fourierdlem10 - Mathbox for Glauco Siliprandi

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Theorem fourierdlem10 46137

Description: Condition on the bounds of a nonempty subinterval. (Contributed by Glauco Siliprandi, 11-Dec-2019.)

**Hypotheses**
Ref	Expression
fourierdlem10.1	⊢ (𝜑 → 𝐴 ∈ ℝ)
fourierdlem10.2	⊢ (𝜑 → 𝐵 ∈ ℝ)
fourierdlem10.3	⊢ (𝜑 → 𝐶 ∈ ℝ)
fourierdlem10.4	⊢ (𝜑 → 𝐷 ∈ ℝ)
fourierdlem10.5	⊢ (𝜑 → 𝐶 < 𝐷)
fourierdlem10.6	⊢ (𝜑 → (𝐶(,)𝐷) ⊆ (𝐴(,)𝐵))

**Assertion**
Ref	Expression
fourierdlem10	⊢ (𝜑 → (𝐴 ≤ 𝐶 ∧ 𝐷 ≤ 𝐵))

**Proof of Theorem fourierdlem10**
Step	Hyp	Ref	Expression
1		fourierdlem10.1	. . 3 ⊢ (𝜑 → 𝐴 ∈ ℝ)
2		fourierdlem10.3	. . 3 ⊢ (𝜑 → 𝐶 ∈ ℝ)
3		fourierdlem10.6	. . . . 5 ⊢ (𝜑 → (𝐶(,)𝐷) ⊆ (𝐴(,)𝐵))
4	3	adantr 480	. . . 4 ⊢ ((𝜑 ∧ 𝐶 < 𝐴) → (𝐶(,)𝐷) ⊆ (𝐴(,)𝐵))
5	2	rexrd 11312	. . . . . . 7 ⊢ (𝜑 → 𝐶 ∈ ℝ^*)
6	5	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝐶 < 𝐴) → 𝐶 ∈ ℝ^*)
7		fourierdlem10.4	. . . . . . . 8 ⊢ (𝜑 → 𝐷 ∈ ℝ)
8	7	rexrd 11312	. . . . . . 7 ⊢ (𝜑 → 𝐷 ∈ ℝ^*)
9	8	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝐶 < 𝐴) → 𝐷 ∈ ℝ^*)
10	2, 1	readdcld 11291	. . . . . . . . 9 ⊢ (𝜑 → (𝐶 + 𝐴) ∈ ℝ)
11	10	rehalfcld 12515	. . . . . . . 8 ⊢ (𝜑 → ((𝐶 + 𝐴) / 2) ∈ ℝ)
12	2, 7	readdcld 11291	. . . . . . . . 9 ⊢ (𝜑 → (𝐶 + 𝐷) ∈ ℝ)
13	12	rehalfcld 12515	. . . . . . . 8 ⊢ (𝜑 → ((𝐶 + 𝐷) / 2) ∈ ℝ)
14	11, 13	ifcld 4571	. . . . . . 7 ⊢ (𝜑 → if(𝐴 ≤ 𝐷, ((𝐶 + 𝐴) / 2), ((𝐶 + 𝐷) / 2)) ∈ ℝ)
15	14	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝐶 < 𝐴) → if(𝐴 ≤ 𝐷, ((𝐶 + 𝐴) / 2), ((𝐶 + 𝐷) / 2)) ∈ ℝ)
16		simplr 768	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝐶 < 𝐴) ∧ 𝐴 ≤ 𝐷) → 𝐶 < 𝐴)
17	2	ad2antrr 726	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝐶 < 𝐴) ∧ 𝐴 ≤ 𝐷) → 𝐶 ∈ ℝ)
18	1	ad2antrr 726	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝐶 < 𝐴) ∧ 𝐴 ≤ 𝐷) → 𝐴 ∈ ℝ)
19		avglt1 12506	. . . . . . . . . 10 ⊢ ((𝐶 ∈ ℝ ∧ 𝐴 ∈ ℝ) → (𝐶 < 𝐴 ↔ 𝐶 < ((𝐶 + 𝐴) / 2)))
20	17, 18, 19	syl2anc 584	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝐶 < 𝐴) ∧ 𝐴 ≤ 𝐷) → (𝐶 < 𝐴 ↔ 𝐶 < ((𝐶 + 𝐴) / 2)))
21	16, 20	mpbid 232	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝐶 < 𝐴) ∧ 𝐴 ≤ 𝐷) → 𝐶 < ((𝐶 + 𝐴) / 2))
22		iftrue 4530	. . . . . . . . 9 ⊢ (𝐴 ≤ 𝐷 → if(𝐴 ≤ 𝐷, ((𝐶 + 𝐴) / 2), ((𝐶 + 𝐷) / 2)) = ((𝐶 + 𝐴) / 2))
23	22	adantl 481	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝐶 < 𝐴) ∧ 𝐴 ≤ 𝐷) → if(𝐴 ≤ 𝐷, ((𝐶 + 𝐴) / 2), ((𝐶 + 𝐷) / 2)) = ((𝐶 + 𝐴) / 2))
24	21, 23	breqtrrd 5170	. . . . . . 7 ⊢ (((𝜑 ∧ 𝐶 < 𝐴) ∧ 𝐴 ≤ 𝐷) → 𝐶 < if(𝐴 ≤ 𝐷, ((𝐶 + 𝐴) / 2), ((𝐶 + 𝐷) / 2)))
25		fourierdlem10.5	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐶 < 𝐷)
26	25	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ¬ 𝐴 ≤ 𝐷) → 𝐶 < 𝐷)
