knoppndvlem10 - Mathbox for Asger C. Ipsen

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Theorem knoppndvlem10 34628

Description: Lemma for knoppndv 34641. (Contributed by Asger C. Ipsen, 15-Jun-2021.) (Revised by Asger C. Ipsen, 5-Jul-2021.)

**Hypotheses**
Ref	Expression
knoppndvlem10.t	⊢ 𝑇 = (𝑥 ∈ ℝ ↦ (abs‘((⌊‘(𝑥 + (1 / 2))) − 𝑥)))
knoppndvlem10.f	⊢ 𝐹 = (𝑦 ∈ ℝ ↦ (𝑛 ∈ ℕ₀ ↦ ((𝐶↑𝑛) · (𝑇‘(((2 · 𝑁)↑𝑛) · 𝑦)))))
knoppndvlem10.a	⊢ 𝐴 = ((((2 · 𝑁)↑-𝐽) / 2) · 𝑀)
knoppndvlem10.b	⊢ 𝐵 = ((((2 · 𝑁)↑-𝐽) / 2) · (𝑀 + 1))
knoppndvlem10.c	⊢ (𝜑 → 𝐶 ∈ (-1(,)1))
knoppndvlem10.j	⊢ (𝜑 → 𝐽 ∈ ℕ₀)
knoppndvlem10.m	⊢ (𝜑 → 𝑀 ∈ ℤ)
knoppndvlem10.n	⊢ (𝜑 → 𝑁 ∈ ℕ)

**Assertion**
Ref	Expression
knoppndvlem10	⊢ (𝜑 → (abs‘(((𝐹‘𝐵)‘𝐽) − ((𝐹‘𝐴)‘𝐽))) = (((abs‘𝐶)↑𝐽) / 2))

Distinct variable groups: 𝐴,𝑛,𝑦 𝑥,𝐴 𝐵,𝑛,𝑦 𝑥,𝐵 𝐶,𝑛,𝑦 𝑛,𝐽 𝑥,𝐽 𝑛,𝑀,𝑦 𝑥,𝑀 𝑛,𝑁,𝑦 𝑥,𝑁 𝑇,𝑛,𝑦 𝜑,𝑛,𝑦 Allowed substitution hints: 𝜑(𝑥) 𝐶(𝑥) 𝑇(𝑥) 𝐹(𝑥,𝑦,𝑛) 𝐽(𝑦)

