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Theorem c1liplem1 25951

Description: Lemma for c1lip1 25952. (Contributed by Stefan O'Rear, 15-Nov-2014.)

**Hypotheses**
Ref	Expression
c1liplem1.a	⊢ (𝜑 → 𝐴 ∈ ℝ)
c1liplem1.b	⊢ (𝜑 → 𝐵 ∈ ℝ)
c1liplem1.le	⊢ (𝜑 → 𝐴 ≤ 𝐵)
c1liplem1.f	⊢ (𝜑 → 𝐹 ∈ (ℂ ↑_pm ℝ))
c1liplem1.dv	⊢ (𝜑 → ((ℝ D 𝐹) ↾ (𝐴[,]𝐵)) ∈ ((𝐴[,]𝐵)–cn→ℝ))
c1liplem1.cn	⊢ (𝜑 → (𝐹 ↾ (𝐴[,]𝐵)) ∈ ((𝐴[,]𝐵)–cn→ℝ))
c1liplem1.k	⊢ 𝐾 = sup((abs “ ((ℝ D 𝐹) “ (𝐴[,]𝐵))), ℝ, < )

**Assertion**
Ref	Expression
c1liplem1	⊢ (𝜑 → (𝐾 ∈ ℝ ∧ ∀𝑥 ∈ (𝐴[,]𝐵)∀𝑦 ∈ (𝐴[,]𝐵)(𝑥 < 𝑦 → (abs‘((𝐹‘𝑦) − (𝐹‘𝑥))) ≤ (𝐾 · (abs‘(𝑦 − 𝑥))))))

Distinct variable groups: 𝜑,𝑥,𝑦 𝑥,𝐴,𝑦 𝑥,𝐵,𝑦 𝑥,𝐹,𝑦 Allowed substitution hints: 𝐾(𝑥,𝑦)

Proof of Theorem c1liplem1 Dummy variables 𝑎 𝑏 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		c1liplem1.k	. . 3 ⊢ 𝐾 = sup((abs “ ((ℝ D 𝐹) “ (𝐴[,]𝐵))), ℝ, < )
2		imassrn 6058	. . . . . 6 ⊢ (abs “ ((ℝ D 𝐹) “ (𝐴[,]𝐵))) ⊆ ran abs
3		absf 15354	. . . . . . 7 ⊢ abs:ℂ⟶ℝ
4		frn 6712	. . . . . . 7 ⊢ (abs:ℂ⟶ℝ → ran abs ⊆ ℝ)
5	3, 4	ax-mp 5	. . . . . 6 ⊢ ran abs ⊆ ℝ
6	2, 5	sstri 3968	. . . . 5 ⊢ (abs “ ((ℝ D 𝐹) “ (𝐴[,]𝐵))) ⊆ ℝ
7	6	a1i 11	. . . 4 ⊢ (𝜑 → (abs “ ((ℝ D 𝐹) “ (𝐴[,]𝐵))) ⊆ ℝ)
8		dvf 25858	. . . . . . . 8 ⊢ (ℝ D 𝐹):dom (ℝ D 𝐹)⟶ℂ
9		ffun 6708	. . . . . . . 8 ⊢ ((ℝ D 𝐹):dom (ℝ D 𝐹)⟶ℂ → Fun (ℝ D 𝐹))
10	8, 9	ax-mp 5	. . . . . . 7 ⊢ Fun (ℝ D 𝐹)
11	10	a1i 11	. . . . . 6 ⊢ (𝜑 → Fun (ℝ D 𝐹))
12		c1liplem1.dv	. . . . . . . 8 ⊢ (𝜑 → ((ℝ D 𝐹) ↾ (𝐴[,]𝐵)) ∈ ((𝐴[,]𝐵)–cn→ℝ))
13		cncff 24835	. . . . . . . 8 ⊢ (((ℝ D 𝐹) ↾ (𝐴[,]𝐵)) ∈ ((𝐴[,]𝐵)–cn→ℝ) → ((ℝ D 𝐹) ↾ (𝐴[,]𝐵)):(𝐴[,]𝐵)⟶ℝ)
14		fdm 6714	. . . . . . . 8 ⊢ (((ℝ D 𝐹) ↾ (𝐴[,]𝐵)):(𝐴[,]𝐵)⟶ℝ → dom ((ℝ D 𝐹) ↾ (𝐴[,]𝐵)) = (𝐴[,]𝐵))
15	12, 13, 14	3syl 18	. . . . . . 7 ⊢ (𝜑 → dom ((ℝ D 𝐹) ↾ (𝐴[,]𝐵)) = (𝐴[,]𝐵))
16		ssdmres 6000	. . . . . . 7 ⊢ ((𝐴[,]𝐵) ⊆ dom (ℝ D 𝐹) ↔ dom ((ℝ D 𝐹) ↾ (𝐴[,]𝐵)) = (𝐴[,]𝐵))
17	15, 16	sylibr 234	. . . . . 6 ⊢ (𝜑 → (𝐴[,]𝐵) ⊆ dom (ℝ D 𝐹))
18		c1liplem1.a	. . . . . . . 8 ⊢ (𝜑 → 𝐴 ∈ ℝ)
19	18	rexrd 11283	. . . . . . 7 ⊢ (𝜑 → 𝐴 ∈ ℝ^*)
20		c1liplem1.b	. . . . . . . 8 ⊢ (𝜑 → 𝐵 ∈ ℝ)
21	20	rexrd 11283	. . . . . . 7 ⊢ (𝜑 → 𝐵 ∈ ℝ^*)
22		c1liplem1.le	. . . . . . 7 ⊢ (𝜑 → 𝐴 ≤ 𝐵)
23		lbicc2 13479	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐴 ≤ 𝐵) → 𝐴 ∈ (𝐴[,]𝐵))
24	19, 21, 22, 23	syl3anc 1373	. . . . . 6 ⊢ (𝜑 → 𝐴 ∈ (𝐴[,]𝐵))
25		funfvima2 7222	. . . . . . 7 ⊢ ((Fun (ℝ D 𝐹) ∧ (𝐴[,]𝐵) ⊆ dom (ℝ D 𝐹)) → (𝐴 ∈ (𝐴[,]𝐵) → ((ℝ D 𝐹)‘𝐴) ∈ ((ℝ D 𝐹) “ (𝐴[,]𝐵))))
26	25	imp 406	. . . . . 6 ⊢ (((Fun (ℝ D 𝐹) ∧ (𝐴[,]𝐵) ⊆ dom (ℝ D 𝐹)) ∧ 𝐴 ∈ (𝐴[,]𝐵)) → ((ℝ D 𝐹)‘𝐴) ∈ ((ℝ D 𝐹) “ (𝐴[,]𝐵)))
27	11, 17, 24, 26	syl21anc 837	. . . . 5 ⊢ (𝜑 → ((ℝ D 𝐹)‘𝐴) ∈ ((ℝ D 𝐹) “ (𝐴[,]𝐵)))
28		ffun 6708	. . . . . . 7 ⊢ (abs:ℂ⟶ℝ → Fun abs)
29	3, 28	ax-mp 5	. . . . . 6 ⊢ Fun abs
30		imassrn 6058	. . . . . . . 8 ⊢ ((ℝ D 𝐹) “ (𝐴[,]𝐵)) ⊆ ran (ℝ D 𝐹)
31		frn 6712	. . . . . . . . 9 ⊢ ((ℝ D 𝐹):dom (ℝ D 𝐹)⟶ℂ → ran (ℝ D 𝐹) ⊆ ℂ)
32	8, 31	ax-mp 5	. . . . . . . 8 ⊢ ran (ℝ D 𝐹) ⊆ ℂ
33	30, 32	sstri 3968	. . . . . . 7 ⊢ ((ℝ D 𝐹) “ (𝐴[,]𝐵)) ⊆ ℂ
34	3	fdmi 6716	. . . . . . 7 ⊢ dom abs = ℂ
35	33, 34	sseqtrri 4008	. . . . . 6 ⊢ ((ℝ D 𝐹) “ (𝐴[,]𝐵)) ⊆ dom abs
36		funfvima2 7222	. . . . . 6 ⊢ ((Fun abs ∧ ((ℝ D 𝐹) “ (𝐴[,]𝐵)) ⊆ dom abs) → (((ℝ D 𝐹)‘𝐴) ∈ ((ℝ D 𝐹) “ (𝐴[,]𝐵)) → (abs‘((ℝ D 𝐹)‘𝐴)) ∈ (abs “ ((ℝ D 𝐹) “ (𝐴[,]𝐵)))))
