dvbdfbdioolem1 - Mathbox for Glauco Siliprandi

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Theorem dvbdfbdioolem1 45849

Description: Given a function with bounded derivative, on an open interval, here is an absolute bound to the difference of the image of two points in the interval. (Contributed by Glauco Siliprandi, 11-Dec-2019.)

**Hypotheses**
Ref	Expression
dvbdfbdioolem1.a	⊢ (𝜑 → 𝐴 ∈ ℝ)
dvbdfbdioolem1.b	⊢ (𝜑 → 𝐵 ∈ ℝ)
dvbdfbdioolem1.f	⊢ (𝜑 → 𝐹:(𝐴(,)𝐵)⟶ℝ)
dvbdfbdioolem1.dmdv	⊢ (𝜑 → dom (ℝ D 𝐹) = (𝐴(,)𝐵))
dvbdfbdioolem1.k	⊢ (𝜑 → 𝐾 ∈ ℝ)
dvbdfbdioolem1.dvbd	⊢ (𝜑 → ∀𝑥 ∈ (𝐴(,)𝐵)(abs‘((ℝ D 𝐹)‘𝑥)) ≤ 𝐾)
dvbdfbdioolem1.c	⊢ (𝜑 → 𝐶 ∈ (𝐴(,)𝐵))
dvbdfbdioolem1.d	⊢ (𝜑 → 𝐷 ∈ (𝐶(,)𝐵))

**Assertion**
Ref	Expression
dvbdfbdioolem1	⊢ (𝜑 → ((abs‘((𝐹‘𝐷) − (𝐹‘𝐶))) ≤ (𝐾 · (𝐷 − 𝐶)) ∧ (abs‘((𝐹‘𝐷) − (𝐹‘𝐶))) ≤ (𝐾 · (𝐵 − 𝐴))))

Distinct variable groups: 𝑥,𝐴 𝑥,𝐵 𝑥,𝐶 𝑥,𝐷 𝑥,𝐹 𝑥,𝐾 𝜑,𝑥

**Proof of Theorem dvbdfbdioolem1**
Step	Hyp	Ref	Expression
1		ioossre 13468	. . . 4 ⊢ (𝐴(,)𝐵) ⊆ ℝ
2		dvbdfbdioolem1.c	. . . 4 ⊢ (𝜑 → 𝐶 ∈ (𝐴(,)𝐵))
3	1, 2	sselid 4006	. . 3 ⊢ (𝜑 → 𝐶 ∈ ℝ)
4		ioossre 13468	. . . 4 ⊢ (𝐶(,)𝐵) ⊆ ℝ
5		dvbdfbdioolem1.d	. . . 4 ⊢ (𝜑 → 𝐷 ∈ (𝐶(,)𝐵))
6	4, 5	sselid 4006	. . 3 ⊢ (𝜑 → 𝐷 ∈ ℝ)
7	3	rexrd 11340	. . . 4 ⊢ (𝜑 → 𝐶 ∈ ℝ^*)
8		dvbdfbdioolem1.b	. . . . 5 ⊢ (𝜑 → 𝐵 ∈ ℝ)
9	8	rexrd 11340	. . . 4 ⊢ (𝜑 → 𝐵 ∈ ℝ^*)
10		ioogtlb 45413	. . . 4 ⊢ ((𝐶 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐷 ∈ (𝐶(,)𝐵)) → 𝐶 < 𝐷)
11	7, 9, 5, 10	syl3anc 1371	. . 3 ⊢ (𝜑 → 𝐶 < 𝐷)
12		dvbdfbdioolem1.a	. . . . . 6 ⊢ (𝜑 → 𝐴 ∈ ℝ)
13	12	rexrd 11340	. . . . 5 ⊢ (𝜑 → 𝐴 ∈ ℝ^*)
14		ioogtlb 45413	. . . . . 6 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ (𝐴(,)𝐵)) → 𝐴 < 𝐶)
15	13, 9, 2, 14	syl3anc 1371	. . . . 5 ⊢ (𝜑 → 𝐴 < 𝐶)
16		iooltub 45428	. . . . . 6 ⊢ ((𝐶 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐷 ∈ (𝐶(,)𝐵)) → 𝐷 < 𝐵)
17	7, 9, 5, 16	syl3anc 1371	. . . . 5 ⊢ (𝜑 → 𝐷 < 𝐵)
18		iccssioo 13476	. . . . 5 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) ∧ (𝐴 < 𝐶 ∧ 𝐷 < 𝐵)) → (𝐶[,]𝐷) ⊆ (𝐴(,)𝐵))
19	13, 9, 15, 17, 18	syl22anc 838	. . . 4 ⊢ (𝜑 → (𝐶[,]𝐷) ⊆ (𝐴(,)𝐵))
20		dvbdfbdioolem1.f	. . . . 5 ⊢ (𝜑 → 𝐹:(𝐴(,)𝐵)⟶ℝ)
21		ax-resscn 11241	. . . . . . 7 ⊢ ℝ ⊆ ℂ
22	21	a1i 11	. . . . . 6 ⊢ (𝜑 → ℝ ⊆ ℂ)
23	20, 22	fssd 6764	. . . . . . 7 ⊢ (𝜑 → 𝐹:(𝐴(,)𝐵)⟶ℂ)
24	1	a1i 11	. . . . . . 7 ⊢ (𝜑 → (𝐴(,)𝐵) ⊆ ℝ)
25		dvbdfbdioolem1.dmdv	. . . . . . 7 ⊢ (𝜑 → dom (ℝ D 𝐹) = (𝐴(,)𝐵))
