fourierdlem10 - Mathbox for Glauco Siliprandi

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Theorem fourierdlem10 47234

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 486	. . . 4 ⊢ ((𝜑 ∧ 𝐶 < 𝐴) → (𝐶(,)𝐷) ⊆ (𝐴(,)𝐵))
5	2	rexrd 11364	. . . . . . 7 ⊢ (𝜑 → 𝐶 ∈ ℝ^*)
6	5	adantr 486	. . . . . 6 ⊢ ((𝜑 ∧ 𝐶 < 𝐴) → 𝐶 ∈ ℝ^*)
7		fourierdlem10.4	. . . . . . . 8 ⊢ (𝜑 → 𝐷 ∈ ℝ)
8	7	rexrd 11364	. . . . . . 7 ⊢ (𝜑 → 𝐷 ∈ ℝ^*)
9	8	adantr 486	. . . . . 6 ⊢ ((𝜑 ∧ 𝐶 < 𝐴) → 𝐷 ∈ ℝ^*)
10	2, 1	readdcld 11343	. . . . . . . . 9 ⊢ (𝜑 → (𝐶 + 𝐴) ∈ ℝ)
11	10	rehalfcld 12598	. . . . . . . 8 ⊢ (𝜑 → ((𝐶 + 𝐴) / 2) ∈ ℝ)
12	2, 7	readdcld 11343	. . . . . . . . 9 ⊢ (𝜑 → (𝐶 + 𝐷) ∈ ℝ)
13	12	rehalfcld 12598	. . . . . . . 8 ⊢ (𝜑 → ((𝐶 + 𝐷) / 2) ∈ ℝ)
14	11, 13	ifcld 4528	. . . . . . 7 ⊢ (𝜑 → if(𝐴 ≤ 𝐷, ((𝐶 + 𝐴) / 2), ((𝐶 + 𝐷) / 2)) ∈ ℝ)
15	14	adantr 486	. . . . . 6 ⊢ ((𝜑 ∧ 𝐶 < 𝐴) → if(𝐴 ≤ 𝐷, ((𝐶 + 𝐴) / 2), ((𝐶 + 𝐷) / 2)) ∈ ℝ)
16		simplr 781	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝐶 < 𝐴) ∧ 𝐴 ≤ 𝐷) → 𝐶 < 𝐴)
17	2	ad2antrr 739	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝐶 < 𝐴) ∧ 𝐴 ≤ 𝐷) → 𝐶 ∈ ℝ)
18	1	ad2antrr 739	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝐶 < 𝐴) ∧ 𝐴 ≤ 𝐷) → 𝐴 ∈ ℝ)
19		avglt1 12589	. . . . . . . . . 10 ⊢ ((𝐶 ∈ ℝ ∧ 𝐴 ∈ ℝ) → (𝐶 < 𝐴 ↔ 𝐶 < ((𝐶 + 𝐴) / 2)))
20	17, 18, 19	syl2anc 596	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝐶 < 𝐴) ∧ 𝐴 ≤ 𝐷) → (𝐶 < 𝐴 ↔ 𝐶 < ((𝐶 + 𝐴) / 2)))
21	16, 20	mpbid 235	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝐶 < 𝐴) ∧ 𝐴 ≤ 𝐷) → 𝐶 < ((𝐶 + 𝐴) / 2))
22		iftrue 4487	. . . . . . . . 9 ⊢ (𝐴 ≤ 𝐷 → if(𝐴 ≤ 𝐷, ((𝐶 + 𝐴) / 2), ((𝐶 + 𝐷) / 2)) = ((𝐶 + 𝐴) / 2))
23	22	adantl 487	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝐶 < 𝐴) ∧ 𝐴 ≤ 𝐷) → if(𝐴 ≤ 𝐷, ((𝐶 + 𝐴) / 2), ((𝐶 + 𝐷) / 2)) = ((𝐶 + 𝐴) / 2))
24	21, 23	breqtrrd 5132	. . . . . . 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 486	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ¬ 𝐴 ≤ 𝐷) → 𝐶 < 𝐷)
27	2	adantr 486	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ ¬ 𝐴 ≤ 𝐷) → 𝐶 ∈ ℝ)
28	7	adantr 486	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ ¬ 𝐴 ≤ 𝐷) → 𝐷 ∈ ℝ)
29		avglt1 12589	. . . . . . . . . . 11 ⊢ ((𝐶 ∈ ℝ ∧ 𝐷 ∈ ℝ) → (𝐶 < 𝐷 ↔ 𝐶 < ((𝐶 + 𝐷) / 2)))
30	27, 28, 29	syl2anc 596	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ¬ 𝐴 ≤ 𝐷) → (𝐶 < 𝐷 ↔ 𝐶 < ((𝐶 + 𝐷) / 2)))
31	26, 30	mpbid 235	. . . . . . . . 9 ⊢ ((𝜑 ∧ ¬ 𝐴 ≤ 𝐷) → 𝐶 < ((𝐶 + 𝐷) / 2))