27	2	adantr 480	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ ¬ 𝐴 ≤ 𝐷) → 𝐶 ∈ ℝ)
28	7	adantr 480	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ ¬ 𝐴 ≤ 𝐷) → 𝐷 ∈ ℝ)
29		avglt1 12506	. . . . . . . . . . 11 ⊢ ((𝐶 ∈ ℝ ∧ 𝐷 ∈ ℝ) → (𝐶 < 𝐷 ↔ 𝐶 < ((𝐶 + 𝐷) / 2)))
30	27, 28, 29	syl2anc 584	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ¬ 𝐴 ≤ 𝐷) → (𝐶 < 𝐷 ↔ 𝐶 < ((𝐶 + 𝐷) / 2)))
31	26, 30	mpbid 232	. . . . . . . . 9 ⊢ ((𝜑 ∧ ¬ 𝐴 ≤ 𝐷) → 𝐶 < ((𝐶 + 𝐷) / 2))
32		iffalse 4533	. . . . . . . . . . 11 ⊢ (¬ 𝐴 ≤ 𝐷 → if(𝐴 ≤ 𝐷, ((𝐶 + 𝐴) / 2), ((𝐶 + 𝐷) / 2)) = ((𝐶 + 𝐷) / 2))
33	32	eqcomd 2742	. . . . . . . . . 10 ⊢ (¬ 𝐴 ≤ 𝐷 → ((𝐶 + 𝐷) / 2) = if(𝐴 ≤ 𝐷, ((𝐶 + 𝐴) / 2), ((𝐶 + 𝐷) / 2)))
34	33	adantl 481	. . . . . . . . 9 ⊢ ((𝜑 ∧ ¬ 𝐴 ≤ 𝐷) → ((𝐶 + 𝐷) / 2) = if(𝐴 ≤ 𝐷, ((𝐶 + 𝐴) / 2), ((𝐶 + 𝐷) / 2)))
35	31, 34	breqtrd 5168	. . . . . . . 8 ⊢ ((𝜑 ∧ ¬ 𝐴 ≤ 𝐷) → 𝐶 < if(𝐴 ≤ 𝐷, ((𝐶 + 𝐴) / 2), ((𝐶 + 𝐷) / 2)))
36	35	adantlr 715	. . . . . . 7 ⊢ (((𝜑 ∧ 𝐶 < 𝐴) ∧ ¬ 𝐴 ≤ 𝐷) → 𝐶 < if(𝐴 ≤ 𝐷, ((𝐶 + 𝐴) / 2), ((𝐶 + 𝐷) / 2)))
37	24, 36	pm2.61dan 812	. . . . . 6 ⊢ ((𝜑 ∧ 𝐶 < 𝐴) → 𝐶 < if(𝐴 ≤ 𝐷, ((𝐶 + 𝐴) / 2), ((𝐶 + 𝐷) / 2)))
38	22	adantl 481	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝐴 ≤ 𝐷) → if(𝐴 ≤ 𝐷, ((𝐶 + 𝐴) / 2), ((𝐶 + 𝐷) / 2)) = ((𝐶 + 𝐴) / 2))
39	10	adantr 480	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝐴 ≤ 𝐷) → (𝐶 + 𝐴) ∈ ℝ)
40	12	adantr 480	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝐴 ≤ 𝐷) → (𝐶 + 𝐷) ∈ ℝ)
41		2rp 13040	. . . . . . . . . . . 12 ⊢ 2 ∈ ℝ⁺
42	41	a1i 11	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝐴 ≤ 𝐷) → 2 ∈ ℝ⁺)
43	1	adantr 480	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝐴 ≤ 𝐷) → 𝐴 ∈ ℝ)
44	7	adantr 480	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝐴 ≤ 𝐷) → 𝐷 ∈ ℝ)
45	2	adantr 480	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝐴 ≤ 𝐷) → 𝐶 ∈ ℝ)
46		simpr 484	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝐴 ≤ 𝐷) → 𝐴 ≤ 𝐷)
47	43, 44, 45, 46	leadd2dd 11879	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝐴 ≤ 𝐷) → (𝐶 + 𝐴) ≤ (𝐶 + 𝐷))
48	39, 40, 42, 47	lediv1dd 13136	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝐴 ≤ 𝐷) → ((𝐶 + 𝐴) / 2) ≤ ((𝐶 + 𝐷) / 2))
49	38, 48	eqbrtrd 5164	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐴 ≤ 𝐷) → if(𝐴 ≤ 𝐷, ((𝐶 + 𝐴) / 2), ((𝐶 + 𝐷) / 2)) ≤ ((𝐶 + 𝐷) / 2))
50	32	adantl 481	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ¬ 𝐴 ≤ 𝐷) → if(𝐴 ≤ 𝐷, ((𝐶 + 𝐴) / 2), ((𝐶 + 𝐷) / 2)) = ((𝐶 + 𝐷) / 2))
51	13	leidd 11830	. . . . . . . . . . 11 ⊢ (𝜑 → ((𝐶 + 𝐷) / 2) ≤ ((𝐶 + 𝐷) / 2))
52	51	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ¬ 𝐴 ≤ 𝐷) → ((𝐶 + 𝐷) / 2) ≤ ((𝐶 + 𝐷) / 2))
