dvbdfbdioolem2 - Mathbox for Glauco Siliprandi

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Theorem dvbdfbdioolem2 47212

Description: A function on an open interval, with bounded derivative, is bounded. (Contributed by Glauco Siliprandi, 11-Dec-2019.)

**Hypotheses**
Ref	Expression
dvbdfbdioolem2.a	⊢ (𝜑 → 𝐴 ∈ ℝ)
dvbdfbdioolem2.b	⊢ (𝜑 → 𝐵 ∈ ℝ)
dvbdfbdioolem2.altb	⊢ (𝜑 → 𝐴 < 𝐵)
dvbdfbdioolem2.f	⊢ (𝜑 → 𝐹:(𝐴(,)𝐵)⟶ℝ)
dvbdfbdioolem2.dmdv	⊢ (𝜑 → dom (ℝ D 𝐹) = (𝐴(,)𝐵))
dvbdfbdioolem2.k	⊢ (𝜑 → 𝐾 ∈ ℝ)
dvbdfbdioolem2.dvbd	⊢ (𝜑 → ∀𝑥 ∈ (𝐴(,)𝐵)(abs‘((ℝ D 𝐹)‘𝑥)) ≤ 𝐾)
dvbdfbdioolem2.m	⊢ 𝑀 = ((abs‘(𝐹‘((𝐴 + 𝐵) / 2))) + (𝐾 · (𝐵 − 𝐴)))

**Assertion**
Ref	Expression
dvbdfbdioolem2	⊢ (𝜑 → ∀𝑥 ∈ (𝐴(,)𝐵)(abs‘(𝐹‘𝑥)) ≤ 𝑀)

Distinct variable groups: 𝑥,𝐴 𝑥,𝐵 𝑥,𝐹 𝑥,𝐾 𝜑,𝑥 Allowed substitution hint: 𝑀(𝑥)

Proof of Theorem dvbdfbdioolem2 Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		dvbdfbdioolem2.f	. . . . . . . 8 ⊢ (𝜑 → 𝐹:(𝐴(,)𝐵)⟶ℝ)
2	1	ffvelcdmda 7116	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → (𝐹‘𝑥) ∈ ℝ)
3	2	recnd 11376	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → (𝐹‘𝑥) ∈ ℂ)
4	3	abscld 15650	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → (abs‘(𝐹‘𝑥)) ∈ ℝ)
5		dvbdfbdioolem2.a	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐴 ∈ ℝ)
6	5	rexrd 11398	. . . . . . . . . 10 ⊢ (𝜑 → 𝐴 ∈ ℝ^*)
7		dvbdfbdioolem2.b	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐵 ∈ ℝ)
8	7	rexrd 11398	. . . . . . . . . 10 ⊢ (𝜑 → 𝐵 ∈ ℝ^*)
9	5, 7	readdcld 11377	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐴 + 𝐵) ∈ ℝ)
10	9	rehalfcld 12633	. . . . . . . . . 10 ⊢ (𝜑 → ((𝐴 + 𝐵) / 2) ∈ ℝ)
11		dvbdfbdioolem2.altb	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐴 < 𝐵)
12		avglt1 12624	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 < 𝐵 ↔ 𝐴 < ((𝐴 + 𝐵) / 2)))
13	5, 7, 12	syl2anc 596	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐴 < 𝐵 ↔ 𝐴 < ((𝐴 + 𝐵) / 2)))
14	11, 13	mpbid 235	. . . . . . . . . 10 ⊢ (𝜑 → 𝐴 < ((𝐴 + 𝐵) / 2))
15		avglt2 12625	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 < 𝐵 ↔ ((𝐴 + 𝐵) / 2) < 𝐵))
16	5, 7, 15	syl2anc 596	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐴 < 𝐵 ↔ ((𝐴 + 𝐵) / 2) < 𝐵))
17	11, 16	mpbid 235	. . . . . . . . . 10 ⊢ (𝜑 → ((𝐴 + 𝐵) / 2) < 𝐵)
18	6, 8, 10, 14, 17	eliood 46783	. . . . . . . . 9 ⊢ (𝜑 → ((𝐴 + 𝐵) / 2) ∈ (𝐴(,)𝐵))
19	1, 18	ffvelcdmd 7117	. . . . . . . 8 ⊢ (𝜑 → (𝐹‘((𝐴 + 𝐵) / 2)) ∈ ℝ)
20	19	recnd 11376	. . . . . . 7 ⊢ (𝜑 → (𝐹‘((𝐴 + 𝐵) / 2)) ∈ ℂ)
21	20	abscld 15650	. . . . . 6 ⊢ (𝜑 → (abs‘(𝐹‘((𝐴 + 𝐵) / 2))) ∈ ℝ)