26		dvcn 25977	. . . . . . 7 ⊢ (((ℝ ⊆ ℂ ∧ 𝐹:(𝐴(,)𝐵)⟶ℂ ∧ (𝐴(,)𝐵) ⊆ ℝ) ∧ dom (ℝ D 𝐹) = (𝐴(,)𝐵)) → 𝐹 ∈ ((𝐴(,)𝐵)–cn→ℂ))
27	22, 23, 24, 25, 26	syl31anc 1373	. . . . . 6 ⊢ (𝜑 → 𝐹 ∈ ((𝐴(,)𝐵)–cn→ℂ))
28		cncfcdm 24943	. . . . . 6 ⊢ ((ℝ ⊆ ℂ ∧ 𝐹 ∈ ((𝐴(,)𝐵)–cn→ℂ)) → (𝐹 ∈ ((𝐴(,)𝐵)–cn→ℝ) ↔ 𝐹:(𝐴(,)𝐵)⟶ℝ))
29	22, 27, 28	syl2anc 583	. . . . 5 ⊢ (𝜑 → (𝐹 ∈ ((𝐴(,)𝐵)–cn→ℝ) ↔ 𝐹:(𝐴(,)𝐵)⟶ℝ))
30	20, 29	mpbird 257	. . . 4 ⊢ (𝜑 → 𝐹 ∈ ((𝐴(,)𝐵)–cn→ℝ))
31		rescncf 24942	. . . 4 ⊢ ((𝐶[,]𝐷) ⊆ (𝐴(,)𝐵) → (𝐹 ∈ ((𝐴(,)𝐵)–cn→ℝ) → (𝐹 ↾ (𝐶[,]𝐷)) ∈ ((𝐶[,]𝐷)–cn→ℝ)))
32	19, 30, 31	sylc 65	. . 3 ⊢ (𝜑 → (𝐹 ↾ (𝐶[,]𝐷)) ∈ ((𝐶[,]𝐷)–cn→ℝ))
33	19, 24	sstrd 4019	. . . . . . 7 ⊢ (𝜑 → (𝐶[,]𝐷) ⊆ ℝ)
34		eqid 2740	. . . . . . . 8 ⊢ (TopOpen‘ℂ_fld) = (TopOpen‘ℂ_fld)
35	34	tgioo2 24844	. . . . . . . 8 ⊢ (topGen‘ran (,)) = ((TopOpen‘ℂ_fld) ↾_t ℝ)
36	34, 35	dvres 25966	. . . . . . 7 ⊢ (((ℝ ⊆ ℂ ∧ 𝐹:(𝐴(,)𝐵)⟶ℂ) ∧ ((𝐴(,)𝐵) ⊆ ℝ ∧ (𝐶[,]𝐷) ⊆ ℝ)) → (ℝ D (𝐹 ↾ (𝐶[,]𝐷))) = ((ℝ D 𝐹) ↾ ((int‘(topGen‘ran (,)))‘(𝐶[,]𝐷))))
37	22, 23, 24, 33, 36	syl22anc 838	. . . . . 6 ⊢ (𝜑 → (ℝ D (𝐹 ↾ (𝐶[,]𝐷))) = ((ℝ D 𝐹) ↾ ((int‘(topGen‘ran (,)))‘(𝐶[,]𝐷))))
38		iccntr 24862	. . . . . . . 8 ⊢ ((𝐶 ∈ ℝ ∧ 𝐷 ∈ ℝ) → ((int‘(topGen‘ran (,)))‘(𝐶[,]𝐷)) = (𝐶(,)𝐷))
39	3, 6, 38	syl2anc 583	. . . . . . 7 ⊢ (𝜑 → ((int‘(topGen‘ran (,)))‘(𝐶[,]𝐷)) = (𝐶(,)𝐷))
40	39	reseq2d 6009	. . . . . 6 ⊢ (𝜑 → ((ℝ D 𝐹) ↾ ((int‘(topGen‘ran (,)))‘(𝐶[,]𝐷))) = ((ℝ D 𝐹) ↾ (𝐶(,)𝐷)))
41	37, 40	eqtrd 2780	. . . . 5 ⊢ (𝜑 → (ℝ D (𝐹 ↾ (𝐶[,]𝐷))) = ((ℝ D 𝐹) ↾ (𝐶(,)𝐷)))
42	41	dmeqd 5930	. . . 4 ⊢ (𝜑 → dom (ℝ D (𝐹 ↾ (𝐶[,]𝐷))) = dom ((ℝ D 𝐹) ↾ (𝐶(,)𝐷)))
43	12, 3, 15	ltled 11438	. . . . . . 7 ⊢ (𝜑 → 𝐴 ≤ 𝐶)
44	6, 8, 17	ltled 11438	. . . . . . 7 ⊢ (𝜑 → 𝐷 ≤ 𝐵)
45		ioossioo 13501	. . . . . . 7 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) ∧ (𝐴 ≤ 𝐶 ∧ 𝐷 ≤ 𝐵)) → (𝐶(,)𝐷) ⊆ (𝐴(,)𝐵))
46	13, 9, 43, 44, 45	syl22anc 838	. . . . . 6 ⊢ (𝜑 → (𝐶(,)𝐷) ⊆ (𝐴(,)𝐵))
47	46, 25	sseqtrrd 4050	. . . . 5 ⊢ (𝜑 → (𝐶(,)𝐷) ⊆ dom (ℝ D 𝐹))
48		ssdmres 6042	. . . . 5 ⊢ ((𝐶(,)𝐷) ⊆ dom (ℝ D 𝐹) ↔ dom ((ℝ D 𝐹) ↾ (𝐶(,)𝐷)) = (𝐶(,)𝐷))
49	47, 48	sylib 218	. . . 4 ⊢ (𝜑 → dom ((ℝ D 𝐹) ↾ (𝐶(,)𝐷)) = (𝐶(,)𝐷))
50	42, 49	eqtrd 2780	. . 3 ⊢ (𝜑 → dom (ℝ D (𝐹 ↾ (𝐶[,]𝐷))) = (𝐶(,)𝐷))
51	3, 6, 11, 32, 50	mvth 26051	. 2 ⊢ (𝜑 → ∃𝑥 ∈ (𝐶(,)𝐷)((ℝ D (𝐹 ↾ (𝐶[,]𝐷)))‘𝑥) = ((((𝐹 ↾ (𝐶[,]𝐷))‘𝐷) − ((𝐹 ↾ (𝐶[,]𝐷))‘𝐶)) / (𝐷 − 𝐶)))
52	41	fveq1d 6922	. . . . . . . . 9 ⊢ (𝜑 → ((ℝ D (𝐹 ↾ (𝐶[,]𝐷)))‘𝑥) = (((ℝ D 𝐹) ↾ (𝐶(,)𝐷))‘𝑥))
