knoppndvlem10 - Mathbox for Asger C. Ipsen

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Theorem knoppndvlem10 36049

Description: Lemma for knoppndv 36062. (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 479	. . . . . . 7 ⊢ ((𝜑 ∧ 2 ∥ 𝑀) → 𝐶 ∈ (-1(,)1))
6		knoppndvlem10.j	. . . . . . . 8 ⊢ (𝜑 → 𝐽 ∈ ℕ₀)
7	6	adantr 479	. . . . . . 7 ⊢ ((𝜑 ∧ 2 ∥ 𝑀) → 𝐽 ∈ ℕ₀)
8		knoppndvlem10.m	. . . . . . . . 9 ⊢ (𝜑 → 𝑀 ∈ ℤ)
9	8	peano2zd 12694	. . . . . . . 8 ⊢ (𝜑 → (𝑀 + 1) ∈ ℤ)
10	9	adantr 479	. . . . . . 7 ⊢ ((𝜑 ∧ 2 ∥ 𝑀) → (𝑀 + 1) ∈ ℤ)
11		knoppndvlem10.n	. . . . . . . 8 ⊢ (𝜑 → 𝑁 ∈ ℕ)
12	11	adantr 479	. . . . . . 7 ⊢ ((𝜑 ∧ 2 ∥ 𝑀) → 𝑁 ∈ ℕ)
13		notnot 142	. . . . . . . . 9 ⊢ (2 ∥ 𝑀 → ¬ ¬ 2 ∥ 𝑀)
14	13	adantl 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 2 ∥ 𝑀) → ¬ ¬ 2 ∥ 𝑀)
15	8	adantr 479	. . . . . . . . 9 ⊢ ((𝜑 ∧ 2 ∥ 𝑀) → 𝑀 ∈ ℤ)
16		oddp1even 16315	. . . . . . . . 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 36048	. . . . . 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 36047	. . . . . 6 ⊢ ((𝜑 ∧ 2 ∥ 𝑀) → ((𝐹‘𝐴)‘𝐽) = 0)
23	19, 22	oveq12d 7431	. . . . 5 ⊢ ((𝜑 ∧ 2 ∥ 𝑀) → (((𝐹‘𝐵)‘𝐽) − ((𝐹‘𝐴)‘𝐽)) = (((𝐶↑𝐽) / 2) − 0))
24	4	knoppndvlem3 36042	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐶 ∈ ℝ ∧ (abs‘𝐶) < 1))
25	24	simpld 493	. . . . . . . . . 10 ⊢ (𝜑 → 𝐶 ∈ ℝ)
26	25	recnd 11267	. . . . . . . . 9 ⊢ (𝜑 → 𝐶 ∈ ℂ)
27	26, 6	expcld 14137	. . . . . . . 8 ⊢ (𝜑 → (𝐶↑𝐽) ∈ ℂ)
28		2cnd 12315	. . . . . . . 8 ⊢ (𝜑 → 2 ∈ ℂ)
29		2ne0 12341	. . . . . . . . 9 ⊢ 2 ≠ 0
30	29	a1i 11	. . . . . . . 8 ⊢ (𝜑 → 2 ≠ 0)
31	27, 28, 30	divcld 12015	. . . . . . 7 ⊢ (𝜑 → ((𝐶↑𝐽) / 2) ∈ ℂ)
32	31	subid1d 11585	. . . . . 6 ⊢ (𝜑 → (((𝐶↑𝐽) / 2) − 0) = ((𝐶↑𝐽) / 2))
33	32	adantr 479	. . . . 5 ⊢ ((𝜑 ∧ 2 ∥ 𝑀) → (((𝐶↑𝐽) / 2) − 0) = ((𝐶↑𝐽) / 2))
34	23, 33	eqtrd 2765	. . . 4 ⊢ ((𝜑 ∧ 2 ∥ 𝑀) → (((𝐹‘𝐵)‘𝐽) − ((𝐹‘𝐴)‘𝐽)) = ((𝐶↑𝐽) / 2))
35	34	fveq2d 6894	. . 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 12609	. . . . . . . . . 10 ⊢ (𝜑 → 𝐽 ∈ ℤ)
38	11, 37, 9	knoppndvlem1 36040	. . . . . . . . 9 ⊢ (𝜑 → ((((2 · 𝑁)↑-𝐽) / 2) · (𝑀 + 1)) ∈ ℝ)
39	36, 38	eqeltrd 2825	. . . . . . . 8 ⊢ (𝜑 → 𝐵 ∈ ℝ)
40	1, 2, 11, 25, 39, 6	knoppcnlem3 36023	. . . . . . 7 ⊢ (𝜑 → ((𝐹‘𝐵)‘𝐽) ∈ ℝ)
41	40	recnd 11267	. . . . . 6 ⊢ (𝜑 → ((𝐹‘𝐵)‘𝐽) ∈ ℂ)
42	20	a1i 11	. . . . . . . . 9 ⊢ (𝜑 → 𝐴 = ((((2 · 𝑁)↑-𝐽) / 2) · 𝑀))
43	11, 37, 8	knoppndvlem1 36040	. . . . . . . . 9 ⊢ (𝜑 → ((((2 · 𝑁)↑-𝐽) / 2) · 𝑀) ∈ ℝ)