22	21	adantr 486	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → (abs‘(𝐹‘((𝐴 + 𝐵) / 2))) ∈ ℝ)
23	4, 22	resubcld 11783	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → ((abs‘(𝐹‘𝑥)) − (abs‘(𝐹‘((𝐴 + 𝐵) / 2)))) ∈ ℝ)
24		dvbdfbdioolem2.k	. . . . . 6 ⊢ (𝜑 → 𝐾 ∈ ℝ)
25	24	adantr 486	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → 𝐾 ∈ ℝ)
26	7	adantr 486	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → 𝐵 ∈ ℝ)
27	5	adantr 486	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → 𝐴 ∈ ℝ)
28	26, 27	resubcld 11783	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → (𝐵 − 𝐴) ∈ ℝ)
29	25, 28	remulcld 11378	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → (𝐾 · (𝐵 − 𝐴)) ∈ ℝ)
30	20	adantr 486	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → (𝐹‘((𝐴 + 𝐵) / 2)) ∈ ℂ)
31	3, 30	subcld 11708	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → ((𝐹‘𝑥) − (𝐹‘((𝐴 + 𝐵) / 2))) ∈ ℂ)
32	31	abscld 15650	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → (abs‘((𝐹‘𝑥) − (𝐹‘((𝐴 + 𝐵) / 2)))) ∈ ℝ)
33	3, 30	abs2difd 15671	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → ((abs‘(𝐹‘𝑥)) − (abs‘(𝐹‘((𝐴 + 𝐵) / 2)))) ≤ (abs‘((𝐹‘𝑥) − (𝐹‘((𝐴 + 𝐵) / 2)))))
34		simpll 779	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) ∧ ((𝐴 + 𝐵) / 2) < 𝑥) → 𝜑)
35	10	rexrd 11398	. . . . . . . . 9 ⊢ (𝜑 → ((𝐴 + 𝐵) / 2) ∈ ℝ^*)
36	35	ad2antrr 739	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) ∧ ((𝐴 + 𝐵) / 2) < 𝑥) → ((𝐴 + 𝐵) / 2) ∈ ℝ^*)
37	8	ad2antrr 739	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) ∧ ((𝐴 + 𝐵) / 2) < 𝑥) → 𝐵 ∈ ℝ^*)
38		elioore 13546	. . . . . . . . . 10 ⊢ (𝑥 ∈ (𝐴(,)𝐵) → 𝑥 ∈ ℝ)
39	38	adantl 487	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → 𝑥 ∈ ℝ)
40	39	adantr 486	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) ∧ ((𝐴 + 𝐵) / 2) < 𝑥) → 𝑥 ∈ ℝ)
41		simpr 490	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) ∧ ((𝐴 + 𝐵) / 2) < 𝑥) → ((𝐴 + 𝐵) / 2) < 𝑥)
42	6	adantr 486	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → 𝐴 ∈ ℝ^*)
43	8	adantr 486	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → 𝐵 ∈ ℝ^*)
44		simpr 490	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → 𝑥 ∈ (𝐴(,)𝐵))
45		iooltub 46795	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝑥 ∈ (𝐴(,)𝐵)) → 𝑥 < 𝐵)
46	42, 43, 44, 45	syl3anc 1399	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → 𝑥 < 𝐵)
47	46	adantr 486	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) ∧ ((𝐴 + 𝐵) / 2) < 𝑥) → 𝑥 < 𝐵)
48	36, 37, 40, 41, 47	eliood 46783	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) ∧ ((𝐴 + 𝐵) / 2) < 𝑥) → 𝑥 ∈ (((𝐴 + 𝐵) / 2)(,)𝐵))