53	50, 52	eqbrtrd 5164	. . . . . . . . 9 ⊢ ((𝜑 ∧ ¬ 𝐴 ≤ 𝐷) → if(𝐴 ≤ 𝐷, ((𝐶 + 𝐴) / 2), ((𝐶 + 𝐷) / 2)) ≤ ((𝐶 + 𝐷) / 2))
54	49, 53	pm2.61dan 812	. . . . . . . 8 ⊢ (𝜑 → if(𝐴 ≤ 𝐷, ((𝐶 + 𝐴) / 2), ((𝐶 + 𝐷) / 2)) ≤ ((𝐶 + 𝐷) / 2))
55		avglt2 12507	. . . . . . . . . 10 ⊢ ((𝐶 ∈ ℝ ∧ 𝐷 ∈ ℝ) → (𝐶 < 𝐷 ↔ ((𝐶 + 𝐷) / 2) < 𝐷))
56	2, 7, 55	syl2anc 584	. . . . . . . . 9 ⊢ (𝜑 → (𝐶 < 𝐷 ↔ ((𝐶 + 𝐷) / 2) < 𝐷))
57	25, 56	mpbid 232	. . . . . . . 8 ⊢ (𝜑 → ((𝐶 + 𝐷) / 2) < 𝐷)
58	14, 13, 7, 54, 57	lelttrd 11420	. . . . . . 7 ⊢ (𝜑 → if(𝐴 ≤ 𝐷, ((𝐶 + 𝐴) / 2), ((𝐶 + 𝐷) / 2)) < 𝐷)
59	58	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝐶 < 𝐴) → if(𝐴 ≤ 𝐷, ((𝐶 + 𝐴) / 2), ((𝐶 + 𝐷) / 2)) < 𝐷)
60	6, 9, 15, 37, 59	eliood 45516	. . . . 5 ⊢ ((𝜑 ∧ 𝐶 < 𝐴) → if(𝐴 ≤ 𝐷, ((𝐶 + 𝐴) / 2), ((𝐶 + 𝐷) / 2)) ∈ (𝐶(,)𝐷))
61	1	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐶 < 𝐴) → 𝐴 ∈ ℝ)
62	11	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐶 < 𝐴) → ((𝐶 + 𝐴) / 2) ∈ ℝ)
63	14	adantr 480	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝐴 ≤ 𝐷) → if(𝐴 ≤ 𝐷, ((𝐶 + 𝐴) / 2), ((𝐶 + 𝐷) / 2)) ∈ ℝ)
64	63, 38	eqled 11365	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝐴 ≤ 𝐷) → if(𝐴 ≤ 𝐷, ((𝐶 + 𝐴) / 2), ((𝐶 + 𝐷) / 2)) ≤ ((𝐶 + 𝐴) / 2))
65	14	adantr 480	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ ¬ 𝐴 ≤ 𝐷) → if(𝐴 ≤ 𝐷, ((𝐶 + 𝐴) / 2), ((𝐶 + 𝐷) / 2)) ∈ ℝ)
66	11	adantr 480	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ ¬ 𝐴 ≤ 𝐷) → ((𝐶 + 𝐴) / 2) ∈ ℝ)
67		simpr 484	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ ¬ 𝐴 ≤ 𝐷) → ¬ 𝐴 ≤ 𝐷)
68	1	adantr 480	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ ¬ 𝐴 ≤ 𝐷) → 𝐴 ∈ ℝ)
69	28, 68	ltnled 11409	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ ¬ 𝐴 ≤ 𝐷) → (𝐷 < 𝐴 ↔ ¬ 𝐴 ≤ 𝐷))
70	67, 69	mpbird 257	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ ¬ 𝐴 ≤ 𝐷) → 𝐷 < 𝐴)
71	12	adantr 480	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝐷 < 𝐴) → (𝐶 + 𝐷) ∈ ℝ)
72	10	adantr 480	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝐷 < 𝐴) → (𝐶 + 𝐴) ∈ ℝ)
73	41	a1i 11	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝐷 < 𝐴) → 2 ∈ ℝ⁺)
74	7	adantr 480	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝐷 < 𝐴) → 𝐷 ∈ ℝ)
75	1	adantr 480	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝐷 < 𝐴) → 𝐴 ∈ ℝ)
76	2	adantr 480	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝐷 < 𝐴) → 𝐶 ∈ ℝ)
77		simpr 484	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝐷 < 𝐴) → 𝐷 < 𝐴)
78	74, 75, 76, 77	ltadd2dd 11421	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝐷 < 𝐴) → (𝐶 + 𝐷) < (𝐶 + 𝐴))
79	71, 72, 73, 78	ltdiv1dd 13135	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝐷 < 𝐴) → ((𝐶 + 𝐷) / 2) < ((𝐶 + 𝐴) / 2))