44	42, 43	eqeltrd 2825	. . . . . . . 8 ⊢ (𝜑 → 𝐴 ∈ ℝ)
45	1, 2, 11, 25, 44, 6	knoppcnlem3 36023	. . . . . . 7 ⊢ (𝜑 → ((𝐹‘𝐴)‘𝐽) ∈ ℝ)
46	45	recnd 11267	. . . . . 6 ⊢ (𝜑 → ((𝐹‘𝐴)‘𝐽) ∈ ℂ)
47	41, 46	abssubd 15427	. . . . 5 ⊢ (𝜑 → (abs‘(((𝐹‘𝐵)‘𝐽) − ((𝐹‘𝐴)‘𝐽))) = (abs‘(((𝐹‘𝐴)‘𝐽) − ((𝐹‘𝐵)‘𝐽))))
48	47	adantr 479	. . . 4 ⊢ ((𝜑 ∧ ¬ 2 ∥ 𝑀) → (abs‘(((𝐹‘𝐵)‘𝐽) − ((𝐹‘𝐴)‘𝐽))) = (abs‘(((𝐹‘𝐴)‘𝐽) − ((𝐹‘𝐵)‘𝐽))))
49	4	adantr 479	. . . . . . . 8 ⊢ ((𝜑 ∧ ¬ 2 ∥ 𝑀) → 𝐶 ∈ (-1(,)1))
50	6	adantr 479	. . . . . . . 8 ⊢ ((𝜑 ∧ ¬ 2 ∥ 𝑀) → 𝐽 ∈ ℕ₀)
51	8	adantr 479	. . . . . . . 8 ⊢ ((𝜑 ∧ ¬ 2 ∥ 𝑀) → 𝑀 ∈ ℤ)
52	11	adantr 479	. . . . . . . 8 ⊢ ((𝜑 ∧ ¬ 2 ∥ 𝑀) → 𝑁 ∈ ℕ)
53		simpr 483	. . . . . . . 8 ⊢ ((𝜑 ∧ ¬ 2 ∥ 𝑀) → ¬ 2 ∥ 𝑀)
54	1, 2, 20, 49, 50, 51, 52, 53	knoppndvlem9 36048	. . . . . . 7 ⊢ ((𝜑 ∧ ¬ 2 ∥ 𝑀) → ((𝐹‘𝐴)‘𝐽) = ((𝐶↑𝐽) / 2))
55	9	adantr 479	. . . . . . . 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 36047	. . . . . . 7 ⊢ ((𝜑 ∧ ¬ 2 ∥ 𝑀) → ((𝐹‘𝐵)‘𝐽) = 0)
59	54, 58	oveq12d 7431	. . . . . 6 ⊢ ((𝜑 ∧ ¬ 2 ∥ 𝑀) → (((𝐹‘𝐴)‘𝐽) − ((𝐹‘𝐵)‘𝐽)) = (((𝐶↑𝐽) / 2) − 0))
60	32	adantr 479	. . . . . 6 ⊢ ((𝜑 ∧ ¬ 2 ∥ 𝑀) → (((𝐶↑𝐽) / 2) − 0) = ((𝐶↑𝐽) / 2))
61	59, 60	eqtrd 2765	. . . . 5 ⊢ ((𝜑 ∧ ¬ 2 ∥ 𝑀) → (((𝐹‘𝐴)‘𝐽) − ((𝐹‘𝐵)‘𝐽)) = ((𝐶↑𝐽) / 2))
62	61	fveq2d 6894	. . . 4 ⊢ ((𝜑 ∧ ¬ 2 ∥ 𝑀) → (abs‘(((𝐹‘𝐴)‘𝐽) − ((𝐹‘𝐵)‘𝐽))) = (abs‘((𝐶↑𝐽) / 2)))
63	48, 62	eqtrd 2765	. . 3 ⊢ ((𝜑 ∧ ¬ 2 ∥ 𝑀) → (abs‘(((𝐹‘𝐵)‘𝐽) − ((𝐹‘𝐴)‘𝐽))) = (abs‘((𝐶↑𝐽) / 2)))
64	35, 63	pm2.61dan 811	. 2 ⊢ (𝜑 → (abs‘(((𝐹‘𝐵)‘𝐽) − ((𝐹‘𝐴)‘𝐽))) = (abs‘((𝐶↑𝐽) / 2)))
65	27, 28, 30	absdivd 15429	. . 3 ⊢ (𝜑 → (abs‘((𝐶↑𝐽) / 2)) = ((abs‘(𝐶↑𝐽)) / (abs‘2)))
66	26, 6	absexpd 15426	. . . 4 ⊢ (𝜑 → (abs‘(𝐶↑𝐽)) = ((abs‘𝐶)↑𝐽))
67		0le2 12339	. . . . . 6 ⊢ 0 ≤ 2
68		2re 12311	. . . . . . 7 ⊢ 2 ∈ ℝ
69	68	absidi 15351	. . . . . 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 7431	. . 3 ⊢ (𝜑 → ((abs‘(𝐶↑𝐽)) / (abs‘2)) = (((abs‘𝐶)↑𝐽) / 2))
73	65, 72	eqtrd 2765	. 2 ⊢ (𝜑 → (abs‘((𝐶↑𝐽) / 2)) = (((abs‘𝐶)↑𝐽) / 2))
74	64, 73	eqtrd 2765	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 394 = wceq 1533 ∈ wcel 2098 ≠ wne 2930 class class class wbr 5144 ↦ cmpt 5227 ‘cfv 6543 (class class class)co 7413 ℝcr 11132 0cc0 11133 1c1 11134 + caddc 11136 · cmul 11138 < clt 11273 ≤ cle 11274 − cmin 11469 -cneg 11470 / cdiv 11896 ℕcn 12237 2c2 12292 ℕ₀cn0 12497 ℤcz 12583 (,)cioo 13351 ⌊cfl 13782 ↑cexp 14053 abscabs 15208 ∥ cdvds 16225

