Step | Hyp | Ref
| Expression |
1 | | 0re 9919 |
. . . 4
⊢ 0 ∈
ℝ |
2 | 1 | ne0ii 3882 |
. . 3
⊢ ℝ
≠ ∅ |
3 | | ral0 4028 |
. . . . 5
⊢
∀𝑥 ∈
∅ ∀𝑦 ∈
(𝐴[,]𝐵)(abs‘((𝐹‘𝑦) − (𝐹‘𝑥))) ≤ (𝑘 · (abs‘(𝑦 − 𝑥))) |
4 | | c1lip1.a |
. . . . . . . . 9
⊢ (𝜑 → 𝐴 ∈ ℝ) |
5 | 4 | rexrd 9968 |
. . . . . . . 8
⊢ (𝜑 → 𝐴 ∈
ℝ*) |
6 | | c1lip1.b |
. . . . . . . . 9
⊢ (𝜑 → 𝐵 ∈ ℝ) |
7 | 6 | rexrd 9968 |
. . . . . . . 8
⊢ (𝜑 → 𝐵 ∈
ℝ*) |
8 | | icc0 12094 |
. . . . . . . 8
⊢ ((𝐴 ∈ ℝ*
∧ 𝐵 ∈
ℝ*) → ((𝐴[,]𝐵) = ∅ ↔ 𝐵 < 𝐴)) |
9 | 5, 7, 8 | syl2anc 691 |
. . . . . . 7
⊢ (𝜑 → ((𝐴[,]𝐵) = ∅ ↔ 𝐵 < 𝐴)) |
10 | 9 | biimpar 501 |
. . . . . 6
⊢ ((𝜑 ∧ 𝐵 < 𝐴) → (𝐴[,]𝐵) = ∅) |
11 | 10 | raleqdv 3121 |
. . . . 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 247 |
. . . 4
⊢ ((𝜑 ∧ 𝐵 < 𝐴) → ∀𝑥 ∈ (𝐴[,]𝐵)∀𝑦 ∈ (𝐴[,]𝐵)(abs‘((𝐹‘𝑦) − (𝐹‘𝑥))) ≤ (𝑘 · (abs‘(𝑦 − 𝑥)))) |
13 | 12 | ralrimivw 2950 |
. . 3
⊢ ((𝜑 ∧ 𝐵 < 𝐴) → ∀𝑘 ∈ ℝ ∀𝑥 ∈ (𝐴[,]𝐵)∀𝑦 ∈ (𝐴[,]𝐵)(abs‘((𝐹‘𝑦) − (𝐹‘𝑥))) ≤ (𝑘 · (abs‘(𝑦 − 𝑥)))) |
14 | | r19.2z 4012 |
. . 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 694 |
. 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 476 |
. . . . 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 2610 |
. . . . 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 23563 |
. . . 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 6556 |
. . . . . . . 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 4595 |
. . . . . . 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 329 |
. . . . . 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 2972 |
. . . . 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 3282 |
. . . 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 4586 |
. . . . . . . . . 10
⊢ (𝑎 = 𝑥 → (𝑎 < 𝑏 ↔ 𝑥 < 𝑏)) |
34 | | fveq2 6103 |
. . . . . . . . . . . . 13
⊢ (𝑎 = 𝑥 → (𝐹‘𝑎) = (𝐹‘𝑥)) |
35 | 34 | oveq2d 6565 |
. . . . . . . . . . . 12
⊢ (𝑎 = 𝑥 → ((𝐹‘𝑏) − (𝐹‘𝑎)) = ((𝐹‘𝑏) − (𝐹‘𝑥))) |
36 | 35 | fveq2d 6107 |
. . . . . . . . . . 11
⊢ (𝑎 = 𝑥 → (abs‘((𝐹‘𝑏) − (𝐹‘𝑎))) = (abs‘((𝐹‘𝑏) − (𝐹‘𝑥)))) |
37 | | oveq2 6557 |
. . . . . . . . . . . . 13
⊢ (𝑎 = 𝑥 → (𝑏 − 𝑎) = (𝑏 − 𝑥)) |
38 | 37 | fveq2d 6107 |
. . . . . . . . . . . 12
⊢ (𝑎 = 𝑥 → (abs‘(𝑏 − 𝑎)) = (abs‘(𝑏 − 𝑥))) |
39 | 38 | oveq2d 6565 |
. . . . . . . . . . 11
⊢ (𝑎 = 𝑥 → (𝑘 · (abs‘(𝑏 − 𝑎))) = (𝑘 · (abs‘(𝑏 − 𝑥)))) |
40 | 36, 39 | breq12d 4596 |
. . . . . . . . . 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 333 |
. . . . . . . . 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 4587 |
. . . . . . . . . 10
⊢ (𝑏 = 𝑦 → (𝑥 < 𝑏 ↔ 𝑥 < 𝑦)) |
43 | | fveq2 6103 |
. . . . . . . . . . . . 13
⊢ (𝑏 = 𝑦 → (𝐹‘𝑏) = (𝐹‘𝑦)) |