37	29, 35, 36	mp2an 692	. . . . 5 ⊢ (((ℝ D 𝐹)‘𝐴) ∈ ((ℝ D 𝐹) “ (𝐴[,]𝐵)) → (abs‘((ℝ D 𝐹)‘𝐴)) ∈ (abs “ ((ℝ D 𝐹) “ (𝐴[,]𝐵))))
38		ne0i 4316	. . . . 5 ⊢ ((abs‘((ℝ D 𝐹)‘𝐴)) ∈ (abs “ ((ℝ D 𝐹) “ (𝐴[,]𝐵))) → (abs “ ((ℝ D 𝐹) “ (𝐴[,]𝐵))) ≠ ∅)
39	27, 37, 38	3syl 18	. . . 4 ⊢ (𝜑 → (abs “ ((ℝ D 𝐹) “ (𝐴[,]𝐵))) ≠ ∅)
40		ax-resscn 11184	. . . . . . . 8 ⊢ ℝ ⊆ ℂ
41		ssid 3981	. . . . . . . 8 ⊢ ℂ ⊆ ℂ
42		cncfss 24841	. . . . . . . 8 ⊢ ((ℝ ⊆ ℂ ∧ ℂ ⊆ ℂ) → ((𝐴[,]𝐵)–cn→ℝ) ⊆ ((𝐴[,]𝐵)–cn→ℂ))
43	40, 41, 42	mp2an 692	. . . . . . 7 ⊢ ((𝐴[,]𝐵)–cn→ℝ) ⊆ ((𝐴[,]𝐵)–cn→ℂ)
44	43, 12	sselid 3956	. . . . . 6 ⊢ (𝜑 → ((ℝ D 𝐹) ↾ (𝐴[,]𝐵)) ∈ ((𝐴[,]𝐵)–cn→ℂ))
45		cniccbdd 25412	. . . . . 6 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ ((ℝ D 𝐹) ↾ (𝐴[,]𝐵)) ∈ ((𝐴[,]𝐵)–cn→ℂ)) → ∃𝑎 ∈ ℝ ∀𝑥 ∈ (𝐴[,]𝐵)(abs‘(((ℝ D 𝐹) ↾ (𝐴[,]𝐵))‘𝑥)) ≤ 𝑎)
46	18, 20, 44, 45	syl3anc 1373	. . . . 5 ⊢ (𝜑 → ∃𝑎 ∈ ℝ ∀𝑥 ∈ (𝐴[,]𝐵)(abs‘(((ℝ D 𝐹) ↾ (𝐴[,]𝐵))‘𝑥)) ≤ 𝑎)
47		fvelima 6943	. . . . . . . . . 10 ⊢ ((Fun abs ∧ 𝑏 ∈ (abs “ ((ℝ D 𝐹) “ (𝐴[,]𝐵)))) → ∃𝑦 ∈ ((ℝ D 𝐹) “ (𝐴[,]𝐵))(abs‘𝑦) = 𝑏)
48	29, 47	mpan 690	. . . . . . . . 9 ⊢ (𝑏 ∈ (abs “ ((ℝ D 𝐹) “ (𝐴[,]𝐵))) → ∃𝑦 ∈ ((ℝ D 𝐹) “ (𝐴[,]𝐵))(abs‘𝑦) = 𝑏)
49		fvres 6894	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑏 ∈ (𝐴[,]𝐵) → (((ℝ D 𝐹) ↾ (𝐴[,]𝐵))‘𝑏) = ((ℝ D 𝐹)‘𝑏))
50	49	adantl 481	. . . . . . . . . . . . . . . . 17 ⊢ ((∀𝑥 ∈ (𝐴[,]𝐵)(abs‘(((ℝ D 𝐹) ↾ (𝐴[,]𝐵))‘𝑥)) ≤ 𝑎 ∧ 𝑏 ∈ (𝐴[,]𝐵)) → (((ℝ D 𝐹) ↾ (𝐴[,]𝐵))‘𝑏) = ((ℝ D 𝐹)‘𝑏))
51	50	fveq2d 6879	. . . . . . . . . . . . . . . 16 ⊢ ((∀𝑥 ∈ (𝐴[,]𝐵)(abs‘(((ℝ D 𝐹) ↾ (𝐴[,]𝐵))‘𝑥)) ≤ 𝑎 ∧ 𝑏 ∈ (𝐴[,]𝐵)) → (abs‘(((ℝ D 𝐹) ↾ (𝐴[,]𝐵))‘𝑏)) = (abs‘((ℝ D 𝐹)‘𝑏)))
52		2fveq3 6880	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 = 𝑏 → (abs‘(((ℝ D 𝐹) ↾ (𝐴[,]𝐵))‘𝑥)) = (abs‘(((ℝ D 𝐹) ↾ (𝐴[,]𝐵))‘𝑏)))
53	52	breq1d 5129	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 = 𝑏 → ((abs‘(((ℝ D 𝐹) ↾ (𝐴[,]𝐵))‘𝑥)) ≤ 𝑎 ↔ (abs‘(((ℝ D 𝐹) ↾ (𝐴[,]𝐵))‘𝑏)) ≤ 𝑎))
54	53	rspccva 3600	. . . . . . . . . . . . . . . 16 ⊢ ((∀𝑥 ∈ (𝐴[,]𝐵)(abs‘(((ℝ D 𝐹) ↾ (𝐴[,]𝐵))‘𝑥)) ≤ 𝑎 ∧ 𝑏 ∈ (𝐴[,]𝐵)) → (abs‘(((ℝ D 𝐹) ↾ (𝐴[,]𝐵))‘𝑏)) ≤ 𝑎)
55	51, 54	eqbrtrrd 5143	. . . . . . . . . . . . . . 15 ⊢ ((∀𝑥 ∈ (𝐴[,]𝐵)(abs‘(((ℝ D 𝐹) ↾ (𝐴[,]𝐵))‘𝑥)) ≤ 𝑎 ∧ 𝑏 ∈ (𝐴[,]𝐵)) → (abs‘((ℝ D 𝐹)‘𝑏)) ≤ 𝑎)
56	55	adantll 714	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑎 ∈ ℝ) ∧ ∀𝑥 ∈ (𝐴[,]𝐵)(abs‘(((ℝ D 𝐹) ↾ (𝐴[,]𝐵))‘𝑥)) ≤ 𝑎) ∧ 𝑏 ∈ (𝐴[,]𝐵)) → (abs‘((ℝ D 𝐹)‘𝑏)) ≤ 𝑎)
57		fveq2 6875	. . . . . . . . . . . . . . 15 ⊢ (((ℝ D 𝐹)‘𝑏) = 𝑦 → (abs‘((ℝ D 𝐹)‘𝑏)) = (abs‘𝑦))
58	57	breq1d 5129	. . . . . . . . . . . . . 14 ⊢ (((ℝ D 𝐹)‘𝑏) = 𝑦 → ((abs‘((ℝ D 𝐹)‘𝑏)) ≤ 𝑎 ↔ (abs‘𝑦) ≤ 𝑎))
59	56, 58	syl5ibcom 245	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑎 ∈ ℝ) ∧ ∀𝑥 ∈ (𝐴[,]𝐵)(abs‘(((ℝ D 𝐹) ↾ (𝐴[,]𝐵))‘𝑥)) ≤ 𝑎) ∧ 𝑏 ∈ (𝐴[,]𝐵)) → (((ℝ D 𝐹)‘𝑏) = 𝑦 → (abs‘𝑦) ≤ 𝑎))
60	59	rexlimdva 3141	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑎 ∈ ℝ) ∧ ∀𝑥 ∈ (𝐴[,]𝐵)(abs‘(((ℝ D 𝐹) ↾ (𝐴[,]𝐵))‘𝑥)) ≤ 𝑎) → (∃𝑏 ∈ (𝐴[,]𝐵)((ℝ D 𝐹)‘𝑏) = 𝑦 → (abs‘𝑦) ≤ 𝑎))
61		fvelima 6943	. . . . . . . . . . . . 13 ⊢ ((Fun (ℝ D 𝐹) ∧ 𝑦 ∈ ((ℝ D 𝐹) “ (𝐴[,]𝐵))) → ∃𝑏 ∈ (𝐴[,]𝐵)((ℝ D 𝐹)‘𝑏) = 𝑦)
62	10, 61	mpan 690	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ ((ℝ D 𝐹) “ (𝐴[,]𝐵)) → ∃𝑏 ∈ (𝐴[,]𝐵)((ℝ D 𝐹)‘𝑏) = 𝑦)
63	60, 62	impel 505	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑎 ∈ ℝ) ∧ ∀𝑥 ∈ (𝐴[,]𝐵)(abs‘(((ℝ D 𝐹) ↾ (𝐴[,]𝐵))‘𝑥)) ≤ 𝑎) ∧ 𝑦 ∈ ((ℝ D 𝐹) “ (𝐴[,]𝐵))) → (abs‘𝑦) ≤ 𝑎)
