knoppndvlem10 - Mathbox for Asger C. Ipsen

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Theorem knoppndvlem10 35397

Description: Lemma for knoppndv 35410. (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 482	. . . . . . 7 ⊢ ((𝜑 ∧ 2 ∥ 𝑀) → 𝐶 ∈ (-1(,)1))
6		knoppndvlem10.j	. . . . . . . 8 ⊢ (𝜑 → 𝐽 ∈ ℕ₀)
7	6	adantr 482	. . . . . . 7 ⊢ ((𝜑 ∧ 2 ∥ 𝑀) → 𝐽 ∈ ℕ₀)
8		knoppndvlem10.m	. . . . . . . . 9 ⊢ (𝜑 → 𝑀 ∈ ℤ)
9	8	peano2zd 12669	. . . . . . . 8 ⊢ (𝜑 → (𝑀 + 1) ∈ ℤ)
10	9	adantr 482	. . . . . . 7 ⊢ ((𝜑 ∧ 2 ∥ 𝑀) → (𝑀 + 1) ∈ ℤ)
11		knoppndvlem10.n	. . . . . . . 8 ⊢ (𝜑 → 𝑁 ∈ ℕ)
12	11	adantr 482	. . . . . . 7 ⊢ ((𝜑 ∧ 2 ∥ 𝑀) → 𝑁 ∈ ℕ)
13		notnot 142	. . . . . . . . 9 ⊢ (2 ∥ 𝑀 → ¬ ¬ 2 ∥ 𝑀)
14	13	adantl 483	. . . . . . . 8 ⊢ ((𝜑 ∧ 2 ∥ 𝑀) → ¬ ¬ 2 ∥ 𝑀)
15	8	adantr 482	. . . . . . . . 9 ⊢ ((𝜑 ∧ 2 ∥ 𝑀) → 𝑀 ∈ ℤ)
16		oddp1even 16287	. . . . . . . . 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 324	. . . . . . 7 ⊢ ((𝜑 ∧ 2 ∥ 𝑀) → ¬ 2 ∥ (𝑀 + 1))
19	1, 2, 3, 5, 7, 10, 12, 18	knoppndvlem9 35396	. . . . . 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 35395	. . . . . 6 ⊢ ((𝜑 ∧ 2 ∥ 𝑀) → ((𝐹‘𝐴)‘𝐽) = 0)
23	19, 22	oveq12d 7427	. . . . 5 ⊢ ((𝜑 ∧ 2 ∥ 𝑀) → (((𝐹‘𝐵)‘𝐽) − ((𝐹‘𝐴)‘𝐽)) = (((𝐶↑𝐽) / 2) − 0))
24	4	knoppndvlem3 35390	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐶 ∈ ℝ ∧ (abs‘𝐶) < 1))
25	24	simpld 496	. . . . . . . . . 10 ⊢ (𝜑 → 𝐶 ∈ ℝ)
26	25	recnd 11242	. . . . . . . . 9 ⊢ (𝜑 → 𝐶 ∈ ℂ)
27	26, 6	expcld 14111	. . . . . . . 8 ⊢ (𝜑 → (𝐶↑𝐽) ∈ ℂ)
28		2cnd 12290	. . . . . . . 8 ⊢ (𝜑 → 2 ∈ ℂ)
29		2ne0 12316	. . . . . . . . 9 ⊢ 2 ≠ 0
30	29	a1i 11	. . . . . . . 8 ⊢ (𝜑 → 2 ≠ 0)
31	27, 28, 30	divcld 11990	. . . . . . 7 ⊢ (𝜑 → ((𝐶↑𝐽) / 2) ∈ ℂ)
32	31	subid1d 11560	. . . . . 6 ⊢ (𝜑 → (((𝐶↑𝐽) / 2) − 0) = ((𝐶↑𝐽) / 2))
33	32	adantr 482	. . . . 5 ⊢ ((𝜑 ∧ 2 ∥ 𝑀) → (((𝐶↑𝐽) / 2) − 0) = ((𝐶↑𝐽) / 2))
34	23, 33	eqtrd 2773	. . . 4 ⊢ ((𝜑 ∧ 2 ∥ 𝑀) → (((𝐹‘𝐵)‘𝐽) − ((𝐹‘𝐴)‘𝐽)) = ((𝐶↑𝐽) / 2))
35	34	fveq2d 6896	. . 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 12584	. . . . . . . . . 10 ⊢ (𝜑 → 𝐽 ∈ ℤ)
38	11, 37, 9	knoppndvlem1 35388	. . . . . . . . 9 ⊢ (𝜑 → ((((2 · 𝑁)↑-𝐽) / 2) · (𝑀 + 1)) ∈ ℝ)
39	36, 38	eqeltrd 2834	. . . . . . . 8 ⊢ (𝜑 → 𝐵 ∈ ℝ)
40	1, 2, 11, 25, 39, 6	knoppcnlem3 35371	. . . . . . 7 ⊢ (𝜑 → ((𝐹‘𝐵)‘𝐽) ∈ ℝ)
41	40	recnd 11242	. . . . . 6 ⊢ (𝜑 → ((𝐹‘𝐵)‘𝐽) ∈ ℂ)
42	20	a1i 11	. . . . . . . . 9 ⊢ (𝜑 → 𝐴 = ((((2 · 𝑁)↑-𝐽) / 2) · 𝑀))