53		fvres 6939	. . . . . . . . 9 ⊢ (𝑥 ∈ (𝐶(,)𝐷) → (((ℝ D 𝐹) ↾ (𝐶(,)𝐷))‘𝑥) = ((ℝ D 𝐹)‘𝑥))
54	52, 53	sylan9eq 2800	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐶(,)𝐷)) → ((ℝ D (𝐹 ↾ (𝐶[,]𝐷)))‘𝑥) = ((ℝ D 𝐹)‘𝑥))
55	54	eqcomd 2746	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐶(,)𝐷)) → ((ℝ D 𝐹)‘𝑥) = ((ℝ D (𝐹 ↾ (𝐶[,]𝐷)))‘𝑥))
56	55	3adant3 1132	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐶(,)𝐷) ∧ ((ℝ D (𝐹 ↾ (𝐶[,]𝐷)))‘𝑥) = ((((𝐹 ↾ (𝐶[,]𝐷))‘𝐷) − ((𝐹 ↾ (𝐶[,]𝐷))‘𝐶)) / (𝐷 − 𝐶))) → ((ℝ D 𝐹)‘𝑥) = ((ℝ D (𝐹 ↾ (𝐶[,]𝐷)))‘𝑥))
57		simp3 1138	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐶(,)𝐷) ∧ ((ℝ D (𝐹 ↾ (𝐶[,]𝐷)))‘𝑥) = ((((𝐹 ↾ (𝐶[,]𝐷))‘𝐷) − ((𝐹 ↾ (𝐶[,]𝐷))‘𝐶)) / (𝐷 − 𝐶))) → ((ℝ D (𝐹 ↾ (𝐶[,]𝐷)))‘𝑥) = ((((𝐹 ↾ (𝐶[,]𝐷))‘𝐷) − ((𝐹 ↾ (𝐶[,]𝐷))‘𝐶)) / (𝐷 − 𝐶)))
58	6	rexrd 11340	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐷 ∈ ℝ^*)
59	3, 6, 11	ltled 11438	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐶 ≤ 𝐷)
60		ubicc2 13525	. . . . . . . . . . 11 ⊢ ((𝐶 ∈ ℝ^* ∧ 𝐷 ∈ ℝ^* ∧ 𝐶 ≤ 𝐷) → 𝐷 ∈ (𝐶[,]𝐷))
61	7, 58, 59, 60	syl3anc 1371	. . . . . . . . . 10 ⊢ (𝜑 → 𝐷 ∈ (𝐶[,]𝐷))
62		fvres 6939	. . . . . . . . . 10 ⊢ (𝐷 ∈ (𝐶[,]𝐷) → ((𝐹 ↾ (𝐶[,]𝐷))‘𝐷) = (𝐹‘𝐷))
63	61, 62	syl 17	. . . . . . . . 9 ⊢ (𝜑 → ((𝐹 ↾ (𝐶[,]𝐷))‘𝐷) = (𝐹‘𝐷))
64		lbicc2 13524	. . . . . . . . . . 11 ⊢ ((𝐶 ∈ ℝ^* ∧ 𝐷 ∈ ℝ^* ∧ 𝐶 ≤ 𝐷) → 𝐶 ∈ (𝐶[,]𝐷))
65	7, 58, 59, 64	syl3anc 1371	. . . . . . . . . 10 ⊢ (𝜑 → 𝐶 ∈ (𝐶[,]𝐷))
66		fvres 6939	. . . . . . . . . 10 ⊢ (𝐶 ∈ (𝐶[,]𝐷) → ((𝐹 ↾ (𝐶[,]𝐷))‘𝐶) = (𝐹‘𝐶))
67	65, 66	syl 17	. . . . . . . . 9 ⊢ (𝜑 → ((𝐹 ↾ (𝐶[,]𝐷))‘𝐶) = (𝐹‘𝐶))
68	63, 67	oveq12d 7466	. . . . . . . 8 ⊢ (𝜑 → (((𝐹 ↾ (𝐶[,]𝐷))‘𝐷) − ((𝐹 ↾ (𝐶[,]𝐷))‘𝐶)) = ((𝐹‘𝐷) − (𝐹‘𝐶)))
69	68	oveq1d 7463	. . . . . . 7 ⊢ (𝜑 → ((((𝐹 ↾ (𝐶[,]𝐷))‘𝐷) − ((𝐹 ↾ (𝐶[,]𝐷))‘𝐶)) / (𝐷 − 𝐶)) = (((𝐹‘𝐷) − (𝐹‘𝐶)) / (𝐷 − 𝐶)))
70	69	3ad2ant1 1133	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐶(,)𝐷) ∧ ((ℝ D (𝐹 ↾ (𝐶[,]𝐷)))‘𝑥) = ((((𝐹 ↾ (𝐶[,]𝐷))‘𝐷) − ((𝐹 ↾ (𝐶[,]𝐷))‘𝐶)) / (𝐷 − 𝐶))) → ((((𝐹 ↾ (𝐶[,]𝐷))‘𝐷) − ((𝐹 ↾ (𝐶[,]𝐷))‘𝐶)) / (𝐷 − 𝐶)) = (((𝐹‘𝐷) − (𝐹‘𝐶)) / (𝐷 − 𝐶)))
71	56, 57, 70	3eqtrd 2784	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐶(,)𝐷) ∧ ((ℝ D (𝐹 ↾ (𝐶[,]𝐷)))‘𝑥) = ((((𝐹 ↾ (𝐶[,]𝐷))‘𝐷) − ((𝐹 ↾ (𝐶[,]𝐷))‘𝐶)) / (𝐷 − 𝐶))) → ((ℝ D 𝐹)‘𝑥) = (((𝐹‘𝐷) − (𝐹‘𝐶)) / (𝐷 − 𝐶)))
72		simp3 1138	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐶(,)𝐷) ∧ ((ℝ D 𝐹)‘𝑥) = (((𝐹‘𝐷) − (𝐹‘𝐶)) / (𝐷 − 𝐶))) → ((ℝ D 𝐹)‘𝑥) = (((𝐹‘𝐷) − (𝐹‘𝐶)) / (𝐷 − 𝐶)))