64		breq1 5122	. . . . . . . . . . 11 ⊢ ((abs‘𝑦) = 𝑏 → ((abs‘𝑦) ≤ 𝑎 ↔ 𝑏 ≤ 𝑎))
65	63, 64	syl5ibcom 245	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑎 ∈ ℝ) ∧ ∀𝑥 ∈ (𝐴[,]𝐵)(abs‘(((ℝ D 𝐹) ↾ (𝐴[,]𝐵))‘𝑥)) ≤ 𝑎) ∧ 𝑦 ∈ ((ℝ D 𝐹) “ (𝐴[,]𝐵))) → ((abs‘𝑦) = 𝑏 → 𝑏 ≤ 𝑎))
66	65	rexlimdva 3141	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑎 ∈ ℝ) ∧ ∀𝑥 ∈ (𝐴[,]𝐵)(abs‘(((ℝ D 𝐹) ↾ (𝐴[,]𝐵))‘𝑥)) ≤ 𝑎) → (∃𝑦 ∈ ((ℝ D 𝐹) “ (𝐴[,]𝐵))(abs‘𝑦) = 𝑏 → 𝑏 ≤ 𝑎))
67	48, 66	syl5 34	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑎 ∈ ℝ) ∧ ∀𝑥 ∈ (𝐴[,]𝐵)(abs‘(((ℝ D 𝐹) ↾ (𝐴[,]𝐵))‘𝑥)) ≤ 𝑎) → (𝑏 ∈ (abs “ ((ℝ D 𝐹) “ (𝐴[,]𝐵))) → 𝑏 ≤ 𝑎))
68	67	ralrimiv 3131	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑎 ∈ ℝ) ∧ ∀𝑥 ∈ (𝐴[,]𝐵)(abs‘(((ℝ D 𝐹) ↾ (𝐴[,]𝐵))‘𝑥)) ≤ 𝑎) → ∀𝑏 ∈ (abs “ ((ℝ D 𝐹) “ (𝐴[,]𝐵)))𝑏 ≤ 𝑎)
69	68	ex 412	. . . . . 6 ⊢ ((𝜑 ∧ 𝑎 ∈ ℝ) → (∀𝑥 ∈ (𝐴[,]𝐵)(abs‘(((ℝ D 𝐹) ↾ (𝐴[,]𝐵))‘𝑥)) ≤ 𝑎 → ∀𝑏 ∈ (abs “ ((ℝ D 𝐹) “ (𝐴[,]𝐵)))𝑏 ≤ 𝑎))
70	69	reximdva 3153	. . . . 5 ⊢ (𝜑 → (∃𝑎 ∈ ℝ ∀𝑥 ∈ (𝐴[,]𝐵)(abs‘(((ℝ D 𝐹) ↾ (𝐴[,]𝐵))‘𝑥)) ≤ 𝑎 → ∃𝑎 ∈ ℝ ∀𝑏 ∈ (abs “ ((ℝ D 𝐹) “ (𝐴[,]𝐵)))𝑏 ≤ 𝑎))
71	46, 70	mpd 15	. . . 4 ⊢ (𝜑 → ∃𝑎 ∈ ℝ ∀𝑏 ∈ (abs “ ((ℝ D 𝐹) “ (𝐴[,]𝐵)))𝑏 ≤ 𝑎)
72	7, 39, 71	suprcld 12203	. . 3 ⊢ (𝜑 → sup((abs “ ((ℝ D 𝐹) “ (𝐴[,]𝐵))), ℝ, < ) ∈ ℝ)
73	1, 72	eqeltrid 2838	. 2 ⊢ (𝜑 → 𝐾 ∈ ℝ)
74		simplrr 777	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) ∧ 𝑥 < 𝑦) → 𝑦 ∈ (𝐴[,]𝐵))
75	74	fvresd 6895	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) ∧ 𝑥 < 𝑦) → ((𝐹 ↾ (𝐴[,]𝐵))‘𝑦) = (𝐹‘𝑦))
76		c1liplem1.cn	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝐹 ↾ (𝐴[,]𝐵)) ∈ ((𝐴[,]𝐵)–cn→ℝ))
77		cncff 24835	. . . . . . . . . . . . . 14 ⊢ ((𝐹 ↾ (𝐴[,]𝐵)) ∈ ((𝐴[,]𝐵)–cn→ℝ) → (𝐹 ↾ (𝐴[,]𝐵)):(𝐴[,]𝐵)⟶ℝ)
78	76, 77	syl 17	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝐹 ↾ (𝐴[,]𝐵)):(𝐴[,]𝐵)⟶ℝ)
79	78	ad2antrr 726	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) ∧ 𝑥 < 𝑦) → (𝐹 ↾ (𝐴[,]𝐵)):(𝐴[,]𝐵)⟶ℝ)
80	79, 74	ffvelcdmd 7074	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) ∧ 𝑥 < 𝑦) → ((𝐹 ↾ (𝐴[,]𝐵))‘𝑦) ∈ ℝ)
81	80	recnd 11261	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) ∧ 𝑥 < 𝑦) → ((𝐹 ↾ (𝐴[,]𝐵))‘𝑦) ∈ ℂ)
82	75, 81	eqeltrrd 2835	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) ∧ 𝑥 < 𝑦) → (𝐹‘𝑦) ∈ ℂ)
83		simplrl 776	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) ∧ 𝑥 < 𝑦) → 𝑥 ∈ (𝐴[,]𝐵))
84	83	fvresd 6895	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) ∧ 𝑥 < 𝑦) → ((𝐹 ↾ (𝐴[,]𝐵))‘𝑥) = (𝐹‘𝑥))
85	79, 83	ffvelcdmd 7074	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) ∧ 𝑥 < 𝑦) → ((𝐹 ↾ (𝐴[,]𝐵))‘𝑥) ∈ ℝ)
86	85	recnd 11261	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) ∧ 𝑥 < 𝑦) → ((𝐹 ↾ (𝐴[,]𝐵))‘𝑥) ∈ ℂ)
87	84, 86	eqeltrrd 2835	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) ∧ 𝑥 < 𝑦) → (𝐹‘𝑥) ∈ ℂ)
88	82, 87	subcld 11592	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) ∧ 𝑥 < 𝑦) → ((𝐹‘𝑦) − (𝐹‘𝑥)) ∈ ℂ)
89		iccssre 13444	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴[,]𝐵) ⊆ ℝ)
90	18, 20, 89	syl2anc 584	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐴[,]𝐵) ⊆ ℝ)
91	90	ad2antrr 726	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) ∧ 𝑥 < 𝑦) → (𝐴[,]𝐵) ⊆ ℝ)
92	91, 74	sseldd 3959	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) ∧ 𝑥 < 𝑦) → 𝑦 ∈ ℝ)
93	91, 83	sseldd 3959	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) ∧ 𝑥 < 𝑦) → 𝑥 ∈ ℝ)