49	5	adantr 486	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (((𝐴 + 𝐵) / 2)(,)𝐵)) → 𝐴 ∈ ℝ)
50	7	adantr 486	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (((𝐴 + 𝐵) / 2)(,)𝐵)) → 𝐵 ∈ ℝ)
51	1	adantr 486	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (((𝐴 + 𝐵) / 2)(,)𝐵)) → 𝐹:(𝐴(,)𝐵)⟶ℝ)
52		dvbdfbdioolem2.dmdv	. . . . . . . . . 10 ⊢ (𝜑 → dom (ℝ D 𝐹) = (𝐴(,)𝐵))
53	52	adantr 486	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (((𝐴 + 𝐵) / 2)(,)𝐵)) → dom (ℝ D 𝐹) = (𝐴(,)𝐵))
54	24	adantr 486	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (((𝐴 + 𝐵) / 2)(,)𝐵)) → 𝐾 ∈ ℝ)
55		dvbdfbdioolem2.dvbd	. . . . . . . . . . 11 ⊢ (𝜑 → ∀𝑥 ∈ (𝐴(,)𝐵)(abs‘((ℝ D 𝐹)‘𝑥)) ≤ 𝐾)
56		2fveq3 6921	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑦 → (abs‘((ℝ D 𝐹)‘𝑥)) = (abs‘((ℝ D 𝐹)‘𝑦)))
57	56	breq1d 5118	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑦 → ((abs‘((ℝ D 𝐹)‘𝑥)) ≤ 𝐾 ↔ (abs‘((ℝ D 𝐹)‘𝑦)) ≤ 𝐾))
58	57	cbvralvw 3244	. . . . . . . . . . 11 ⊢ (∀𝑥 ∈ (𝐴(,)𝐵)(abs‘((ℝ D 𝐹)‘𝑥)) ≤ 𝐾 ↔ ∀𝑦 ∈ (𝐴(,)𝐵)(abs‘((ℝ D 𝐹)‘𝑦)) ≤ 𝐾)
59	55, 58	sylib 221	. . . . . . . . . 10 ⊢ (𝜑 → ∀𝑦 ∈ (𝐴(,)𝐵)(abs‘((ℝ D 𝐹)‘𝑦)) ≤ 𝐾)
60	59	adantr 486	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (((𝐴 + 𝐵) / 2)(,)𝐵)) → ∀𝑦 ∈ (𝐴(,)𝐵)(abs‘((ℝ D 𝐹)‘𝑦)) ≤ 𝐾)
61	18	adantr 486	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (((𝐴 + 𝐵) / 2)(,)𝐵)) → ((𝐴 + 𝐵) / 2) ∈ (𝐴(,)𝐵))
62		simpr 490	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (((𝐴 + 𝐵) / 2)(,)𝐵)) → 𝑥 ∈ (((𝐴 + 𝐵) / 2)(,)𝐵))
63	49, 50, 51, 53, 54, 60, 61, 62	dvbdfbdioolem1 47211	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (((𝐴 + 𝐵) / 2)(,)𝐵)) → ((abs‘((𝐹‘𝑥) − (𝐹‘((𝐴 + 𝐵) / 2)))) ≤ (𝐾 · (𝑥 − ((𝐴 + 𝐵) / 2))) ∧ (abs‘((𝐹‘𝑥) − (𝐹‘((𝐴 + 𝐵) / 2)))) ≤ (𝐾 · (𝐵 − 𝐴))))
64	63	simprd 501	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (((𝐴 + 𝐵) / 2)(,)𝐵)) → (abs‘((𝐹‘𝑥) − (𝐹‘((𝐴 + 𝐵) / 2)))) ≤ (𝐾 · (𝐵 − 𝐴)))
65	34, 48, 64	syl2anc 596	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) ∧ ((𝐴 + 𝐵) / 2) < 𝑥) → (abs‘((𝐹‘𝑥) − (𝐹‘((𝐴 + 𝐵) / 2)))) ≤ (𝐾 · (𝐵 − 𝐴)))
66		fveq2 6916	. . . . . . . . . . . . . 14 ⊢ (((𝐴 + 𝐵) / 2) = 𝑥 → (𝐹‘((𝐴 + 𝐵) / 2)) = (𝐹‘𝑥))
67	66	eqcomd 2769	. . . . . . . . . . . . 13 ⊢ (((𝐴 + 𝐵) / 2) = 𝑥 → (𝐹‘𝑥) = (𝐹‘((𝐴 + 𝐵) / 2)))
68	67	adantl 487	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ ((𝐴 + 𝐵) / 2) = 𝑥) → (𝐹‘𝑥) = (𝐹‘((𝐴 + 𝐵) / 2)))
69	20	adantr 486	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ ((𝐴 + 𝐵) / 2) = 𝑥) → (𝐹‘((𝐴 + 𝐵) / 2)) ∈ ℂ)
70	68, 69	eqeltrd 2864	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ ((𝐴 + 𝐵) / 2) = 𝑥) → (𝐹‘𝑥) ∈ ℂ)
