knoppndvlem10 - Mathbox for Asger C. Ipsen

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Theorem knoppndvlem10 36514

Description: Lemma for knoppndv 36527. (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 12602	. . . . . . . 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 16274	. . . . . . . . 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 36513	. . . . . 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 36512	. . . . . 6 ⊢ ((𝜑 ∧ 2 ∥ 𝑀) → ((𝐹‘𝐴)‘𝐽) = 0)
23	19, 22	oveq12d 7371	. . . . 5 ⊢ ((𝜑 ∧ 2 ∥ 𝑀) → (((𝐹‘𝐵)‘𝐽) − ((𝐹‘𝐴)‘𝐽)) = (((𝐶↑𝐽) / 2) − 0))
24	4	knoppndvlem3 36507	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐶 ∈ ℝ ∧ (abs‘𝐶) < 1))
25	24	simpld 494	. . . . . . . . . 10 ⊢ (𝜑 → 𝐶 ∈ ℝ)
26	25	recnd 11162	. . . . . . . . 9 ⊢ (𝜑 → 𝐶 ∈ ℂ)
27	26, 6	expcld 14072	. . . . . . . 8 ⊢ (𝜑 → (𝐶↑𝐽) ∈ ℂ)
28		2cnd 12225	. . . . . . . 8 ⊢ (𝜑 → 2 ∈ ℂ)
29		2ne0 12251	. . . . . . . . 9 ⊢ 2 ≠ 0
30	29	a1i 11	. . . . . . . 8 ⊢ (𝜑 → 2 ≠ 0)
31	27, 28, 30	divcld 11919	. . . . . . 7 ⊢ (𝜑 → ((𝐶↑𝐽) / 2) ∈ ℂ)
32	31	subid1d 11483	. . . . . 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 2764	. . . 4 ⊢ ((𝜑 ∧ 2 ∥ 𝑀) → (((𝐹‘𝐵)‘𝐽) − ((𝐹‘𝐴)‘𝐽)) = ((𝐶↑𝐽) / 2))
35	34	fveq2d 6830	. . 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 12516	. . . . . . . . . 10 ⊢ (𝜑 → 𝐽 ∈ ℤ)
38	11, 37, 9	knoppndvlem1 36505	. . . . . . . . 9 ⊢ (𝜑 → ((((2 · 𝑁)↑-𝐽) / 2) · (𝑀 + 1)) ∈ ℝ)
39	36, 38	eqeltrd 2828	. . . . . . . 8 ⊢ (𝜑 → 𝐵 ∈ ℝ)
40	1, 2, 11, 25, 39, 6	knoppcnlem3 36488	. . . . . . 7 ⊢ (𝜑 → ((𝐹‘𝐵)‘𝐽) ∈ ℝ)
41	40	recnd 11162	. . . . . 6 ⊢ (𝜑 → ((𝐹‘𝐵)‘𝐽) ∈ ℂ)
42	20	a1i 11	. . . . . . . . 9 ⊢ (𝜑 → 𝐴 = ((((2 · 𝑁)↑-𝐽) / 2) · 𝑀))
43	11, 37, 8	knoppndvlem1 36505	. . . . . . . . 9 ⊢ (𝜑 → ((((2 · 𝑁)↑-𝐽) / 2) · 𝑀) ∈ ℝ)
44	42, 43	eqeltrd 2828	. . . . . . . 8 ⊢ (𝜑 → 𝐴 ∈ ℝ)
45	1, 2, 11, 25, 44, 6	knoppcnlem3 36488	. . . . . . 7 ⊢ (𝜑 → ((𝐹‘𝐴)‘𝐽) ∈ ℝ)
46	45	recnd 11162	. . . . . 6 ⊢ (𝜑 → ((𝐹‘𝐴)‘𝐽) ∈ ℂ)
47	41, 46	abssubd 15382	. . . . 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 36513	. . . . . . 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 232	. . . . . . . 8 ⊢ ((𝜑 ∧ ¬ 2 ∥ 𝑀) → 2 ∥ (𝑀 + 1))
58	1, 2, 3, 49, 50, 55, 52, 57	knoppndvlem8 36512	. . . . . . 7 ⊢ ((𝜑 ∧ ¬ 2 ∥ 𝑀) → ((𝐹‘𝐵)‘𝐽) = 0)
59	54, 58	oveq12d 7371	. . . . . 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 2764	. . . . 5 ⊢ ((𝜑 ∧ ¬ 2 ∥ 𝑀) → (((𝐹‘𝐴)‘𝐽) − ((𝐹‘𝐵)‘𝐽)) = ((𝐶↑𝐽) / 2))
62	61	fveq2d 6830	. . . 4 ⊢ ((𝜑 ∧ ¬ 2 ∥ 𝑀) → (abs‘(((𝐹‘𝐴)‘𝐽) − ((𝐹‘𝐵)‘𝐽))) = (abs‘((𝐶↑𝐽) / 2)))
63	48, 62	eqtrd 2764	. . 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 15384	. . 3 ⊢ (𝜑 → (abs‘((𝐶↑𝐽) / 2)) = ((abs‘(𝐶↑𝐽)) / (abs‘2)))
66	26, 6	absexpd 15381	. . . 4 ⊢ (𝜑 → (abs‘(𝐶↑𝐽)) = ((abs‘𝐶)↑𝐽))
67		0le2 12249	. . . . . 6 ⊢ 0 ≤ 2
68		2re 12221	. . . . . . 7 ⊢ 2 ∈ ℝ
69	68	absidi 15304	. . . . . 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 7371	. . 3 ⊢ (𝜑 → ((abs‘(𝐶↑𝐽)) / (abs‘2)) = (((abs‘𝐶)↑𝐽) / 2))
73	65, 72	eqtrd 2764	. 2 ⊢ (𝜑 → (abs‘((𝐶↑𝐽) / 2)) = (((abs‘𝐶)↑𝐽) / 2))
74	64, 73	eqtrd 2764	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 206 ∧ wa 395 = wceq 1540 ∈ wcel 2109 ≠ wne 2925 class class class wbr 5095 ↦ cmpt 5176 ‘cfv 6486 (class class class)co 7353 ℝcr 11027 0cc0 11028 1c1 11029 + caddc 11031 · cmul 11033 < clt 11168 ≤ cle 11169 − cmin 11366 -cneg 11367 / cdiv 11796 ℕcn 12147 2c2 12202 ℕ₀cn0 12403 ℤcz 12490 (,)cioo 13267 ⌊cfl 13713 ↑cexp 13987 abscabs 15160 ∥ cdvds 16182