94	92, 93	resubcld 11663	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) ∧ 𝑥 < 𝑦) → (𝑦 − 𝑥) ∈ ℝ)
95	94	recnd 11261	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) ∧ 𝑥 < 𝑦) → (𝑦 − 𝑥) ∈ ℂ)
96		simpr 484	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) ∧ 𝑥 < 𝑦) → 𝑥 < 𝑦)
97		difrp 13045	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) → (𝑥 < 𝑦 ↔ (𝑦 − 𝑥) ∈ ℝ⁺))
98	93, 92, 97	syl2anc 584	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) ∧ 𝑥 < 𝑦) → (𝑥 < 𝑦 ↔ (𝑦 − 𝑥) ∈ ℝ⁺))
99	96, 98	mpbid 232	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) ∧ 𝑥 < 𝑦) → (𝑦 − 𝑥) ∈ ℝ⁺)
100	99	rpne0d 13054	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) ∧ 𝑥 < 𝑦) → (𝑦 − 𝑥) ≠ 0)
101	88, 95, 100	absdivd 15472	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) ∧ 𝑥 < 𝑦) → (abs‘(((𝐹‘𝑦) − (𝐹‘𝑥)) / (𝑦 − 𝑥))) = ((abs‘((𝐹‘𝑦) − (𝐹‘𝑥))) / (abs‘(𝑦 − 𝑥))))
102	6	a1i 11	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) ∧ 𝑥 < 𝑦) → (abs “ ((ℝ D 𝐹) “ (𝐴[,]𝐵))) ⊆ ℝ)
103	39	ad2antrr 726	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) ∧ 𝑥 < 𝑦) → (abs “ ((ℝ D 𝐹) “ (𝐴[,]𝐵))) ≠ ∅)
104	71	ad2antrr 726	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) ∧ 𝑥 < 𝑦) → ∃𝑎 ∈ ℝ ∀𝑏 ∈ (abs “ ((ℝ D 𝐹) “ (𝐴[,]𝐵)))𝑏 ≤ 𝑎)
105	29	a1i 11	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) ∧ 𝑥 < 𝑦) → Fun abs)
106	88, 95, 100	divcld 12015	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) ∧ 𝑥 < 𝑦) → (((𝐹‘𝑦) − (𝐹‘𝑥)) / (𝑦 − 𝑥)) ∈ ℂ)
107	106, 34	eleqtrrdi 2845	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) ∧ 𝑥 < 𝑦) → (((𝐹‘𝑦) − (𝐹‘𝑥)) / (𝑦 − 𝑥)) ∈ dom abs)
108	93	rexrd 11283	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) ∧ 𝑥 < 𝑦) → 𝑥 ∈ ℝ^*)
109	92	rexrd 11283	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) ∧ 𝑥 < 𝑦) → 𝑦 ∈ ℝ^*)
110	93, 92, 96	ltled 11381	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) ∧ 𝑥 < 𝑦) → 𝑥 ≤ 𝑦)
111		ubicc2 13480	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 ∈ ℝ^* ∧ 𝑦 ∈ ℝ^* ∧ 𝑥 ≤ 𝑦) → 𝑦 ∈ (𝑥[,]𝑦))
112	108, 109, 110, 111	syl3anc 1373	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) ∧ 𝑥 < 𝑦) → 𝑦 ∈ (𝑥[,]𝑦))
113	112	fvresd 6895	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) ∧ 𝑥 < 𝑦) → ((𝐹 ↾ (𝑥[,]𝑦))‘𝑦) = (𝐹‘𝑦))
114		lbicc2 13479	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 ∈ ℝ^* ∧ 𝑦 ∈ ℝ^* ∧ 𝑥 ≤ 𝑦) → 𝑥 ∈ (𝑥[,]𝑦))
115	108, 109, 110, 114	syl3anc 1373	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) ∧ 𝑥 < 𝑦) → 𝑥 ∈ (𝑥[,]𝑦))
116	115	fvresd 6895	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) ∧ 𝑥 < 𝑦) → ((𝐹 ↾ (𝑥[,]𝑦))‘𝑥) = (𝐹‘𝑥))
117	113, 116	oveq12d 7421	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) ∧ 𝑥 < 𝑦) → (((𝐹 ↾ (𝑥[,]𝑦))‘𝑦) − ((𝐹 ↾ (𝑥[,]𝑦))‘𝑥)) = ((𝐹‘𝑦) − (𝐹‘𝑥)))
118	117	oveq1d 7418	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) ∧ 𝑥 < 𝑦) → ((((𝐹 ↾ (𝑥[,]𝑦))‘𝑦) − ((𝐹 ↾ (𝑥[,]𝑦))‘𝑥)) / (𝑦 − 𝑥)) = (((𝐹‘𝑦) − (𝐹‘𝑥)) / (𝑦 − 𝑥)))
119		iccss2 13432	. . . . . . . . . . . . . . . 16 ⊢ ((𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵)) → (𝑥[,]𝑦) ⊆ (𝐴[,]𝐵))
120	119	ad2antlr 727	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) ∧ 𝑥 < 𝑦) → (𝑥[,]𝑦) ⊆ (𝐴[,]𝐵))
121	120	resabs1d 5995	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) ∧ 𝑥 < 𝑦) → ((𝐹 ↾ (𝐴[,]𝐵)) ↾ (𝑥[,]𝑦)) = (𝐹 ↾ (𝑥[,]𝑦)))
122	76	ad2antrr 726	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) ∧ 𝑥 < 𝑦) → (𝐹 ↾ (𝐴[,]𝐵)) ∈ ((𝐴[,]𝐵)–cn→ℝ))