71	70, 68	subeq0bd 11781	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ((𝐴 + 𝐵) / 2) = 𝑥) → ((𝐹‘𝑥) − (𝐹‘((𝐴 + 𝐵) / 2))) = 0)
72	71	abs00bd 15501	. . . . . . . . 9 ⊢ ((𝜑 ∧ ((𝐴 + 𝐵) / 2) = 𝑥) → (abs‘((𝐹‘𝑥) − (𝐹‘((𝐴 + 𝐵) / 2)))) = 0)
73	24	adantr 486	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ((𝐴 + 𝐵) / 2) = 𝑥) → 𝐾 ∈ ℝ)
74	7	adantr 486	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ ((𝐴 + 𝐵) / 2) = 𝑥) → 𝐵 ∈ ℝ)
75	5	adantr 486	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ ((𝐴 + 𝐵) / 2) = 𝑥) → 𝐴 ∈ ℝ)
76	74, 75	resubcld 11783	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ((𝐴 + 𝐵) / 2) = 𝑥) → (𝐵 − 𝐴) ∈ ℝ)
77		0red 11350	. . . . . . . . . . . 12 ⊢ (𝜑 → 0 ∈ ℝ)
78		ioossre 13578	. . . . . . . . . . . . . . . 16 ⊢ (𝐴(,)𝐵) ⊆ ℝ
79		dvfre 26326	. . . . . . . . . . . . . . . 16 ⊢ ((𝐹:(𝐴(,)𝐵)⟶ℝ ∧ (𝐴(,)𝐵) ⊆ ℝ) → (ℝ D 𝐹):dom (ℝ D 𝐹)⟶ℝ)
80	1, 78, 79	sylancl 598	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (ℝ D 𝐹):dom (ℝ D 𝐹)⟶ℝ)
81	18, 52	eleqtrrd 2867	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → ((𝐴 + 𝐵) / 2) ∈ dom (ℝ D 𝐹))
82	80, 81	ffvelcdmd 7117	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ((ℝ D 𝐹)‘((𝐴 + 𝐵) / 2)) ∈ ℝ)
83	82	recnd 11376	. . . . . . . . . . . . 13 ⊢ (𝜑 → ((ℝ D 𝐹)‘((𝐴 + 𝐵) / 2)) ∈ ℂ)
84	83	abscld 15650	. . . . . . . . . . . 12 ⊢ (𝜑 → (abs‘((ℝ D 𝐹)‘((𝐴 + 𝐵) / 2))) ∈ ℝ)
85	83	absge0d 15658	. . . . . . . . . . . 12 ⊢ (𝜑 → 0 ≤ (abs‘((ℝ D 𝐹)‘((𝐴 + 𝐵) / 2))))
86		2fveq3 6921	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = ((𝐴 + 𝐵) / 2) → (abs‘((ℝ D 𝐹)‘𝑥)) = (abs‘((ℝ D 𝐹)‘((𝐴 + 𝐵) / 2))))
87	86	breq1d 5118	. . . . . . . . . . . . . 14 ⊢ (𝑥 = ((𝐴 + 𝐵) / 2) → ((abs‘((ℝ D 𝐹)‘𝑥)) ≤ 𝐾 ↔ (abs‘((ℝ D 𝐹)‘((𝐴 + 𝐵) / 2))) ≤ 𝐾))
88	87	rspccva 3579	. . . . . . . . . . . . 13 ⊢ ((∀𝑥 ∈ (𝐴(,)𝐵)(abs‘((ℝ D 𝐹)‘𝑥)) ≤ 𝐾 ∧ ((𝐴 + 𝐵) / 2) ∈ (𝐴(,)𝐵)) → (abs‘((ℝ D 𝐹)‘((𝐴 + 𝐵) / 2))) ≤ 𝐾)
89	55, 18, 88	syl2anc 596	. . . . . . . . . . . 12 ⊢ (𝜑 → (abs‘((ℝ D 𝐹)‘((𝐴 + 𝐵) / 2))) ≤ 𝐾)
90	77, 84, 24, 85, 89	letrd 11506	. . . . . . . . . . 11 ⊢ (𝜑 → 0 ≤ 𝐾)
91	90	adantr 486	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ((𝐴 + 𝐵) / 2) = 𝑥) → 0 ≤ 𝐾)
92	7, 5	resubcld 11783	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐵 − 𝐴) ∈ ℝ)
93	5, 7	posdifd 11943	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝐴 < 𝐵 ↔ 0 < (𝐵 − 𝐴)))
94	11, 93	mpbid 235	. . . . . . . . . . . 12 ⊢ (𝜑 → 0 < (𝐵 − 𝐴))
95	77, 92, 94	ltled 11497	. . . . . . . . . . 11 ⊢ (𝜑 → 0 ≤ (𝐵 − 𝐴))
96	95	adantr 486	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ((𝐴 + 𝐵) / 2) = 𝑥) → 0 ≤ (𝐵 − 𝐴))
