Theorem c1lip1 26037

Description: C^1 functions are Lipschitz continuous on closed intervals. (Contributed by Stefan O'Rear, 16-Nov-2014.)

Hypotheses

Ref	Expression
c1lip1.a	⊢ (𝜑 → 𝐴 ∈ ℝ)
c1lip1.b	⊢ (𝜑 → 𝐵 ∈ ℝ)
c1lip1.f	⊢ (𝜑 → 𝐹 ∈ (ℂ ↑pm ℝ))
c1lip1.dv	⊢ (𝜑 → ((ℝ D 𝐹) ↾ (𝐴[,]𝐵)) ∈ ((𝐴[,]𝐵)–cn→ℝ))
c1lip1.cn	⊢ (𝜑 → (𝐹 ↾ (𝐴[,]𝐵)) ∈ ((𝐴[,]𝐵)–cn→ℝ))

Assertion

Ref	Expression
c1lip1	⊢ (𝜑 → ∃𝑘 ∈ ℝ ∀𝑥 ∈ (𝐴[,]𝐵)∀𝑦 ∈ (𝐴[,]𝐵)(abs‘((𝐹‘𝑦) − (𝐹‘𝑥))) ≤ (𝑘 · (abs‘(𝑦 − 𝑥))))

Distinct variable groups:   𝜑,𝑥,𝑦,𝑘   𝑥,𝐴,𝑦,𝑘   𝑥,𝐵,𝑦,𝑘   𝑥,𝐹,𝑦,𝑘

Proof of Theorem c1lip1

Dummy variables 𝑎 𝑏 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		0re 11264	. . . 4 ⊢ 0 ∈ ℝ
2	1	ne0ii 4343	. . 3 ⊢ ℝ ≠ ∅
3		ral0 4512	. . . . 5 ⊢ ∀𝑥 ∈ ∅ ∀𝑦 ∈ (𝐴[,]𝐵)(abs‘((𝐹‘𝑦) − (𝐹‘𝑥))) ≤ (𝑘 · (abs‘(𝑦 − 𝑥)))
4		c1lip1.a	. . . . . . . . 9 ⊢ (𝜑 → 𝐴 ∈ ℝ)
5	4	rexrd 11312	. . . . . . . 8 ⊢ (𝜑 → 𝐴 ∈ ℝ*)
6		c1lip1.b	. . . . . . . . 9 ⊢ (𝜑 → 𝐵 ∈ ℝ)
7	6	rexrd 11312	. . . . . . . 8 ⊢ (𝜑 → 𝐵 ∈ ℝ*)
8		icc0 13436	. . . . . . . 8 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → ((𝐴[,]𝐵) = ∅ ↔ 𝐵 < 𝐴))
9	5, 7, 8	syl2anc 584	. . . . . . 7 ⊢ (𝜑 → ((𝐴[,]𝐵) = ∅ ↔ 𝐵 < 𝐴))
10	9	biimpar 477	. . . . . 6 ⊢ ((𝜑 ∧ 𝐵 < 𝐴) → (𝐴[,]𝐵) = ∅)
11	10	raleqdv 3325	. . . . 5 ⊢ ((𝜑 ∧ 𝐵 < 𝐴) → (∀𝑥 ∈ (𝐴[,]𝐵)∀𝑦 ∈ (𝐴[,]𝐵)(abs‘((𝐹‘𝑦) − (𝐹‘𝑥))) ≤ (𝑘 · (abs‘(𝑦 − 𝑥))) ↔ ∀𝑥 ∈ ∅ ∀𝑦 ∈ (𝐴[,]𝐵)(abs‘((𝐹‘𝑦) − (𝐹‘𝑥))) ≤ (𝑘 · (abs‘(𝑦 − 𝑥)))))
12	3, 11	mpbiri 258	. . . 4 ⊢ ((𝜑 ∧ 𝐵 < 𝐴) → ∀𝑥 ∈ (𝐴[,]𝐵)∀𝑦 ∈ (𝐴[,]𝐵)(abs‘((𝐹‘𝑦) − (𝐹‘𝑥))) ≤ (𝑘 · (abs‘(𝑦 − 𝑥))))
13	12	ralrimivw 3149	. . 3 ⊢ ((𝜑 ∧ 𝐵 < 𝐴) → ∀𝑘 ∈ ℝ ∀𝑥 ∈ (𝐴[,]𝐵)∀𝑦 ∈ (𝐴[,]𝐵)(abs‘((𝐹‘𝑦) − (𝐹‘𝑥))) ≤ (𝑘 · (abs‘(𝑦 − 𝑥))))
14		r19.2z 4494	. . 3 ⊢ ((ℝ ≠ ∅ ∧ ∀𝑘 ∈ ℝ ∀𝑥 ∈ (𝐴[,]𝐵)∀𝑦 ∈ (𝐴[,]𝐵)(abs‘((𝐹‘𝑦) − (𝐹‘𝑥))) ≤ (𝑘 · (abs‘(𝑦 − 𝑥)))) → ∃𝑘 ∈ ℝ ∀𝑥 ∈ (𝐴[,]𝐵)∀𝑦 ∈ (𝐴[,]𝐵)(abs‘((𝐹‘𝑦) − (𝐹‘𝑥))) ≤ (𝑘 · (abs‘(𝑦 − 𝑥))))
15	2, 13, 14	sylancr 587	. 2 ⊢ ((𝜑 ∧ 𝐵 < 𝐴) → ∃𝑘 ∈ ℝ ∀𝑥 ∈ (𝐴[,]𝐵)∀𝑦 ∈ (𝐴[,]𝐵)(abs‘((𝐹‘𝑦) − (𝐹‘𝑥))) ≤ (𝑘 · (abs‘(𝑦 − 𝑥))))
16	4	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ 𝐴 ≤ 𝐵) → 𝐴 ∈ ℝ)
17	6	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ 𝐴 ≤ 𝐵) → 𝐵 ∈ ℝ)
18		simpr 484	. . . . 5 ⊢ ((𝜑 ∧ 𝐴 ≤ 𝐵) → 𝐴 ≤ 𝐵)
19		c1lip1.f	. . . . . 6 ⊢ (𝜑 → 𝐹 ∈ (ℂ ↑pm ℝ))
20	19	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ 𝐴 ≤ 𝐵) → 𝐹 ∈ (ℂ ↑pm ℝ))
21		c1lip1.dv	. . . . . 6 ⊢ (𝜑 → ((ℝ D 𝐹) ↾ (𝐴[,]𝐵)) ∈ ((𝐴[,]𝐵)–cn→ℝ))
22	21	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ 𝐴 ≤ 𝐵) → ((ℝ D 𝐹) ↾ (𝐴[,]𝐵)) ∈ ((𝐴[,]𝐵)–cn→ℝ))
23		c1lip1.cn	. . . . . 6 ⊢ (𝜑 → (𝐹 ↾ (𝐴[,]𝐵)) ∈ ((𝐴[,]𝐵)–cn→ℝ))
24	23	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ 𝐴 ≤ 𝐵) → (𝐹 ↾ (𝐴[,]𝐵)) ∈ ((𝐴[,]𝐵)–cn→ℝ))
25		eqid 2736	. . . . 5 ⊢ sup((abs “ ((ℝ D 𝐹) “ (𝐴[,]𝐵))), ℝ, < ) = sup((abs “ ((ℝ D 𝐹) “ (𝐴[,]𝐵))), ℝ, < )
26	16, 17, 18, 20, 22, 24, 25	c1liplem1 26036	. . . 4 ⊢ ((𝜑 ∧ 𝐴 ≤ 𝐵) → (sup((abs “ ((ℝ D 𝐹) “ (𝐴[,]𝐵))), ℝ, < ) ∈ ℝ ∧ ∀𝑎 ∈ (𝐴[,]𝐵)∀𝑏 ∈ (𝐴[,]𝐵)(𝑎 < 𝑏 → (abs‘((𝐹‘𝑏) − (𝐹‘𝑎))) ≤ (sup((abs “ ((ℝ D 𝐹) “ (𝐴[,]𝐵))), ℝ, < ) · (abs‘(𝑏 − 𝑎))))))