**Proof of Theorem knoppndvlem10**
Step	Hyp	Ref	Expression
1		knoppndvlem10.t	. . . . . . 7 ⊢ 𝑇 = (𝑥 ∈ ℝ ↦ (abs‘((⌊‘(𝑥 + (1 / 2))) − 𝑥)))
2		knoppndvlem10.f	. . . . . . 7 ⊢ 𝐹 = (𝑦 ∈ ℝ ↦ (𝑛 ∈ ℕ₀ ↦ ((𝐶↑𝑛) · (𝑇‘(((2 · 𝑁)↑𝑛) · 𝑦)))))
3		knoppndvlem10.b	. . . . . . 7 ⊢ 𝐵 = ((((2 · 𝑁)↑-𝐽) / 2) · (𝑀 + 1))
4		knoppndvlem10.c	. . . . . . . 8 ⊢ (𝜑 → 𝐶 ∈ (-1(,)1))
5	4	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ 2 ∥ 𝑀) → 𝐶 ∈ (-1(,)1))
6		knoppndvlem10.j	. . . . . . . 8 ⊢ (𝜑 → 𝐽 ∈ ℕ₀)
7	6	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ 2 ∥ 𝑀) → 𝐽 ∈ ℕ₀)
8		knoppndvlem10.m	. . . . . . . . 9 ⊢ (𝜑 → 𝑀 ∈ ℤ)
9	8	peano2zd 12358	. . . . . . . 8 ⊢ (𝜑 → (𝑀 + 1) ∈ ℤ)
10	9	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ 2 ∥ 𝑀) → (𝑀 + 1) ∈ ℤ)
11		knoppndvlem10.n	. . . . . . . 8 ⊢ (𝜑 → 𝑁 ∈ ℕ)
12	11	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ 2 ∥ 𝑀) → 𝑁 ∈ ℕ)
13		notnot 142	. . . . . . . . 9 ⊢ (2 ∥ 𝑀 → ¬ ¬ 2 ∥ 𝑀)
14	13	adantl 481	. . . . . . . 8 ⊢ ((𝜑 ∧ 2 ∥ 𝑀) → ¬ ¬ 2 ∥ 𝑀)
15	8	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ 2 ∥ 𝑀) → 𝑀 ∈ ℤ)
16		oddp1even 15981	. . . . . . . . 9 ⊢ (𝑀 ∈ ℤ → (¬ 2 ∥ 𝑀 ↔ 2 ∥ (𝑀 + 1)))
17	15, 16	syl 17	. . . . . . . 8 ⊢ ((𝜑 ∧ 2 ∥ 𝑀) → (¬ 2 ∥ 𝑀 ↔ 2 ∥ (𝑀 + 1)))
18	14, 17	mtbid 323	. . . . . . 7 ⊢ ((𝜑 ∧ 2 ∥ 𝑀) → ¬ 2 ∥ (𝑀 + 1))
19	1, 2, 3, 5, 7, 10, 12, 18	knoppndvlem9 34627	. . . . . 6 ⊢ ((𝜑 ∧ 2 ∥ 𝑀) → ((𝐹‘𝐵)‘𝐽) = ((𝐶↑𝐽) / 2))
20		knoppndvlem10.a	. . . . . . 7 ⊢ 𝐴 = ((((2 · 𝑁)↑-𝐽) / 2) · 𝑀)
21	14	notnotrd 133	. . . . . . 7 ⊢ ((𝜑 ∧ 2 ∥ 𝑀) → 2 ∥ 𝑀)
22	1, 2, 20, 5, 7, 15, 12, 21	knoppndvlem8 34626	. . . . . 6 ⊢ ((𝜑 ∧ 2 ∥ 𝑀) → ((𝐹‘𝐴)‘𝐽) = 0)
23	19, 22	oveq12d 7273	. . . . 5 ⊢ ((𝜑 ∧ 2 ∥ 𝑀) → (((𝐹‘𝐵)‘𝐽) − ((𝐹‘𝐴)‘𝐽)) = (((𝐶↑𝐽) / 2) − 0))
24	4	knoppndvlem3 34621	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐶 ∈ ℝ ∧ (abs‘𝐶) < 1))
25	24	simpld 494	. . . . . . . . . 10 ⊢ (𝜑 → 𝐶 ∈ ℝ)
26	25	recnd 10934	. . . . . . . . 9 ⊢ (𝜑 → 𝐶 ∈ ℂ)
27	26, 6	expcld 13792	. . . . . . . 8 ⊢ (𝜑 → (𝐶↑𝐽) ∈ ℂ)