97	73, 76, 91, 96	mulge0d 11933	. . . . . . . . 9 ⊢ ((𝜑 ∧ ((𝐴 + 𝐵) / 2) = 𝑥) → 0 ≤ (𝐾 · (𝐵 − 𝐴)))
98	72, 97	eqbrtrd 5133	. . . . . . . 8 ⊢ ((𝜑 ∧ ((𝐴 + 𝐵) / 2) = 𝑥) → (abs‘((𝐹‘𝑥) − (𝐹‘((𝐴 + 𝐵) / 2)))) ≤ (𝐾 · (𝐵 − 𝐴)))
99	98	ad4ant14 765	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) ∧ ¬ ((𝐴 + 𝐵) / 2) < 𝑥) ∧ ((𝐴 + 𝐵) / 2) = 𝑥) → (abs‘((𝐹‘𝑥) − (𝐹‘((𝐴 + 𝐵) / 2)))) ≤ (𝐾 · (𝐵 − 𝐴)))
100		simpll 779	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) ∧ ¬ ((𝐴 + 𝐵) / 2) < 𝑥) ∧ ¬ ((𝐴 + 𝐵) / 2) = 𝑥) → (𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)))
101	39	ad2antrr 739	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) ∧ ¬ ((𝐴 + 𝐵) / 2) < 𝑥) ∧ ¬ ((𝐴 + 𝐵) / 2) = 𝑥) → 𝑥 ∈ ℝ)
102	10	ad3antrrr 743	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) ∧ ¬ ((𝐴 + 𝐵) / 2) < 𝑥) ∧ ¬ ((𝐴 + 𝐵) / 2) = 𝑥) → ((𝐴 + 𝐵) / 2) ∈ ℝ)
103	39	adantr 486	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) ∧ ¬ ((𝐴 + 𝐵) / 2) < 𝑥) → 𝑥 ∈ ℝ)
104	10	ad2antrr 739	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) ∧ ¬ ((𝐴 + 𝐵) / 2) < 𝑥) → ((𝐴 + 𝐵) / 2) ∈ ℝ)
105		simpr 490	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) ∧ ¬ ((𝐴 + 𝐵) / 2) < 𝑥) → ¬ ((𝐴 + 𝐵) / 2) < 𝑥)
106	103, 104, 105	nltled 11499	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) ∧ ¬ ((𝐴 + 𝐵) / 2) < 𝑥) → 𝑥 ≤ ((𝐴 + 𝐵) / 2))
107	106	adantr 486	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) ∧ ¬ ((𝐴 + 𝐵) / 2) < 𝑥) ∧ ¬ ((𝐴 + 𝐵) / 2) = 𝑥) → 𝑥 ≤ ((𝐴 + 𝐵) / 2))
108		neqne 2967	. . . . . . . . . 10 ⊢ (¬ ((𝐴 + 𝐵) / 2) = 𝑥 → ((𝐴 + 𝐵) / 2) ≠ 𝑥)
109	108	adantl 487	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) ∧ ¬ ((𝐴 + 𝐵) / 2) < 𝑥) ∧ ¬ ((𝐴 + 𝐵) / 2) = 𝑥) → ((𝐴 + 𝐵) / 2) ≠ 𝑥)
110	101, 102, 107, 109	leneltd 11503	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) ∧ ¬ ((𝐴 + 𝐵) / 2) < 𝑥) ∧ ¬ ((𝐴 + 𝐵) / 2) = 𝑥) → 𝑥 < ((𝐴 + 𝐵) / 2))
111	3, 30	abssubd 15667	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → (abs‘((𝐹‘𝑥) − (𝐹‘((𝐴 + 𝐵) / 2)))) = (abs‘((𝐹‘((𝐴 + 𝐵) / 2)) − (𝐹‘𝑥))))
112	111	adantr 486	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) ∧ 𝑥 < ((𝐴 + 𝐵) / 2)) → (abs‘((𝐹‘𝑥) − (𝐹‘((𝐴 + 𝐵) / 2)))) = (abs‘((𝐹‘((𝐴 + 𝐵) / 2)) − (𝐹‘𝑥))))
113	5	ad2antrr 739	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) ∧ 𝑥 < ((𝐴 + 𝐵) / 2)) → 𝐴 ∈ ℝ)
114	7	ad2antrr 739	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) ∧ 𝑥 < ((𝐴 + 𝐵) / 2)) → 𝐵 ∈ ℝ)
115	1	ad2antrr 739	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) ∧ 𝑥 < ((𝐴 + 𝐵) / 2)) → 𝐹:(𝐴(,)𝐵)⟶ℝ)
116	52	ad2antrr 739	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) ∧ 𝑥 < ((𝐴 + 𝐵) / 2)) → dom (ℝ D 𝐹) = (𝐴(,)𝐵))