27		oveq1 7439	. . . . . . . 8 ⊢ (𝑘 = sup((abs “ ((ℝ D 𝐹) “ (𝐴[,]𝐵))), ℝ, < ) → (𝑘 · (abs‘(𝑏 − 𝑎))) = (sup((abs “ ((ℝ D 𝐹) “ (𝐴[,]𝐵))), ℝ, < ) · (abs‘(𝑏 − 𝑎))))
28	27	breq2d 5154	. . . . . . 7 ⊢ (𝑘 = sup((abs “ ((ℝ D 𝐹) “ (𝐴[,]𝐵))), ℝ, < ) → ((abs‘((𝐹‘𝑏) − (𝐹‘𝑎))) ≤ (𝑘 · (abs‘(𝑏 − 𝑎))) ↔ (abs‘((𝐹‘𝑏) − (𝐹‘𝑎))) ≤ (sup((abs “ ((ℝ D 𝐹) “ (𝐴[,]𝐵))), ℝ, < ) · (abs‘(𝑏 − 𝑎)))))
29	28	imbi2d 340	. . . . . 6 ⊢ (𝑘 = sup((abs “ ((ℝ D 𝐹) “ (𝐴[,]𝐵))), ℝ, < ) → ((𝑎 < 𝑏 → (abs‘((𝐹‘𝑏) − (𝐹‘𝑎))) ≤ (𝑘 · (abs‘(𝑏 − 𝑎)))) ↔ (𝑎 < 𝑏 → (abs‘((𝐹‘𝑏) − (𝐹‘𝑎))) ≤ (sup((abs “ ((ℝ D 𝐹) “ (𝐴[,]𝐵))), ℝ, < ) · (abs‘(𝑏 − 𝑎))))))
30	29	2ralbidv 3220	. . . . 5 ⊢ (𝑘 = sup((abs “ ((ℝ D 𝐹) “ (𝐴[,]𝐵))), ℝ, < ) → (∀𝑎 ∈ (𝐴[,]𝐵)∀𝑏 ∈ (𝐴[,]𝐵)(𝑎 < 𝑏 → (abs‘((𝐹‘𝑏) − (𝐹‘𝑎))) ≤ (𝑘 · (abs‘(𝑏 − 𝑎)))) ↔ ∀𝑎 ∈ (𝐴[,]𝐵)∀𝑏 ∈ (𝐴[,]𝐵)(𝑎 < 𝑏 → (abs‘((𝐹‘𝑏) − (𝐹‘𝑎))) ≤ (sup((abs “ ((ℝ D 𝐹) “ (𝐴[,]𝐵))), ℝ, < ) · (abs‘(𝑏 − 𝑎))))))
31	30	rspcev 3621	. . . 4 ⊢ ((sup((abs “ ((ℝ D 𝐹) “ (𝐴[,]𝐵))), ℝ, < ) ∈ ℝ ∧ ∀𝑎 ∈ (𝐴[,]𝐵)∀𝑏 ∈ (𝐴[,]𝐵)(𝑎 < 𝑏 → (abs‘((𝐹‘𝑏) − (𝐹‘𝑎))) ≤ (sup((abs “ ((ℝ D 𝐹) “ (𝐴[,]𝐵))), ℝ, < ) · (abs‘(𝑏 − 𝑎))))) → ∃𝑘 ∈ ℝ ∀𝑎 ∈ (𝐴[,]𝐵)∀𝑏 ∈ (𝐴[,]𝐵)(𝑎 < 𝑏 → (abs‘((𝐹‘𝑏) − (𝐹‘𝑎))) ≤ (𝑘 · (abs‘(𝑏 − 𝑎)))))
32	26, 31	syl 17	. . 3 ⊢ ((𝜑 ∧ 𝐴 ≤ 𝐵) → ∃𝑘 ∈ ℝ ∀𝑎 ∈ (𝐴[,]𝐵)∀𝑏 ∈ (𝐴[,]𝐵)(𝑎 < 𝑏 → (abs‘((𝐹‘𝑏) − (𝐹‘𝑎))) ≤ (𝑘 · (abs‘(𝑏 − 𝑎)))))
33		breq1 5145	. . . . . . . . . 10 ⊢ (𝑎 = 𝑥 → (𝑎 < 𝑏 ↔ 𝑥 < 𝑏))
34		fveq2 6905	. . . . . . . . . . . . 13 ⊢ (𝑎 = 𝑥 → (𝐹‘𝑎) = (𝐹‘𝑥))
35	34	oveq2d 7448	. . . . . . . . . . . 12 ⊢ (𝑎 = 𝑥 → ((𝐹‘𝑏) − (𝐹‘𝑎)) = ((𝐹‘𝑏) − (𝐹‘𝑥)))
36	35	fveq2d 6909	. . . . . . . . . . 11 ⊢ (𝑎 = 𝑥 → (abs‘((𝐹‘𝑏) − (𝐹‘𝑎))) = (abs‘((𝐹‘𝑏) − (𝐹‘𝑥))))
37		oveq2 7440	. . . . . . . . . . . . 13 ⊢ (𝑎 = 𝑥 → (𝑏 − 𝑎) = (𝑏 − 𝑥))
38	37	fveq2d 6909	. . . . . . . . . . . 12 ⊢ (𝑎 = 𝑥 → (abs‘(𝑏 − 𝑎)) = (abs‘(𝑏 − 𝑥)))
39	38	oveq2d 7448	. . . . . . . . . . 11 ⊢ (𝑎 = 𝑥 → (𝑘 · (abs‘(𝑏 − 𝑎))) = (𝑘 · (abs‘(𝑏 − 𝑥))))
40	36, 39	breq12d 5155	. . . . . . . . . 10 ⊢ (𝑎 = 𝑥 → ((abs‘((𝐹‘𝑏) − (𝐹‘𝑎))) ≤ (𝑘 · (abs‘(𝑏 − 𝑎))) ↔ (abs‘((𝐹‘𝑏) − (𝐹‘𝑥))) ≤ (𝑘 · (abs‘(𝑏 − 𝑥)))))
41	33, 40	imbi12d 344	. . . . . . . . 9 ⊢ (𝑎 = 𝑥 → ((𝑎 < 𝑏 → (abs‘((𝐹‘𝑏) − (𝐹‘𝑎))) ≤ (𝑘 · (abs‘(𝑏 − 𝑎)))) ↔ (𝑥 < 𝑏 → (abs‘((𝐹‘𝑏) − (𝐹‘𝑥))) ≤ (𝑘 · (abs‘(𝑏 − 𝑥))))))
42		breq2 5146	. . . . . . . . . 10 ⊢ (𝑏 = 𝑦 → (𝑥 < 𝑏 ↔ 𝑥 < 𝑦))
43		fveq2 6905	. . . . . . . . . . . 12 ⊢ (𝑏 = 𝑦 → (𝐹‘𝑏) = (𝐹‘𝑦))
44	43	fvoveq1d 7454	. . . . . . . . . . 11 ⊢ (𝑏 = 𝑦 → (abs‘((𝐹‘𝑏) − (𝐹‘𝑥))) = (abs‘((𝐹‘𝑦) − (𝐹‘𝑥))))
45		fvoveq1 7455	. . . . . . . . . . . 12 ⊢ (𝑏 = 𝑦 → (abs‘(𝑏 − 𝑥)) = (abs‘(𝑦 − 𝑥)))
46	45	oveq2d 7448	. . . . . . . . . . 11 ⊢ (𝑏 = 𝑦 → (𝑘 · (abs‘(𝑏 − 𝑥))) = (𝑘 · (abs‘(𝑦 − 𝑥))))
47	44, 46	breq12d 5155	. . . . . . . . . 10 ⊢ (𝑏 = 𝑦 → ((abs‘((𝐹‘𝑏) − (𝐹‘𝑥))) ≤ (𝑘 · (abs‘(𝑏 − 𝑥))) ↔ (abs‘((𝐹‘𝑦) − (𝐹‘𝑥))) ≤ (𝑘 · (abs‘(𝑦 − 𝑥)))))
48	42, 47	imbi12d 344	. . . . . . . . 9 ⊢ (𝑏 = 𝑦 → ((𝑥 < 𝑏 → (abs‘((𝐹‘𝑏) − (𝐹‘𝑥))) ≤ (𝑘 · (abs‘(𝑏 − 𝑥)))) ↔ (𝑥 < 𝑦 → (abs‘((𝐹‘𝑦) − (𝐹‘𝑥))) ≤ (𝑘 · (abs‘(𝑦 − 𝑥))))))