28		2cnd 11981	. . . . . . . 8 ⊢ (𝜑 → 2 ∈ ℂ)
29		2ne0 12007	. . . . . . . . 9 ⊢ 2 ≠ 0
30	29	a1i 11	. . . . . . . 8 ⊢ (𝜑 → 2 ≠ 0)
31	27, 28, 30	divcld 11681	. . . . . . 7 ⊢ (𝜑 → ((𝐶↑𝐽) / 2) ∈ ℂ)
32	31	subid1d 11251	. . . . . 6 ⊢ (𝜑 → (((𝐶↑𝐽) / 2) − 0) = ((𝐶↑𝐽) / 2))
33	32	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ 2 ∥ 𝑀) → (((𝐶↑𝐽) / 2) − 0) = ((𝐶↑𝐽) / 2))
34	23, 33	eqtrd 2778	. . . 4 ⊢ ((𝜑 ∧ 2 ∥ 𝑀) → (((𝐹‘𝐵)‘𝐽) − ((𝐹‘𝐴)‘𝐽)) = ((𝐶↑𝐽) / 2))
35	34	fveq2d 6760	. . 3 ⊢ ((𝜑 ∧ 2 ∥ 𝑀) → (abs‘(((𝐹‘𝐵)‘𝐽) − ((𝐹‘𝐴)‘𝐽))) = (abs‘((𝐶↑𝐽) / 2)))
36	3	a1i 11	. . . . . . . . 9 ⊢ (𝜑 → 𝐵 = ((((2 · 𝑁)↑-𝐽) / 2) · (𝑀 + 1)))
37	6	nn0zd 12353	. . . . . . . . . 10 ⊢ (𝜑 → 𝐽 ∈ ℤ)
38	11, 37, 9	knoppndvlem1 34619	. . . . . . . . 9 ⊢ (𝜑 → ((((2 · 𝑁)↑-𝐽) / 2) · (𝑀 + 1)) ∈ ℝ)
39	36, 38	eqeltrd 2839	. . . . . . . 8 ⊢ (𝜑 → 𝐵 ∈ ℝ)
40	1, 2, 11, 25, 39, 6	knoppcnlem3 34602	. . . . . . 7 ⊢ (𝜑 → ((𝐹‘𝐵)‘𝐽) ∈ ℝ)
41	40	recnd 10934	. . . . . 6 ⊢ (𝜑 → ((𝐹‘𝐵)‘𝐽) ∈ ℂ)
42	20	a1i 11	. . . . . . . . 9 ⊢ (𝜑 → 𝐴 = ((((2 · 𝑁)↑-𝐽) / 2) · 𝑀))
43	11, 37, 8	knoppndvlem1 34619	. . . . . . . . 9 ⊢ (𝜑 → ((((2 · 𝑁)↑-𝐽) / 2) · 𝑀) ∈ ℝ)
44	42, 43	eqeltrd 2839	. . . . . . . 8 ⊢ (𝜑 → 𝐴 ∈ ℝ)
45	1, 2, 11, 25, 44, 6	knoppcnlem3 34602	. . . . . . 7 ⊢ (𝜑 → ((𝐹‘𝐴)‘𝐽) ∈ ℝ)
46	45	recnd 10934	. . . . . 6 ⊢ (𝜑 → ((𝐹‘𝐴)‘𝐽) ∈ ℂ)
47	41, 46	abssubd 15093	. . . . 5 ⊢ (𝜑 → (abs‘(((𝐹‘𝐵)‘𝐽) − ((𝐹‘𝐴)‘𝐽))) = (abs‘(((𝐹‘𝐴)‘𝐽) − ((𝐹‘𝐵)‘𝐽))))
48	47	adantr 480	. . . 4 ⊢ ((𝜑 ∧ ¬ 2 ∥ 𝑀) → (abs‘(((𝐹‘𝐵)‘𝐽) − ((𝐹‘𝐴)‘𝐽))) = (abs‘(((𝐹‘𝐴)‘𝐽) − ((𝐹‘𝐵)‘𝐽))))
49	4	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ ¬ 2 ∥ 𝑀) → 𝐶 ∈ (-1(,)1))
50	6	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ ¬ 2 ∥ 𝑀) → 𝐽 ∈ ℕ₀)
51	8	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ ¬ 2 ∥ 𝑀) → 𝑀 ∈ ℤ)
52	11	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ ¬ 2 ∥ 𝑀) → 𝑁 ∈ ℕ)
53		simpr 484	. . . . . . . 8 ⊢ ((𝜑 ∧ ¬ 2 ∥ 𝑀) → ¬ 2 ∥ 𝑀)
54	1, 2, 20, 49, 50, 51, 52, 53	knoppndvlem9 34627	. . . . . . 7 ⊢ ((𝜑 ∧ ¬ 2 ∥ 𝑀) → ((𝐹‘𝐴)‘𝐽) = ((𝐶↑𝐽) / 2))