80	70, 79	syldan 591	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ ¬ 𝐴 ≤ 𝐷) → ((𝐶 + 𝐷) / 2) < ((𝐶 + 𝐴) / 2))
81	50, 80	eqbrtrd 5164	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ ¬ 𝐴 ≤ 𝐷) → if(𝐴 ≤ 𝐷, ((𝐶 + 𝐴) / 2), ((𝐶 + 𝐷) / 2)) < ((𝐶 + 𝐴) / 2))
82	65, 66, 81	ltled 11410	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ ¬ 𝐴 ≤ 𝐷) → if(𝐴 ≤ 𝐷, ((𝐶 + 𝐴) / 2), ((𝐶 + 𝐷) / 2)) ≤ ((𝐶 + 𝐴) / 2))
83	64, 82	pm2.61dan 812	. . . . . . . . . 10 ⊢ (𝜑 → if(𝐴 ≤ 𝐷, ((𝐶 + 𝐴) / 2), ((𝐶 + 𝐷) / 2)) ≤ ((𝐶 + 𝐴) / 2))
84	83	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐶 < 𝐴) → if(𝐴 ≤ 𝐷, ((𝐶 + 𝐴) / 2), ((𝐶 + 𝐷) / 2)) ≤ ((𝐶 + 𝐴) / 2))
85		simpr 484	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝐶 < 𝐴) → 𝐶 < 𝐴)
86	2	adantr 480	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝐶 < 𝐴) → 𝐶 ∈ ℝ)
87		avglt2 12507	. . . . . . . . . . 11 ⊢ ((𝐶 ∈ ℝ ∧ 𝐴 ∈ ℝ) → (𝐶 < 𝐴 ↔ ((𝐶 + 𝐴) / 2) < 𝐴))
88	86, 61, 87	syl2anc 584	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝐶 < 𝐴) → (𝐶 < 𝐴 ↔ ((𝐶 + 𝐴) / 2) < 𝐴))
89	85, 88	mpbid 232	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐶 < 𝐴) → ((𝐶 + 𝐴) / 2) < 𝐴)
90	15, 62, 61, 84, 89	lelttrd 11420	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐶 < 𝐴) → if(𝐴 ≤ 𝐷, ((𝐶 + 𝐴) / 2), ((𝐶 + 𝐷) / 2)) < 𝐴)
91	15, 61, 90	ltnsymd 11411	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐶 < 𝐴) → ¬ 𝐴 < if(𝐴 ≤ 𝐷, ((𝐶 + 𝐴) / 2), ((𝐶 + 𝐷) / 2)))
92	91	intn3an2d 1481	. . . . . 6 ⊢ ((𝜑 ∧ 𝐶 < 𝐴) → ¬ (if(𝐴 ≤ 𝐷, ((𝐶 + 𝐴) / 2), ((𝐶 + 𝐷) / 2)) ∈ ℝ ∧ 𝐴 < if(𝐴 ≤ 𝐷, ((𝐶 + 𝐴) / 2), ((𝐶 + 𝐷) / 2)) ∧ if(𝐴 ≤ 𝐷, ((𝐶 + 𝐴) / 2), ((𝐶 + 𝐷) / 2)) < 𝐵))
93	1	rexrd 11312	. . . . . . . 8 ⊢ (𝜑 → 𝐴 ∈ ℝ^*)
94	93	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐶 < 𝐴) → 𝐴 ∈ ℝ^*)
95		fourierdlem10.2	. . . . . . . . 9 ⊢ (𝜑 → 𝐵 ∈ ℝ)
96	95	rexrd 11312	. . . . . . . 8 ⊢ (𝜑 → 𝐵 ∈ ℝ^*)
97	96	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐶 < 𝐴) → 𝐵 ∈ ℝ^*)
98		elioo2 13429	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) → (if(𝐴 ≤ 𝐷, ((𝐶 + 𝐴) / 2), ((𝐶 + 𝐷) / 2)) ∈ (𝐴(,)𝐵) ↔ (if(𝐴 ≤ 𝐷, ((𝐶 + 𝐴) / 2), ((𝐶 + 𝐷) / 2)) ∈ ℝ ∧ 𝐴 < if(𝐴 ≤ 𝐷, ((𝐶 + 𝐴) / 2), ((𝐶 + 𝐷) / 2)) ∧ if(𝐴 ≤ 𝐷, ((𝐶 + 𝐴) / 2), ((𝐶 + 𝐷) / 2)) < 𝐵)))
99	94, 97, 98	syl2anc 584	. . . . . 6 ⊢ ((𝜑 ∧ 𝐶 < 𝐴) → (if(𝐴 ≤ 𝐷, ((𝐶 + 𝐴) / 2), ((𝐶 + 𝐷) / 2)) ∈ (𝐴(,)𝐵) ↔ (if(𝐴 ≤ 𝐷, ((𝐶 + 𝐴) / 2), ((𝐶 + 𝐷) / 2)) ∈ ℝ ∧ 𝐴 < if(𝐴 ≤ 𝐷, ((𝐶 + 𝐴) / 2), ((𝐶 + 𝐷) / 2)) ∧ if(𝐴 ≤ 𝐷, ((𝐶 + 𝐴) / 2), ((𝐶 + 𝐷) / 2)) < 𝐵)))
100	92, 99	mtbird 325	. . . . 5 ⊢ ((𝜑 ∧ 𝐶 < 𝐴) → ¬ if(𝐴 ≤ 𝐷, ((𝐶 + 𝐴) / 2), ((𝐶 + 𝐷) / 2)) ∈ (𝐴(,)𝐵))