32		iffalse 4490	. . . . . . . . . . 11 ⊢ (¬ 𝐴 ≤ 𝐷 → if(𝐴 ≤ 𝐷, ((𝐶 + 𝐴) / 2), ((𝐶 + 𝐷) / 2)) = ((𝐶 + 𝐷) / 2))
33	32	eqcomd 2766	. . . . . . . . . 10 ⊢ (¬ 𝐴 ≤ 𝐷 → ((𝐶 + 𝐷) / 2) = if(𝐴 ≤ 𝐷, ((𝐶 + 𝐴) / 2), ((𝐶 + 𝐷) / 2)))
34	33	adantl 487	. . . . . . . . 9 ⊢ ((𝜑 ∧ ¬ 𝐴 ≤ 𝐷) → ((𝐶 + 𝐷) / 2) = if(𝐴 ≤ 𝐷, ((𝐶 + 𝐴) / 2), ((𝐶 + 𝐷) / 2)))
35	31, 34	breqtrd 5130	. . . . . . . 8 ⊢ ((𝜑 ∧ ¬ 𝐴 ≤ 𝐷) → 𝐶 < if(𝐴 ≤ 𝐷, ((𝐶 + 𝐴) / 2), ((𝐶 + 𝐷) / 2)))
36	35	adantlr 728	. . . . . . 7 ⊢ (((𝜑 ∧ 𝐶 < 𝐴) ∧ ¬ 𝐴 ≤ 𝐷) → 𝐶 < if(𝐴 ≤ 𝐷, ((𝐶 + 𝐴) / 2), ((𝐶 + 𝐷) / 2)))
37	24, 36	pm2.61dan 825	. . . . . 6 ⊢ ((𝜑 ∧ 𝐶 < 𝐴) → 𝐶 < if(𝐴 ≤ 𝐷, ((𝐶 + 𝐴) / 2), ((𝐶 + 𝐷) / 2)))
38	22	adantl 487	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝐴 ≤ 𝐷) → if(𝐴 ≤ 𝐷, ((𝐶 + 𝐴) / 2), ((𝐶 + 𝐷) / 2)) = ((𝐶 + 𝐴) / 2))
39	10	adantr 486	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝐴 ≤ 𝐷) → (𝐶 + 𝐴) ∈ ℝ)
40	12	adantr 486	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝐴 ≤ 𝐷) → (𝐶 + 𝐷) ∈ ℝ)
41		2rp 13130	. . . . . . . . . . . 12 ⊢ 2 ∈ ℝ⁺
42	41	a1i 11	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝐴 ≤ 𝐷) → 2 ∈ ℝ⁺)
43	1	adantr 486	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝐴 ≤ 𝐷) → 𝐴 ∈ ℝ)
44	7	adantr 486	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝐴 ≤ 𝐷) → 𝐷 ∈ ℝ)
45	2	adantr 486	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝐴 ≤ 𝐷) → 𝐶 ∈ ℝ)
46		simpr 490	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝐴 ≤ 𝐷) → 𝐴 ≤ 𝐷)
47	43, 44, 45, 46	leadd2dd 11936	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝐴 ≤ 𝐷) → (𝐶 + 𝐴) ≤ (𝐶 + 𝐷))
48	39, 40, 42, 47	lediv1dd 13227	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝐴 ≤ 𝐷) → ((𝐶 + 𝐴) / 2) ≤ ((𝐶 + 𝐷) / 2))
49	38, 48	eqbrtrd 5126	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐴 ≤ 𝐷) → if(𝐴 ≤ 𝐷, ((𝐶 + 𝐴) / 2), ((𝐶 + 𝐷) / 2)) ≤ ((𝐶 + 𝐷) / 2))
50	32	adantl 487	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ¬ 𝐴 ≤ 𝐷) → if(𝐴 ≤ 𝐷, ((𝐶 + 𝐴) / 2), ((𝐶 + 𝐷) / 2)) = ((𝐶 + 𝐷) / 2))
51	13	leidd 11887	. . . . . . . . . . 11 ⊢ (𝜑 → ((𝐶 + 𝐷) / 2) ≤ ((𝐶 + 𝐷) / 2))
52	51	adantr 486	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ¬ 𝐴 ≤ 𝐷) → ((𝐶 + 𝐷) / 2) ≤ ((𝐶 + 𝐷) / 2))
53	50, 52	eqbrtrd 5126	. . . . . . . . 9 ⊢ ((𝜑 ∧ ¬ 𝐴 ≤ 𝐷) → if(𝐴 ≤ 𝐷, ((𝐶 + 𝐴) / 2), ((𝐶 + 𝐷) / 2)) ≤ ((𝐶 + 𝐷) / 2))
54	49, 53	pm2.61dan 825	. . . . . . . 8 ⊢ (𝜑 → if(𝐴 ≤ 𝐷, ((𝐶 + 𝐴) / 2), ((𝐶 + 𝐷) / 2)) ≤ ((𝐶 + 𝐷) / 2))
55		avglt2 12590	. . . . . . . . . 10 ⊢ ((𝐶 ∈ ℝ ∧ 𝐷 ∈ ℝ) → (𝐶 < 𝐷 ↔ ((𝐶 + 𝐷) / 2) < 𝐷))
