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Theorem c1lip1 25949

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 11125	. . . 4 ⊢ 0 ∈ ℝ
2	1	ne0ii 4293	. . 3 ⊢ ℝ ≠ ∅
3		ral0 4448	. . . . 5 ⊢ ∀𝑥 ∈ ∅ ∀𝑦 ∈ (𝐴[,]𝐵)(abs‘((𝐹‘𝑦) − (𝐹‘𝑥))) ≤ (𝑘 · (abs‘(𝑦 − 𝑥)))
4		c1lip1.a	. . . . . . . . 9 ⊢ (𝜑 → 𝐴 ∈ ℝ)
5	4	rexrd 11173	. . . . . . . 8 ⊢ (𝜑 → 𝐴 ∈ ℝ^*)
6		c1lip1.b	. . . . . . . . 9 ⊢ (𝜑 → 𝐵 ∈ ℝ)
7	6	rexrd 11173	. . . . . . . 8 ⊢ (𝜑 → 𝐵 ∈ ℝ^*)
8		icc0 13300	. . . . . . . 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 3293	. . . . 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 3129	. . 3 ⊢ ((𝜑 ∧ 𝐵 < 𝐴) → ∀𝑘 ∈ ℝ ∀𝑥 ∈ (𝐴[,]𝐵)∀𝑦 ∈ (𝐴[,]𝐵)(abs‘((𝐹‘𝑦) − (𝐹‘𝑥))) ≤ (𝑘 · (abs‘(𝑦 − 𝑥))))
14		r19.2z 4449	. . 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 2733	. . . . 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 25948	. . . 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 7362	. . . . . . . 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 5107	. . . . . . 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 3197	. . . . 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 3573	. . . 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 5098	. . . . . . . . . 10 ⊢ (𝑎 = 𝑥 → (𝑎 < 𝑏 ↔ 𝑥 < 𝑏))
34		fveq2 6831	. . . . . . . . . . . . 13 ⊢ (𝑎 = 𝑥 → (𝐹‘𝑎) = (𝐹‘𝑥))
35	34	oveq2d 7371	. . . . . . . . . . . 12 ⊢ (𝑎 = 𝑥 → ((𝐹‘𝑏) − (𝐹‘𝑎)) = ((𝐹‘𝑏) − (𝐹‘𝑥)))
36	35	fveq2d 6835	. . . . . . . . . . 11 ⊢ (𝑎 = 𝑥 → (abs‘((𝐹‘𝑏) − (𝐹‘𝑎))) = (abs‘((𝐹‘𝑏) − (𝐹‘𝑥))))
37		oveq2 7363	. . . . . . . . . . . . 13 ⊢ (𝑎 = 𝑥 → (𝑏 − 𝑎) = (𝑏 − 𝑥))
38	37	fveq2d 6835	. . . . . . . . . . . 12 ⊢ (𝑎 = 𝑥 → (abs‘(𝑏 − 𝑎)) = (abs‘(𝑏 − 𝑥)))
39	38	oveq2d 7371	. . . . . . . . . . 11 ⊢ (𝑎 = 𝑥 → (𝑘 · (abs‘(𝑏 − 𝑎))) = (𝑘 · (abs‘(𝑏 − 𝑥))))
40	36, 39	breq12d 5108	. . . . . . . . . 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 5099	. . . . . . . . . 10 ⊢ (𝑏 = 𝑦 → (𝑥 < 𝑏 ↔ 𝑥 < 𝑦))
43		fveq2 6831	. . . . . . . . . . . 12 ⊢ (𝑏 = 𝑦 → (𝐹‘𝑏) = (𝐹‘𝑦))
44	43	fvoveq1d 7377	. . . . . . . . . . 11 ⊢ (𝑏 = 𝑦 → (abs‘((𝐹‘𝑏) − (𝐹‘𝑥))) = (abs‘((𝐹‘𝑦) − (𝐹‘𝑥))))
45		fvoveq1 7378	. . . . . . . . . . . 12 ⊢ (𝑏 = 𝑦 → (abs‘(𝑏 − 𝑥)) = (abs‘(𝑦 − 𝑥)))
46	45	oveq2d 7371	. . . . . . . . . . 11 ⊢ (𝑏 = 𝑦 → (𝑘 · (abs‘(𝑏 − 𝑥))) = (𝑘 · (abs‘(𝑦 − 𝑥))))
47	44, 46	breq12d 5108	. . . . . . . . . 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 3584	. . . . . . . 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 12237	. . . . . . . . . 10 ⊢ 0 ≤ 0
55		fvres 6850	. . . . . . . . . . . . . . . 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 24833	. . . . . . . . . . . . . . . . . 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 7023	. . . . . . . . . . . . . . . 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 2834	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝐴 ≤ 𝐵) ∧ 𝑘 ∈ ℝ) ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) → (𝐹‘𝑥) ∈ ℝ)