49	41, 48	rspc2v 3632	. . . . . . . 8 ⊢ ((𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵)) → (∀𝑎 ∈ (𝐴[,]𝐵)∀𝑏 ∈ (𝐴[,]𝐵)(𝑎 < 𝑏 → (abs‘((𝐹‘𝑏) − (𝐹‘𝑎))) ≤ (𝑘 · (abs‘(𝑏 − 𝑎)))) → (𝑥 < 𝑦 → (abs‘((𝐹‘𝑦) − (𝐹‘𝑥))) ≤ (𝑘 · (abs‘(𝑦 − 𝑥))))))
50	49	ad2antlr 727	. . . . . . 7 ⊢ (((((𝜑 ∧ 𝐴 ≤ 𝐵) ∧ 𝑘 ∈ ℝ) ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) ∧ 𝑥 < 𝑦) → (∀𝑎 ∈ (𝐴[,]𝐵)∀𝑏 ∈ (𝐴[,]𝐵)(𝑎 < 𝑏 → (abs‘((𝐹‘𝑏) − (𝐹‘𝑎))) ≤ (𝑘 · (abs‘(𝑏 − 𝑎)))) → (𝑥 < 𝑦 → (abs‘((𝐹‘𝑦) − (𝐹‘𝑥))) ≤ (𝑘 · (abs‘(𝑦 − 𝑥))))))
51		pm2.27 42	. . . . . . . 8 ⊢ (𝑥 < 𝑦 → ((𝑥 < 𝑦 → (abs‘((𝐹‘𝑦) − (𝐹‘𝑥))) ≤ (𝑘 · (abs‘(𝑦 − 𝑥)))) → (abs‘((𝐹‘𝑦) − (𝐹‘𝑥))) ≤ (𝑘 · (abs‘(𝑦 − 𝑥)))))
52	51	adantl 481	. . . . . . 7 ⊢ (((((𝜑 ∧ 𝐴 ≤ 𝐵) ∧ 𝑘 ∈ ℝ) ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) ∧ 𝑥 < 𝑦) → ((𝑥 < 𝑦 → (abs‘((𝐹‘𝑦) − (𝐹‘𝑥))) ≤ (𝑘 · (abs‘(𝑦 − 𝑥)))) → (abs‘((𝐹‘𝑦) − (𝐹‘𝑥))) ≤ (𝑘 · (abs‘(𝑦 − 𝑥)))))
53	50, 52	syld 47	. . . . . 6 ⊢ (((((𝜑 ∧ 𝐴 ≤ 𝐵) ∧ 𝑘 ∈ ℝ) ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) ∧ 𝑥 < 𝑦) → (∀𝑎 ∈ (𝐴[,]𝐵)∀𝑏 ∈ (𝐴[,]𝐵)(𝑎 < 𝑏 → (abs‘((𝐹‘𝑏) − (𝐹‘𝑎))) ≤ (𝑘 · (abs‘(𝑏 − 𝑎)))) → (abs‘((𝐹‘𝑦) − (𝐹‘𝑥))) ≤ (𝑘 · (abs‘(𝑦 − 𝑥)))))
54		0le0 12368	. . . . . . . . . 10 ⊢ 0 ≤ 0
55		fvres 6924	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 ∈ (𝐴[,]𝐵) → ((𝐹 ↾ (𝐴[,]𝐵))‘𝑥) = (𝐹‘𝑥))
56	55	ad2antrl 728	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝐴 ≤ 𝐵) ∧ 𝑘 ∈ ℝ) ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) → ((𝐹 ↾ (𝐴[,]𝐵))‘𝑥) = (𝐹‘𝑥))
57		cncff 24920	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐹 ↾ (𝐴[,]𝐵)) ∈ ((𝐴[,]𝐵)–cn→ℝ) → (𝐹 ↾ (𝐴[,]𝐵)):(𝐴[,]𝐵)⟶ℝ)
58	23, 57	syl 17	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (𝐹 ↾ (𝐴[,]𝐵)):(𝐴[,]𝐵)⟶ℝ)
59	58	ad2antrr 726	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝐴 ≤ 𝐵) ∧ 𝑘 ∈ ℝ) → (𝐹 ↾ (𝐴[,]𝐵)):(𝐴[,]𝐵)⟶ℝ)
60		simpl 482	. . . . . . . . . . . . . . . 16 ⊢ ((𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵)) → 𝑥 ∈ (𝐴[,]𝐵))
61		ffvelcdm 7100	. . . . . . . . . . . . . . . 16 ⊢ (((𝐹 ↾ (𝐴[,]𝐵)):(𝐴[,]𝐵)⟶ℝ ∧ 𝑥 ∈ (𝐴[,]𝐵)) → ((𝐹 ↾ (𝐴[,]𝐵))‘𝑥) ∈ ℝ)
62	59, 60, 61	syl2an 596	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝐴 ≤ 𝐵) ∧ 𝑘 ∈ ℝ) ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) → ((𝐹 ↾ (𝐴[,]𝐵))‘𝑥) ∈ ℝ)
63	56, 62	eqeltrrd 2841	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝐴 ≤ 𝐵) ∧ 𝑘 ∈ ℝ) ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) → (𝐹‘𝑥) ∈ ℝ)
64	63	recnd 11290	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝐴 ≤ 𝐵) ∧ 𝑘 ∈ ℝ) ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) → (𝐹‘𝑥) ∈ ℂ)
65	64	subidd 11609	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝐴 ≤ 𝐵) ∧ 𝑘 ∈ ℝ) ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) → ((𝐹‘𝑥) − (𝐹‘𝑥)) = 0)
66	65	abs00bd 15331	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝐴 ≤ 𝐵) ∧ 𝑘 ∈ ℝ) ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) → (abs‘((𝐹‘𝑥) − (𝐹‘𝑥))) = 0)
67		iccssre 13470	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴[,]𝐵) ⊆ ℝ)
68	4, 6, 67	syl2anc 584	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → (𝐴[,]𝐵) ⊆ ℝ)