123		rescncf 24839	. . . . . . . . . . . . . . 15 ⊢ ((𝑥[,]𝑦) ⊆ (𝐴[,]𝐵) → ((𝐹 ↾ (𝐴[,]𝐵)) ∈ ((𝐴[,]𝐵)–cn→ℝ) → ((𝐹 ↾ (𝐴[,]𝐵)) ↾ (𝑥[,]𝑦)) ∈ ((𝑥[,]𝑦)–cn→ℝ)))
124	120, 122, 123	sylc 65	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) ∧ 𝑥 < 𝑦) → ((𝐹 ↾ (𝐴[,]𝐵)) ↾ (𝑥[,]𝑦)) ∈ ((𝑥[,]𝑦)–cn→ℝ))
125	121, 124	eqeltrrd 2835	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) ∧ 𝑥 < 𝑦) → (𝐹 ↾ (𝑥[,]𝑦)) ∈ ((𝑥[,]𝑦)–cn→ℝ))
126	40	a1i 11	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) ∧ 𝑥 < 𝑦) → ℝ ⊆ ℂ)
127		c1liplem1.f	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → 𝐹 ∈ (ℂ ↑_pm ℝ))
128	127	ad2antrr 726	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) ∧ 𝑥 < 𝑦) → 𝐹 ∈ (ℂ ↑_pm ℝ))
129		cnex 11208	. . . . . . . . . . . . . . . . . . . 20 ⊢ ℂ ∈ V
130		reex 11218	. . . . . . . . . . . . . . . . . . . 20 ⊢ ℝ ∈ V
131	129, 130	elpm2 8886	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝐹 ∈ (ℂ ↑_pm ℝ) ↔ (𝐹:dom 𝐹⟶ℂ ∧ dom 𝐹 ⊆ ℝ))
132	131	simplbi 497	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐹 ∈ (ℂ ↑_pm ℝ) → 𝐹:dom 𝐹⟶ℂ)
133	128, 132	syl 17	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) ∧ 𝑥 < 𝑦) → 𝐹:dom 𝐹⟶ℂ)
134	131	simprbi 496	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐹 ∈ (ℂ ↑_pm ℝ) → dom 𝐹 ⊆ ℝ)
135	128, 134	syl 17	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) ∧ 𝑥 < 𝑦) → dom 𝐹 ⊆ ℝ)
136		iccssre 13444	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) → (𝑥[,]𝑦) ⊆ ℝ)
137	93, 92, 136	syl2anc 584	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) ∧ 𝑥 < 𝑦) → (𝑥[,]𝑦) ⊆ ℝ)
138		eqid 2735	. . . . . . . . . . . . . . . . . 18 ⊢ (TopOpen‘ℂ_fld) = (TopOpen‘ℂ_fld)
139		tgioo4 24742	. . . . . . . . . . . . . . . . . 18 ⊢ (topGen‘ran (,)) = ((TopOpen‘ℂ_fld) ↾_t ℝ)
140	138, 139	dvres 25862	. . . . . . . . . . . . . . . . 17 ⊢ (((ℝ ⊆ ℂ ∧ 𝐹:dom 𝐹⟶ℂ) ∧ (dom 𝐹 ⊆ ℝ ∧ (𝑥[,]𝑦) ⊆ ℝ)) → (ℝ D (𝐹 ↾ (𝑥[,]𝑦))) = ((ℝ D 𝐹) ↾ ((int‘(topGen‘ran (,)))‘(𝑥[,]𝑦))))
141	126, 133, 135, 137, 140	syl22anc 838	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) ∧ 𝑥 < 𝑦) → (ℝ D (𝐹 ↾ (𝑥[,]𝑦))) = ((ℝ D 𝐹) ↾ ((int‘(topGen‘ran (,)))‘(𝑥[,]𝑦))))
142		iccntr 24759	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) → ((int‘(topGen‘ran (,)))‘(𝑥[,]𝑦)) = (𝑥(,)𝑦))
143	93, 92, 142	syl2anc 584	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) ∧ 𝑥 < 𝑦) → ((int‘(topGen‘ran (,)))‘(𝑥[,]𝑦)) = (𝑥(,)𝑦))
144	143	reseq2d 5966	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) ∧ 𝑥 < 𝑦) → ((ℝ D 𝐹) ↾ ((int‘(topGen‘ran (,)))‘(𝑥[,]𝑦))) = ((ℝ D 𝐹) ↾ (𝑥(,)𝑦)))
145	141, 144	eqtrd 2770	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) ∧ 𝑥 < 𝑦) → (ℝ D (𝐹 ↾ (𝑥[,]𝑦))) = ((ℝ D 𝐹) ↾ (𝑥(,)𝑦)))
146	145	dmeqd 5885	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) ∧ 𝑥 < 𝑦) → dom (ℝ D (𝐹 ↾ (𝑥[,]𝑦))) = dom ((ℝ D 𝐹) ↾ (𝑥(,)𝑦)))
147		ioossicc 13448	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥(,)𝑦) ⊆ (𝑥[,]𝑦)
148	147, 120	sstrid 3970	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) ∧ 𝑥 < 𝑦) → (𝑥(,)𝑦) ⊆ (𝐴[,]𝐵))
149	17	ad2antrr 726	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) ∧ 𝑥 < 𝑦) → (𝐴[,]𝐵) ⊆ dom (ℝ D 𝐹))
150	148, 149	sstrd 3969	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) ∧ 𝑥 < 𝑦) → (𝑥(,)𝑦) ⊆ dom (ℝ D 𝐹))
151		ssdmres 6000	. . . . . . . . . . . . . . 15 ⊢ ((𝑥(,)𝑦) ⊆ dom (ℝ D 𝐹) ↔ dom ((ℝ D 𝐹) ↾ (𝑥(,)𝑦)) = (𝑥(,)𝑦))