73	72	eqcomd 2746	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐶(,)𝐷) ∧ ((ℝ D 𝐹)‘𝑥) = (((𝐹‘𝐷) − (𝐹‘𝐶)) / (𝐷 − 𝐶))) → (((𝐹‘𝐷) − (𝐹‘𝐶)) / (𝐷 − 𝐶)) = ((ℝ D 𝐹)‘𝑥))
74	19, 61	sseldd 4009	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝐷 ∈ (𝐴(,)𝐵))
75	20, 74	ffvelcdmd 7119	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝐹‘𝐷) ∈ ℝ)
76	20, 2	ffvelcdmd 7119	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝐹‘𝐶) ∈ ℝ)
77	75, 76	resubcld 11718	. . . . . . . . . . . . 13 ⊢ (𝜑 → ((𝐹‘𝐷) − (𝐹‘𝐶)) ∈ ℝ)
78	77	recnd 11318	. . . . . . . . . . . 12 ⊢ (𝜑 → ((𝐹‘𝐷) − (𝐹‘𝐶)) ∈ ℂ)
79	78	3ad2ant1 1133	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐶(,)𝐷) ∧ ((ℝ D 𝐹)‘𝑥) = (((𝐹‘𝐷) − (𝐹‘𝐶)) / (𝐷 − 𝐶))) → ((𝐹‘𝐷) − (𝐹‘𝐶)) ∈ ℂ)
80		dvfre 26009	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐹:(𝐴(,)𝐵)⟶ℝ ∧ (𝐴(,)𝐵) ⊆ ℝ) → (ℝ D 𝐹):dom (ℝ D 𝐹)⟶ℝ)
81	20, 24, 80	syl2anc 583	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (ℝ D 𝐹):dom (ℝ D 𝐹)⟶ℝ)
82	25	feq2d 6733	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → ((ℝ D 𝐹):dom (ℝ D 𝐹)⟶ℝ ↔ (ℝ D 𝐹):(𝐴(,)𝐵)⟶ℝ))
83	81, 82	mpbid 232	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (ℝ D 𝐹):(𝐴(,)𝐵)⟶ℝ)
84	83	adantr 480	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐶(,)𝐷)) → (ℝ D 𝐹):(𝐴(,)𝐵)⟶ℝ)
85	46	sselda 4008	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐶(,)𝐷)) → 𝑥 ∈ (𝐴(,)𝐵))
86	84, 85	ffvelcdmd 7119	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐶(,)𝐷)) → ((ℝ D 𝐹)‘𝑥) ∈ ℝ)
87	86	recnd 11318	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐶(,)𝐷)) → ((ℝ D 𝐹)‘𝑥) ∈ ℂ)
88	87	3adant3 1132	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐶(,)𝐷) ∧ ((ℝ D 𝐹)‘𝑥) = (((𝐹‘𝐷) − (𝐹‘𝐶)) / (𝐷 − 𝐶))) → ((ℝ D 𝐹)‘𝑥) ∈ ℂ)
89	6, 3	resubcld 11718	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝐷 − 𝐶) ∈ ℝ)
90	89	recnd 11318	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐷 − 𝐶) ∈ ℂ)
91	90	3ad2ant1 1133	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐶(,)𝐷) ∧ ((ℝ D 𝐹)‘𝑥) = (((𝐹‘𝐷) − (𝐹‘𝐶)) / (𝐷 − 𝐶))) → (𝐷 − 𝐶) ∈ ℂ)
92	3, 6	posdifd 11877	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝐶 < 𝐷 ↔ 0 < (𝐷 − 𝐶)))
93	11, 92	mpbid 232	. . . . . . . . . . . . 13 ⊢ (𝜑 → 0 < (𝐷 − 𝐶))
94	93	gt0ne0d 11854	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐷 − 𝐶) ≠ 0)
95	94	3ad2ant1 1133	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐶(,)𝐷) ∧ ((ℝ D 𝐹)‘𝑥) = (((𝐹‘𝐷) − (𝐹‘𝐶)) / (𝐷 − 𝐶))) → (𝐷 − 𝐶) ≠ 0)