69	68	ad3antrrr 730	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ 𝐴 ≤ 𝐵) ∧ 𝑘 ∈ ℝ) ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) → (𝐴[,]𝐵) ⊆ ℝ)
70		simprl 770	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ 𝐴 ≤ 𝐵) ∧ 𝑘 ∈ ℝ) ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) → 𝑥 ∈ (𝐴[,]𝐵))
71	69, 70	sseldd 3983	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝐴 ≤ 𝐵) ∧ 𝑘 ∈ ℝ) ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) → 𝑥 ∈ ℝ)
72	71	recnd 11290	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝐴 ≤ 𝐵) ∧ 𝑘 ∈ ℝ) ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) → 𝑥 ∈ ℂ)
73	72	subidd 11609	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝐴 ≤ 𝐵) ∧ 𝑘 ∈ ℝ) ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) → (𝑥 − 𝑥) = 0)
74	73	abs00bd 15331	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝐴 ≤ 𝐵) ∧ 𝑘 ∈ ℝ) ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) → (abs‘(𝑥 − 𝑥)) = 0)
75	74	oveq2d 7448	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝐴 ≤ 𝐵) ∧ 𝑘 ∈ ℝ) ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) → (𝑘 · (abs‘(𝑥 − 𝑥))) = (𝑘 · 0))
76		simplr 768	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝐴 ≤ 𝐵) ∧ 𝑘 ∈ ℝ) ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) → 𝑘 ∈ ℝ)
77	76	recnd 11290	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝐴 ≤ 𝐵) ∧ 𝑘 ∈ ℝ) ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) → 𝑘 ∈ ℂ)
78	77	mul01d 11461	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝐴 ≤ 𝐵) ∧ 𝑘 ∈ ℝ) ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) → (𝑘 · 0) = 0)
79	75, 78	eqtrd 2776	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝐴 ≤ 𝐵) ∧ 𝑘 ∈ ℝ) ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) → (𝑘 · (abs‘(𝑥 − 𝑥))) = 0)
80	66, 79	breq12d 5155	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝐴 ≤ 𝐵) ∧ 𝑘 ∈ ℝ) ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) → ((abs‘((𝐹‘𝑥) − (𝐹‘𝑥))) ≤ (𝑘 · (abs‘(𝑥 − 𝑥))) ↔ 0 ≤ 0))
81	54, 80	mpbiri 258	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝐴 ≤ 𝐵) ∧ 𝑘 ∈ ℝ) ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) → (abs‘((𝐹‘𝑥) − (𝐹‘𝑥))) ≤ (𝑘 · (abs‘(𝑥 − 𝑥))))
82		fveq2 6905	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑦 → (𝐹‘𝑥) = (𝐹‘𝑦))
83	82	fvoveq1d 7454	. . . . . . . . . 10 ⊢ (𝑥 = 𝑦 → (abs‘((𝐹‘𝑥) − (𝐹‘𝑥))) = (abs‘((𝐹‘𝑦) − (𝐹‘𝑥))))
84		fvoveq1 7455	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑦 → (abs‘(𝑥 − 𝑥)) = (abs‘(𝑦 − 𝑥)))
85	84	oveq2d 7448	. . . . . . . . . 10 ⊢ (𝑥 = 𝑦 → (𝑘 · (abs‘(𝑥 − 𝑥))) = (𝑘 · (abs‘(𝑦 − 𝑥))))
86	83, 85	breq12d 5155	. . . . . . . . 9 ⊢ (𝑥 = 𝑦 → ((abs‘((𝐹‘𝑥) − (𝐹‘𝑥))) ≤ (𝑘 · (abs‘(𝑥 − 𝑥))) ↔ (abs‘((𝐹‘𝑦) − (𝐹‘𝑥))) ≤ (𝑘 · (abs‘(𝑦 − 𝑥)))))
87	81, 86	syl5ibcom 245	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝐴 ≤ 𝐵) ∧ 𝑘 ∈ ℝ) ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) → (𝑥 = 𝑦 → (abs‘((𝐹‘𝑦) − (𝐹‘𝑥))) ≤ (𝑘 · (abs‘(𝑦 − 𝑥)))))
88	87	imp 406	. . . . . . 7 ⊢ (((((𝜑 ∧ 𝐴 ≤ 𝐵) ∧ 𝑘 ∈ ℝ) ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) ∧ 𝑥 = 𝑦) → (abs‘((𝐹‘𝑦) − (𝐹‘𝑥))) ≤ (𝑘 · (abs‘(𝑦 − 𝑥))))