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-rep 5281 ax-sep 5295 ax-nul 5302 ax-pow 5360 ax-pr 5424 ax-un 7735 ax-cnex 11189 ax-resscn 11190 ax-1cn 11191 ax-icn 11192 ax-addcl 11193 ax-addrcl 11194 ax-mulcl 11195 ax-mulrcl 11196 ax-mulcom 11197 ax-addass 11198 ax-mulass 11199 ax-distr 11200 ax-i2m1 11201 ax-1ne0 11202 ax-1rid 11203 ax-rnegex 11204 ax-rrecex 11205 ax-cnre 11206 ax-pre-lttri 11207 ax-pre-lttrn 11208 ax-pre-ltadd 11209 ax-pre-mulgt0 11210 ax-pre-sup 11211

This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3364 df-reu 3365 df-rab 3420 df-v 3465 df-sbc 3771 df-csb 3887 df-dif 3944 df-un 3946 df-in 3948 df-ss 3958 df-pss 3961 df-nul 4320 df-if 4526 df-pw 4601 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4905 df-iun 4994 df-br 5145 df-opab 5207 df-mpt 5228 df-tr 5262 df-id 5571 df-eprel 5577 df-po 5585 df-so 5586 df-fr 5628 df-we 5630 df-xp 5679 df-rel 5680 df-cnv 5681 df-co 5682 df-dm 5683 df-rn 5684 df-res 5685 df-ima 5686 df-pred 6301 df-ord 6368 df-on 6369 df-lim 6370 df-suc 6371 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-riota 7369 df-ov 7416 df-oprab 7417 df-mpo 7418 df-om 7866 df-1st 7987 df-2nd 7988 df-frecs 8280 df-wrecs 8311 df-recs 8385 df-rdg 8424 df-er 8718 df-en 8958 df-dom 8959 df-sdom 8960 df-sup 9460 df-inf 9461 df-pnf 11275 df-mnf 11276 df-xr 11277 df-ltxr 11278 df-le 11279 df-sub 11471 df-neg 11472 df-div 11897 df-nn 12238 df-2 12300 df-3 12301 df-n0 12498 df-z 12584 df-uz 12848 df-rp 13002 df-ioo 13355 df-ico 13357 df-fl 13784 df-seq 13994 df-exp 14054 df-cj 15073 df-re 15074 df-im 15075 df-sqrt 15209 df-abs 15210 df-dvds 16226

This theorem is referenced by: knoppndvlem15 36054

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