knoppndvlem22 - Mathbox for Asger C. Ipsen

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Theorem knoppndvlem22 33332

Description: Lemma for knoppndv 33333. (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 33330	. . 3 ⊢ (𝜑 → (1 − (1 / (((2 · 𝑁) · (abs‘𝐶)) − 1))) ∈ ℝ⁺)
7	1, 2, 3, 4, 6, 5	knoppndvlem18 33328	. 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 2772	. . 3 ⊢ (1 − (1 / (((2 · 𝑁) · (abs‘𝐶)) − 1))) = (1 − (1 / (((2 · 𝑁) · (abs‘𝐶)) − 1)))
12	1	adantr 473	. . 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 473	. . 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 473	. . 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 473	. . 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 758	. . 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 473	. . 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 473	. . 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 768	. . 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 769	. . 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 33331	. 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 3230	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 387 ∧ w3a 1068 = wceq 1507 ∈ wcel 2048 ≠ wne 2961 ∃wrex 3083 class class class wbr 4923 ↦ cmpt 5002 ‘cfv 6182 (class class class)co 6970 ℝcr 10326 1c1 10328 + caddc 10330 · cmul 10332 < clt 10466 ≤ cle 10467 − cmin 10662 -cneg 10663 / cdiv 11090 ℕcn 11431 2c2 11488 ℕ₀cn0 11700 ℝ⁺crp 12197 (,)cioo 12547 ⌊cfl 12968 ↑cexp 13237 abscabs 14444 Σcsu 14893

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1758 ax-4 1772 ax-5 1869 ax-6 1928 ax-7 1964 ax-8 2050 ax-9 2057 ax-10 2077 ax-11 2091 ax-12 2104 ax-13 2299 ax-ext 2745 ax-rep 5043 ax-sep 5054 ax-nul 5061 ax-pow 5113 ax-pr 5180 ax-un 7273 ax-inf2 8890 ax-cnex 10383 ax-resscn 10384 ax-1cn 10385 ax-icn 10386 ax-addcl 10387 ax-addrcl 10388 ax-mulcl 10389 ax-mulrcl 10390 ax-mulcom 10391 ax-addass 10392 ax-mulass 10393 ax-distr 10394 ax-i2m1 10395 ax-1ne0 10396 ax-1rid 10397 ax-rnegex 10398 ax-rrecex 10399 ax-cnre 10400 ax-pre-lttri 10401 ax-pre-lttrn 10402 ax-pre-ltadd 10403 ax-pre-mulgt0 10404 ax-pre-sup 10405 ax-addf 10406 ax-mulf 10407

This theorem depends on definitions: df-bi 199 df-an 388 df-or 834 df-3or 1069 df-3an 1070 df-tru 1510 df-fal 1520 df-ex 1743 df-nf 1747 df-sb 2014 df-mo 2544 df-eu 2580 df-clab 2754 df-cleq 2765 df-clel 2840 df-nfc 2912 df-ne 2962 df-nel 3068 df-ral 3087 df-rex 3088 df-reu 3089 df-rmo 3090 df-rab 3091 df-v 3411 df-sbc 3678 df-csb 3783 df-dif 3828 df-un 3830 df-in 3832 df-ss 3839 df-pss 3841 df-nul 4174 df-if 4345 df-pw 4418 df-sn 4436 df-pr 4438 df-tp 4440 df-op 4442 df-uni 4707 df-int 4744 df-iun 4788 df-iin 4789 df-br 4924 df-opab 4986 df-mpt 5003 df-tr 5025 df-id 5305 df-eprel 5310 df-po 5319 df-so 5320 df-fr 5359 df-se 5360 df-we 5361 df-xp 5406 df-rel 5407 df-cnv 5408 df-co 5409 df-dm 5410 df-rn 5411 df-res 5412 df-ima 5413 df-pred 5980 df-ord 6026 df-on 6027 df-lim 6028 df-suc 6029 df-iota 6146 df-fun 6184 df-fn 6185 df-f 6186 df-f1 6187 df-fo 6188 df-f1o 6189 df-fv 6190 df-isom 6191 df-riota 6931 df-ov 6973 df-oprab 6974 df-mpo 6975 df-of 7221 df-om 7391 df-1st 7494 df-2nd 7495 df-supp 7627 df-wrecs 7743 df-recs 7805 df-rdg 7843 df-1o 7897 df-2o 7898 df-oadd 7901 df-er 8081 df-map 8200 df-pm 8201 df-ixp 8252 df-en 8299 df-dom 8300 df-sdom 8301 df-fin 8302 df-fsupp 8621 df-fi 8662 df-sup 8693 df-inf 8694 df-oi 8761 df-card 9154 df-cda 9380 df-pnf 10468 df-mnf 10469 df-xr 10470 df-ltxr 10471 df-le 10472 df-sub 10664 df-neg 10665 df-div 11091 df-nn 11432 df-2 11496 df-3 11497 df-4 11498 df-5 11499 df-6 11500 df-7 11501 df-8 11502 df-9 11503 df-n0 11701 df-z 11787 df-dec 11905 df-uz 12052 df-q 12156 df-rp 12198 df-xneg 12317 df-xadd 12318 df-xmul 12319 df-ioo 12551 df-ico 12553 df-icc 12554 df-fz 12702 df-fzo 12843 df-fl 12970 df-seq 13178 df-exp 13238 df-hash 13499 df-cj 14309 df-re 14310 df-im 14311 df-sqrt 14445 df-abs 14446 df-limsup 14679 df-clim 14696 df-rlim 14697 df-sum 14894 df-dvds 15458 df-struct 16331 df-ndx 16332 df-slot 16333 df-base 16335 df-sets 16336 df-ress 16337 df-plusg 16424 df-mulr 16425 df-starv 16426 df-sca 16427 df-vsca 16428 df-ip 16429 df-tset 16430 df-ple 16431 df-ds 16433 df-unif 16434 df-hom 16435 df-cco 16436 df-rest 16542 df-topn 16543 df-0g 16561 df-gsum 16562 df-topgen 16563 df-pt 16564 df-prds 16567 df-xrs 16621 df-qtop 16626 df-imas 16627 df-xps 16629 df-mre 16705 df-mrc 16706 df-acs 16708 df-mgm 17700 df-sgrp 17742 df-mnd 17753 df-submnd 17794 df-mulg 18002 df-cntz 18208 df-cmn 18658 df-psmet 20229 df-xmet 20230 df-met 20231 df-bl 20232 df-mopn 20233 df-cnfld 20238 df-top 21196 df-topon 21213 df-topsp 21235 df-bases 21248 df-cn 21529 df-cnp 21530 df-tx 21864 df-hmeo 22057 df-xms 22623 df-ms 22624 df-tms 22625 df-cncf 23179 df-ulm 24658

This theorem is referenced by: knoppndv 33333

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