89	88	a1d 25	. . . . . 6 ⊢ (((((𝜑 ∧ 𝐴 ≤ 𝐵) ∧ 𝑘 ∈ ℝ) ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) ∧ 𝑥 = 𝑦) → (∀𝑎 ∈ (𝐴[,]𝐵)∀𝑏 ∈ (𝐴[,]𝐵)(𝑎 < 𝑏 → (abs‘((𝐹‘𝑏) − (𝐹‘𝑎))) ≤ (𝑘 · (abs‘(𝑏 − 𝑎)))) → (abs‘((𝐹‘𝑦) − (𝐹‘𝑥))) ≤ (𝑘 · (abs‘(𝑦 − 𝑥)))))
90		breq1 5145	. . . . . . . . . . 11 ⊢ (𝑎 = 𝑦 → (𝑎 < 𝑏 ↔ 𝑦 < 𝑏))
91		fveq2 6905	. . . . . . . . . . . . . 14 ⊢ (𝑎 = 𝑦 → (𝐹‘𝑎) = (𝐹‘𝑦))
92	91	oveq2d 7448	. . . . . . . . . . . . 13 ⊢ (𝑎 = 𝑦 → ((𝐹‘𝑏) − (𝐹‘𝑎)) = ((𝐹‘𝑏) − (𝐹‘𝑦)))
93	92	fveq2d 6909	. . . . . . . . . . . 12 ⊢ (𝑎 = 𝑦 → (abs‘((𝐹‘𝑏) − (𝐹‘𝑎))) = (abs‘((𝐹‘𝑏) − (𝐹‘𝑦))))
94		oveq2 7440	. . . . . . . . . . . . . 14 ⊢ (𝑎 = 𝑦 → (𝑏 − 𝑎) = (𝑏 − 𝑦))
95	94	fveq2d 6909	. . . . . . . . . . . . 13 ⊢ (𝑎 = 𝑦 → (abs‘(𝑏 − 𝑎)) = (abs‘(𝑏 − 𝑦)))
96	95	oveq2d 7448	. . . . . . . . . . . 12 ⊢ (𝑎 = 𝑦 → (𝑘 · (abs‘(𝑏 − 𝑎))) = (𝑘 · (abs‘(𝑏 − 𝑦))))
97	93, 96	breq12d 5155	. . . . . . . . . . 11 ⊢ (𝑎 = 𝑦 → ((abs‘((𝐹‘𝑏) − (𝐹‘𝑎))) ≤ (𝑘 · (abs‘(𝑏 − 𝑎))) ↔ (abs‘((𝐹‘𝑏) − (𝐹‘𝑦))) ≤ (𝑘 · (abs‘(𝑏 − 𝑦)))))
98	90, 97	imbi12d 344	. . . . . . . . . 10 ⊢ (𝑎 = 𝑦 → ((𝑎 < 𝑏 → (abs‘((𝐹‘𝑏) − (𝐹‘𝑎))) ≤ (𝑘 · (abs‘(𝑏 − 𝑎)))) ↔ (𝑦 < 𝑏 → (abs‘((𝐹‘𝑏) − (𝐹‘𝑦))) ≤ (𝑘 · (abs‘(𝑏 − 𝑦))))))
99		breq2 5146	. . . . . . . . . . 11 ⊢ (𝑏 = 𝑥 → (𝑦 < 𝑏 ↔ 𝑦 < 𝑥))
100		fveq2 6905	. . . . . . . . . . . . 13 ⊢ (𝑏 = 𝑥 → (𝐹‘𝑏) = (𝐹‘𝑥))
101	100	fvoveq1d 7454	. . . . . . . . . . . 12 ⊢ (𝑏 = 𝑥 → (abs‘((𝐹‘𝑏) − (𝐹‘𝑦))) = (abs‘((𝐹‘𝑥) − (𝐹‘𝑦))))
102		fvoveq1 7455	. . . . . . . . . . . . 13 ⊢ (𝑏 = 𝑥 → (abs‘(𝑏 − 𝑦)) = (abs‘(𝑥 − 𝑦)))
103	102	oveq2d 7448	. . . . . . . . . . . 12 ⊢ (𝑏 = 𝑥 → (𝑘 · (abs‘(𝑏 − 𝑦))) = (𝑘 · (abs‘(𝑥 − 𝑦))))
104	101, 103	breq12d 5155	. . . . . . . . . . 11 ⊢ (𝑏 = 𝑥 → ((abs‘((𝐹‘𝑏) − (𝐹‘𝑦))) ≤ (𝑘 · (abs‘(𝑏 − 𝑦))) ↔ (abs‘((𝐹‘𝑥) − (𝐹‘𝑦))) ≤ (𝑘 · (abs‘(𝑥 − 𝑦)))))
105	99, 104	imbi12d 344	. . . . . . . . . 10 ⊢ (𝑏 = 𝑥 → ((𝑦 < 𝑏 → (abs‘((𝐹‘𝑏) − (𝐹‘𝑦))) ≤ (𝑘 · (abs‘(𝑏 − 𝑦)))) ↔ (𝑦 < 𝑥 → (abs‘((𝐹‘𝑥) − (𝐹‘𝑦))) ≤ (𝑘 · (abs‘(𝑥 − 𝑦))))))
106	98, 105	rspc2v 3632	. . . . . . . . 9 ⊢ ((𝑦 ∈ (𝐴[,]𝐵) ∧ 𝑥 ∈ (𝐴[,]𝐵)) → (∀𝑎 ∈ (𝐴[,]𝐵)∀𝑏 ∈ (𝐴[,]𝐵)(𝑎 < 𝑏 → (abs‘((𝐹‘𝑏) − (𝐹‘𝑎))) ≤ (𝑘 · (abs‘(𝑏 − 𝑎)))) → (𝑦 < 𝑥 → (abs‘((𝐹‘𝑥) − (𝐹‘𝑦))) ≤ (𝑘 · (abs‘(𝑥 − 𝑦))))))
107	106	ancoms 458	. . . . . . . 8 ⊢ ((𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵)) → (∀𝑎 ∈ (𝐴[,]𝐵)∀𝑏 ∈ (𝐴[,]𝐵)(𝑎 < 𝑏 → (abs‘((𝐹‘𝑏) − (𝐹‘𝑎))) ≤ (𝑘 · (abs‘(𝑏 − 𝑎)))) → (𝑦 < 𝑥 → (abs‘((𝐹‘𝑥) − (𝐹‘𝑦))) ≤ (𝑘 · (abs‘(𝑥 − 𝑦))))))
108	107	ad2antlr 727	. . . . . . 7 ⊢ (((((𝜑 ∧ 𝐴 ≤ 𝐵) ∧ 𝑘 ∈ ℝ) ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) ∧ 𝑦 < 𝑥) → (∀𝑎 ∈ (𝐴[,]𝐵)∀𝑏 ∈ (𝐴[,]𝐵)(𝑎 < 𝑏 → (abs‘((𝐹‘𝑏) − (𝐹‘𝑎))) ≤ (𝑘 · (abs‘(𝑏 − 𝑎)))) → (𝑦 < 𝑥 → (abs‘((𝐹‘𝑥) − (𝐹‘𝑦))) ≤ (𝑘 · (abs‘(𝑥 − 𝑦))))))
109		simpr 484	. . . . . . . 8 ⊢ (((((𝜑 ∧ 𝐴 ≤ 𝐵) ∧ 𝑘 ∈ ℝ) ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) ∧ 𝑦 < 𝑥) → 𝑦 < 𝑥)
110		fvres 6924	. . . . . . . . . . . . . . 15 ⊢ (𝑦 ∈ (𝐴[,]𝐵) → ((𝐹 ↾ (𝐴[,]𝐵))‘𝑦) = (𝐹‘𝑦))