64	63	recnd 11151	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝐴 ≤ 𝐵) ∧ 𝑘 ∈ ℝ) ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) → (𝐹‘𝑥) ∈ ℂ)
65	64	subidd 11471	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝐴 ≤ 𝐵) ∧ 𝑘 ∈ ℝ) ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) → ((𝐹‘𝑥) − (𝐹‘𝑥)) = 0)
66	65	abs00bd 15205	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝐴 ≤ 𝐵) ∧ 𝑘 ∈ ℝ) ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) → (abs‘((𝐹‘𝑥) − (𝐹‘𝑥))) = 0)
67		iccssre 13336	. . . . . . . . . . . . . . . . . . 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 3931	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝐴 ≤ 𝐵) ∧ 𝑘 ∈ ℝ) ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) → 𝑥 ∈ ℝ)
72	71	recnd 11151	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝐴 ≤ 𝐵) ∧ 𝑘 ∈ ℝ) ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) → 𝑥 ∈ ℂ)
73	72	subidd 11471	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝐴 ≤ 𝐵) ∧ 𝑘 ∈ ℝ) ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) → (𝑥 − 𝑥) = 0)
74	73	abs00bd 15205	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝐴 ≤ 𝐵) ∧ 𝑘 ∈ ℝ) ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) → (abs‘(𝑥 − 𝑥)) = 0)
75	74	oveq2d 7371	. . . . . . . . . . . 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 11151	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝐴 ≤ 𝐵) ∧ 𝑘 ∈ ℝ) ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) → 𝑘 ∈ ℂ)
78	77	mul01d 11323	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝐴 ≤ 𝐵) ∧ 𝑘 ∈ ℝ) ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) → (𝑘 · 0) = 0)
79	75, 78	eqtrd 2768	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝐴 ≤ 𝐵) ∧ 𝑘 ∈ ℝ) ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) → (𝑘 · (abs‘(𝑥 − 𝑥))) = 0)
80	66, 79	breq12d 5108	. . . . . . . . . 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 6831	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑦 → (𝐹‘𝑥) = (𝐹‘𝑦))
83	82	fvoveq1d 7377	. . . . . . . . . 10 ⊢ (𝑥 = 𝑦 → (abs‘((𝐹‘𝑥) − (𝐹‘𝑥))) = (abs‘((𝐹‘𝑦) − (𝐹‘𝑥))))
84		fvoveq1 7378	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑦 → (abs‘(𝑥 − 𝑥)) = (abs‘(𝑦 − 𝑥)))
85	84	oveq2d 7371	. . . . . . . . . 10 ⊢ (𝑥 = 𝑦 → (𝑘 · (abs‘(𝑥 − 𝑥))) = (𝑘 · (abs‘(𝑦 − 𝑥))))
86	83, 85	breq12d 5108	. . . . . . . . 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 5098	. . . . . . . . . . 11 ⊢ (𝑎 = 𝑦 → (𝑎 < 𝑏 ↔ 𝑦 < 𝑏))
91		fveq2 6831	. . . . . . . . . . . . . 14 ⊢ (𝑎 = 𝑦 → (𝐹‘𝑎) = (𝐹‘𝑦))
92	91	oveq2d 7371	. . . . . . . . . . . . 13 ⊢ (𝑎 = 𝑦 → ((𝐹‘𝑏) − (𝐹‘𝑎)) = ((𝐹‘𝑏) − (𝐹‘𝑦)))
93	92	fveq2d 6835	. . . . . . . . . . . 12 ⊢ (𝑎 = 𝑦 → (abs‘((𝐹‘𝑏) − (𝐹‘𝑎))) = (abs‘((𝐹‘𝑏) − (𝐹‘𝑦))))
94		oveq2 7363	. . . . . . . . . . . . . 14 ⊢ (𝑎 = 𝑦 → (𝑏 − 𝑎) = (𝑏 − 𝑦))
95	94	fveq2d 6835	. . . . . . . . . . . . 13 ⊢ (𝑎 = 𝑦 → (abs‘(𝑏 − 𝑎)) = (abs‘(𝑏 − 𝑦)))
96	95	oveq2d 7371	. . . . . . . . . . . 12 ⊢ (𝑎 = 𝑦 → (𝑘 · (abs‘(𝑏 − 𝑎))) = (𝑘 · (abs‘(𝑏 − 𝑦))))