101		nelss 4048	. . . . 5 ⊢ ((if(𝐴 ≤ 𝐷, ((𝐶 + 𝐴) / 2), ((𝐶 + 𝐷) / 2)) ∈ (𝐶(,)𝐷) ∧ ¬ if(𝐴 ≤ 𝐷, ((𝐶 + 𝐴) / 2), ((𝐶 + 𝐷) / 2)) ∈ (𝐴(,)𝐵)) → ¬ (𝐶(,)𝐷) ⊆ (𝐴(,)𝐵))
102	60, 100, 101	syl2anc 584	. . . 4 ⊢ ((𝜑 ∧ 𝐶 < 𝐴) → ¬ (𝐶(,)𝐷) ⊆ (𝐴(,)𝐵))
103	4, 102	pm2.65da 816	. . 3 ⊢ (𝜑 → ¬ 𝐶 < 𝐴)
104	1, 2, 103	nltled 11412	. 2 ⊢ (𝜑 → 𝐴 ≤ 𝐶)
105	3	adantr 480	. . . 4 ⊢ ((𝜑 ∧ 𝐵 < 𝐷) → (𝐶(,)𝐷) ⊆ (𝐴(,)𝐵))
106	5	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝐵 < 𝐷) → 𝐶 ∈ ℝ^*)
107	8	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝐵 < 𝐷) → 𝐷 ∈ ℝ^*)
108	95, 7	readdcld 11291	. . . . . . . . 9 ⊢ (𝜑 → (𝐵 + 𝐷) ∈ ℝ)
109	108	rehalfcld 12515	. . . . . . . 8 ⊢ (𝜑 → ((𝐵 + 𝐷) / 2) ∈ ℝ)
110	109, 13	ifcld 4571	. . . . . . 7 ⊢ (𝜑 → if(𝐶 ≤ 𝐵, ((𝐵 + 𝐷) / 2), ((𝐶 + 𝐷) / 2)) ∈ ℝ)
111	110	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝐵 < 𝐷) → if(𝐶 ≤ 𝐵, ((𝐵 + 𝐷) / 2), ((𝐶 + 𝐷) / 2)) ∈ ℝ)
112	2	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐶 ≤ 𝐵) → 𝐶 ∈ ℝ)
113	13	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐶 ≤ 𝐵) → ((𝐶 + 𝐷) / 2) ∈ ℝ)
114	110	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐶 ≤ 𝐵) → if(𝐶 ≤ 𝐵, ((𝐵 + 𝐷) / 2), ((𝐶 + 𝐷) / 2)) ∈ ℝ)
115	2, 7, 29	syl2anc 584	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐶 < 𝐷 ↔ 𝐶 < ((𝐶 + 𝐷) / 2)))
116	25, 115	mpbid 232	. . . . . . . . . 10 ⊢ (𝜑 → 𝐶 < ((𝐶 + 𝐷) / 2))
117	116	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐶 ≤ 𝐵) → 𝐶 < ((𝐶 + 𝐷) / 2))
118	12	adantr 480	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝐶 ≤ 𝐵) → (𝐶 + 𝐷) ∈ ℝ)
119	108	adantr 480	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝐶 ≤ 𝐵) → (𝐵 + 𝐷) ∈ ℝ)
120	41	a1i 11	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝐶 ≤ 𝐵) → 2 ∈ ℝ⁺)
121	95	adantr 480	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝐶 ≤ 𝐵) → 𝐵 ∈ ℝ)
122	7	adantr 480	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝐶 ≤ 𝐵) → 𝐷 ∈ ℝ)
123		simpr 484	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝐶 ≤ 𝐵) → 𝐶 ≤ 𝐵)
124	112, 121, 122, 123	leadd1dd 11878	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝐶 ≤ 𝐵) → (𝐶 + 𝐷) ≤ (𝐵 + 𝐷))
125	118, 119, 120, 124	lediv1dd 13136	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝐶 ≤ 𝐵) → ((𝐶 + 𝐷) / 2) ≤ ((𝐵 + 𝐷) / 2))
126		iftrue 4530	. . . . . . . . . . 11 ⊢ (𝐶 ≤ 𝐵 → if(𝐶 ≤ 𝐵, ((𝐵 + 𝐷) / 2), ((𝐶 + 𝐷) / 2)) = ((𝐵 + 𝐷) / 2))
127	126	adantl 481	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝐶 ≤ 𝐵) → if(𝐶 ≤ 𝐵, ((𝐵 + 𝐷) / 2), ((𝐶 + 𝐷) / 2)) = ((𝐵 + 𝐷) / 2))
128	125, 127	breqtrrd 5170	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐶 ≤ 𝐵) → ((𝐶 + 𝐷) / 2) ≤ if(𝐶 ≤ 𝐵, ((𝐵 + 𝐷) / 2), ((𝐶 + 𝐷) / 2)))
129	112, 113, 114, 117, 128	ltletrd 11422	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐶 ≤ 𝐵) → 𝐶 < if(𝐶 ≤ 𝐵, ((𝐵 + 𝐷) / 2), ((𝐶 + 𝐷) / 2)))