111	110	ad2antll 729	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝐴 ≤ 𝐵) ∧ 𝑘 ∈ ℝ) ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) → ((𝐹 ↾ (𝐴[,]𝐵))‘𝑦) = (𝐹‘𝑦))
112		simpr 484	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵)) → 𝑦 ∈ (𝐴[,]𝐵))
113		ffvelcdm 7100	. . . . . . . . . . . . . . 15 ⊢ (((𝐹 ↾ (𝐴[,]𝐵)):(𝐴[,]𝐵)⟶ℝ ∧ 𝑦 ∈ (𝐴[,]𝐵)) → ((𝐹 ↾ (𝐴[,]𝐵))‘𝑦) ∈ ℝ)
114	59, 112, 113	syl2an 596	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝐴 ≤ 𝐵) ∧ 𝑘 ∈ ℝ) ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) → ((𝐹 ↾ (𝐴[,]𝐵))‘𝑦) ∈ ℝ)
115	111, 114	eqeltrrd 2841	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝐴 ≤ 𝐵) ∧ 𝑘 ∈ ℝ) ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) → (𝐹‘𝑦) ∈ ℝ)
116	115	recnd 11290	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝐴 ≤ 𝐵) ∧ 𝑘 ∈ ℝ) ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) → (𝐹‘𝑦) ∈ ℂ)
117	64, 116	abssubd 15493	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝐴 ≤ 𝐵) ∧ 𝑘 ∈ ℝ) ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) → (abs‘((𝐹‘𝑥) − (𝐹‘𝑦))) = (abs‘((𝐹‘𝑦) − (𝐹‘𝑥))))
118	117	adantr 480	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ 𝐴 ≤ 𝐵) ∧ 𝑘 ∈ ℝ) ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) ∧ 𝑦 < 𝑥) → (abs‘((𝐹‘𝑥) − (𝐹‘𝑦))) = (abs‘((𝐹‘𝑦) − (𝐹‘𝑥))))
119	68	ad2antrr 726	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝐴 ≤ 𝐵) ∧ 𝑘 ∈ ℝ) → (𝐴[,]𝐵) ⊆ ℝ)
120	119	sseld 3981	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝐴 ≤ 𝐵) ∧ 𝑘 ∈ ℝ) → (𝑥 ∈ (𝐴[,]𝐵) → 𝑥 ∈ ℝ))
121	119	sseld 3981	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝐴 ≤ 𝐵) ∧ 𝑘 ∈ ℝ) → (𝑦 ∈ (𝐴[,]𝐵) → 𝑦 ∈ ℝ))
122	120, 121	anim12d 609	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝐴 ≤ 𝐵) ∧ 𝑘 ∈ ℝ) → ((𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵)) → (𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ)))
123	122	imp 406	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝐴 ≤ 𝐵) ∧ 𝑘 ∈ ℝ) ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) → (𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ))
124		recn 11246	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ ℝ → 𝑥 ∈ ℂ)
125		recn 11246	. . . . . . . . . . . . . 14 ⊢ (𝑦 ∈ ℝ → 𝑦 ∈ ℂ)
126		abssub 15366	. . . . . . . . . . . . . 14 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ) → (abs‘(𝑥 − 𝑦)) = (abs‘(𝑦 − 𝑥)))
127	124, 125, 126	syl2an 596	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) → (abs‘(𝑥 − 𝑦)) = (abs‘(𝑦 − 𝑥)))
128	123, 127	syl 17	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝐴 ≤ 𝐵) ∧ 𝑘 ∈ ℝ) ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) → (abs‘(𝑥 − 𝑦)) = (abs‘(𝑦 − 𝑥)))
129	128	adantr 480	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ 𝐴 ≤ 𝐵) ∧ 𝑘 ∈ ℝ) ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) ∧ 𝑦 < 𝑥) → (abs‘(𝑥 − 𝑦)) = (abs‘(𝑦 − 𝑥)))
130	129	oveq2d 7448	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ 𝐴 ≤ 𝐵) ∧ 𝑘 ∈ ℝ) ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) ∧ 𝑦 < 𝑥) → (𝑘 · (abs‘(𝑥 − 𝑦))) = (𝑘 · (abs‘(𝑦 − 𝑥))))