56	2, 7, 55	syl2anc 596	. . . . . . . . 9 ⊢ (𝜑 → (𝐶 < 𝐷 ↔ ((𝐶 + 𝐷) / 2) < 𝐷))
57	25, 56	mpbid 235	. . . . . . . 8 ⊢ (𝜑 → ((𝐶 + 𝐷) / 2) < 𝐷)
58	14, 13, 7, 54, 57	lelttrd 11473	. . . . . . 7 ⊢ (𝜑 → if(𝐴 ≤ 𝐷, ((𝐶 + 𝐴) / 2), ((𝐶 + 𝐷) / 2)) < 𝐷)
59	58	adantr 486	. . . . . 6 ⊢ ((𝜑 ∧ 𝐶 < 𝐴) → if(𝐴 ≤ 𝐷, ((𝐶 + 𝐴) / 2), ((𝐶 + 𝐷) / 2)) < 𝐷)
60	6, 9, 15, 37, 59	eliood 46617	. . . . 5 ⊢ ((𝜑 ∧ 𝐶 < 𝐴) → if(𝐴 ≤ 𝐷, ((𝐶 + 𝐴) / 2), ((𝐶 + 𝐷) / 2)) ∈ (𝐶(,)𝐷))
61	1	adantr 486	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐶 < 𝐴) → 𝐴 ∈ ℝ)
62	11	adantr 486	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐶 < 𝐴) → ((𝐶 + 𝐴) / 2) ∈ ℝ)
63	14	adantr 486	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝐴 ≤ 𝐷) → if(𝐴 ≤ 𝐷, ((𝐶 + 𝐴) / 2), ((𝐶 + 𝐷) / 2)) ∈ ℝ)
64	63, 38	eqled 11418	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝐴 ≤ 𝐷) → if(𝐴 ≤ 𝐷, ((𝐶 + 𝐴) / 2), ((𝐶 + 𝐷) / 2)) ≤ ((𝐶 + 𝐴) / 2))
65	14	adantr 486	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ ¬ 𝐴 ≤ 𝐷) → if(𝐴 ≤ 𝐷, ((𝐶 + 𝐴) / 2), ((𝐶 + 𝐷) / 2)) ∈ ℝ)
66	11	adantr 486	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ ¬ 𝐴 ≤ 𝐷) → ((𝐶 + 𝐴) / 2) ∈ ℝ)
67		simpr 490	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ ¬ 𝐴 ≤ 𝐷) → ¬ 𝐴 ≤ 𝐷)
68	1	adantr 486	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ ¬ 𝐴 ≤ 𝐷) → 𝐴 ∈ ℝ)
69	28, 68	ltnled 11462	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ ¬ 𝐴 ≤ 𝐷) → (𝐷 < 𝐴 ↔ ¬ 𝐴 ≤ 𝐷))
70	67, 69	mpbird 260	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ ¬ 𝐴 ≤ 𝐷) → 𝐷 < 𝐴)
71	12	adantr 486	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝐷 < 𝐴) → (𝐶 + 𝐷) ∈ ℝ)
72	10	adantr 486	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝐷 < 𝐴) → (𝐶 + 𝐴) ∈ ℝ)
73	41	a1i 11	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝐷 < 𝐴) → 2 ∈ ℝ⁺)
74	7	adantr 486	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝐷 < 𝐴) → 𝐷 ∈ ℝ)
75	1	adantr 486	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝐷 < 𝐴) → 𝐴 ∈ ℝ)
76	2	adantr 486	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝐷 < 𝐴) → 𝐶 ∈ ℝ)
77		simpr 490	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝐷 < 𝐴) → 𝐷 < 𝐴)
78	74, 75, 76, 77	ltadd2dd 11474	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝐷 < 𝐴) → (𝐶 + 𝐷) < (𝐶 + 𝐴))
79	71, 72, 73, 78	ltdiv1dd 13226	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝐷 < 𝐴) → ((𝐶 + 𝐷) / 2) < ((𝐶 + 𝐴) / 2))