117	24	ad2antrr 739	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) ∧ 𝑥 < ((𝐴 + 𝐵) / 2)) → 𝐾 ∈ ℝ)
118	59	ad2antrr 739	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) ∧ 𝑥 < ((𝐴 + 𝐵) / 2)) → ∀𝑦 ∈ (𝐴(,)𝐵)(abs‘((ℝ D 𝐹)‘𝑦)) ≤ 𝐾)
119	44	adantr 486	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) ∧ 𝑥 < ((𝐴 + 𝐵) / 2)) → 𝑥 ∈ (𝐴(,)𝐵))
120	38	rexrd 11398	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ (𝐴(,)𝐵) → 𝑥 ∈ ℝ^*)
121	120	ad2antlr 740	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) ∧ 𝑥 < ((𝐴 + 𝐵) / 2)) → 𝑥 ∈ ℝ^*)
122	8	ad2antrr 739	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) ∧ 𝑥 < ((𝐴 + 𝐵) / 2)) → 𝐵 ∈ ℝ^*)
123	10	ad2antrr 739	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) ∧ 𝑥 < ((𝐴 + 𝐵) / 2)) → ((𝐴 + 𝐵) / 2) ∈ ℝ)
124		simpr 490	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) ∧ 𝑥 < ((𝐴 + 𝐵) / 2)) → 𝑥 < ((𝐴 + 𝐵) / 2))
125	17	ad2antrr 739	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) ∧ 𝑥 < ((𝐴 + 𝐵) / 2)) → ((𝐴 + 𝐵) / 2) < 𝐵)
126	121, 122, 123, 124, 125	eliood 46783	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) ∧ 𝑥 < ((𝐴 + 𝐵) / 2)) → ((𝐴 + 𝐵) / 2) ∈ (𝑥(,)𝐵))
127	113, 114, 115, 116, 117, 118, 119, 126	dvbdfbdioolem1 47211	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) ∧ 𝑥 < ((𝐴 + 𝐵) / 2)) → ((abs‘((𝐹‘((𝐴 + 𝐵) / 2)) − (𝐹‘𝑥))) ≤ (𝐾 · (((𝐴 + 𝐵) / 2) − 𝑥)) ∧ (abs‘((𝐹‘((𝐴 + 𝐵) / 2)) − (𝐹‘𝑥))) ≤ (𝐾 · (𝐵 − 𝐴))))
128	127	simprd 501	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) ∧ 𝑥 < ((𝐴 + 𝐵) / 2)) → (abs‘((𝐹‘((𝐴 + 𝐵) / 2)) − (𝐹‘𝑥))) ≤ (𝐾 · (𝐵 − 𝐴)))
129	112, 128	eqbrtrd 5133	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) ∧ 𝑥 < ((𝐴 + 𝐵) / 2)) → (abs‘((𝐹‘𝑥) − (𝐹‘((𝐴 + 𝐵) / 2)))) ≤ (𝐾 · (𝐵 − 𝐴)))
130	100, 110, 129	syl2anc 596	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) ∧ ¬ ((𝐴 + 𝐵) / 2) < 𝑥) ∧ ¬ ((𝐴 + 𝐵) / 2) = 𝑥) → (abs‘((𝐹‘𝑥) − (𝐹‘((𝐴 + 𝐵) / 2)))) ≤ (𝐾 · (𝐵 − 𝐴)))
131	99, 130	pm2.61dan 825	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) ∧ ¬ ((𝐴 + 𝐵) / 2) < 𝑥) → (abs‘((𝐹‘𝑥) − (𝐹‘((𝐴 + 𝐵) / 2)))) ≤ (𝐾 · (𝐵 − 𝐴)))
132	65, 131	pm2.61dan 825	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → (abs‘((𝐹‘𝑥) − (𝐹‘((𝐴 + 𝐵) / 2)))) ≤ (𝐾 · (𝐵 − 𝐴)))
133	23, 32, 29, 33, 132	letrd 11506	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → ((abs‘(𝐹‘𝑥)) − (abs‘(𝐹‘((𝐴 + 𝐵) / 2)))) ≤ (𝐾 · (𝐵 − 𝐴)))
134	23, 29, 22, 133	leadd1dd 11970	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → (((abs‘(𝐹‘𝑥)) − (abs‘(𝐹‘((𝐴 + 𝐵) / 2)))) + (abs‘(𝐹‘((𝐴 + 𝐵) / 2)))) ≤ ((𝐾 · (𝐵 − 𝐴)) + (abs‘(𝐹‘((𝐴 + 𝐵) / 2)))))
