knoppndvlem22 - Mathbox for Asger C. Ipsen

	Mathbox for Asger C. Ipsen		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > Mathboxes > knoppndvlem22		Structured version Visualization version GIF version

Theorem knoppndvlem22 34762

Description: Lemma for knoppndv 34763. (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 34760	. . 3 ⊢ (𝜑 → (1 − (1 / (((2 · 𝑁) · (abs‘𝐶)) − 1))) ∈ ℝ⁺)
7	1, 2, 3, 4, 6, 5	knoppndvlem18 34758	. 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 2736	. . 3 ⊢ (1 − (1 / (((2 · 𝑁) · (abs‘𝐶)) − 1))) = (1 − (1 / (((2 · 𝑁) · (abs‘𝐶)) − 1)))
12	1	adantr 482	. . 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 482	. . 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 482	. . 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 482	. . 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 769	. . 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 482	. . 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 482	. . 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 779	. . 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 780	. . 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 34761	. 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 3155	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 397 ∧ w3a 1087 = wceq 1539 ∈ wcel 2104 ≠ wne 2941 ∃wrex 3071 class class class wbr 5081 ↦ cmpt 5164 ‘cfv 6458 (class class class)co 7307 ℝcr 10920 1c1 10922 + caddc 10924 · cmul 10926 < clt 11059 ≤ cle 11060 − cmin 11255 -cneg 11256 / cdiv 11682 ℕcn 12023 2c2 12078 ℕ₀cn0 12283 ℝ⁺crp 12780 (,)cioo 13129 ⌊cfl 13560 ↑cexp 13832 abscabs 14994 Σcsu 15446

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 1911 ax-6 1969 ax-7 2009 ax-8 2106 ax-9 2114 ax-10 2135 ax-11 2152 ax-12 2169 ax-ext 2707 ax-rep 5218 ax-sep 5232 ax-nul 5239 ax-pow 5297 ax-pr 5361 ax-un 7620 ax-inf2 9447 ax-cnex 10977 ax-resscn 10978 ax-1cn 10979 ax-icn 10980 ax-addcl 10981 ax-addrcl 10982 ax-mulcl 10983 ax-mulrcl 10984 ax-mulcom 10985 ax-addass 10986 ax-mulass 10987 ax-distr 10988 ax-i2m1 10989 ax-1ne0 10990 ax-1rid 10991 ax-rnegex 10992 ax-rrecex 10993 ax-cnre 10994 ax-pre-lttri 10995 ax-pre-lttrn 10996 ax-pre-ltadd 10997 ax-pre-mulgt0 10998 ax-pre-sup 10999 ax-addf 11000 ax-mulf 11001

This theorem depends on definitions: df-bi 206 df-an 398 df-or 846 df-3or 1088 df-3an 1089 df-tru 1542 df-fal 1552 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2887 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-rmo 3304 df-reu 3305 df-rab 3306 df-v 3439 df-sbc 3722 df-csb 3838 df-dif 3895 df-un 3897 df-in 3899 df-ss 3909 df-pss 3911 df-nul 4263 df-if 4466 df-pw 4541 df-sn 4566 df-pr 4568 df-tp 4570 df-op 4572 df-uni 4845 df-int 4887 df-iun 4933 df-iin 4934 df-br 5082 df-opab 5144 df-mpt 5165 df-tr 5199 df-id 5500 df-eprel 5506 df-po 5514 df-so 5515 df-fr 5555 df-se 5556 df-we 5557 df-xp 5606 df-rel 5607 df-cnv 5608 df-co 5609 df-dm 5610 df-rn 5611 df-res 5612 df-ima 5613 df-pred 6217 df-ord 6284 df-on 6285 df-lim 6286 df-suc 6287 df-iota 6410 df-fun 6460 df-fn 6461 df-f 6462 df-f1 6463 df-fo 6464 df-f1o 6465 df-fv 6466 df-isom 6467 df-riota 7264 df-ov 7310 df-oprab 7311 df-mpo 7312 df-of 7565 df-om 7745 df-1st 7863 df-2nd 7864 df-supp 8009 df-frecs 8128 df-wrecs 8159 df-recs 8233 df-rdg 8272 df-1o 8328 df-2o 8329 df-er 8529 df-map 8648 df-pm 8649 df-ixp 8717 df-en 8765 df-dom 8766 df-sdom 8767 df-fin 8768 df-fsupp 9177 df-fi 9218 df-sup 9249 df-inf 9250 df-oi 9317 df-card 9745 df-pnf 11061 df-mnf 11062 df-xr 11063 df-ltxr 11064 df-le 11065 df-sub 11257 df-neg 11258 df-div 11683 df-nn 12024 df-2 12086 df-3 12087 df-4 12088 df-5 12089 df-6 12090 df-7 12091 df-8 12092 df-9 12093 df-n0 12284 df-z 12370 df-dec 12488 df-uz 12633 df-q 12739 df-rp 12781 df-xneg 12898 df-xadd 12899 df-xmul 12900 df-ioo 13133 df-ico 13135 df-icc 13136 df-fz 13290 df-fzo 13433 df-fl 13562 df-seq 13772 df-exp 13833 df-hash 14095 df-cj 14859 df-re 14860 df-im 14861 df-sqrt 14995 df-abs 14996 df-limsup 15229 df-clim 15246 df-rlim 15247 df-sum 15447 df-dvds 16013 df-struct 16897 df-sets 16914 df-slot 16932 df-ndx 16944 df-base 16962 df-ress 16991 df-plusg 17024 df-mulr 17025 df-starv 17026 df-sca 17027 df-vsca 17028 df-ip 17029 df-tset 17030 df-ple 17031 df-ds 17033 df-unif 17034 df-hom 17035 df-cco 17036 df-rest 17182 df-topn 17183 df-0g 17201 df-gsum 17202 df-topgen 17203 df-pt 17204 df-prds 17207 df-xrs 17262 df-qtop 17267 df-imas 17268 df-xps 17270 df-mre 17344 df-mrc 17345 df-acs 17347 df-mgm 18375 df-sgrp 18424 df-mnd 18435 df-submnd 18480 df-mulg 18750 df-cntz 18972 df-cmn 19437 df-psmet 20638 df-xmet 20639 df-met 20640 df-bl 20641 df-mopn 20642 df-cnfld 20647 df-top 22092 df-topon 22109 df-topsp 22131 df-bases 22145 df-cn 22427 df-cnp 22428 df-tx 22762 df-hmeo 22955 df-xms 23522 df-ms 23523 df-tms 23524 df-cncf 24090 df-ulm 25585

This theorem is referenced by: knoppndv 34763

W3C validator