dvbdfbdioolem1 - Mathbox for Glauco Siliprandi

	Mathbox for Glauco Siliprandi		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > Mathboxes > dvbdfbdioolem1		Structured version Visualization version GIF version

Theorem dvbdfbdioolem1 44159

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 13325	. . . 4 ⊢ (𝐴(,)𝐵) ⊆ ℝ
2		dvbdfbdioolem1.c	. . . 4 ⊢ (𝜑 → 𝐶 ∈ (𝐴(,)𝐵))
3	1, 2	sselid 3942	. . 3 ⊢ (𝜑 → 𝐶 ∈ ℝ)
4		ioossre 13325	. . . 4 ⊢ (𝐶(,)𝐵) ⊆ ℝ
5		dvbdfbdioolem1.d	. . . 4 ⊢ (𝜑 → 𝐷 ∈ (𝐶(,)𝐵))
6	4, 5	sselid 3942	. . 3 ⊢ (𝜑 → 𝐷 ∈ ℝ)
7	3	rexrd 11205	. . . 4 ⊢ (𝜑 → 𝐶 ∈ ℝ^*)
8		dvbdfbdioolem1.b	. . . . 5 ⊢ (𝜑 → 𝐵 ∈ ℝ)
9	8	rexrd 11205	. . . 4 ⊢ (𝜑 → 𝐵 ∈ ℝ^*)
10		ioogtlb 43723	. . . 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 11205	. . . . 5 ⊢ (𝜑 → 𝐴 ∈ ℝ^*)
14		ioogtlb 43723	. . . . . 6 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ (𝐴(,)𝐵)) → 𝐴 < 𝐶)
15	13, 9, 2, 14	syl3anc 1371	. . . . 5 ⊢ (𝜑 → 𝐴 < 𝐶)
16		iooltub 43738	. . . . . 6 ⊢ ((𝐶 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐷 ∈ (𝐶(,)𝐵)) → 𝐷 < 𝐵)
17	7, 9, 5, 16	syl3anc 1371	. . . . 5 ⊢ (𝜑 → 𝐷 < 𝐵)
18		iccssioo 13333	. . . . 5 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) ∧ (𝐴 < 𝐶 ∧ 𝐷 < 𝐵)) → (𝐶[,]𝐷) ⊆ (𝐴(,)𝐵))
19	13, 9, 15, 17, 18	syl22anc 837	. . . 4 ⊢ (𝜑 → (𝐶[,]𝐷) ⊆ (𝐴(,)𝐵))
20		dvbdfbdioolem1.f	. . . . 5 ⊢ (𝜑 → 𝐹:(𝐴(,)𝐵)⟶ℝ)
21		ax-resscn 11108	. . . . . . 7 ⊢ ℝ ⊆ ℂ
22	21	a1i 11	. . . . . 6 ⊢ (𝜑 → ℝ ⊆ ℂ)
23	20, 22	fssd 6686	. . . . . . 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 25285	. . . . . . 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 24261	. . . . . 6 ⊢ ((ℝ ⊆ ℂ ∧ 𝐹 ∈ ((𝐴(,)𝐵)–cn→ℂ)) → (𝐹 ∈ ((𝐴(,)𝐵)–cn→ℝ) ↔ 𝐹:(𝐴(,)𝐵)⟶ℝ))
29	22, 27, 28	syl2anc 584	. . . . 5 ⊢ (𝜑 → (𝐹 ∈ ((𝐴(,)𝐵)–cn→ℝ) ↔ 𝐹:(𝐴(,)𝐵)⟶ℝ))
30	20, 29	mpbird 256	. . . 4 ⊢ (𝜑 → 𝐹 ∈ ((𝐴(,)𝐵)–cn→ℝ))
31		rescncf 24260	. . . 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 3954	. . . . . . 7 ⊢ (𝜑 → (𝐶[,]𝐷) ⊆ ℝ)
34		eqid 2736	. . . . . . . 8 ⊢ (TopOpen‘ℂ_fld) = (TopOpen‘ℂ_fld)
35	34	tgioo2 24166	. . . . . . . 8 ⊢ (topGen‘ran (,)) = ((TopOpen‘ℂ_fld) ↾_t ℝ)
36	34, 35	dvres 25275	. . . . . . 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 837	. . . . . 6 ⊢ (𝜑 → (ℝ D (𝐹 ↾ (𝐶[,]𝐷))) = ((ℝ D 𝐹) ↾ ((int‘(topGen‘ran (,)))‘(𝐶[,]𝐷))))
38		iccntr 24184	. . . . . . . 8 ⊢ ((𝐶 ∈ ℝ ∧ 𝐷 ∈ ℝ) → ((int‘(topGen‘ran (,)))‘(𝐶[,]𝐷)) = (𝐶(,)𝐷))
39	3, 6, 38	syl2anc 584	. . . . . . 7 ⊢ (𝜑 → ((int‘(topGen‘ran (,)))‘(𝐶[,]𝐷)) = (𝐶(,)𝐷))
40	39	reseq2d 5937	. . . . . 6 ⊢ (𝜑 → ((ℝ D 𝐹) ↾ ((int‘(topGen‘ran (,)))‘(𝐶[,]𝐷))) = ((ℝ D 𝐹) ↾ (𝐶(,)𝐷)))