97	93, 96	breq12d 5108	. . . . . . . . . . 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 5099	. . . . . . . . . . 11 ⊢ (𝑏 = 𝑥 → (𝑦 < 𝑏 ↔ 𝑦 < 𝑥))
100		fveq2 6831	. . . . . . . . . . . . 13 ⊢ (𝑏 = 𝑥 → (𝐹‘𝑏) = (𝐹‘𝑥))
101	100	fvoveq1d 7377	. . . . . . . . . . . 12 ⊢ (𝑏 = 𝑥 → (abs‘((𝐹‘𝑏) − (𝐹‘𝑦))) = (abs‘((𝐹‘𝑥) − (𝐹‘𝑦))))
102		fvoveq1 7378	. . . . . . . . . . . . 13 ⊢ (𝑏 = 𝑥 → (abs‘(𝑏 − 𝑦)) = (abs‘(𝑥 − 𝑦)))
103	102	oveq2d 7371	. . . . . . . . . . . 12 ⊢ (𝑏 = 𝑥 → (𝑘 · (abs‘(𝑏 − 𝑦))) = (𝑘 · (abs‘(𝑥 − 𝑦))))
104	101, 103	breq12d 5108	. . . . . . . . . . 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 3584	. . . . . . . . 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 6850	. . . . . . . . . . . . . . 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 7023	. . . . . . . . . . . . . . 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 2834	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝐴 ≤ 𝐵) ∧ 𝑘 ∈ ℝ) ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) → (𝐹‘𝑦) ∈ ℝ)
116	115	recnd 11151	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝐴 ≤ 𝐵) ∧ 𝑘 ∈ ℝ) ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) → (𝐹‘𝑦) ∈ ℂ)
117	64, 116	abssubd 15370	. . . . . . . . . . 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 3929	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝐴 ≤ 𝐵) ∧ 𝑘 ∈ ℝ) → (𝑥 ∈ (𝐴[,]𝐵) → 𝑥 ∈ ℝ))
121	119	sseld 3929	. . . . . . . . . . . . . . 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 11107	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ ℝ → 𝑥 ∈ ℂ)
125		recn 11107	. . . . . . . . . . . . . 14 ⊢ (𝑦 ∈ ℝ → 𝑦 ∈ ℂ)
126		abssub 15241	. . . . . . . . . . . . . 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 7371	. . . . . . . . . 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 5108	. . . . . . . . 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 11208	. . . . . . 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 1434	. . . . 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 3188	. . . 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 3146	. . 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 11232	1 ⊢ (𝜑 → ∃𝑘 ∈ ℝ ∀𝑥 ∈ (𝐴[,]𝐵)∀𝑦 ∈ (𝐴[,]𝐵)(abs‘((𝐹‘𝑦) − (𝐹‘𝑥))) ≤ (𝑘 · (abs‘(𝑦 − 𝑥))))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∨ w3o 1085 = wceq 1541 ∈ wcel 2113 ≠ wne 2929 ∀wral 3048 ∃wrex 3057 ⊆ wss 3898 ∅c0 4282 class class class wbr 5095 ↾ cres 5623 “ cima 5624 ⟶wf 6485 ‘cfv 6489 (class class class)co 7355 ↑_pm cpm 8760 supcsup 9335 ℂcc 11015 ℝcr 11016 0cc0 11017 · cmul 11022 ℝ^*cxr 11156 < clt 11157 ≤ cle 11158 − cmin 11355 [,]cicc 13255 abscabs 15148 –cn→ccncf 24816 D cdv 25811

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1968 ax-7 2009 ax-8 2115 ax-9 2123 ax-10 2146 ax-11 2162 ax-12 2182 ax-ext 2705 ax-rep 5221 ax-sep 5238 ax-nul 5248 ax-pow 5307 ax-pr 5374 ax-un 7677 ax-cnex 11073 ax-resscn 11074 ax-1cn 11075 ax-icn 11076 ax-addcl 11077 ax-addrcl 11078 ax-mulcl 11079 ax-mulrcl 11080 ax-mulcom 11081 ax-addass 11082 ax-mulass 11083 ax-distr 11084 ax-i2m1 11085 ax-1ne0 11086 ax-1rid 11087 ax-rnegex 11088 ax-rrecex 11089 ax-cnre 11090 ax-pre-lttri 11091 ax-pre-lttrn 11092 ax-pre-ltadd 11093 ax-pre-mulgt0 11094 ax-pre-sup 11095 ax-addf 11096