43	11, 37, 8	knoppndvlem1 35388	. . . . . . . . 9 ⊢ (𝜑 → ((((2 · 𝑁)↑-𝐽) / 2) · 𝑀) ∈ ℝ)
44	42, 43	eqeltrd 2834	. . . . . . . 8 ⊢ (𝜑 → 𝐴 ∈ ℝ)
45	1, 2, 11, 25, 44, 6	knoppcnlem3 35371	. . . . . . 7 ⊢ (𝜑 → ((𝐹‘𝐴)‘𝐽) ∈ ℝ)
46	45	recnd 11242	. . . . . 6 ⊢ (𝜑 → ((𝐹‘𝐴)‘𝐽) ∈ ℂ)
47	41, 46	abssubd 15400	. . . . 5 ⊢ (𝜑 → (abs‘(((𝐹‘𝐵)‘𝐽) − ((𝐹‘𝐴)‘𝐽))) = (abs‘(((𝐹‘𝐴)‘𝐽) − ((𝐹‘𝐵)‘𝐽))))
48	47	adantr 482	. . . 4 ⊢ ((𝜑 ∧ ¬ 2 ∥ 𝑀) → (abs‘(((𝐹‘𝐵)‘𝐽) − ((𝐹‘𝐴)‘𝐽))) = (abs‘(((𝐹‘𝐴)‘𝐽) − ((𝐹‘𝐵)‘𝐽))))
49	4	adantr 482	. . . . . . . 8 ⊢ ((𝜑 ∧ ¬ 2 ∥ 𝑀) → 𝐶 ∈ (-1(,)1))
50	6	adantr 482	. . . . . . . 8 ⊢ ((𝜑 ∧ ¬ 2 ∥ 𝑀) → 𝐽 ∈ ℕ₀)
51	8	adantr 482	. . . . . . . 8 ⊢ ((𝜑 ∧ ¬ 2 ∥ 𝑀) → 𝑀 ∈ ℤ)
52	11	adantr 482	. . . . . . . 8 ⊢ ((𝜑 ∧ ¬ 2 ∥ 𝑀) → 𝑁 ∈ ℕ)
53		simpr 486	. . . . . . . 8 ⊢ ((𝜑 ∧ ¬ 2 ∥ 𝑀) → ¬ 2 ∥ 𝑀)
54	1, 2, 20, 49, 50, 51, 52, 53	knoppndvlem9 35396	. . . . . . 7 ⊢ ((𝜑 ∧ ¬ 2 ∥ 𝑀) → ((𝐹‘𝐴)‘𝐽) = ((𝐶↑𝐽) / 2))
55	9	adantr 482	. . . . . . . 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 35395	. . . . . . 7 ⊢ ((𝜑 ∧ ¬ 2 ∥ 𝑀) → ((𝐹‘𝐵)‘𝐽) = 0)
59	54, 58	oveq12d 7427	. . . . . 6 ⊢ ((𝜑 ∧ ¬ 2 ∥ 𝑀) → (((𝐹‘𝐴)‘𝐽) − ((𝐹‘𝐵)‘𝐽)) = (((𝐶↑𝐽) / 2) − 0))
60	32	adantr 482	. . . . . 6 ⊢ ((𝜑 ∧ ¬ 2 ∥ 𝑀) → (((𝐶↑𝐽) / 2) − 0) = ((𝐶↑𝐽) / 2))
61	59, 60	eqtrd 2773	. . . . 5 ⊢ ((𝜑 ∧ ¬ 2 ∥ 𝑀) → (((𝐹‘𝐴)‘𝐽) − ((𝐹‘𝐵)‘𝐽)) = ((𝐶↑𝐽) / 2))
62	61	fveq2d 6896	. . . 4 ⊢ ((𝜑 ∧ ¬ 2 ∥ 𝑀) → (abs‘(((𝐹‘𝐴)‘𝐽) − ((𝐹‘𝐵)‘𝐽))) = (abs‘((𝐶↑𝐽) / 2)))
63	48, 62	eqtrd 2773	. . 3 ⊢ ((𝜑 ∧ ¬ 2 ∥ 𝑀) → (abs‘(((𝐹‘𝐵)‘𝐽) − ((𝐹‘𝐴)‘𝐽))) = (abs‘((𝐶↑𝐽) / 2)))
64	35, 63	pm2.61dan 812	. 2 ⊢ (𝜑 → (abs‘(((𝐹‘𝐵)‘𝐽) − ((𝐹‘𝐴)‘𝐽))) = (abs‘((𝐶↑𝐽) / 2)))
65	27, 28, 30	absdivd 15402	. . 3 ⊢ (𝜑 → (abs‘((𝐶↑𝐽) / 2)) = ((abs‘(𝐶↑𝐽)) / (abs‘2)))
66	26, 6	absexpd 15399	. . . 4 ⊢ (𝜑 → (abs‘(𝐶↑𝐽)) = ((abs‘𝐶)↑𝐽))
67		0le2 12314	. . . . . 6 ⊢ 0 ≤ 2
68		2re 12286	. . . . . . 7 ⊢ 2 ∈ ℝ
69	68	absidi 15324	. . . . . 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 7427	. . 3 ⊢ (𝜑 → ((abs‘(𝐶↑𝐽)) / (abs‘2)) = (((abs‘𝐶)↑𝐽) / 2))
73	65, 72	eqtrd 2773	. 2 ⊢ (𝜑 → (abs‘((𝐶↑𝐽) / 2)) = (((abs‘𝐶)↑𝐽) / 2))
74	64, 73	eqtrd 2773	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 397 = wceq 1542 ∈ wcel 2107 ≠ wne 2941 class class class wbr 5149 ↦ cmpt 5232 ‘cfv 6544 (class class class)co 7409 ℝcr 11109 0cc0 11110 1c1 11111 + caddc 11113 · cmul 11115 < clt 11248 ≤ cle 11249 − cmin 11444 -cneg 11445 / cdiv 11871 ℕcn 12212 2c2 12267 ℕ₀cn0 12472 ℤcz 12558 (,)cioo 13324 ⌊cfl 13755 ↑cexp 14027 abscabs 15181 ∥ cdvds 16197

