knoppndvlem10 - Mathbox for Asger C. Ipsen

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Theorem knoppndvlem10 36482

Description: Lemma for knoppndv 36495. (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 12617	. . . . . . . 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 16290	. . . . . . . . 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 36481	. . . . . 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 36480	. . . . . 6 ⊢ ((𝜑 ∧ 2 ∥ 𝑀) → ((𝐹‘𝐴)‘𝐽) = 0)
23	19, 22	oveq12d 7387	. . . . 5 ⊢ ((𝜑 ∧ 2 ∥ 𝑀) → (((𝐹‘𝐵)‘𝐽) − ((𝐹‘𝐴)‘𝐽)) = (((𝐶↑𝐽) / 2) − 0))
24	4	knoppndvlem3 36475	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐶 ∈ ℝ ∧ (abs‘𝐶) < 1))
25	24	simpld 494	. . . . . . . . . 10 ⊢ (𝜑 → 𝐶 ∈ ℝ)
26	25	recnd 11178	. . . . . . . . 9 ⊢ (𝜑 → 𝐶 ∈ ℂ)
27	26, 6	expcld 14087	. . . . . . . 8 ⊢ (𝜑 → (𝐶↑𝐽) ∈ ℂ)
28		2cnd 12240	. . . . . . . 8 ⊢ (𝜑 → 2 ∈ ℂ)
29		2ne0 12266	. . . . . . . . 9 ⊢ 2 ≠ 0
30	29	a1i 11	. . . . . . . 8 ⊢ (𝜑 → 2 ≠ 0)
31	27, 28, 30	divcld 11934	. . . . . . 7 ⊢ (𝜑 → ((𝐶↑𝐽) / 2) ∈ ℂ)
32	31	subid1d 11498	. . . . . 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 6844	. . 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 12531	. . . . . . . . . 10 ⊢ (𝜑 → 𝐽 ∈ ℤ)
38	11, 37, 9	knoppndvlem1 36473	. . . . . . . . 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 36456	. . . . . . 7 ⊢ (𝜑 → ((𝐹‘𝐵)‘𝐽) ∈ ℝ)
41	40	recnd 11178	. . . . . 6 ⊢ (𝜑 → ((𝐹‘𝐵)‘𝐽) ∈ ℂ)
42	20	a1i 11	. . . . . . . . 9 ⊢ (𝜑 → 𝐴 = ((((2 · 𝑁)↑-𝐽) / 2) · 𝑀))
43	11, 37, 8	knoppndvlem1 36473	. . . . . . . . 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 36456	. . . . . . 7 ⊢ (𝜑 → ((𝐹‘𝐴)‘𝐽) ∈ ℝ)
46	45	recnd 11178	. . . . . 6 ⊢ (𝜑 → ((𝐹‘𝐴)‘𝐽) ∈ ℂ)
47	41, 46	abssubd 15398	. . . . 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 36481	. . . . . . 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 36480	. . . . . . 7 ⊢ ((𝜑 ∧ ¬ 2 ∥ 𝑀) → ((𝐹‘𝐵)‘𝐽) = 0)
59	54, 58	oveq12d 7387	. . . . . 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 6844	. . . 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 15400	. . 3 ⊢ (𝜑 → (abs‘((𝐶↑𝐽) / 2)) = ((abs‘(𝐶↑𝐽)) / (abs‘2)))
66	26, 6	absexpd 15397	. . . 4 ⊢ (𝜑 → (abs‘(𝐶↑𝐽)) = ((abs‘𝐶)↑𝐽))
67		0le2 12264	. . . . . 6 ⊢ 0 ≤ 2
68		2re 12236	. . . . . . 7 ⊢ 2 ∈ ℝ
69	68	absidi 15320	. . . . . 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 7387	. . 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 5102 ↦ cmpt 5183 ‘cfv 6499 (class class class)co 7369 ℝcr 11043 0cc0 11044 1c1 11045 + caddc 11047 · cmul 11049 < clt 11184 ≤ cle 11185 − cmin 11381 -cneg 11382 / cdiv 11811 ℕcn 12162 2c2 12217 ℕ₀cn0 12418 ℤcz 12505 (,)cioo 13282 ⌊cfl 13728 ↑cexp 14002 abscabs 15176 ∥ cdvds 16198

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 5229 ax-sep 5246 ax-nul 5256 ax-pow 5315 ax-pr 5382 ax-un 7691 ax-cnex 11100 ax-resscn 11101 ax-1cn 11102 ax-icn 11103 ax-addcl 11104 ax-addrcl 11105 ax-mulcl 11106 ax-mulrcl 11107 ax-mulcom 11108 ax-addass 11109 ax-mulass 11110 ax-distr 11111 ax-i2m1 11112 ax-1ne0 11113 ax-1rid 11114 ax-rnegex 11115 ax-rrecex 11116 ax-cnre 11117 ax-pre-lttri 11118 ax-pre-lttrn 11119 ax-pre-ltadd 11120 ax-pre-mulgt0 11121 ax-pre-sup 11122

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 3351 df-reu 3352 df-rab 3403 df-v 3446 df-sbc 3751 df-csb 3860 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-pss 3931 df-nul 4293 df-if 4485 df-pw 4561 df-sn 4586 df-pr 4588 df-op 4592 df-uni 4868 df-iun 4953 df-br 5103 df-opab 5165 df-mpt 5184 df-tr 5210 df-id 5526 df-eprel 5531 df-po 5539 df-so 5540 df-fr 5584 df-we 5586 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-pred 6262 df-ord 6323 df-on 6324 df-lim 6325 df-suc 6326 df-iota 6452 df-fun 6501 df-fn 6502 df-f 6503 df-f1 6504 df-fo 6505 df-f1o 6506 df-fv 6507 df-riota 7326 df-ov 7372 df-oprab 7373 df-mpo 7374 df-om 7823 df-1st 7947 df-2nd 7948 df-frecs 8237 df-wrecs 8268 df-recs 8317 df-rdg 8355 df-er 8648 df-en 8896 df-dom 8897 df-sdom 8898 df-sup 9369 df-inf 9370 df-pnf 11186 df-mnf 11187 df-xr 11188 df-ltxr 11189 df-le 11190 df-sub 11383 df-neg 11384 df-div 11812 df-nn 12163 df-2 12225 df-3 12226 df-n0 12419 df-z 12506 df-uz 12770 df-rp 12928 df-ioo 13286 df-ico 13288 df-fl 13730 df-seq 13943 df-exp 14003 df-cj 15041 df-re 15042 df-im 15043 df-sqrt 15177 df-abs 15178 df-dvds 16199

This theorem is referenced by: knoppndvlem15 36487

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