55	9	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ ¬ 2 ∥ 𝑀) → (𝑀 + 1) ∈ ℤ)
56	51, 16	syl 17	. . . . . . . . 9 ⊢ ((𝜑 ∧ ¬ 2 ∥ 𝑀) → (¬ 2 ∥ 𝑀 ↔ 2 ∥ (𝑀 + 1)))
57	53, 56	mpbid 231	. . . . . . . 8 ⊢ ((𝜑 ∧ ¬ 2 ∥ 𝑀) → 2 ∥ (𝑀 + 1))
58	1, 2, 3, 49, 50, 55, 52, 57	knoppndvlem8 34626	. . . . . . 7 ⊢ ((𝜑 ∧ ¬ 2 ∥ 𝑀) → ((𝐹‘𝐵)‘𝐽) = 0)
59	54, 58	oveq12d 7273	. . . . . 6 ⊢ ((𝜑 ∧ ¬ 2 ∥ 𝑀) → (((𝐹‘𝐴)‘𝐽) − ((𝐹‘𝐵)‘𝐽)) = (((𝐶↑𝐽) / 2) − 0))
60	32	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ ¬ 2 ∥ 𝑀) → (((𝐶↑𝐽) / 2) − 0) = ((𝐶↑𝐽) / 2))
61	59, 60	eqtrd 2778	. . . . 5 ⊢ ((𝜑 ∧ ¬ 2 ∥ 𝑀) → (((𝐹‘𝐴)‘𝐽) − ((𝐹‘𝐵)‘𝐽)) = ((𝐶↑𝐽) / 2))
62	61	fveq2d 6760	. . . 4 ⊢ ((𝜑 ∧ ¬ 2 ∥ 𝑀) → (abs‘(((𝐹‘𝐴)‘𝐽) − ((𝐹‘𝐵)‘𝐽))) = (abs‘((𝐶↑𝐽) / 2)))
63	48, 62	eqtrd 2778	. . 3 ⊢ ((𝜑 ∧ ¬ 2 ∥ 𝑀) → (abs‘(((𝐹‘𝐵)‘𝐽) − ((𝐹‘𝐴)‘𝐽))) = (abs‘((𝐶↑𝐽) / 2)))
64	35, 63	pm2.61dan 809	. 2 ⊢ (𝜑 → (abs‘(((𝐹‘𝐵)‘𝐽) − ((𝐹‘𝐴)‘𝐽))) = (abs‘((𝐶↑𝐽) / 2)))
65	27, 28, 30	absdivd 15095	. . 3 ⊢ (𝜑 → (abs‘((𝐶↑𝐽) / 2)) = ((abs‘(𝐶↑𝐽)) / (abs‘2)))
66	26, 6	absexpd 15092	. . . 4 ⊢ (𝜑 → (abs‘(𝐶↑𝐽)) = ((abs‘𝐶)↑𝐽))
67		0le2 12005	. . . . . 6 ⊢ 0 ≤ 2
68		2re 11977	. . . . . . 7 ⊢ 2 ∈ ℝ
69	68	absidi 15017	. . . . . 6 ⊢ (0 ≤ 2 → (abs‘2) = 2)
70	67, 69	ax-mp 5	. . . . 5 ⊢ (abs‘2) = 2
71	70	a1i 11	. . . 4 ⊢ (𝜑 → (abs‘2) = 2)
72	66, 71	oveq12d 7273	. . 3 ⊢ (𝜑 → ((abs‘(𝐶↑𝐽)) / (abs‘2)) = (((abs‘𝐶)↑𝐽) / 2))
73	65, 72	eqtrd 2778	. 2 ⊢ (𝜑 → (abs‘((𝐶↑𝐽) / 2)) = (((abs‘𝐶)↑𝐽) / 2))
74	64, 73	eqtrd 2778	1 ⊢ (𝜑 → (abs‘(((𝐹‘𝐵)‘𝐽) − ((𝐹‘𝐴)‘𝐽))) = (((abs‘𝐶)↑𝐽) / 2))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 395 = wceq 1539 ∈ wcel 2108 ≠ wne 2942 class class class wbr 5070 ↦ cmpt 5153 ‘cfv 6418 (class class class)co 7255 ℝcr 10801 0cc0 10802 1c1 10803 + caddc 10805 · cmul 10807 < clt 10940 ≤ cle 10941 − cmin 11135 -cneg 11136 / cdiv 11562 ℕcn 11903 2c2 11958 ℕ₀cn0 12163 ℤcz 12249 (,)cioo 13008 ⌊cfl 13438 ↑cexp 13710 abscabs 14873 ∥ cdvds 15891