96	79, 88, 91, 95	divmul3d 12104	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐶(,)𝐷) ∧ ((ℝ D 𝐹)‘𝑥) = (((𝐹‘𝐷) − (𝐹‘𝐶)) / (𝐷 − 𝐶))) → ((((𝐹‘𝐷) − (𝐹‘𝐶)) / (𝐷 − 𝐶)) = ((ℝ D 𝐹)‘𝑥) ↔ ((𝐹‘𝐷) − (𝐹‘𝐶)) = (((ℝ D 𝐹)‘𝑥) · (𝐷 − 𝐶))))
97	73, 96	mpbid 232	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐶(,)𝐷) ∧ ((ℝ D 𝐹)‘𝑥) = (((𝐹‘𝐷) − (𝐹‘𝐶)) / (𝐷 − 𝐶))) → ((𝐹‘𝐷) − (𝐹‘𝐶)) = (((ℝ D 𝐹)‘𝑥) · (𝐷 − 𝐶)))
98	97	fveq2d 6924	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐶(,)𝐷) ∧ ((ℝ D 𝐹)‘𝑥) = (((𝐹‘𝐷) − (𝐹‘𝐶)) / (𝐷 − 𝐶))) → (abs‘((𝐹‘𝐷) − (𝐹‘𝐶))) = (abs‘(((ℝ D 𝐹)‘𝑥) · (𝐷 − 𝐶))))
99	90	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐶(,)𝐷)) → (𝐷 − 𝐶) ∈ ℂ)
100	87, 99	absmuld 15503	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐶(,)𝐷)) → (abs‘(((ℝ D 𝐹)‘𝑥) · (𝐷 − 𝐶))) = ((abs‘((ℝ D 𝐹)‘𝑥)) · (abs‘(𝐷 − 𝐶))))
101	100	3adant3 1132	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐶(,)𝐷) ∧ ((ℝ D 𝐹)‘𝑥) = (((𝐹‘𝐷) − (𝐹‘𝐶)) / (𝐷 − 𝐶))) → (abs‘(((ℝ D 𝐹)‘𝑥) · (𝐷 − 𝐶))) = ((abs‘((ℝ D 𝐹)‘𝑥)) · (abs‘(𝐷 − 𝐶))))
102	98, 101	eqtrd 2780	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐶(,)𝐷) ∧ ((ℝ D 𝐹)‘𝑥) = (((𝐹‘𝐷) − (𝐹‘𝐶)) / (𝐷 − 𝐶))) → (abs‘((𝐹‘𝐷) − (𝐹‘𝐶))) = ((abs‘((ℝ D 𝐹)‘𝑥)) · (abs‘(𝐷 − 𝐶))))
103	3, 6, 59	abssubge0d 15480	. . . . . . . . 9 ⊢ (𝜑 → (abs‘(𝐷 − 𝐶)) = (𝐷 − 𝐶))
104	103	oveq2d 7464	. . . . . . . 8 ⊢ (𝜑 → ((abs‘((ℝ D 𝐹)‘𝑥)) · (abs‘(𝐷 − 𝐶))) = ((abs‘((ℝ D 𝐹)‘𝑥)) · (𝐷 − 𝐶)))
105	104	3ad2ant1 1133	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐶(,)𝐷) ∧ ((ℝ D 𝐹)‘𝑥) = (((𝐹‘𝐷) − (𝐹‘𝐶)) / (𝐷 − 𝐶))) → ((abs‘((ℝ D 𝐹)‘𝑥)) · (abs‘(𝐷 − 𝐶))) = ((abs‘((ℝ D 𝐹)‘𝑥)) · (𝐷 − 𝐶)))
106	102, 105	eqtrd 2780	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐶(,)𝐷) ∧ ((ℝ D 𝐹)‘𝑥) = (((𝐹‘𝐷) − (𝐹‘𝐶)) / (𝐷 − 𝐶))) → (abs‘((𝐹‘𝐷) − (𝐹‘𝐶))) = ((abs‘((ℝ D 𝐹)‘𝑥)) · (𝐷 − 𝐶)))
107	87	abscld 15485	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐶(,)𝐷)) → (abs‘((ℝ D 𝐹)‘𝑥)) ∈ ℝ)
108		dvbdfbdioolem1.k	. . . . . . . . 9 ⊢ (𝜑 → 𝐾 ∈ ℝ)
109	108	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐶(,)𝐷)) → 𝐾 ∈ ℝ)
110	89	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐶(,)𝐷)) → (𝐷 − 𝐶) ∈ ℝ)
111		0red 11293	. . . . . . . . . 10 ⊢ (𝜑 → 0 ∈ ℝ)
112	111, 89, 93	ltled 11438	. . . . . . . . 9 ⊢ (𝜑 → 0 ≤ (𝐷 − 𝐶))
113	112	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐶(,)𝐷)) → 0 ≤ (𝐷 − 𝐶))