135	4	recnd 11376	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → (abs‘(𝐹‘𝑥)) ∈ ℂ)
136	22	recnd 11376	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → (abs‘(𝐹‘((𝐴 + 𝐵) / 2))) ∈ ℂ)
137	135, 136	npcand 11712	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → (((abs‘(𝐹‘𝑥)) − (abs‘(𝐹‘((𝐴 + 𝐵) / 2)))) + (abs‘(𝐹‘((𝐴 + 𝐵) / 2)))) = (abs‘(𝐹‘𝑥)))
138	137	eqcomd 2769	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → (abs‘(𝐹‘𝑥)) = (((abs‘(𝐹‘𝑥)) − (abs‘(𝐹‘((𝐴 + 𝐵) / 2)))) + (abs‘(𝐹‘((𝐴 + 𝐵) / 2)))))
139		dvbdfbdioolem2.m	. . . . 5 ⊢ 𝑀 = ((abs‘(𝐹‘((𝐴 + 𝐵) / 2))) + (𝐾 · (𝐵 − 𝐴)))
140	21	recnd 11376	. . . . . 6 ⊢ (𝜑 → (abs‘(𝐹‘((𝐴 + 𝐵) / 2))) ∈ ℂ)
141	24	recnd 11376	. . . . . . 7 ⊢ (𝜑 → 𝐾 ∈ ℂ)
142	7	recnd 11376	. . . . . . . 8 ⊢ (𝜑 → 𝐵 ∈ ℂ)
143	5	recnd 11376	. . . . . . . 8 ⊢ (𝜑 → 𝐴 ∈ ℂ)
144	142, 143	subcld 11708	. . . . . . 7 ⊢ (𝜑 → (𝐵 − 𝐴) ∈ ℂ)
145	141, 144	mulcld 11368	. . . . . 6 ⊢ (𝜑 → (𝐾 · (𝐵 − 𝐴)) ∈ ℂ)
146	140, 145	addcomd 11551	. . . . 5 ⊢ (𝜑 → ((abs‘(𝐹‘((𝐴 + 𝐵) / 2))) + (𝐾 · (𝐵 − 𝐴))) = ((𝐾 · (𝐵 − 𝐴)) + (abs‘(𝐹‘((𝐴 + 𝐵) / 2)))))
147	139, 146	eqtrid 2810	. . . 4 ⊢ (𝜑 → 𝑀 = ((𝐾 · (𝐵 − 𝐴)) + (abs‘(𝐹‘((𝐴 + 𝐵) / 2)))))
148	147	adantr 486	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → 𝑀 = ((𝐾 · (𝐵 − 𝐴)) + (abs‘(𝐹‘((𝐴 + 𝐵) / 2)))))
149	134, 138, 148	3brtr4d 5143	. 2 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → (abs‘(𝐹‘𝑥)) ≤ 𝑀)
150	149	ralrimiva 3158	1 ⊢ (𝜑 → ∀𝑥 ∈ (𝐴(,)𝐵)(abs‘(𝐹‘𝑥)) ≤ 𝑀)

Colors of variables: wff setvar class

This proof depends on syntax axioms: ¬ wn 3 → wi 4 ↔ wb 209 ∧ wa 401 = wceq 1570 ∈ wcel 2147 ≠ wne 2959 ∀wral 3080 ⊆ wss 3900 class class class wbr 5108 dom cdm 5658 ⟶wf 6562 ‘cfv 6566 (class class class)co 7451 ℂcc 11237 ℝcr 11238 0cc0 11239 + caddc 11242 · cmul 11244 ℝ^*cxr 11381 < clt 11382 ≤ cle 11383 − cmin 11580 / cdiv 12013 2c2 12437 (,)cioo 13516 abscabs 15444 D cdv 26238

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1945 ax-6 2002 ax-7 2043 ax-8 2149 ax-9 2157 ax-10 2180 ax-11 2197 ax-12 2216 ax-ext 2735 ax-rep 5238 ax-sep 5255 ax-nul 5266 ax-pow 5334 ax-pr 5398 ax-un 7783 ax-cnex 11295 ax-resscn 11296 ax-1cn 11297 ax-icn 11298 ax-addcl 11299 ax-addrcl 11300 ax-mulcl 11301 ax-mulrcl 11302 ax-mulcom 11303 ax-addass 11304 ax-mulass 11305 ax-distr 11306 ax-i2m1 11307 ax-1ne0 11308 ax-1rid 11309 ax-rnegex 11310 ax-rrecex 11311 ax-cnre 11312 ax-pre-lttri 11313 ax-pre-lttrn 11314 ax-pre-ltadd 11315 ax-pre-mulgt0 11316 ax-pre-sup 11317 ax-addf 11318