130	116	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ ¬ 𝐶 ≤ 𝐵) → 𝐶 < ((𝐶 + 𝐷) / 2))
131		iffalse 4533	. . . . . . . . . . 11 ⊢ (¬ 𝐶 ≤ 𝐵 → if(𝐶 ≤ 𝐵, ((𝐵 + 𝐷) / 2), ((𝐶 + 𝐷) / 2)) = ((𝐶 + 𝐷) / 2))
132	131	eqcomd 2742	. . . . . . . . . 10 ⊢ (¬ 𝐶 ≤ 𝐵 → ((𝐶 + 𝐷) / 2) = if(𝐶 ≤ 𝐵, ((𝐵 + 𝐷) / 2), ((𝐶 + 𝐷) / 2)))
133	132	adantl 481	. . . . . . . . 9 ⊢ ((𝜑 ∧ ¬ 𝐶 ≤ 𝐵) → ((𝐶 + 𝐷) / 2) = if(𝐶 ≤ 𝐵, ((𝐵 + 𝐷) / 2), ((𝐶 + 𝐷) / 2)))
134	130, 133	breqtrd 5168	. . . . . . . 8 ⊢ ((𝜑 ∧ ¬ 𝐶 ≤ 𝐵) → 𝐶 < if(𝐶 ≤ 𝐵, ((𝐵 + 𝐷) / 2), ((𝐶 + 𝐷) / 2)))
135	129, 134	pm2.61dan 812	. . . . . . 7 ⊢ (𝜑 → 𝐶 < if(𝐶 ≤ 𝐵, ((𝐵 + 𝐷) / 2), ((𝐶 + 𝐷) / 2)))
136	135	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝐵 < 𝐷) → 𝐶 < if(𝐶 ≤ 𝐵, ((𝐵 + 𝐷) / 2), ((𝐶 + 𝐷) / 2)))
137	126	adantl 481	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝐵 < 𝐷) ∧ 𝐶 ≤ 𝐵) → if(𝐶 ≤ 𝐵, ((𝐵 + 𝐷) / 2), ((𝐶 + 𝐷) / 2)) = ((𝐵 + 𝐷) / 2))
138		simpr 484	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝐵 < 𝐷) → 𝐵 < 𝐷)
139	95	adantr 480	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝐵 < 𝐷) → 𝐵 ∈ ℝ)
140	7	adantr 480	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝐵 < 𝐷) → 𝐷 ∈ ℝ)
141		avglt2 12507	. . . . . . . . . . 11 ⊢ ((𝐵 ∈ ℝ ∧ 𝐷 ∈ ℝ) → (𝐵 < 𝐷 ↔ ((𝐵 + 𝐷) / 2) < 𝐷))
142	139, 140, 141	syl2anc 584	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝐵 < 𝐷) → (𝐵 < 𝐷 ↔ ((𝐵 + 𝐷) / 2) < 𝐷))
143	138, 142	mpbid 232	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐵 < 𝐷) → ((𝐵 + 𝐷) / 2) < 𝐷)
144	143	adantr 480	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝐵 < 𝐷) ∧ 𝐶 ≤ 𝐵) → ((𝐵 + 𝐷) / 2) < 𝐷)
145	137, 144	eqbrtrd 5164	. . . . . . 7 ⊢ (((𝜑 ∧ 𝐵 < 𝐷) ∧ 𝐶 ≤ 𝐵) → if(𝐶 ≤ 𝐵, ((𝐵 + 𝐷) / 2), ((𝐶 + 𝐷) / 2)) < 𝐷)
146	131	adantl 481	. . . . . . . . 9 ⊢ ((𝜑 ∧ ¬ 𝐶 ≤ 𝐵) → if(𝐶 ≤ 𝐵, ((𝐵 + 𝐷) / 2), ((𝐶 + 𝐷) / 2)) = ((𝐶 + 𝐷) / 2))
147	57	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ ¬ 𝐶 ≤ 𝐵) → ((𝐶 + 𝐷) / 2) < 𝐷)
148	146, 147	eqbrtrd 5164	. . . . . . . 8 ⊢ ((𝜑 ∧ ¬ 𝐶 ≤ 𝐵) → if(𝐶 ≤ 𝐵, ((𝐵 + 𝐷) / 2), ((𝐶 + 𝐷) / 2)) < 𝐷)
149	148	adantlr 715	. . . . . . 7 ⊢ (((𝜑 ∧ 𝐵 < 𝐷) ∧ ¬ 𝐶 ≤ 𝐵) → if(𝐶 ≤ 𝐵, ((𝐵 + 𝐷) / 2), ((𝐶 + 𝐷) / 2)) < 𝐷)
150	145, 149	pm2.61dan 812	. . . . . 6 ⊢ ((𝜑 ∧ 𝐵 < 𝐷) → if(𝐶 ≤ 𝐵, ((𝐵 + 𝐷) / 2), ((𝐶 + 𝐷) / 2)) < 𝐷)
151	106, 107, 111, 136, 150	eliood 45516	. . . . 5 ⊢ ((𝜑 ∧ 𝐵 < 𝐷) → if(𝐶 ≤ 𝐵, ((𝐵 + 𝐷) / 2), ((𝐶 + 𝐷) / 2)) ∈ (𝐶(,)𝐷))
152	109	adantr 480	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝐵 < 𝐷) → ((𝐵 + 𝐷) / 2) ∈ ℝ)
153		avglt1 12506	. . . . . . . . . . . . . 14 ⊢ ((𝐵 ∈ ℝ ∧ 𝐷 ∈ ℝ) → (𝐵 < 𝐷 ↔ 𝐵 < ((𝐵 + 𝐷) / 2)))