41	37, 40	eqtrd 2776	. . . . 5 ⊢ (𝜑 → (ℝ D (𝐹 ↾ (𝐶[,]𝐷))) = ((ℝ D 𝐹) ↾ (𝐶(,)𝐷)))
42	41	dmeqd 5861	. . . 4 ⊢ (𝜑 → dom (ℝ D (𝐹 ↾ (𝐶[,]𝐷))) = dom ((ℝ D 𝐹) ↾ (𝐶(,)𝐷)))
43	12, 3, 15	ltled 11303	. . . . . . 7 ⊢ (𝜑 → 𝐴 ≤ 𝐶)
44	6, 8, 17	ltled 11303	. . . . . . 7 ⊢ (𝜑 → 𝐷 ≤ 𝐵)
45		ioossioo 13358	. . . . . . 7 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) ∧ (𝐴 ≤ 𝐶 ∧ 𝐷 ≤ 𝐵)) → (𝐶(,)𝐷) ⊆ (𝐴(,)𝐵))
46	13, 9, 43, 44, 45	syl22anc 837	. . . . . 6 ⊢ (𝜑 → (𝐶(,)𝐷) ⊆ (𝐴(,)𝐵))
47	46, 25	sseqtrrd 3985	. . . . 5 ⊢ (𝜑 → (𝐶(,)𝐷) ⊆ dom (ℝ D 𝐹))
48		ssdmres 5960	. . . . 5 ⊢ ((𝐶(,)𝐷) ⊆ dom (ℝ D 𝐹) ↔ dom ((ℝ D 𝐹) ↾ (𝐶(,)𝐷)) = (𝐶(,)𝐷))
49	47, 48	sylib 217	. . . 4 ⊢ (𝜑 → dom ((ℝ D 𝐹) ↾ (𝐶(,)𝐷)) = (𝐶(,)𝐷))
50	42, 49	eqtrd 2776	. . 3 ⊢ (𝜑 → dom (ℝ D (𝐹 ↾ (𝐶[,]𝐷))) = (𝐶(,)𝐷))
51	3, 6, 11, 32, 50	mvth 25356	. 2 ⊢ (𝜑 → ∃𝑥 ∈ (𝐶(,)𝐷)((ℝ D (𝐹 ↾ (𝐶[,]𝐷)))‘𝑥) = ((((𝐹 ↾ (𝐶[,]𝐷))‘𝐷) − ((𝐹 ↾ (𝐶[,]𝐷))‘𝐶)) / (𝐷 − 𝐶)))
52	41	fveq1d 6844	. . . . . . . . 9 ⊢ (𝜑 → ((ℝ D (𝐹 ↾ (𝐶[,]𝐷)))‘𝑥) = (((ℝ D 𝐹) ↾ (𝐶(,)𝐷))‘𝑥))
53		fvres 6861	. . . . . . . . 9 ⊢ (𝑥 ∈ (𝐶(,)𝐷) → (((ℝ D 𝐹) ↾ (𝐶(,)𝐷))‘𝑥) = ((ℝ D 𝐹)‘𝑥))
54	52, 53	sylan9eq 2796	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐶(,)𝐷)) → ((ℝ D (𝐹 ↾ (𝐶[,]𝐷)))‘𝑥) = ((ℝ D 𝐹)‘𝑥))
55	54	eqcomd 2742	. . . . . . 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 11205	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐷 ∈ ℝ^*)
59	3, 6, 11	ltled 11303	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐶 ≤ 𝐷)
60		ubicc2 13382	. . . . . . . . . . 11 ⊢ ((𝐶 ∈ ℝ^* ∧ 𝐷 ∈ ℝ^* ∧ 𝐶 ≤ 𝐷) → 𝐷 ∈ (𝐶[,]𝐷))
61	7, 58, 59, 60	syl3anc 1371	. . . . . . . . . 10 ⊢ (𝜑 → 𝐷 ∈ (𝐶[,]𝐷))
62		fvres 6861	. . . . . . . . . 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 13381	. . . . . . . . . . 11 ⊢ ((𝐶 ∈ ℝ^* ∧ 𝐷 ∈ ℝ^* ∧ 𝐶 ≤ 𝐷) → 𝐶 ∈ (𝐶[,]𝐷))
65	7, 58, 59, 64	syl3anc 1371	. . . . . . . . . 10 ⊢ (𝜑 → 𝐶 ∈ (𝐶[,]𝐷))
66		fvres 6861	. . . . . . . . . 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 7375	. . . . . . . 8 ⊢ (𝜑 → (((𝐹 ↾ (𝐶[,]𝐷))‘𝐷) − ((𝐹 ↾ (𝐶[,]𝐷))‘𝐶)) = ((𝐹‘𝐷) − (𝐹‘𝐶)))
69	68	oveq1d 7372	. . . . . . 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 2780	. . . . 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 2742	. . . . . . . . . 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 3945	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝐷 ∈ (𝐴(,)𝐵))
75	20, 74	ffvelcdmd 7036	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝐹‘𝐷) ∈ ℝ)
76	20, 2	ffvelcdmd 7036	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝐹‘𝐶) ∈ ℝ)
77	75, 76	resubcld 11583	. . . . . . . . . . . . 13 ⊢ (𝜑 → ((𝐹‘𝐷) − (𝐹‘𝐶)) ∈ ℝ)