131	118, 130	breq12d 5155	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝐴 ≤ 𝐵) ∧ 𝑘 ∈ ℝ) ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) ∧ 𝑦 < 𝑥) → ((abs‘((𝐹‘𝑥) − (𝐹‘𝑦))) ≤ (𝑘 · (abs‘(𝑥 − 𝑦))) ↔ (abs‘((𝐹‘𝑦) − (𝐹‘𝑥))) ≤ (𝑘 · (abs‘(𝑦 − 𝑥)))))
132	131	biimpd 229	. . . . . . . 8 ⊢ (((((𝜑 ∧ 𝐴 ≤ 𝐵) ∧ 𝑘 ∈ ℝ) ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) ∧ 𝑦 < 𝑥) → ((abs‘((𝐹‘𝑥) − (𝐹‘𝑦))) ≤ (𝑘 · (abs‘(𝑥 − 𝑦))) → (abs‘((𝐹‘𝑦) − (𝐹‘𝑥))) ≤ (𝑘 · (abs‘(𝑦 − 𝑥)))))
133	109, 132	embantd 59	. . . . . . 7 ⊢ (((((𝜑 ∧ 𝐴 ≤ 𝐵) ∧ 𝑘 ∈ ℝ) ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) ∧ 𝑦 < 𝑥) → ((𝑦 < 𝑥 → (abs‘((𝐹‘𝑥) − (𝐹‘𝑦))) ≤ (𝑘 · (abs‘(𝑥 − 𝑦)))) → (abs‘((𝐹‘𝑦) − (𝐹‘𝑥))) ≤ (𝑘 · (abs‘(𝑦 − 𝑥)))))
134	108, 133	syld 47	. . . . . 6 ⊢ (((((𝜑 ∧ 𝐴 ≤ 𝐵) ∧ 𝑘 ∈ ℝ) ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) ∧ 𝑦 < 𝑥) → (∀𝑎 ∈ (𝐴[,]𝐵)∀𝑏 ∈ (𝐴[,]𝐵)(𝑎 < 𝑏 → (abs‘((𝐹‘𝑏) − (𝐹‘𝑎))) ≤ (𝑘 · (abs‘(𝑏 − 𝑎)))) → (abs‘((𝐹‘𝑦) − (𝐹‘𝑥))) ≤ (𝑘 · (abs‘(𝑦 − 𝑥)))))
135		lttri4 11346	. . . . . . 7 ⊢ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) → (𝑥 < 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 < 𝑥))
136	123, 135	syl 17	. . . . . 6 ⊢ ((((𝜑 ∧ 𝐴 ≤ 𝐵) ∧ 𝑘 ∈ ℝ) ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) → (𝑥 < 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 < 𝑥))
137	53, 89, 134, 136	mpjao3dan 1433	. . . . 5 ⊢ ((((𝜑 ∧ 𝐴 ≤ 𝐵) ∧ 𝑘 ∈ ℝ) ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) → (∀𝑎 ∈ (𝐴[,]𝐵)∀𝑏 ∈ (𝐴[,]𝐵)(𝑎 < 𝑏 → (abs‘((𝐹‘𝑏) − (𝐹‘𝑎))) ≤ (𝑘 · (abs‘(𝑏 − 𝑎)))) → (abs‘((𝐹‘𝑦) − (𝐹‘𝑥))) ≤ (𝑘 · (abs‘(𝑦 − 𝑥)))))
138	137	ralrimdvva 3210	. . . 4 ⊢ (((𝜑 ∧ 𝐴 ≤ 𝐵) ∧ 𝑘 ∈ ℝ) → (∀𝑎 ∈ (𝐴[,]𝐵)∀𝑏 ∈ (𝐴[,]𝐵)(𝑎 < 𝑏 → (abs‘((𝐹‘𝑏) − (𝐹‘𝑎))) ≤ (𝑘 · (abs‘(𝑏 − 𝑎)))) → ∀𝑥 ∈ (𝐴[,]𝐵)∀𝑦 ∈ (𝐴[,]𝐵)(abs‘((𝐹‘𝑦) − (𝐹‘𝑥))) ≤ (𝑘 · (abs‘(𝑦 − 𝑥)))))
139	138	reximdva 3167	. . 3 ⊢ ((𝜑 ∧ 𝐴 ≤ 𝐵) → (∃𝑘 ∈ ℝ ∀𝑎 ∈ (𝐴[,]𝐵)∀𝑏 ∈ (𝐴[,]𝐵)(𝑎 < 𝑏 → (abs‘((𝐹‘𝑏) − (𝐹‘𝑎))) ≤ (𝑘 · (abs‘(𝑏 − 𝑎)))) → ∃𝑘 ∈ ℝ ∀𝑥 ∈ (𝐴[,]𝐵)∀𝑦 ∈ (𝐴[,]𝐵)(abs‘((𝐹‘𝑦) − (𝐹‘𝑥))) ≤ (𝑘 · (abs‘(𝑦 − 𝑥)))))
140	32, 139	mpd 15	. 2 ⊢ ((𝜑 ∧ 𝐴 ≤ 𝐵) → ∃𝑘 ∈ ℝ ∀𝑥 ∈ (𝐴[,]𝐵)∀𝑦 ∈ (𝐴[,]𝐵)(abs‘((𝐹‘𝑦) − (𝐹‘𝑥))) ≤ (𝑘 · (abs‘(𝑦 − 𝑥))))
141	15, 140, 6, 4	ltlecasei 11370	1 ⊢ (𝜑 → ∃𝑘 ∈ ℝ ∀𝑥 ∈ (𝐴[,]𝐵)∀𝑦 ∈ (𝐴[,]𝐵)(abs‘((𝐹‘𝑦) − (𝐹‘𝑥))) ≤ (𝑘 · (abs‘(𝑦 − 𝑥))))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∨ w3o 1085   = wceq 1539   ∈ wcel 2107   ≠ wne 2939  ∀wral 3060  ∃wrex 3069   ⊆ wss 3950  ∅c0 4332   class class class wbr 5142   ↾ cres 5686   “ cima 5687  ⟶wf 6556  ‘cfv 6560  (class class class)co 7432   ↑_pm cpm 8868  supcsup 9481  ℂcc 11154  ℝcr 11155  0cc0 11156   · cmul 11161  ℝ^*cxr 11295   < clt 11296   ≤ cle 11297   − cmin 11493  [,]cicc 13391  abscabs 15274  –cn→ccncf 24903   D cdv 25899

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-rep 5278  ax-sep 5295  ax-nul 5305  ax-pow 5364  ax-pr 5431  ax-un 7756  ax-cnex 11212  ax-resscn 11213  ax-1cn 11214  ax-icn 11215  ax-addcl 11216  ax-addrcl 11217  ax-mulcl 11218  ax-mulrcl 11219  ax-mulcom 11220  ax-addass 11221  ax-mulass 11222  ax-distr 11223  ax-i2m1 11224  ax-1ne0 11225  ax-1rid 11226  ax-rnegex 11227  ax-rrecex 11228  ax-cnre 11229  ax-pre-lttri 11230  ax-pre-lttrn 11231  ax-pre-ltadd 11232  ax-pre-mulgt0 11233  ax-pre-sup 11234  ax-addf 11235