80	70, 79	syldan 603	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ ¬ 𝐴 ≤ 𝐷) → ((𝐶 + 𝐷) / 2) < ((𝐶 + 𝐴) / 2))
81	50, 80	eqbrtrd 5126	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ ¬ 𝐴 ≤ 𝐷) → if(𝐴 ≤ 𝐷, ((𝐶 + 𝐴) / 2), ((𝐶 + 𝐷) / 2)) < ((𝐶 + 𝐴) / 2))
82	65, 66, 81	ltled 11463	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ ¬ 𝐴 ≤ 𝐷) → if(𝐴 ≤ 𝐷, ((𝐶 + 𝐴) / 2), ((𝐶 + 𝐷) / 2)) ≤ ((𝐶 + 𝐴) / 2))
83	64, 82	pm2.61dan 825	. . . . . . . . . 10 ⊢ (𝜑 → if(𝐴 ≤ 𝐷, ((𝐶 + 𝐴) / 2), ((𝐶 + 𝐷) / 2)) ≤ ((𝐶 + 𝐴) / 2))
84	83	adantr 486	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐶 < 𝐴) → if(𝐴 ≤ 𝐷, ((𝐶 + 𝐴) / 2), ((𝐶 + 𝐷) / 2)) ≤ ((𝐶 + 𝐴) / 2))
85		simpr 490	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝐶 < 𝐴) → 𝐶 < 𝐴)
86	2	adantr 486	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝐶 < 𝐴) → 𝐶 ∈ ℝ)
87		avglt2 12590	. . . . . . . . . . 11 ⊢ ((𝐶 ∈ ℝ ∧ 𝐴 ∈ ℝ) → (𝐶 < 𝐴 ↔ ((𝐶 + 𝐴) / 2) < 𝐴))
88	86, 61, 87	syl2anc 596	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝐶 < 𝐴) → (𝐶 < 𝐴 ↔ ((𝐶 + 𝐴) / 2) < 𝐴))
89	85, 88	mpbid 235	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐶 < 𝐴) → ((𝐶 + 𝐴) / 2) < 𝐴)
90	15, 62, 61, 84, 89	lelttrd 11473	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐶 < 𝐴) → if(𝐴 ≤ 𝐷, ((𝐶 + 𝐴) / 2), ((𝐶 + 𝐷) / 2)) < 𝐴)
91	15, 61, 90	ltnsymd 11464	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐶 < 𝐴) → ¬ 𝐴 < if(𝐴 ≤ 𝐷, ((𝐶 + 𝐴) / 2), ((𝐶 + 𝐷) / 2)))
92	91	intn3an2d 1511	. . . . . 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 11364	. . . . . . . 8 ⊢ (𝜑 → 𝐴 ∈ ℝ^*)
94	93	adantr 486	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐶 < 𝐴) → 𝐴 ∈ ℝ^*)
95		fourierdlem10.2	. . . . . . . . 9 ⊢ (𝜑 → 𝐵 ∈ ℝ)
96	95	rexrd 11364	. . . . . . . 8 ⊢ (𝜑 → 𝐵 ∈ ℝ^*)
97	96	adantr 486	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐶 < 𝐴) → 𝐵 ∈ ℝ^*)
98		elioo2 13522	. . . . . . 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 596	. . . . . 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 328	. . . . 5 ⊢ ((𝜑 ∧ 𝐶 < 𝐴) → ¬ if(𝐴 ≤ 𝐷, ((𝐶 + 𝐴) / 2), ((𝐶 + 𝐷) / 2)) ∈ (𝐴(,)𝐵))
101		nelss 3975	. . . . 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 596	. . . 4 ⊢ ((𝜑 ∧ 𝐶 < 𝐴) → ¬ (𝐶(,)𝐷) ⊆ (𝐴(,)𝐵))
103	4, 102	pm2.65da 829	. . 3 ⊢ (𝜑 → ¬ 𝐶 < 𝐴)
104	1, 2, 103	nltled 11465	. 2 ⊢ (𝜑 → 𝐴 ≤ 𝐶)
105	3	adantr 486	. . . 4 ⊢ ((𝜑 ∧ 𝐵 < 𝐷) → (𝐶(,)𝐷) ⊆ (𝐴(,)𝐵))
106	5	adantr 486	. . . . . 6 ⊢ ((𝜑 ∧ 𝐵 < 𝐷) → 𝐶 ∈ ℝ^*)
107	8	adantr 486	. . . . . 6 ⊢ ((𝜑 ∧ 𝐵 < 𝐷) → 𝐷 ∈ ℝ^*)