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2068 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2725 df-clel 2808 df-nfc 2882 df-ne 2930 df-nel 3034 df-ral 3049 df-rex 3058 df-rmo 3347 df-reu 3348 df-rab 3397 df-v 3439 df-sbc 3738 df-csb 3847 df-dif 3901 df-un 3903 df-in 3905 df-ss 3915 df-pss 3918 df-nul 4283 df-if 4477 df-pw 4553 df-sn 4578 df-pr 4580 df-tp 4582 df-op 4584 df-uni 4861 df-int 4900 df-iun 4945 df-iin 4946 df-br 5096 df-opab 5158 df-mpt 5177 df-tr 5203 df-id 5516 df-eprel 5521 df-po 5529 df-so 5530 df-fr 5574 df-se 5575 df-we 5576 df-xp 5627 df-rel 5628 df-cnv 5629 df-co 5630 df-dm 5631 df-rn 5632 df-res 5633 df-ima 5634 df-pred 6256 df-ord 6317 df-on 6318 df-lim 6319 df-suc 6320 df-iota 6445 df-fun 6491 df-fn 6492 df-f 6493 df-f1 6494 df-fo 6495 df-f1o 6496 df-fv 6497 df-isom 6498 df-riota 7312 df-ov 7358 df-oprab 7359 df-mpo 7360 df-of 7619 df-om 7806 df-1st 7930 df-2nd 7931 df-supp 8100 df-frecs 8220 df-wrecs 8251 df-recs 8300 df-rdg 8338 df-1o 8394 df-2o 8395 df-er 8631 df-map 8761 df-pm 8762 df-ixp 8832 df-en 8880 df-dom 8881 df-sdom 8882 df-fin 8883 df-fsupp 9257 df-fi 9306 df-sup 9337 df-inf 9338 df-oi 9407 df-card 9843 df-pnf 11159 df-mnf 11160 df-xr 11161 df-ltxr 11162 df-le 11163 df-sub 11357 df-neg 11358 df-div 11786 df-nn 12137 df-2 12199 df-3 12200 df-4 12201 df-5 12202 df-6 12203 df-7 12204 df-8 12205 df-9 12206 df-n0 12393 df-z 12480 df-dec 12599 df-uz 12743 df-q 12853 df-rp 12897 df-xneg 13017 df-xadd 13018 df-xmul 13019 df-ioo 13256 df-ico 13258 df-icc 13259 df-fz 13415 df-fzo 13562 df-seq 13916 df-exp 13976 df-hash 14245 df-cj 15013 df-re 15014 df-im 15015 df-sqrt 15149 df-abs 15150 df-struct 17065 df-sets 17082 df-slot 17100 df-ndx 17112 df-base 17128 df-ress 17149 df-plusg 17181 df-mulr 17182 df-starv 17183 df-sca 17184 df-vsca 17185 df-ip 17186 df-tset 17187 df-ple 17188 df-ds 17190 df-unif 17191 df-hom 17192 df-cco 17193 df-rest 17333 df-topn 17334 df-0g 17352 df-gsum 17353 df-topgen 17354 df-pt 17355 df-prds 17358 df-xrs 17414 df-qtop 17419 df-imas 17420 df-xps 17422 df-mre 17496 df-mrc 17497 df-acs 17499 df-mgm 18556 df-sgrp 18635 df-mnd 18651 df-submnd 18700 df-mulg 18989 df-cntz 19237 df-cmn 19702 df-psmet 21292 df-xmet 21293 df-met 21294 df-bl 21295 df-mopn 21296 df-fbas 21297 df-fg 21298 df-cnfld 21301 df-top 22829 df-topon 22846 df-topsp 22868 df-bases 22881 df-cld 22954 df-ntr 22955 df-cls 22956 df-nei 23033 df-lp 23071 df-perf 23072 df-cn 23162 df-cnp 23163 df-haus 23250 df-cmp 23322 df-tx 23497 df-hmeo 23690 df-fil 23781 df-fm 23873 df-flim 23874 df-flf 23875 df-xms 24255 df-ms 24256 df-tms 24257 df-cncf 24818 df-limc 25814 df-dv 25815

This theorem is referenced by: c1lip2 25950

W3C validator