78	77	recnd 11183	. . . . . . . . . . . 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 25315	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐹:(𝐴(,)𝐵)⟶ℝ ∧ (𝐴(,)𝐵) ⊆ ℝ) → (ℝ D 𝐹):dom (ℝ D 𝐹)⟶ℝ)
81	20, 24, 80	syl2anc 584	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (ℝ D 𝐹):dom (ℝ D 𝐹)⟶ℝ)
82	25	feq2d 6654	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → ((ℝ D 𝐹):dom (ℝ D 𝐹)⟶ℝ ↔ (ℝ D 𝐹):(𝐴(,)𝐵)⟶ℝ))
83	81, 82	mpbid 231	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (ℝ D 𝐹):(𝐴(,)𝐵)⟶ℝ)
84	83	adantr 481	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐶(,)𝐷)) → (ℝ D 𝐹):(𝐴(,)𝐵)⟶ℝ)
85	46	sselda 3944	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐶(,)𝐷)) → 𝑥 ∈ (𝐴(,)𝐵))
86	84, 85	ffvelcdmd 7036	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐶(,)𝐷)) → ((ℝ D 𝐹)‘𝑥) ∈ ℝ)
87	86	recnd 11183	. . . . . . . . . . . 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 11583	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝐷 − 𝐶) ∈ ℝ)
90	89	recnd 11183	. . . . . . . . . . . 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 11742	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝐶 < 𝐷 ↔ 0 < (𝐷 − 𝐶)))
93	11, 92	mpbid 231	. . . . . . . . . . . . 13 ⊢ (𝜑 → 0 < (𝐷 − 𝐶))
94	93	gt0ne0d 11719	. . . . . . . . . . . 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 11965	. . . . . . . . . 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 231	. . . . . . . . 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 6846	. . . . . . . 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 481	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐶(,)𝐷)) → (𝐷 − 𝐶) ∈ ℂ)
100	87, 99	absmuld 15339	. . . . . . . . 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 2776	. . . . . . 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 15316	. . . . . . . . 9 ⊢ (𝜑 → (abs‘(𝐷 − 𝐶)) = (𝐷 − 𝐶))
104	103	oveq2d 7373	. . . . . . . 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 2776	. . . . . 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 15321	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐶(,)𝐷)) → (abs‘((ℝ D 𝐹)‘𝑥)) ∈ ℝ)
108		dvbdfbdioolem1.k	. . . . . . . . 9 ⊢ (𝜑 → 𝐾 ∈ ℝ)
109	108	adantr 481	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐶(,)𝐷)) → 𝐾 ∈ ℝ)
110	89	adantr 481	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐶(,)𝐷)) → (𝐷 − 𝐶) ∈ ℝ)
111		0red 11158	. . . . . . . . . 10 ⊢ (𝜑 → 0 ∈ ℝ)
112	111, 89, 93	ltled 11303	. . . . . . . . 9 ⊢ (𝜑 → 0 ≤ (𝐷 − 𝐶))
113	112	adantr 481	. . . . . . . 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 481	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐶(,)𝐷)) → ∀𝑥 ∈ (𝐴(,)𝐵)(abs‘((ℝ D 𝐹)‘𝑥)) ≤ 𝐾)