108	95, 7	readdcld 11343	. . . . . . . . 9 ⊢ (𝜑 → (𝐵 + 𝐷) ∈ ℝ)
109	108	rehalfcld 12598	. . . . . . . 8 ⊢ (𝜑 → ((𝐵 + 𝐷) / 2) ∈ ℝ)
110	109, 13	ifcld 4528	. . . . . . 7 ⊢ (𝜑 → if(𝐶 ≤ 𝐵, ((𝐵 + 𝐷) / 2), ((𝐶 + 𝐷) / 2)) ∈ ℝ)
111	110	adantr 486	. . . . . 6 ⊢ ((𝜑 ∧ 𝐵 < 𝐷) → if(𝐶 ≤ 𝐵, ((𝐵 + 𝐷) / 2), ((𝐶 + 𝐷) / 2)) ∈ ℝ)
112	2	adantr 486	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐶 ≤ 𝐵) → 𝐶 ∈ ℝ)
113	13	adantr 486	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐶 ≤ 𝐵) → ((𝐶 + 𝐷) / 2) ∈ ℝ)
114	110	adantr 486	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐶 ≤ 𝐵) → if(𝐶 ≤ 𝐵, ((𝐵 + 𝐷) / 2), ((𝐶 + 𝐷) / 2)) ∈ ℝ)
115	2, 7, 29	syl2anc 596	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐶 < 𝐷 ↔ 𝐶 < ((𝐶 + 𝐷) / 2)))
116	25, 115	mpbid 235	. . . . . . . . . 10 ⊢ (𝜑 → 𝐶 < ((𝐶 + 𝐷) / 2))
117	116	adantr 486	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐶 ≤ 𝐵) → 𝐶 < ((𝐶 + 𝐷) / 2))
118	12	adantr 486	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝐶 ≤ 𝐵) → (𝐶 + 𝐷) ∈ ℝ)
119	108	adantr 486	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝐶 ≤ 𝐵) → (𝐵 + 𝐷) ∈ ℝ)
120	41	a1i 11	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝐶 ≤ 𝐵) → 2 ∈ ℝ⁺)
121	95	adantr 486	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝐶 ≤ 𝐵) → 𝐵 ∈ ℝ)
122	7	adantr 486	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝐶 ≤ 𝐵) → 𝐷 ∈ ℝ)
123		simpr 490	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝐶 ≤ 𝐵) → 𝐶 ≤ 𝐵)
124	112, 121, 122, 123	leadd1dd 11935	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝐶 ≤ 𝐵) → (𝐶 + 𝐷) ≤ (𝐵 + 𝐷))
125	118, 119, 120, 124	lediv1dd 13227	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝐶 ≤ 𝐵) → ((𝐶 + 𝐷) / 2) ≤ ((𝐵 + 𝐷) / 2))
126		iftrue 4487	. . . . . . . . . . 11 ⊢ (𝐶 ≤ 𝐵 → if(𝐶 ≤ 𝐵, ((𝐵 + 𝐷) / 2), ((𝐶 + 𝐷) / 2)) = ((𝐵 + 𝐷) / 2))
127	126	adantl 487	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝐶 ≤ 𝐵) → if(𝐶 ≤ 𝐵, ((𝐵 + 𝐷) / 2), ((𝐶 + 𝐷) / 2)) = ((𝐵 + 𝐷) / 2))
128	125, 127	breqtrrd 5132	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐶 ≤ 𝐵) → ((𝐶 + 𝐷) / 2) ≤ if(𝐶 ≤ 𝐵, ((𝐵 + 𝐷) / 2), ((𝐶 + 𝐷) / 2)))
129	112, 113, 114, 117, 128	ltletrd 11475	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐶 ≤ 𝐵) → 𝐶 < if(𝐶 ≤ 𝐵, ((𝐵 + 𝐷) / 2), ((𝐶 + 𝐷) / 2)))
130	116	adantr 486	. . . . . . . . 9 ⊢ ((𝜑 ∧ ¬ 𝐶 ≤ 𝐵) → 𝐶 < ((𝐶 + 𝐷) / 2))
131		iffalse 4490	. . . . . . . . . . 11 ⊢ (¬ 𝐶 ≤ 𝐵 → if(𝐶 ≤ 𝐵, ((𝐵 + 𝐷) / 2), ((𝐶 + 𝐷) / 2)) = ((𝐶 + 𝐷) / 2))