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 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2701 ax-rep 5221 ax-sep 5238 ax-nul 5248 ax-pow 5307 ax-pr 5374 ax-un 7675 ax-cnex 11084 ax-resscn 11085 ax-1cn 11086 ax-icn 11087 ax-addcl 11088 ax-addrcl 11089 ax-mulcl 11090 ax-mulrcl 11091 ax-mulcom 11092 ax-addass 11093 ax-mulass 11094 ax-distr 11095 ax-i2m1 11096 ax-1ne0 11097 ax-1rid 11098 ax-rnegex 11099 ax-rrecex 11100 ax-cnre 11101 ax-pre-lttri 11102 ax-pre-lttrn 11103 ax-pre-ltadd 11104 ax-pre-mulgt0 11105 ax-pre-sup 11106

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2533 df-eu 2562 df-clab 2708 df-cleq 2721 df-clel 2803 df-nfc 2878 df-ne 2926 df-nel 3030 df-ral 3045 df-rex 3054 df-rmo 3345 df-reu 3346 df-rab 3397 df-v 3440 df-sbc 3745 df-csb 3854 df-dif 3908 df-un 3910 df-in 3912 df-ss 3922 df-pss 3925 df-nul 4287 df-if 4479 df-pw 4555 df-sn 4580 df-pr 4582 df-op 4586 df-uni 4862 df-iun 4946 df-br 5096 df-opab 5158 df-mpt 5177 df-tr 5203 df-id 5518 df-eprel 5523 df-po 5531 df-so 5532 df-fr 5576 df-we 5578 df-xp 5629 df-rel 5630 df-cnv 5631 df-co 5632 df-dm 5633 df-rn 5634 df-res 5635 df-ima 5636 df-pred 6253 df-ord 6314 df-on 6315 df-lim 6316 df-suc 6317 df-iota 6442 df-fun 6488 df-fn 6489 df-f 6490 df-f1 6491 df-fo 6492 df-f1o 6493 df-fv 6494 df-riota 7310 df-ov 7356 df-oprab 7357 df-mpo 7358 df-om 7807 df-1st 7931 df-2nd 7932 df-frecs 8221 df-wrecs 8252 df-recs 8301 df-rdg 8339 df-er 8632 df-en 8880 df-dom 8881 df-sdom 8882 df-sup 9351 df-inf 9352 df-pnf 11170 df-mnf 11171 df-xr 11172 df-ltxr 11173 df-le 11174 df-sub 11368 df-neg 11369 df-div 11797 df-nn 12148 df-2 12210 df-3 12211 df-n0 12404 df-z 12491 df-uz 12755 df-rp 12913 df-ioo 13271 df-ico 13273 df-fl 13715 df-seq 13928 df-exp 13988 df-cj 15025 df-re 15026 df-im 15027 df-sqrt 15161 df-abs 15162 df-dvds 16183

This theorem is referenced by: knoppndvlem15 36519

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