116		rspa 3231	. . . . . . . . 9 ⊢ ((∀𝑥 ∈ (𝐴(,)𝐵)(abs‘((ℝ D 𝐹)‘𝑥)) ≤ 𝐾 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → (abs‘((ℝ D 𝐹)‘𝑥)) ≤ 𝐾)
117	115, 85, 116	syl2anc 584	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐶(,)𝐷)) → (abs‘((ℝ D 𝐹)‘𝑥)) ≤ 𝐾)
118	107, 109, 110, 113, 117	lemul1ad 12094	. . . . . . 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 5127	. . . . 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 15321	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐶(,)𝐷)) → (abs‘(𝐷 − 𝐶)) ∈ ℝ)
123	8, 12	resubcld 11583	. . . . . . . . 9 ⊢ (𝜑 → (𝐵 − 𝐴) ∈ ℝ)
124	123	adantr 481	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐶(,)𝐷)) → (𝐵 − 𝐴) ∈ ℝ)
125	87	absge0d 15329	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐶(,)𝐷)) → 0 ≤ (abs‘((ℝ D 𝐹)‘𝑥)))
126	99	absge0d 15329	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐶(,)𝐷)) → 0 ≤ (abs‘(𝐷 − 𝐶)))
127	6, 12, 8, 3, 44, 43	le2subd 11775	. . . . . . . . . 10 ⊢ (𝜑 → (𝐷 − 𝐶) ≤ (𝐵 − 𝐴))
128	103, 127	eqbrtrd 5127	. . . . . . . . 9 ⊢ (𝜑 → (abs‘(𝐷 − 𝐶)) ≤ (𝐵 − 𝐴))
129	128	adantr 481	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐶(,)𝐷)) → (abs‘(𝐷 − 𝐶)) ≤ (𝐵 − 𝐴))
130	107, 109, 122, 124, 125, 126, 117, 129	lemul12ad 12097	. . . . . . 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 5127	. . . . 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 512	. . 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 3156	. 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 205 ∧ wa 396 ∧ w3a 1087 = wceq 1541 ∈ wcel 2106 ≠ wne 2943 ∀wral 3064 ∃wrex 3073 ⊆ wss 3910 class class class wbr 5105 dom cdm 5633 ran crn 5634 ↾ cres 5635 ⟶wf 6492 ‘cfv 6496 (class class class)co 7357 ℂcc 11049 ℝcr 11050 0cc0 11051 · cmul 11056 ℝ^*cxr 11188 < clt 11189 ≤ cle 11190 − cmin 11385 / cdiv 11812 (,)cioo 13264 [,]cicc 13267 abscabs 15119 TopOpenctopn 17303 topGenctg 17319 ℂ_fldccnfld 20796 intcnt 22368 –cn→ccncf 24239 D cdv 25227

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2707 ax-rep 5242 ax-sep 5256 ax-nul 5263 ax-pow 5320 ax-pr 5384 ax-un 7672 ax-cnex 11107 ax-resscn 11108 ax-1cn 11109 ax-icn 11110 ax-addcl 11111 ax-addrcl 11112 ax-mulcl 11113 ax-mulrcl 11114 ax-mulcom 11115 ax-addass 11116 ax-mulass 11117 ax-distr 11118 ax-i2m1 11119 ax-1ne0 11120 ax-1rid 11121 ax-rnegex 11122 ax-rrecex 11123 ax-cnre 11124 ax-pre-lttri 11125 ax-pre-lttrn 11126 ax-pre-ltadd 11127 ax-pre-mulgt0 11128 ax-pre-sup 11129 ax-addf 11130 ax-mulf 11131