132	131	eqcomd 2766	. . . . . . . . . 10 ⊢ (¬ 𝐶 ≤ 𝐵 → ((𝐶 + 𝐷) / 2) = if(𝐶 ≤ 𝐵, ((𝐵 + 𝐷) / 2), ((𝐶 + 𝐷) / 2)))
133	132	adantl 487	. . . . . . . . 9 ⊢ ((𝜑 ∧ ¬ 𝐶 ≤ 𝐵) → ((𝐶 + 𝐷) / 2) = if(𝐶 ≤ 𝐵, ((𝐵 + 𝐷) / 2), ((𝐶 + 𝐷) / 2)))
134	130, 133	breqtrd 5130	. . . . . . . 8 ⊢ ((𝜑 ∧ ¬ 𝐶 ≤ 𝐵) → 𝐶 < if(𝐶 ≤ 𝐵, ((𝐵 + 𝐷) / 2), ((𝐶 + 𝐷) / 2)))
135	129, 134	pm2.61dan 825	. . . . . . 7 ⊢ (𝜑 → 𝐶 < if(𝐶 ≤ 𝐵, ((𝐵 + 𝐷) / 2), ((𝐶 + 𝐷) / 2)))
136	135	adantr 486	. . . . . 6 ⊢ ((𝜑 ∧ 𝐵 < 𝐷) → 𝐶 < if(𝐶 ≤ 𝐵, ((𝐵 + 𝐷) / 2), ((𝐶 + 𝐷) / 2)))
137	126	adantl 487	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝐵 < 𝐷) ∧ 𝐶 ≤ 𝐵) → if(𝐶 ≤ 𝐵, ((𝐵 + 𝐷) / 2), ((𝐶 + 𝐷) / 2)) = ((𝐵 + 𝐷) / 2))
138		simpr 490	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝐵 < 𝐷) → 𝐵 < 𝐷)
139	95	adantr 486	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝐵 < 𝐷) → 𝐵 ∈ ℝ)
140	7	adantr 486	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝐵 < 𝐷) → 𝐷 ∈ ℝ)
141		avglt2 12590	. . . . . . . . . . 11 ⊢ ((𝐵 ∈ ℝ ∧ 𝐷 ∈ ℝ) → (𝐵 < 𝐷 ↔ ((𝐵 + 𝐷) / 2) < 𝐷))
142	139, 140, 141	syl2anc 596	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝐵 < 𝐷) → (𝐵 < 𝐷 ↔ ((𝐵 + 𝐷) / 2) < 𝐷))
143	138, 142	mpbid 235	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐵 < 𝐷) → ((𝐵 + 𝐷) / 2) < 𝐷)
144	143	adantr 486	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝐵 < 𝐷) ∧ 𝐶 ≤ 𝐵) → ((𝐵 + 𝐷) / 2) < 𝐷)
145	137, 144	eqbrtrd 5126	. . . . . . 7 ⊢ (((𝜑 ∧ 𝐵 < 𝐷) ∧ 𝐶 ≤ 𝐵) → if(𝐶 ≤ 𝐵, ((𝐵 + 𝐷) / 2), ((𝐶 + 𝐷) / 2)) < 𝐷)
146	131	adantl 487	. . . . . . . . 9 ⊢ ((𝜑 ∧ ¬ 𝐶 ≤ 𝐵) → if(𝐶 ≤ 𝐵, ((𝐵 + 𝐷) / 2), ((𝐶 + 𝐷) / 2)) = ((𝐶 + 𝐷) / 2))
147	57	adantr 486	. . . . . . . . 9 ⊢ ((𝜑 ∧ ¬ 𝐶 ≤ 𝐵) → ((𝐶 + 𝐷) / 2) < 𝐷)
148	146, 147	eqbrtrd 5126	. . . . . . . 8 ⊢ ((𝜑 ∧ ¬ 𝐶 ≤ 𝐵) → if(𝐶 ≤ 𝐵, ((𝐵 + 𝐷) / 2), ((𝐶 + 𝐷) / 2)) < 𝐷)
149	148	adantlr 728	. . . . . . 7 ⊢ (((𝜑 ∧ 𝐵 < 𝐷) ∧ ¬ 𝐶 ≤ 𝐵) → if(𝐶 ≤ 𝐵, ((𝐵 + 𝐷) / 2), ((𝐶 + 𝐷) / 2)) < 𝐷)
150	145, 149	pm2.61dan 825	. . . . . 6 ⊢ ((𝜑 ∧ 𝐵 < 𝐷) → if(𝐶 ≤ 𝐵, ((𝐵 + 𝐷) / 2), ((𝐶 + 𝐷) / 2)) < 𝐷)
151	106, 107, 111, 136, 150	eliood 46617	. . . . 5 ⊢ ((𝜑 ∧ 𝐵 < 𝐷) → if(𝐶 ≤ 𝐵, ((𝐵 + 𝐷) / 2), ((𝐶 + 𝐷) / 2)) ∈ (𝐶(,)𝐷))
152	109	adantr 486	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝐵 < 𝐷) → ((𝐵 + 𝐷) / 2) ∈ ℝ)
153		avglt1 12589	. . . . . . . . . . . . . 14 ⊢ ((𝐵 ∈ ℝ ∧ 𝐷 ∈ ℝ) → (𝐵 < 𝐷 ↔ 𝐵 < ((𝐵 + 𝐷) / 2)))
154	139, 140, 153	syl2anc 596	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝐵 < 𝐷) → (𝐵 < 𝐷 ↔ 𝐵 < ((𝐵 + 𝐷) / 2)))