152	150, 151	sylib 218	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) ∧ 𝑥 < 𝑦) → dom ((ℝ D 𝐹) ↾ (𝑥(,)𝑦)) = (𝑥(,)𝑦))
153	146, 152	eqtrd 2770	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) ∧ 𝑥 < 𝑦) → dom (ℝ D (𝐹 ↾ (𝑥[,]𝑦))) = (𝑥(,)𝑦))
154	93, 92, 96, 125, 153	mvth 25947	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) ∧ 𝑥 < 𝑦) → ∃𝑎 ∈ (𝑥(,)𝑦)((ℝ D (𝐹 ↾ (𝑥[,]𝑦)))‘𝑎) = ((((𝐹 ↾ (𝑥[,]𝑦))‘𝑦) − ((𝐹 ↾ (𝑥[,]𝑦))‘𝑥)) / (𝑦 − 𝑥)))
155	145	fveq1d 6877	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) ∧ 𝑥 < 𝑦) → ((ℝ D (𝐹 ↾ (𝑥[,]𝑦)))‘𝑎) = (((ℝ D 𝐹) ↾ (𝑥(,)𝑦))‘𝑎))
156	155	adantrr 717	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) ∧ (𝑥 < 𝑦 ∧ 𝑎 ∈ (𝑥(,)𝑦))) → ((ℝ D (𝐹 ↾ (𝑥[,]𝑦)))‘𝑎) = (((ℝ D 𝐹) ↾ (𝑥(,)𝑦))‘𝑎))
157		fvres 6894	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑎 ∈ (𝑥(,)𝑦) → (((ℝ D 𝐹) ↾ (𝑥(,)𝑦))‘𝑎) = ((ℝ D 𝐹)‘𝑎))
158	157	ad2antll 729	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) ∧ (𝑥 < 𝑦 ∧ 𝑎 ∈ (𝑥(,)𝑦))) → (((ℝ D 𝐹) ↾ (𝑥(,)𝑦))‘𝑎) = ((ℝ D 𝐹)‘𝑎))
159	156, 158	eqtrd 2770	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) ∧ (𝑥 < 𝑦 ∧ 𝑎 ∈ (𝑥(,)𝑦))) → ((ℝ D (𝐹 ↾ (𝑥[,]𝑦)))‘𝑎) = ((ℝ D 𝐹)‘𝑎))
160	10	a1i 11	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) ∧ (𝑥 < 𝑦 ∧ 𝑎 ∈ (𝑥(,)𝑦))) → Fun (ℝ D 𝐹))
161	17	ad2antrr 726	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) ∧ (𝑥 < 𝑦 ∧ 𝑎 ∈ (𝑥(,)𝑦))) → (𝐴[,]𝐵) ⊆ dom (ℝ D 𝐹))
162	148	sseld 3957	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) ∧ 𝑥 < 𝑦) → (𝑎 ∈ (𝑥(,)𝑦) → 𝑎 ∈ (𝐴[,]𝐵)))
163	162	impr 454	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) ∧ (𝑥 < 𝑦 ∧ 𝑎 ∈ (𝑥(,)𝑦))) → 𝑎 ∈ (𝐴[,]𝐵))
164		funfvima2 7222	. . . . . . . . . . . . . . . . . 18 ⊢ ((Fun (ℝ D 𝐹) ∧ (𝐴[,]𝐵) ⊆ dom (ℝ D 𝐹)) → (𝑎 ∈ (𝐴[,]𝐵) → ((ℝ D 𝐹)‘𝑎) ∈ ((ℝ D 𝐹) “ (𝐴[,]𝐵))))
165	164	imp 406	. . . . . . . . . . . . . . . . 17 ⊢ (((Fun (ℝ D 𝐹) ∧ (𝐴[,]𝐵) ⊆ dom (ℝ D 𝐹)) ∧ 𝑎 ∈ (𝐴[,]𝐵)) → ((ℝ D 𝐹)‘𝑎) ∈ ((ℝ D 𝐹) “ (𝐴[,]𝐵)))
166	160, 161, 163, 165	syl21anc 837	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) ∧ (𝑥 < 𝑦 ∧ 𝑎 ∈ (𝑥(,)𝑦))) → ((ℝ D 𝐹)‘𝑎) ∈ ((ℝ D 𝐹) “ (𝐴[,]𝐵)))
167	159, 166	eqeltrd 2834	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) ∧ (𝑥 < 𝑦 ∧ 𝑎 ∈ (𝑥(,)𝑦))) → ((ℝ D (𝐹 ↾ (𝑥[,]𝑦)))‘𝑎) ∈ ((ℝ D 𝐹) “ (𝐴[,]𝐵)))
168		eleq1 2822	. . . . . . . . . . . . . . 15 ⊢ (((ℝ D (𝐹 ↾ (𝑥[,]𝑦)))‘𝑎) = ((((𝐹 ↾ (𝑥[,]𝑦))‘𝑦) − ((𝐹 ↾ (𝑥[,]𝑦))‘𝑥)) / (𝑦 − 𝑥)) → (((ℝ D (𝐹 ↾ (𝑥[,]𝑦)))‘𝑎) ∈ ((ℝ D 𝐹) “ (𝐴[,]𝐵)) ↔ ((((𝐹 ↾ (𝑥[,]𝑦))‘𝑦) − ((𝐹 ↾ (𝑥[,]𝑦))‘𝑥)) / (𝑦 − 𝑥)) ∈ ((ℝ D 𝐹) “ (𝐴[,]𝐵))))
169	167, 168	syl5ibcom 245	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) ∧ (𝑥 < 𝑦 ∧ 𝑎 ∈ (𝑥(,)𝑦))) → (((ℝ D (𝐹 ↾ (𝑥[,]𝑦)))‘𝑎) = ((((𝐹 ↾ (𝑥[,]𝑦))‘𝑦) − ((𝐹 ↾ (𝑥[,]𝑦))‘𝑥)) / (𝑦 − 𝑥)) → ((((𝐹 ↾ (𝑥[,]𝑦))‘𝑦) − ((𝐹 ↾ (𝑥[,]𝑦))‘𝑥)) / (𝑦 − 𝑥)) ∈ ((ℝ D 𝐹) “ (𝐴[,]𝐵))))
170	169	expr 456	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) ∧ 𝑥 < 𝑦) → (𝑎 ∈ (𝑥(,)𝑦) → (((ℝ D (𝐹 ↾ (𝑥[,]𝑦)))‘𝑎) = ((((𝐹 ↾ (𝑥[,]𝑦))‘𝑦) − ((𝐹 ↾ (𝑥[,]𝑦))‘𝑥)) / (𝑦 − 𝑥)) → ((((𝐹 ↾ (𝑥[,]𝑦))‘𝑦) − ((𝐹 ↾ (𝑥[,]𝑦))‘𝑥)) / (𝑦 − 𝑥)) ∈ ((ℝ D 𝐹) “ (𝐴[,]𝐵)))))
171	170	rexlimdv 3139	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) ∧ 𝑥 < 𝑦) → (∃𝑎 ∈ (𝑥(,)𝑦)((ℝ D (𝐹 ↾ (𝑥[,]𝑦)))‘𝑎) = ((((𝐹 ↾ (𝑥[,]𝑦))‘𝑦) − ((𝐹 ↾ (𝑥[,]𝑦))‘𝑥)) / (𝑦 − 𝑥)) → ((((𝐹 ↾ (𝑥[,]𝑦))‘𝑦) − ((𝐹 ↾ (𝑥[,]𝑦))‘𝑥)) / (𝑦 − 𝑥)) ∈ ((ℝ D 𝐹) “ (𝐴[,]𝐵))))