154	139, 140, 153	syl2anc 584	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝐵 < 𝐷) → (𝐵 < 𝐷 ↔ 𝐵 < ((𝐵 + 𝐷) / 2)))
155	138, 154	mpbid 232	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝐵 < 𝐷) → 𝐵 < ((𝐵 + 𝐷) / 2))
156	139, 152, 155	ltled 11410	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝐵 < 𝐷) → 𝐵 ≤ ((𝐵 + 𝐷) / 2))
157	156	adantr 480	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝐵 < 𝐷) ∧ 𝐶 ≤ 𝐵) → 𝐵 ≤ ((𝐵 + 𝐷) / 2))
158	157, 137	breqtrrd 5170	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝐵 < 𝐷) ∧ 𝐶 ≤ 𝐵) → 𝐵 ≤ if(𝐶 ≤ 𝐵, ((𝐵 + 𝐷) / 2), ((𝐶 + 𝐷) / 2)))
159	95	adantr 480	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ ¬ 𝐶 ≤ 𝐵) → 𝐵 ∈ ℝ)
160	13	adantr 480	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ ¬ 𝐶 ≤ 𝐵) → ((𝐶 + 𝐷) / 2) ∈ ℝ)
161	2	adantr 480	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ ¬ 𝐶 ≤ 𝐵) → 𝐶 ∈ ℝ)
162		simpr 484	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ ¬ 𝐶 ≤ 𝐵) → ¬ 𝐶 ≤ 𝐵)
163	159, 161	ltnled 11409	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ ¬ 𝐶 ≤ 𝐵) → (𝐵 < 𝐶 ↔ ¬ 𝐶 ≤ 𝐵))
164	162, 163	mpbird 257	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ ¬ 𝐶 ≤ 𝐵) → 𝐵 < 𝐶)
165	159, 161, 160, 164, 130	lttrd 11423	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ ¬ 𝐶 ≤ 𝐵) → 𝐵 < ((𝐶 + 𝐷) / 2))
166	159, 160, 165	ltled 11410	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ ¬ 𝐶 ≤ 𝐵) → 𝐵 ≤ ((𝐶 + 𝐷) / 2))
167	166, 133	breqtrd 5168	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ¬ 𝐶 ≤ 𝐵) → 𝐵 ≤ if(𝐶 ≤ 𝐵, ((𝐵 + 𝐷) / 2), ((𝐶 + 𝐷) / 2)))
168	167	adantlr 715	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝐵 < 𝐷) ∧ ¬ 𝐶 ≤ 𝐵) → 𝐵 ≤ if(𝐶 ≤ 𝐵, ((𝐵 + 𝐷) / 2), ((𝐶 + 𝐷) / 2)))
169	158, 168	pm2.61dan 812	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐵 < 𝐷) → 𝐵 ≤ if(𝐶 ≤ 𝐵, ((𝐵 + 𝐷) / 2), ((𝐶 + 𝐷) / 2)))
170	139, 111, 169	lensymd 11413	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐵 < 𝐷) → ¬ if(𝐶 ≤ 𝐵, ((𝐵 + 𝐷) / 2), ((𝐶 + 𝐷) / 2)) < 𝐵)
171	170	intn3an3d 1482	. . . . . 6 ⊢ ((𝜑 ∧ 𝐵 < 𝐷) → ¬ (if(𝐶 ≤ 𝐵, ((𝐵 + 𝐷) / 2), ((𝐶 + 𝐷) / 2)) ∈ ℝ ∧ 𝐴 < if(𝐶 ≤ 𝐵, ((𝐵 + 𝐷) / 2), ((𝐶 + 𝐷) / 2)) ∧ if(𝐶 ≤ 𝐵, ((𝐵 + 𝐷) / 2), ((𝐶 + 𝐷) / 2)) < 𝐵))
172	93	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐵 < 𝐷) → 𝐴 ∈ ℝ^*)
173	96	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐵 < 𝐷) → 𝐵 ∈ ℝ^*)
174		elioo2 13429	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) → (if(𝐶 ≤ 𝐵, ((𝐵 + 𝐷) / 2), ((𝐶 + 𝐷) / 2)) ∈ (𝐴(,)𝐵) ↔ (if(𝐶 ≤ 𝐵, ((𝐵 + 𝐷) / 2), ((𝐶 + 𝐷) / 2)) ∈ ℝ ∧ 𝐴 < if(𝐶 ≤ 𝐵, ((𝐵 + 𝐷) / 2), ((𝐶 + 𝐷) / 2)) ∧ if(𝐶 ≤ 𝐵, ((𝐵 + 𝐷) / 2), ((𝐶 + 𝐷) / 2)) < 𝐵)))