155	138, 154	mpbid 235	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝐵 < 𝐷) → 𝐵 < ((𝐵 + 𝐷) / 2))
156	139, 152, 155	ltled 11463	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝐵 < 𝐷) → 𝐵 ≤ ((𝐵 + 𝐷) / 2))
157	156	adantr 486	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝐵 < 𝐷) ∧ 𝐶 ≤ 𝐵) → 𝐵 ≤ ((𝐵 + 𝐷) / 2))
158	157, 137	breqtrrd 5132	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝐵 < 𝐷) ∧ 𝐶 ≤ 𝐵) → 𝐵 ≤ if(𝐶 ≤ 𝐵, ((𝐵 + 𝐷) / 2), ((𝐶 + 𝐷) / 2)))
159	95	adantr 486	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ ¬ 𝐶 ≤ 𝐵) → 𝐵 ∈ ℝ)
160	13	adantr 486	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ ¬ 𝐶 ≤ 𝐵) → ((𝐶 + 𝐷) / 2) ∈ ℝ)
161	2	adantr 486	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ ¬ 𝐶 ≤ 𝐵) → 𝐶 ∈ ℝ)
162		simpr 490	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ ¬ 𝐶 ≤ 𝐵) → ¬ 𝐶 ≤ 𝐵)
163	159, 161	ltnled 11462	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ ¬ 𝐶 ≤ 𝐵) → (𝐵 < 𝐶 ↔ ¬ 𝐶 ≤ 𝐵))
164	162, 163	mpbird 260	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ ¬ 𝐶 ≤ 𝐵) → 𝐵 < 𝐶)
165	159, 161, 160, 164, 130	lttrd 11476	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ ¬ 𝐶 ≤ 𝐵) → 𝐵 < ((𝐶 + 𝐷) / 2))
166	159, 160, 165	ltled 11463	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ ¬ 𝐶 ≤ 𝐵) → 𝐵 ≤ ((𝐶 + 𝐷) / 2))
167	166, 133	breqtrd 5130	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ¬ 𝐶 ≤ 𝐵) → 𝐵 ≤ if(𝐶 ≤ 𝐵, ((𝐵 + 𝐷) / 2), ((𝐶 + 𝐷) / 2)))
168	167	adantlr 728	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝐵 < 𝐷) ∧ ¬ 𝐶 ≤ 𝐵) → 𝐵 ≤ if(𝐶 ≤ 𝐵, ((𝐵 + 𝐷) / 2), ((𝐶 + 𝐷) / 2)))
169	158, 168	pm2.61dan 825	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐵 < 𝐷) → 𝐵 ≤ if(𝐶 ≤ 𝐵, ((𝐵 + 𝐷) / 2), ((𝐶 + 𝐷) / 2)))
170	139, 111, 169	lensymd 11466	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐵 < 𝐷) → ¬ if(𝐶 ≤ 𝐵, ((𝐵 + 𝐷) / 2), ((𝐶 + 𝐷) / 2)) < 𝐵)
171	170	intn3an3d 1512	. . . . . 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 486	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐵 < 𝐷) → 𝐴 ∈ ℝ^*)
173	96	adantr 486	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐵 < 𝐷) → 𝐵 ∈ ℝ^*)
174		elioo2 13522	. . . . . . 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 596	. . . . . 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 328	. . . . 5 ⊢ ((𝜑 ∧ 𝐵 < 𝐷) → ¬ if(𝐶 ≤ 𝐵, ((𝐵 + 𝐷) / 2), ((𝐶 + 𝐷) / 2)) ∈ (𝐴(,)𝐵))
177		nelss 3975	. . . . 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 596	. . . 4 ⊢ ((𝜑 ∧ 𝐵 < 𝐷) → ¬ (𝐶(,)𝐷) ⊆ (𝐴(,)𝐵))