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ne 2940  df-nel 3046  df-ral 3061  df-rex 3070  df-rmo 3379  df-reu 3380  df-rab 3436  df-v 3481  df-sbc 3788  df-csb 3899  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-pss 3970  df-nul 4333  df-if 4525  df-pw 4601  df-sn 4626  df-pr 4628  df-tp 4630  df-op 4632  df-uni 4907  df-int 4946  df-iun 4992  df-iin 4993  df-br 5143  df-opab 5205  df-mpt 5225  df-tr 5259  df-id 5577  df-eprel 5583  df-po 5591  df-so 5592  df-fr 5636  df-se 5637  df-we 5638  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-rn 5695  df-res 5696  df-ima 5697  df-pred 6320  df-ord 6386  df-on 6387  df-lim 6388  df-suc 6389  df-iota 6513  df-fun 6562  df-fn 6563  df-f 6564  df-f1 6565  df-fo 6566  df-f1o 6567  df-fv 6568  df-isom 6569  df-riota 7389  df-ov 7435  df-oprab 7436  df-mpo 7437  df-of 7698  df-om 7889  df-1st 8015  df-2nd 8016  df-supp 8187  df-frecs 8307  df-wrecs 8338  df-recs 8412  df-rdg 8451  df-1o 8507  df-2o 8508  df-er 8746  df-map 8869  df-pm 8870  df-ixp 8939  df-en 8987  df-dom 8988  df-sdom 8989  df-fin 8990  df-fsupp 9403  df-fi 9452  df-sup 9483  df-inf 9484  df-oi 9551  df-card 9980  df-pnf 11298  df-mnf 11299  df-xr 11300  df-ltxr 11301  df-le 11302  df-sub 11495  df-neg 11496  df-div 11922  df-nn 12268  df-2 12330  df-3 12331  df-4 12332  df-5 12333  df-6 12334  df-7 12335  df-8 12336  df-9 12337  df-n0 12529  df-z 12616  df-dec 12736  df-uz 12880  df-q 12992  df-rp 13036  df-xneg 13155  df-xadd 13156  df-xmul 13157  df-ioo 13392  df-ico 13394  df-icc 13395  df-fz 13549  df-fzo 13696  df-seq 14044  df-exp 14104  df-hash 14371  df-cj 15139  df-re 15140  df-im 15141  df-sqrt 15275  df-abs 15276  df-struct 17185  df-sets 17202  df-slot 17220  df-ndx 17232  df-base 17249  df-ress 17276  df-plusg 17311  df-mulr 17312  df-starv 17313  df-sca 17314  df-vsca 17315  df-ip 17316  df-tset 17317  df-ple 17318  df-ds 17320  df-unif 17321  df-hom 17322  df-cco 17323  df-rest 17468  df-topn 17469  df-0g 17487  df-gsum 17488  df-topgen 17489  df-pt 17490  df-prds 17493  df-xrs 17548  df-qtop 17553  df-imas 17554  df-xps 17556  df-mre 17630  df-mrc 17631  df-acs 17633  df-mgm 18654  df-sgrp 18733  df-mnd 18749  df-submnd 18798  df-mulg 19087  df-cntz 19336  df-cmn 19801  df-psmet 21357  df-xmet 21358  df-met 21359  df-bl 21360  df-mopn 21361  df-fbas 21362  df-fg 21363  df-cnfld 21366  df-top 22901  df-topon 22918  df-topsp 22940  df-bases 22954  df-cld 23028  df-ntr 23029  df-cls 23030  df-nei 23107  df-lp 23145  df-perf 23146  df-cn 23236  df-cnp 23237  df-haus 23324  df-cmp 23396  df-tx 23571  df-hmeo 23764  df-fil 23855  df-fm 23947  df-flim 23948  df-flf 23949  df-xms 24331  df-ms 24332  df-tms 24333  df-cncf 24905  df-limc 25902  df-dv 25903

This theorem is referenced by:  c1lip2  26038