172	154, 171	mpd 15	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) ∧ 𝑥 < 𝑦) → ((((𝐹 ↾ (𝑥[,]𝑦))‘𝑦) − ((𝐹 ↾ (𝑥[,]𝑦))‘𝑥)) / (𝑦 − 𝑥)) ∈ ((ℝ D 𝐹) “ (𝐴[,]𝐵)))
173	118, 172	eqeltrrd 2835	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) ∧ 𝑥 < 𝑦) → (((𝐹‘𝑦) − (𝐹‘𝑥)) / (𝑦 − 𝑥)) ∈ ((ℝ D 𝐹) “ (𝐴[,]𝐵)))
174		funfvima 7221	. . . . . . . . . . 11 ⊢ ((Fun abs ∧ (((𝐹‘𝑦) − (𝐹‘𝑥)) / (𝑦 − 𝑥)) ∈ dom abs) → ((((𝐹‘𝑦) − (𝐹‘𝑥)) / (𝑦 − 𝑥)) ∈ ((ℝ D 𝐹) “ (𝐴[,]𝐵)) → (abs‘(((𝐹‘𝑦) − (𝐹‘𝑥)) / (𝑦 − 𝑥))) ∈ (abs “ ((ℝ D 𝐹) “ (𝐴[,]𝐵)))))
175	174	imp 406	. . . . . . . . . 10 ⊢ (((Fun abs ∧ (((𝐹‘𝑦) − (𝐹‘𝑥)) / (𝑦 − 𝑥)) ∈ dom abs) ∧ (((𝐹‘𝑦) − (𝐹‘𝑥)) / (𝑦 − 𝑥)) ∈ ((ℝ D 𝐹) “ (𝐴[,]𝐵))) → (abs‘(((𝐹‘𝑦) − (𝐹‘𝑥)) / (𝑦 − 𝑥))) ∈ (abs “ ((ℝ D 𝐹) “ (𝐴[,]𝐵))))
176	105, 107, 173, 175	syl21anc 837	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) ∧ 𝑥 < 𝑦) → (abs‘(((𝐹‘𝑦) − (𝐹‘𝑥)) / (𝑦 − 𝑥))) ∈ (abs “ ((ℝ D 𝐹) “ (𝐴[,]𝐵))))
177	102, 103, 104, 176	suprubd 12202	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) ∧ 𝑥 < 𝑦) → (abs‘(((𝐹‘𝑦) − (𝐹‘𝑥)) / (𝑦 − 𝑥))) ≤ sup((abs “ ((ℝ D 𝐹) “ (𝐴[,]𝐵))), ℝ, < ))
178	177, 1	breqtrrdi 5161	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) ∧ 𝑥 < 𝑦) → (abs‘(((𝐹‘𝑦) − (𝐹‘𝑥)) / (𝑦 − 𝑥))) ≤ 𝐾)
179	101, 178	eqbrtrrd 5143	. . . . . 6 ⊢ (((𝜑 ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) ∧ 𝑥 < 𝑦) → ((abs‘((𝐹‘𝑦) − (𝐹‘𝑥))) / (abs‘(𝑦 − 𝑥))) ≤ 𝐾)
180	88	abscld 15453	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) ∧ 𝑥 < 𝑦) → (abs‘((𝐹‘𝑦) − (𝐹‘𝑥))) ∈ ℝ)
181	73	ad2antrr 726	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) ∧ 𝑥 < 𝑦) → 𝐾 ∈ ℝ)
182	95, 100	absrpcld 15465	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) ∧ 𝑥 < 𝑦) → (abs‘(𝑦 − 𝑥)) ∈ ℝ⁺)
183	180, 181, 182	ledivmuld 13102	. . . . . 6 ⊢ (((𝜑 ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) ∧ 𝑥 < 𝑦) → (((abs‘((𝐹‘𝑦) − (𝐹‘𝑥))) / (abs‘(𝑦 − 𝑥))) ≤ 𝐾 ↔ (abs‘((𝐹‘𝑦) − (𝐹‘𝑥))) ≤ ((abs‘(𝑦 − 𝑥)) · 𝐾)))
184	179, 183	mpbid 232	. . . . 5 ⊢ (((𝜑 ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) ∧ 𝑥 < 𝑦) → (abs‘((𝐹‘𝑦) − (𝐹‘𝑥))) ≤ ((abs‘(𝑦 − 𝑥)) · 𝐾))
185	182	rpcnd 13051	. . . . . 6 ⊢ (((𝜑 ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) ∧ 𝑥 < 𝑦) → (abs‘(𝑦 − 𝑥)) ∈ ℂ)
186	181	recnd 11261	. . . . . 6 ⊢ (((𝜑 ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) ∧ 𝑥 < 𝑦) → 𝐾 ∈ ℂ)
187	185, 186	mulcomd 11254	. . . . 5 ⊢ (((𝜑 ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) ∧ 𝑥 < 𝑦) → ((abs‘(𝑦 − 𝑥)) · 𝐾) = (𝐾 · (abs‘(𝑦 − 𝑥))))
188	184, 187	breqtrd 5145	. . . 4 ⊢ (((𝜑 ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) ∧ 𝑥 < 𝑦) → (abs‘((𝐹‘𝑦) − (𝐹‘𝑥))) ≤ (𝐾 · (abs‘(𝑦 − 𝑥))))
189	188	ex 412	. . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) → (𝑥 < 𝑦 → (abs‘((𝐹‘𝑦) − (𝐹‘𝑥))) ≤ (𝐾 · (abs‘(𝑦 − 𝑥)))))
190	189	ralrimivva 3187	. 2 ⊢ (𝜑 → ∀𝑥 ∈ (𝐴[,]𝐵)∀𝑦 ∈ (𝐴[,]𝐵)(𝑥 < 𝑦 → (abs‘((𝐹‘𝑦) − (𝐹‘𝑥))) ≤ (𝐾 · (abs‘(𝑦 − 𝑥)))))
191	73, 190	jca 511	1 ⊢ (𝜑 → (𝐾 ∈ ℝ ∧ ∀𝑥 ∈ (𝐴[,]𝐵)∀𝑦 ∈ (𝐴[,]𝐵)(𝑥 < 𝑦 → (abs‘((𝐹‘𝑦) − (𝐹‘𝑥))) ≤ (𝐾 · (abs‘(𝑦 − 𝑥))))))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1540 ∈ wcel 2108 ≠ wne 2932 ∀wral 3051 ∃wrex 3060 ⊆ wss 3926 ∅c0 4308 class class class wbr 5119 dom cdm 5654 ran crn 5655 ↾ cres 5656 “ cima 5657 Fun wfun 6524 ⟶wf 6526 ‘cfv 6530 (class class class)co 7403 ↑_pm cpm 8839 supcsup 9450 ℂcc 11125 ℝcr 11126 · cmul 11132 ℝ^*cxr 11266 < clt 11267 ≤ cle 11268 − cmin 11464 / cdiv 11892 ℝ⁺crp 13006 (,)cioo 13360 [,]cicc 13363 abscabs 15251 TopOpenctopn 17433 topGenctg 17449 ℂ_fldccnfld 21313 intcnt 22953 –cn→ccncf 24818 D cdv 25814