114		dvbdfbdioolem1.dvbd	. . . . . . . . . 10 ⊢ (𝜑 → ∀𝑥 ∈ (𝐴(,)𝐵)(abs‘((ℝ D 𝐹)‘𝑥)) ≤ 𝐾)
115	114	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐶(,)𝐷)) → ∀𝑥 ∈ (𝐴(,)𝐵)(abs‘((ℝ D 𝐹)‘𝑥)) ≤ 𝐾)
116		rspa 3254	. . . . . . . . 9 ⊢ ((∀𝑥 ∈ (𝐴(,)𝐵)(abs‘((ℝ D 𝐹)‘𝑥)) ≤ 𝐾 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → (abs‘((ℝ D 𝐹)‘𝑥)) ≤ 𝐾)
117	115, 85, 116	syl2anc 583	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐶(,)𝐷)) → (abs‘((ℝ D 𝐹)‘𝑥)) ≤ 𝐾)
118	107, 109, 110, 113, 117	lemul1ad 12234	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐶(,)𝐷)) → ((abs‘((ℝ D 𝐹)‘𝑥)) · (𝐷 − 𝐶)) ≤ (𝐾 · (𝐷 − 𝐶)))
119	118	3adant3 1132	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐶(,)𝐷) ∧ ((ℝ D 𝐹)‘𝑥) = (((𝐹‘𝐷) − (𝐹‘𝐶)) / (𝐷 − 𝐶))) → ((abs‘((ℝ D 𝐹)‘𝑥)) · (𝐷 − 𝐶)) ≤ (𝐾 · (𝐷 − 𝐶)))
120	106, 119	eqbrtrd 5188	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐶(,)𝐷) ∧ ((ℝ D 𝐹)‘𝑥) = (((𝐹‘𝐷) − (𝐹‘𝐶)) / (𝐷 − 𝐶))) → (abs‘((𝐹‘𝐷) − (𝐹‘𝐶))) ≤ (𝐾 · (𝐷 − 𝐶)))
121	71, 120	syld3an3 1409	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐶(,)𝐷) ∧ ((ℝ D (𝐹 ↾ (𝐶[,]𝐷)))‘𝑥) = ((((𝐹 ↾ (𝐶[,]𝐷))‘𝐷) − ((𝐹 ↾ (𝐶[,]𝐷))‘𝐶)) / (𝐷 − 𝐶))) → (abs‘((𝐹‘𝐷) − (𝐹‘𝐶))) ≤ (𝐾 · (𝐷 − 𝐶)))
122	99	abscld 15485	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐶(,)𝐷)) → (abs‘(𝐷 − 𝐶)) ∈ ℝ)
123	8, 12	resubcld 11718	. . . . . . . . 9 ⊢ (𝜑 → (𝐵 − 𝐴) ∈ ℝ)
124	123	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐶(,)𝐷)) → (𝐵 − 𝐴) ∈ ℝ)
125	87	absge0d 15493	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐶(,)𝐷)) → 0 ≤ (abs‘((ℝ D 𝐹)‘𝑥)))
126	99	absge0d 15493	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐶(,)𝐷)) → 0 ≤ (abs‘(𝐷 − 𝐶)))
127	6, 12, 8, 3, 44, 43	le2subd 11910	. . . . . . . . . 10 ⊢ (𝜑 → (𝐷 − 𝐶) ≤ (𝐵 − 𝐴))
128	103, 127	eqbrtrd 5188	. . . . . . . . 9 ⊢ (𝜑 → (abs‘(𝐷 − 𝐶)) ≤ (𝐵 − 𝐴))
129	128	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐶(,)𝐷)) → (abs‘(𝐷 − 𝐶)) ≤ (𝐵 − 𝐴))
130	107, 109, 122, 124, 125, 126, 117, 129	lemul12ad 12237	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐶(,)𝐷)) → ((abs‘((ℝ D 𝐹)‘𝑥)) · (abs‘(𝐷 − 𝐶))) ≤ (𝐾 · (𝐵 − 𝐴)))
131	130	3adant3 1132	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐶(,)𝐷) ∧ ((ℝ D 𝐹)‘𝑥) = (((𝐹‘𝐷) − (𝐹‘𝐶)) / (𝐷 − 𝐶))) → ((abs‘((ℝ D 𝐹)‘𝑥)) · (abs‘(𝐷 − 𝐶))) ≤ (𝐾 · (𝐵 − 𝐴)))
132	102, 131	eqbrtrd 5188	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐶(,)𝐷) ∧ ((ℝ D 𝐹)‘𝑥) = (((𝐹‘𝐷) − (𝐹‘𝐶)) / (𝐷 − 𝐶))) → (abs‘((𝐹‘𝐷) − (𝐹‘𝐶))) ≤ (𝐾 · (𝐵 − 𝐴)))