This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3065 df-rex 3074 df-rmo 3353 df-reu 3354 df-rab 3408 df-v 3447 df-sbc 3740 df-csb 3856 df-dif 3913 df-un 3915 df-in 3917 df-ss 3927 df-pss 3929 df-nul 4283 df-if 4487 df-pw 4562 df-sn 4587 df-pr 4589 df-tp 4591 df-op 4593 df-uni 4866 df-int 4908 df-iun 4956 df-iin 4957 df-br 5106 df-opab 5168 df-mpt 5189 df-tr 5223 df-id 5531 df-eprel 5537 df-po 5545 df-so 5546 df-fr 5588 df-se 5589 df-we 5590 df-xp 5639 df-rel 5640 df-cnv 5641 df-co 5642 df-dm 5643 df-rn 5644 df-res 5645 df-ima 5646 df-pred 6253 df-ord 6320 df-on 6321 df-lim 6322 df-suc 6323 df-iota 6448 df-fun 6498 df-fn 6499 df-f 6500 df-f1 6501 df-fo 6502 df-f1o 6503 df-fv 6504 df-isom 6505 df-riota 7313 df-ov 7360 df-oprab 7361 df-mpo 7362 df-of 7617 df-om 7803 df-1st 7921 df-2nd 7922 df-supp 8093 df-frecs 8212 df-wrecs 8243 df-recs 8317 df-rdg 8356 df-1o 8412 df-2o 8413 df-er 8648 df-map 8767 df-pm 8768 df-ixp 8836 df-en 8884 df-dom 8885 df-sdom 8886 df-fin 8887 df-fsupp 9306 df-fi 9347 df-sup 9378 df-inf 9379 df-oi 9446 df-card 9875 df-pnf 11191 df-mnf 11192 df-xr 11193 df-ltxr 11194 df-le 11195 df-sub 11387 df-neg 11388 df-div 11813 df-nn 12154 df-2 12216 df-3 12217 df-4 12218 df-5 12219 df-6 12220 df-7 12221 df-8 12222 df-9 12223 df-n0 12414 df-z 12500 df-dec 12619 df-uz 12764 df-q 12874 df-rp 12916 df-xneg 13033 df-xadd 13034 df-xmul 13035 df-ioo 13268 df-ico 13270 df-icc 13271 df-fz 13425 df-fzo 13568 df-seq 13907 df-exp 13968 df-hash 14231 df-cj 14984 df-re 14985 df-im 14986 df-sqrt 15120 df-abs 15121 df-struct 17019 df-sets 17036 df-slot 17054 df-ndx 17066 df-base 17084 df-ress 17113 df-plusg 17146 df-mulr 17147 df-starv 17148 df-sca 17149 df-vsca 17150 df-ip 17151 df-tset 17152 df-ple 17153 df-ds 17155 df-unif 17156 df-hom 17157 df-cco 17158 df-rest 17304 df-topn 17305 df-0g 17323 df-gsum 17324 df-topgen 17325 df-pt 17326 df-prds 17329 df-xrs 17384 df-qtop 17389 df-imas 17390 df-xps 17392 df-mre 17466 df-mrc 17467 df-acs 17469 df-mgm 18497 df-sgrp 18546 df-mnd 18557 df-submnd 18602 df-mulg 18873 df-cntz 19097 df-cmn 19564 df-psmet 20788 df-xmet 20789 df-met 20790 df-bl 20791 df-mopn 20792 df-fbas 20793 df-fg 20794 df-cnfld 20797 df-top 22243 df-topon 22260 df-topsp 22282 df-bases 22296 df-cld 22370 df-ntr 22371 df-cls 22372 df-nei 22449 df-lp 22487 df-perf 22488 df-cn 22578 df-cnp 22579 df-haus 22666 df-cmp 22738 df-tx 22913 df-hmeo 23106 df-fil 23197 df-fm 23289 df-flim 23290 df-flf 23291 df-xms 23673 df-ms 23674 df-tms 23675 df-cncf 24241 df-limc 25230 df-dv 25231

This theorem is referenced by: dvbdfbdioolem2 44160 ioodvbdlimc1lem1 44162 ioodvbdlimc1lem2 44163 ioodvbdlimc2lem 44165

W3C validator