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5286 ax-sep 5300 ax-nul 5307 ax-pow 5364 ax-pr 5428 ax-un 7725 ax-cnex 11166 ax-resscn 11167 ax-1cn 11168 ax-icn 11169 ax-addcl 11170 ax-addrcl 11171 ax-mulcl 11172 ax-mulrcl 11173 ax-mulcom 11174 ax-addass 11175 ax-mulass 11176 ax-distr 11177 ax-i2m1 11178 ax-1ne0 11179 ax-1rid 11180 ax-rnegex 11181 ax-rrecex 11182 ax-cnre 11183 ax-pre-lttri 11184 ax-pre-lttrn 11185 ax-pre-ltadd 11186 ax-pre-mulgt0 11187 ax-pre-sup 11188

This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-rmo 3377 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4910 df-iun 5000 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-id 5575 df-eprel 5581 df-po 5589 df-so 5590 df-fr 5632 df-we 5634 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-rn 5688 df-res 5689 df-ima 5690 df-pred 6301 df-ord 6368 df-on 6369 df-lim 6370 df-suc 6371 df-iota 6496 df-fun 6546 df-fn 6547 df-f 6548 df-f1 6549 df-fo 6550 df-f1o 6551 df-fv 6552 df-riota 7365 df-ov 7412 df-oprab 7413 df-mpo 7414 df-om 7856 df-1st 7975 df-2nd 7976 df-frecs 8266 df-wrecs 8297 df-recs 8371 df-rdg 8410 df-er 8703 df-en 8940 df-dom 8941 df-sdom 8942 df-sup 9437 df-inf 9438 df-pnf 11250 df-mnf 11251 df-xr 11252 df-ltxr 11253 df-le 11254 df-sub 11446 df-neg 11447 df-div 11872 df-nn 12213 df-2 12275 df-3 12276 df-n0 12473 df-z 12559 df-uz 12823 df-rp 12975 df-ioo 13328 df-ico 13330 df-fl 13757 df-seq 13967 df-exp 14028 df-cj 15046 df-re 15047 df-im 15048 df-sqrt 15182 df-abs 15183 df-dvds 16198

This theorem is referenced by: knoppndvlem15 35402

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