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2102 df-mo 2567 df-eu 2597 df-clab 2742 df-cleq 2755 df-clel 2839 df-nfc 2913 df-ne 2960 df-nel 3066 df-ral 3081 df-rex 3091 df-rmo 3369 df-reu 3370 df-rab 3417 df-v 3456 df-sbc 3743 df-csb 3852 df-ss 3901 df-pss 4017 df-dif 4050 df-in 4094 df-un 4154 df-nul 4282 df-if 4486 df-pw 4562 df-sn 4589 df-pr 4591 df-tp 4593 df-op 4841 df-uni 4873 df-int 4913 df-iun 4958 df-iin 4959 df-br 5109 df-opab 5174 df-mpt 5193 df-tr 5219 df-id 5553 df-eprel 5558 df-po 5566 df-so 5567 df-fr 5611 df-se 5612 df-we 5613 df-xp 5664 df-rel 5665 df-cnv 5666 df-co 5667 df-dm 5668 df-rn 5669 df-res 5670 df-ima 5671 df-pred 6330 df-ord 6391 df-on 6392 df-lim 6393 df-suc 6394 df-iota 6520 df-fun 6568 df-fn 6569 df-f 6570 df-f1 6571 df-fo 6572 df-f1o 6573 df-fv 6574 df-isom 6575 df-riota 7408 df-ov 7454 df-oprab 7455 df-mpo 7456 df-of 7725 df-om 7912 df-1st 8035 df-2nd 8036 df-supp 8215 df-frecs 8337 df-wrecs 8368 df-recs 8417 df-rdg 8456 df-1o 8514 df-2o 8515 df-er 8755 df-map 8887 df-pm 8888 df-ixp 8964 df-en 9012 df-dom 9013 df-sdom 9014 df-fin 9015 df-fsupp 9392 df-fi 9440 df-sup 9471 df-inf 9472 df-oi 9541 df-card 10057 df-pnf 11384 df-mnf 11385 df-xr 11386 df-ltxr 11387 df-le 11388 df-sub 11582 df-neg 11583 df-div 12014 df-nn 12376 df-2 12445 df-3 12446 df-4 12447 df-5 12448 df-6 12449 df-7 12450 df-8 12451 df-9 12452 df-n0 12647 df-z 12734 df-dec 12855 df-uz 13006 df-q 13116 df-rp 13161 df-xneg 13281 df-xadd 13282 df-xmul 13283 df-ioo 13520 df-ico 13522 df-icc 13523 df-fz 13680 df-fzo 13829 df-seq 14185 df-exp 14245 df-hash 14516 df-cj 15309 df-re 15310 df-im 15311 df-sqrt 15445 df-abs 15446 df-struct 17369 df-sets 17386 df-slot 17404 df-ndx 17416 df-base 17432 df-ress 17453 df-plusg 17486 df-mulr 17487 df-starv 17488 df-sca 17489 df-vsca 17490 df-ip 17491 df-tset 17492 df-ple 17493 df-ds 17495 df-unif 17496 df-hom 17497 df-cco 17498 df-euf 17499 df-rest 17643 df-topn 17644 df-0g 17662 df-gsum 17663 df-topgen 17664 df-pt 17665 df-prds 17668 df-xrs 17724 df-qtop 17729 df-imas 17730 df-xps 17732 df-mre 17806 df-mrc 17807 df-acs 17809 df-mgm 18866 df-sgrp 18958 df-mnd 18974 df-submnd 19029 df-mulg 19328 df-cntz 19582 df-cmn 20047 df-psmet 21721 df-xmet 21722 df-met 21723 df-bl 21724 df-mopn 21725 df-fbas 21726 df-fg 21727 df-cnfld 21730 df-top 23267 df-topon 23284 df-topsp 23306 df-bases 23319 df-cld 23392 df-ntr 23393 df-cls 23394 df-nei 23471 df-lp 23509 df-perf 23510 df-cn 23600 df-cnp 23601 df-haus 23688 df-cmp 23760 df-tx 23936 df-hmeo 24129 df-fil 24220 df-fm 24312 df-flim 24313 df-flf 24314 df-xms 24694 df-ms 24695 df-tms 24696 df-cncf 25254 df-limc 26241 df-dv 26242

This theorem is used by: dvbdfbdioo 47213

W3C validator