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1799 ax-4 1813 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2110 ax-9 2118 ax-10 2139 ax-11 2156 ax-12 2173 ax-ext 2709 ax-rep 5205 ax-sep 5218 ax-nul 5225 ax-pow 5283 ax-pr 5347 ax-un 7566 ax-cnex 10858 ax-resscn 10859 ax-1cn 10860 ax-icn 10861 ax-addcl 10862 ax-addrcl 10863 ax-mulcl 10864 ax-mulrcl 10865 ax-mulcom 10866 ax-addass 10867 ax-mulass 10868 ax-distr 10869 ax-i2m1 10870 ax-1ne0 10871 ax-1rid 10872 ax-rnegex 10873 ax-rrecex 10874 ax-cnre 10875 ax-pre-lttri 10876 ax-pre-lttrn 10877 ax-pre-ltadd 10878 ax-pre-mulgt0 10879 ax-pre-sup 10880

This theorem depends on definitions: df-bi 206 df-an 396 df-or 844 df-3or 1086 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1784 df-nf 1788 df-sb 2069 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2817 df-nfc 2888 df-ne 2943 df-nel 3049 df-ral 3068 df-rex 3069 df-reu 3070 df-rmo 3071 df-rab 3072 df-v 3424 df-sbc 3712 df-csb 3829 df-dif 3886 df-un 3888 df-in 3890 df-ss 3900 df-pss 3902 df-nul 4254 df-if 4457 df-pw 4532 df-sn 4559 df-pr 4561 df-tp 4563 df-op 4565 df-uni 4837 df-iun 4923 df-br 5071 df-opab 5133 df-mpt 5154 df-tr 5188 df-id 5480 df-eprel 5486 df-po 5494 df-so 5495 df-fr 5535 df-we 5537 df-xp 5586 df-rel 5587 df-cnv 5588 df-co 5589 df-dm 5590 df-rn 5591 df-res 5592 df-ima 5593 df-pred 6191 df-ord 6254 df-on 6255 df-lim 6256 df-suc 6257 df-iota 6376 df-fun 6420 df-fn 6421 df-f 6422 df-f1 6423 df-fo 6424 df-f1o 6425 df-fv 6426 df-riota 7212 df-ov 7258 df-oprab 7259 df-mpo 7260 df-om 7688 df-1st 7804 df-2nd 7805 df-frecs 8068 df-wrecs 8099 df-recs 8173 df-rdg 8212 df-er 8456 df-en 8692 df-dom 8693 df-sdom 8694 df-sup 9131 df-inf 9132 df-pnf 10942 df-mnf 10943 df-xr 10944 df-ltxr 10945 df-le 10946 df-sub 11137 df-neg 11138 df-div 11563 df-nn 11904 df-2 11966 df-3 11967 df-n0 12164 df-z 12250 df-uz 12512 df-rp 12660 df-ioo 13012 df-ico 13014 df-fl 13440 df-seq 13650 df-exp 13711 df-cj 14738 df-re 14739 df-im 14740 df-sqrt 14874 df-abs 14875 df-dvds 15892

This theorem is referenced by: knoppndvlem15 34633

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