179	105, 178	pm2.65da 829	. . 3 ⊢ (𝜑 → ¬ 𝐵 < 𝐷)
180	7, 95, 179	nltled 11465	. 2 ⊢ (𝜑 → 𝐷 ≤ 𝐵)
181	104, 180	jca 521	1 ⊢ (𝜑 → (𝐴 ≤ 𝐶 ∧ 𝐷 ≤ 𝐵))

Colors of variables: wff setvar class

This proof depends on syntax axioms: ¬ wn 3 → wi 4 ↔ wb 209 ∧ wa 401 ∧ w3a 1103 = wceq 1569 ∈ wcel 2144 ⊆ wss 3896 ifcif 4481 class class class wbr 5102 (class class class)co 7422 ℝcr 11204 + caddc 11208 ℝ^*cxr 11347 < clt 11348 ≤ cle 11349 / cdiv 11978 2c2 12402 ℝ⁺crp 13125 (,)cioo 13481

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1827 ax-4 1841 ax-5 1942 ax-6 1999 ax-7 2040 ax-8 2146 ax-9 2154 ax-10 2177 ax-11 2193 ax-12 2212 ax-ext 2732 ax-sep 5248 ax-nul 5259 ax-pow 5326 ax-pr 5390 ax-un 7753 ax-cnex 11261 ax-resscn 11262 ax-1cn 11263 ax-icn 11264 ax-addcl 11265 ax-addrcl 11266 ax-mulcl 11267 ax-mulrcl 11268 ax-mulcom 11269 ax-addass 11270 ax-mulass 11271 ax-distr 11272 ax-i2m1 11273 ax-1ne0 11274 ax-1rid 11275 ax-rnegex 11276 ax-rrecex 11277 ax-cnre 11278 ax-pre-lttri 11279 ax-pre-lttrn 11280 ax-pre-ltadd 11281 ax-pre-mulgt0 11282

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1572 df-fal 1582 df-ex 1812 df-nf 1816 df-sb 2099 df-mo 2564 df-eu 2594 df-clab 2739 df-cleq 2752 df-clel 2835 df-nfc 2909 df-ne 2956 df-nel 3062 df-ral 3077 df-rex 3087 df-rmo 3365 df-reu 3366 df-rab 3413 df-v 3452 df-sbc 3739 df-csb 3848 df-ss 3897 df-pss 4013 df-dif 4046 df-in 4090 df-un 4150 df-nul 4278 df-if 4482 df-pw 4558 df-sn 4585 df-pr 4587 df-op 4835 df-uni 4867 df-iun 4952 df-br 5103 df-opab 5167 df-mpt 5186 df-tr 5212 df-id 5545 df-eprel 5550 df-po 5558 df-so 5559 df-fr 5603 df-we 5605 df-xp 5656 df-rel 5657 df-cnv 5658 df-co 5659 df-dm 5660 df-rn 5661 df-res 5662 df-ima 5663 df-pred 6306 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6496 df-fun 6543 df-fn 6544 df-f 6545 df-f1 6546 df-fo 6547 df-f1o 6548 df-fv 6549 df-riota 7379 df-ov 7425 df-oprab 7426 df-mpo 7427 df-om 7880 df-1st 8003 df-2nd 8004 df-frecs 8304 df-wrecs 8335 df-recs 8384 df-rdg 8423 df-er 8722 df-en 8979 df-dom 8980 df-sdom 8981 df-pnf 11350 df-mnf 11351 df-xr 11352 df-ltxr 11353 df-le 11354 df-sub 11548 df-neg 11549 df-div 11979 df-nn 12341 df-2 12410 df-rp 13126 df-ioo 13485

This theorem is used by: fourierdlem32 47256 fourierdlem33 47257 fourierdlem46 47269 fourierdlem50 47273 fourierdlem72 47295 fourierdlem76 47299 fourierdlem89 47312 fourierdlem91 47314 fourierdlem103 47326 fourierdlem104 47327

W3C validator