133	71, 132	syld3an3 1409	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐶(,)𝐷) ∧ ((ℝ D (𝐹 ↾ (𝐶[,]𝐷)))‘𝑥) = ((((𝐹 ↾ (𝐶[,]𝐷))‘𝐷) − ((𝐹 ↾ (𝐶[,]𝐷))‘𝐶)) / (𝐷 − 𝐶))) → (abs‘((𝐹‘𝐷) − (𝐹‘𝐶))) ≤ (𝐾 · (𝐵 − 𝐴)))
134	121, 133	jca 511	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐶(,)𝐷) ∧ ((ℝ D (𝐹 ↾ (𝐶[,]𝐷)))‘𝑥) = ((((𝐹 ↾ (𝐶[,]𝐷))‘𝐷) − ((𝐹 ↾ (𝐶[,]𝐷))‘𝐶)) / (𝐷 − 𝐶))) → ((abs‘((𝐹‘𝐷) − (𝐹‘𝐶))) ≤ (𝐾 · (𝐷 − 𝐶)) ∧ (abs‘((𝐹‘𝐷) − (𝐹‘𝐶))) ≤ (𝐾 · (𝐵 − 𝐴))))
135	134	rexlimdv3a 3165	. 2 ⊢ (𝜑 → (∃𝑥 ∈ (𝐶(,)𝐷)((ℝ D (𝐹 ↾ (𝐶[,]𝐷)))‘𝑥) = ((((𝐹 ↾ (𝐶[,]𝐷))‘𝐷) − ((𝐹 ↾ (𝐶[,]𝐷))‘𝐶)) / (𝐷 − 𝐶)) → ((abs‘((𝐹‘𝐷) − (𝐹‘𝐶))) ≤ (𝐾 · (𝐷 − 𝐶)) ∧ (abs‘((𝐹‘𝐷) − (𝐹‘𝐶))) ≤ (𝐾 · (𝐵 − 𝐴)))))
136	51, 135	mpd 15	1 ⊢ (𝜑 → ((abs‘((𝐹‘𝐷) − (𝐹‘𝐶))) ≤ (𝐾 · (𝐷 − 𝐶)) ∧ (abs‘((𝐹‘𝐷) − (𝐹‘𝐶))) ≤ (𝐾 · (𝐵 − 𝐴))))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∧ w3a 1087 = wceq 1537 ∈ wcel 2108 ≠ wne 2946 ∀wral 3067 ∃wrex 3076 ⊆ wss 3976 class class class wbr 5166 dom cdm 5700 ran crn 5701 ↾ cres 5702 ⟶wf 6569 ‘cfv 6573 (class class class)co 7448 ℂcc 11182 ℝcr 11183 0cc0 11184 · cmul 11189 ℝ^*cxr 11323 < clt 11324 ≤ cle 11325 − cmin 11520 / cdiv 11947 (,)cioo 13407 [,]cicc 13410 abscabs 15283 TopOpenctopn 17481 topGenctg 17497 ℂ_fldccnfld 21387 intcnt 23046 –cn→ccncf 24921 D cdv 25918

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 ax-5 1909 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2158 ax-12 2178 ax-ext 2711 ax-rep 5303 ax-sep 5317 ax-nul 5324 ax-pow 5383 ax-pr 5447 ax-un 7770 ax-cnex 11240 ax-resscn 11241 ax-1cn 11242 ax-icn 11243 ax-addcl 11244 ax-addrcl 11245 ax-mulcl 11246 ax-mulrcl 11247 ax-mulcom 11248 ax-addass 11249 ax-mulass 11250 ax-distr 11251 ax-i2m1 11252 ax-1ne0 11253 ax-1rid 11254 ax-rnegex 11255 ax-rrecex 11256 ax-cnre 11257 ax-pre-lttri 11258 ax-pre-lttrn 11259 ax-pre-ltadd 11260 ax-pre-mulgt0 11261 ax-pre-sup 11262 ax-addf 11263

This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-3or 1088 df-3an 1089 df-tru 1540 df-fal 1550 df-ex 1778 df-nf 1782 df-sb 2065 df-mo 2543 df-eu 2572 df-clab 2718 df-cleq 2732 df-clel 2819 df-nfc 2895 df-ne 2947 df-nel 3053 df-ral 3068 df-rex 3077 df-rmo 3388 df-reu 3389 df-rab 3444 df-v 3490 df-sbc 3805 df-csb 3922 df-dif 3979 df-un 3981 df-in 3983 df-ss 3993 df-pss 3996 df-nul 4353 df-if 4549 df-pw 4624 df-sn 4649 df-pr 4651 df-tp 4653 df-op 4655 df-uni 4932 