175	172, 173, 174	syl2anc 584	. . . . . 6 ⊢ ((𝜑 ∧ 𝐵 < 𝐷) → (if(𝐶 ≤ 𝐵, ((𝐵 + 𝐷) / 2), ((𝐶 + 𝐷) / 2)) ∈ (𝐴(,)𝐵) ↔ (if(𝐶 ≤ 𝐵, ((𝐵 + 𝐷) / 2), ((𝐶 + 𝐷) / 2)) ∈ ℝ ∧ 𝐴 < if(𝐶 ≤ 𝐵, ((𝐵 + 𝐷) / 2), ((𝐶 + 𝐷) / 2)) ∧ if(𝐶 ≤ 𝐵, ((𝐵 + 𝐷) / 2), ((𝐶 + 𝐷) / 2)) < 𝐵)))
176	171, 175	mtbird 325	. . . . 5 ⊢ ((𝜑 ∧ 𝐵 < 𝐷) → ¬ if(𝐶 ≤ 𝐵, ((𝐵 + 𝐷) / 2), ((𝐶 + 𝐷) / 2)) ∈ (𝐴(,)𝐵))
177		nelss 4048	. . . . 5 ⊢ ((if(𝐶 ≤ 𝐵, ((𝐵 + 𝐷) / 2), ((𝐶 + 𝐷) / 2)) ∈ (𝐶(,)𝐷) ∧ ¬ if(𝐶 ≤ 𝐵, ((𝐵 + 𝐷) / 2), ((𝐶 + 𝐷) / 2)) ∈ (𝐴(,)𝐵)) → ¬ (𝐶(,)𝐷) ⊆ (𝐴(,)𝐵))
178	151, 176, 177	syl2anc 584	. . . 4 ⊢ ((𝜑 ∧ 𝐵 < 𝐷) → ¬ (𝐶(,)𝐷) ⊆ (𝐴(,)𝐵))
179	105, 178	pm2.65da 816	. . 3 ⊢ (𝜑 → ¬ 𝐵 < 𝐷)
180	7, 95, 179	nltled 11412	. 2 ⊢ (𝜑 → 𝐷 ≤ 𝐵)
181	104, 180	jca 511	1 ⊢ (𝜑 → (𝐴 ≤ 𝐶 ∧ 𝐷 ≤ 𝐵))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 206 ∧ wa 395 ∧ w3a 1086 = wceq 1539 ∈ wcel 2107 ⊆ wss 3950 ifcif 4524 class class class wbr 5142 (class class class)co 7432 ℝcr 11155 + caddc 11159 ℝ^*cxr 11295 < clt 11296 ≤ cle 11297 / cdiv 11921 2c2 12322 ℝ⁺crp 13035 (,)cioo 13388

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2707 ax-sep 5295 ax-nul 5305 ax-pow 5364 ax-pr 5431 ax-un 7756 ax-cnex 11212 ax-resscn 11213 ax-1cn 11214 ax-icn 11215 ax-addcl 11216 ax-addrcl 11217 ax-mulcl 11218 ax-mulrcl 11219 ax-mulcom 11220 ax-addass 11221 ax-mulass 11222 ax-distr 11223 ax-i2m1 11224 ax-1ne0 11225 ax-1rid 11226 ax-rnegex 11227 ax-rrecex 11228 ax-cnre 11229 ax-pre-lttri 11230 ax-pre-lttrn 11231 ax-pre-ltadd 11232 ax-pre-mulgt0 11233

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-rmo 3379 df-reu 3380 df-rab 3436 df-v 3481 df-sbc 3788 df-csb 3899 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-pss 3970 df-nul 4333 df-if 4525 df-pw 4601 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4907 df-iun 4992 df-br 5143 df-opab 5205 df-mpt 5225 df-tr 5259 df-id 5577 df-eprel 5583 df-po 5591 df-so 5592 df-fr 5636 df-we 5638 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-rn 5695 df-res 5696 df-ima 5697 df-pred 6320 df-ord 6386 df-on 6387 df-lim 6388 df-suc 6389 df-iota 6513 df-fun 6562 df-fn 6563 df-f 6564 df-f1 6565 df-fo 6566 df-f1o 6567 df-fv 6568 df-riota 7389 df-ov 7435 df-oprab 7436 df-mpo 7437 df-om 7889 df-1st 8015 df-2nd 8016 df-frecs 8307 df-wrecs 8338 df-recs 8412 df-rdg 8451 df-er 8746 df-en 8987 df-dom 8988 df-sdom 8989 df-pnf 11298 df-mnf 11299 df-xr 11300 df-ltxr 11301 df-le 11302 df-sub 11495 df-neg 11496 df-div 11922 df-nn 12268 df-2 12330 df-rp 13036 df-ioo 13392

This theorem is referenced by: fourierdlem32 46159 fourierdlem33 46160 fourierdlem46 46172 fourierdlem50 46176 fourierdlem72 46198 fourierdlem76 46202 fourierdlem89 46215 fourierdlem91 46217 fourierdlem103 46229 fourierdlem104 46230

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