcsu 15593

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2701 ax-rep 5218 ax-sep 5235 ax-nul 5245 ax-pow 5304 ax-pr 5371 ax-un 7671 ax-inf2 9537 ax-cnex 11065 ax-resscn 11066 ax-1cn 11067 ax-icn 11068 ax-addcl 11069 ax-addrcl 11070 ax-mulcl 11071 ax-mulrcl 11072 ax-mulcom 11073 ax-addass 11074 ax-mulass 11075 ax-distr 11076 ax-i2m1 11077 ax-1ne0 11078 ax-1rid 11079 ax-rnegex 11080 ax-rrecex 11081 ax-cnre 11082 ax-pre-lttri 11083 ax-pre-lttrn 11084 ax-pre-ltadd 11085 ax-pre-mulgt0 11086 ax-pre-sup 11087 ax-addf 11088

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2533 df-eu 2562 df-clab 2708 df-cleq 2721 df-clel 2803 df-nfc 2878 df-ne 2926 df-nel 3030 df-ral 3045 df-rex 3054 df-rmo 3343 df-reu 3344 df-rab 3395 df-v 3438 df-sbc 3743 df-csb 3852 df-dif 3906 df-un 3908 df-in 3910 df-ss 3920 df-pss 3923 df-nul 4285 df-if 4477 df-pw 4553 df-sn 4578 df-pr 4580 df-tp 4582 df-op 4584 df-uni 4859 df-int 4897 df-iun 4943 df-iin 4944 df-br 5093 df-opab 5155 df-mpt 5174 df-tr 5200 df-id 5514 df-eprel 5519 df-po 5527 df-so 5528 df-fr 5572 df-se 5573 df-we 5574 df-xp 5625 df-rel 5626 df-cnv 5627 df-co 5628 df-dm 5629 df-rn 5630 df-res 5631 df-ima 5632 df-pred 6249 df-ord 6310 df-on 6311 df-lim 6312 df-suc 6313 df-iota 6438 df-fun 6484 df-fn 6485 df-f 6486 df-f1 6487 df-fo 6488 df-f1o 6489 df-fv 6490 df-isom 6491 df-riota 7306 df-ov 7352 df-oprab 7353 df-mpo 7354 df-of 7613 df-om 7800 df-1st 7924 df-2nd 7925 df-supp 8094 df-frecs 8214 df-wrecs 8245 df-recs 8294 df-rdg 8332 df-1o 8388 df-2o 8389 df-er 8625 df-map 8755 df-pm 8756 df-ixp 8825 df-en 8873 df-dom 8874 df-sdom 8875 df-fin 8876 df-fsupp 9252 df-fi 9301 df-sup 9332 df-inf 9333 df-oi 9402 df-card 9835 df-pnf 11151 df-mnf 11152 df-xr 11153 df-ltxr 11154 df-le 11155 df-sub 11349 df-neg 11350 df-div 11778 df-nn 12129 df-2 12191 df-3 12192 df-4 12193 df-5 12194 df-6 12195 df-7 12196 df-8 12197 df-9 12198 df-n0 12385 df-z 12472 df-dec 12592 df-uz 12736 df-q 12850 df-rp 12894 df-xneg 13014 df-xadd 13015 df-xmul 13016 df-ioo 13252 df-ico 13254 df-icc 13255 df-fz 13411 df-fzo 13558 df-fl 13696 df-seq 13909 df-exp 13969 df-hash 14238 df-cj 15006 df-re 15007 df-im 15008 df-sqrt 15142 df-abs 15143 df-limsup 15378 df-clim 15395 df-rlim 15396 df-sum 15594 df-dvds 16164 df-struct 17058 df-sets 17075 df-slot 17093 df-ndx 17105 df-base 17121 df-ress 17142 df-plusg 17174 df-mulr 17175 df-starv 17176 df-sca 17177 df-vsca 17178 df-ip 17179 df-tset 17180 df-ple 17181 df-ds 17183 df-unif 17184 df-hom 17185 df-cco 17186 df-rest 17326 df-topn 17327 df-0g 17345 df-gsum 17346 df-topgen 17347 df-pt 17348 df-prds 17351 df-xrs 17406 df-qtop 17411 df-imas 17412 df-xps 17414 df-mre 17488 df-mrc 17489 df-acs 17491 df-mgm 18514 df-sgrp 18593 df-mnd 18609 df-submnd 18658 df-mulg 18947 df-cntz 19196 df-cmn 19661 df-psmet 21253 df-xmet 21254 df-met 21255 df-bl 21256 df-mopn 21257 df-cnfld 21262 df-top 22779 df-topon 22796 df-topsp 22818 df-bases 22831 df-cn 23112 df-cnp 23113 df-tx 23447 df-hmeo 23640 df-xms 24206 df-ms 24207 df-tms 24208 df-cncf 24769 df-ulm 26284

This theorem is referenced by: knoppndv 36508

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