df-int 4971 df-iun 5017 df-iin 5018 df-br 5167 df-opab 5229 df-mpt 5250 df-tr 5284 df-id 5593 df-eprel 5599 df-po 5607 df-so 5608 df-fr 5652 df-se 5653 df-we 5654 df-xp 5706 df-rel 5707 df-cnv 5708 df-co 5709 df-dm 5710 df-rn 5711 df-res 5712 df-ima 5713 df-pred 6332 df-ord 6398 df-on 6399 df-lim 6400 df-suc 6401 df-iota 6525 df-fun 6575 df-fn 6576 df-f 6577 df-f1 6578 df-fo 6579 df-f1o 6580 df-fv 6581 df-isom 6582 df-riota 7404 df-ov 7451 df-oprab 7452 df-mpo 7453 df-of 7714 df-om 7904 df-1st 8030 df-2nd 8031 df-supp 8202 df-frecs 8322 df-wrecs 8353 df-recs 8427 df-rdg 8466 df-1o 8522 df-2o 8523 df-er 8763 df-map 8886 df-pm 8887 df-ixp 8956 df-en 9004 df-dom 9005 df-sdom 9006 df-fin 9007 df-fsupp 9432 df-fi 9480 df-sup 9511 df-inf 9512 df-oi 9579 df-card 10008 df-pnf 11326 df-mnf 11327 df-xr 11328 df-ltxr 11329 df-le 11330 df-sub 11522 df-neg 11523 df-div 11948 df-nn 12294 df-2 12356 df-3 12357 df-4 12358 df-5 12359 df-6 12360 df-7 12361 df-8 12362 df-9 12363 df-n0 12554 df-z 12640 df-dec 12759 df-uz 12904 df-q 13014 df-rp 13058 df-xneg 13175 df-xadd 13176 df-xmul 13177 df-ioo 13411 df-ico 13413 df-icc 13414 df-fz 13568 df-fzo 13712 df-seq 14053 df-exp 14113 df-hash 14380 df-cj 15148 df-re 15149 df-im 15150 df-sqrt 15284 df-abs 15285 df-struct 17194 df-sets 17211 df-slot 17229 df-ndx 17241 df-base 17259 df-ress 17288 df-plusg 17324 df-mulr 17325 df-starv 17326 df-sca 17327 df-vsca 17328 df-ip 17329 df-tset 17330 df-ple 17331 df-ds 17333 df-unif 17334 df-hom 17335 df-cco 17336 df-rest 17482 df-topn 17483 df-0g 17501 df-gsum 17502 df-topgen 17503 df-pt 17504 df-prds 17507 df-xrs 17562 df-qtop 17567 df-imas 17568 df-xps 17570 df-mre 17644 df-mrc 17645 df-acs 17647 df-mgm 18678 df-sgrp 18757 df-mnd 18773 df-submnd 18819 df-mulg 19108 df-cntz 19357 df-cmn 19824 df-psmet 21379 df-xmet 21380 df-met 21381 df-bl 21382 df-mopn 21383 df-fbas 21384 df-fg 21385 df-cnfld 21388 df-top 22921 df-topon 22938 df-topsp 22960 df-bases 22974 df-cld 23048 df-ntr 23049 df-cls 23050 df-nei 23127 df-lp 23165 df-perf 23166 df-cn 23256 df-cnp 23257 df-haus 23344 df-cmp 23416 df-tx 23591 df-hmeo 23784 df-fil 23875 df-fm 23967 df-flim 23968 df-flf 23969 df-xms 24351 df-ms 24352 df-tms 24353 df-cncf 24923 df-limc 25921 df-dv 25922

This theorem is referenced by: dvbdfbdioolem2 45850 ioodvbdlimc1lem1 45852 ioodvbdlimc1lem2 45853 ioodvbdlimc2lem 45855

W3C validator