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2707 ax-rep 5249 ax-sep 5266 ax-nul 5276 ax-pow 5335 ax-pr 5402 ax-un 7727 ax-cnex 11183 ax-resscn 11184 ax-1cn 11185 ax-icn 11186 ax-addcl 11187 ax-addrcl 11188 ax-mulcl 11189 ax-mulrcl 11190 ax-mulcom 11191 ax-addass 11192 ax-mulass 11193 ax-distr 11194 ax-i2m1 11195 ax-1ne0 11196 ax-1rid 11197 ax-rnegex 11198 ax-rrecex 11199 ax-cnre 11200 ax-pre-lttri 11201 ax-pre-lttrn 11202 ax-pre-ltadd 11203 ax-pre-mulgt0 11204 ax-pre-sup 11205 ax-addf 11206

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2727 df-clel 2809 df-nfc 2885 df-ne 2933 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3359 df-reu 3360 df-rab 3416 df-v 3461 df-sbc 3766 df-csb 3875 df-dif 3929 df-un 3931 df-in 3933 df-ss 3943 df-pss 3946 df-nul 4309 df-if 4501 df-pw 4577 df-sn 4602 df-pr 4604 df-tp 4606 df-op 4608 df-uni 4884 df-int 4923 df-iun 4969 df-iin 4970 df-br 5120 df-opab 5182 df-mpt 5202 df-tr 5230 df-id 5548 df-eprel 5553 df-po 5561 df-so 5562 df-fr 5606 df-se 5607 df-we 5608 df-xp 5660 df-rel 5661 df-cnv 5662 df-co 5663 df-dm 5664 df-rn 5665 df-res 5666 df-ima 5667 df-pred 6290 df-ord 6355 df-on 6356 df-lim 6357 df-suc 6358 df-iota 6483 df-fun 6532 df-fn 6533 df-f 6534 df-f1 6535 df-fo 6536 df-f1o 6537 df-fv 6538 df-isom 6539 df-riota 7360 df-ov 7406 df-oprab 7407 df-mpo 7408 df-of 7669 df-om 7860 df-1st 7986 df-2nd 7987 df-supp 8158 df-frecs 8278 df-wrecs 8309 df-recs 8383 df-rdg 8422 df-1o 8478 df-2o 8479 df-er 8717 df-map 8840 df-pm 8841 df-ixp 8910 df-en 8958 df-dom 8959 df-sdom 8960 df-fin 8961 df-fsupp 9372 df-fi 9421 df-sup 9452 df-inf 9453 df-oi 9522 df-card 9951 df-pnf 11269 df-mnf 11270 df-xr 11271 df-ltxr 11272 df-le 11273 df-sub 11466 df-neg 11467 df-div 11893 df-nn 12239 df-2 12301 df-3 12302 df-4 12303 df-5 12304 df-6 12305 df-7 12306 df-8 12307 df-9 12308 df-n0 12500 df-z 12587 df-dec 12707 df-uz 12851 df-q 12963 df-rp 13007 df-xneg 13126 df-xadd 13127 df-xmul 13128 df-ioo 13364 df-ico 13366 df-icc 13367 df-fz 13523 df-fzo 13670 df-seq 14018 df-exp 14078 df-hash 14347 df-cj 15116 df-re 15117 df-im 15118 df-sqrt 15252 df-abs 15253 df-struct 17164 df-sets 17181 df-slot 17199 df-ndx 17211 df-base 17227 df-ress 17250 df-plusg 17282 df-mulr 17283 df-starv 17284 df-sca 17285 df-vsca 17286 df-ip 17287 df-tset 17288 df-ple 17289 df-ds 17291 df-unif 17292 df-hom 17293 df-cco 17294 df-rest 17434 df-topn 17435 df-0g 17453 df-gsum 17454 df-topgen 17455 df-pt 17456 df-prds 17459 df-xrs 17514 df-qtop 17519 df-imas 17520 df-xps 17522 df-mre 17596 df-mrc 17597 df-acs 17599 df-mgm 18616 df-sgrp 18695 df-mnd 18711 df-submnd 18760 df-mulg 19049 df-cntz 19298 df-cmn 19761 df-psmet 21305 df-xmet 21306 df-met 21307 df-bl 21308 df-mopn 21309 df-fbas 21310 df-fg 21311 df-cnfld 21314 df-top 22830 df-topon 22847 df-topsp 22869 df-bases 22882 df-cld 22955 df-ntr 22956 df-cls 22957 df-nei 23034 df-lp 23072 df-perf 23073 df-cn 23163 df-cnp 23164 df-haus 23251 df-cmp 23323 df-tx 23498 df-hmeo 23691 df-fil 23782 df-fm 23874 df-flim 23875 df-flf 23876 df-xms 24257 df-ms 24258 df-tms 24259 df-cncf 24820 df-limc 25817 df-dv 25818

This theorem is referenced by: c1lip1 25952

W3C validator