44 | 43 | oveq1d 6564 |
. . . . . . . . . . . 12
⊢ (𝑏 = 𝑦 → ((𝐹‘𝑏) − (𝐹‘𝑥)) = ((𝐹‘𝑦) − (𝐹‘𝑥))) |
45 | 44 | fveq2d 6107 |
. . . . . . . . . . 11
⊢ (𝑏 = 𝑦 → (abs‘((𝐹‘𝑏) − (𝐹‘𝑥))) = (abs‘((𝐹‘𝑦) − (𝐹‘𝑥)))) |
46 | | oveq1 6556 |
. . . . . . . . . . . . 13
⊢ (𝑏 = 𝑦 → (𝑏 − 𝑥) = (𝑦 − 𝑥)) |
47 | 46 | fveq2d 6107 |
. . . . . . . . . . . 12
⊢ (𝑏 = 𝑦 → (abs‘(𝑏 − 𝑥)) = (abs‘(𝑦 − 𝑥))) |
48 | 47 | oveq2d 6565 |
. . . . . . . . . . 11
⊢ (𝑏 = 𝑦 → (𝑘 · (abs‘(𝑏 − 𝑥))) = (𝑘 · (abs‘(𝑦 − 𝑥)))) |
49 | 45, 48 | breq12d 4596 |
. . . . . . . . . 10
⊢ (𝑏 = 𝑦 → ((abs‘((𝐹‘𝑏) − (𝐹‘𝑥))) ≤ (𝑘 · (abs‘(𝑏 − 𝑥))) ↔ (abs‘((𝐹‘𝑦) − (𝐹‘𝑥))) ≤ (𝑘 · (abs‘(𝑦 − 𝑥))))) |
50 | 42, 49 | imbi12d 333 |
. . . . . . . . 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)))))) |
51 | 41, 50 | rspc2v 3293 |
. . . . . . . 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)))))) |
52 | 51 | ad2antlr 759 |
. . . . . . 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)))))) |
53 | | pm2.27 41 |
. . . . . . . 8
⊢ (𝑥 < 𝑦 → ((𝑥 < 𝑦 → (abs‘((𝐹‘𝑦) − (𝐹‘𝑥))) ≤ (𝑘 · (abs‘(𝑦 − 𝑥)))) → (abs‘((𝐹‘𝑦) − (𝐹‘𝑥))) ≤ (𝑘 · (abs‘(𝑦 − 𝑥))))) |
54 | 53 | 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))))) |
55 | 52, 54 | syld 46 |
. . . . . 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))))) |
56 | | 0le0 10987 |
. . . . . . . . . 10
⊢ 0 ≤
0 |
57 | | fvres 6117 |
. . . . . . . . . . . . . . . 16
⊢ (𝑥 ∈ (𝐴[,]𝐵) → ((𝐹 ↾ (𝐴[,]𝐵))‘𝑥) = (𝐹‘𝑥)) |
58 | 57 | ad2antrl 760 |
. . . . . . . . . . . . . . 15
⊢ ((((𝜑 ∧ 𝐴 ≤ 𝐵) ∧ 𝑘 ∈ ℝ) ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) → ((𝐹 ↾ (𝐴[,]𝐵))‘𝑥) = (𝐹‘𝑥)) |
59 | | cncff 22504 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝐹 ↾ (𝐴[,]𝐵)) ∈ ((𝐴[,]𝐵)–cn→ℝ) → (𝐹 ↾ (𝐴[,]𝐵)):(𝐴[,]𝐵)⟶ℝ) |
60 | 23, 59 | syl 17 |
. . . . . . . . . . . . . . . . 17
⊢ (𝜑 → (𝐹 ↾ (𝐴[,]𝐵)):(𝐴[,]𝐵)⟶ℝ) |
61 | 60 | ad2antrr 758 |
. . . . . . . . . . . . . . . 16
⊢ (((𝜑 ∧ 𝐴 ≤ 𝐵) ∧ 𝑘 ∈ ℝ) → (𝐹 ↾ (𝐴[,]𝐵)):(𝐴[,]𝐵)⟶ℝ) |
62 | | simpl 472 |
. . . . . . . . . . . . . . . 16
⊢ ((𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵)) → 𝑥 ∈ (𝐴[,]𝐵)) |
63 | | ffvelrn 6265 |
. . . . . . . . . . . . . . . 16
⊢ (((𝐹 ↾ (𝐴[,]𝐵)):(𝐴[,]𝐵)⟶ℝ ∧ 𝑥 ∈ (𝐴[,]𝐵)) → ((𝐹 ↾ (𝐴[,]𝐵))‘𝑥) ∈ ℝ) |
64 | 61, 62, 63 | syl2an 493 |
. . . . . . . . . . . . . . 15
⊢ ((((𝜑 ∧ 𝐴 ≤ 𝐵) ∧ 𝑘 ∈ ℝ) ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) → ((𝐹 ↾ (𝐴[,]𝐵))‘𝑥) ∈ ℝ) |
65 | 58, 64 | eqeltrrd 2689 |
. . . . . . . . . . . . . 14
⊢ ((((𝜑 ∧ 𝐴 ≤ 𝐵) ∧ 𝑘 ∈ ℝ) ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) → (𝐹‘𝑥) ∈ ℝ) |
66 | 65 | recnd 9947 |
. . . . . . . . . . . . 13
⊢ ((((𝜑 ∧ 𝐴 ≤ 𝐵) ∧ 𝑘 ∈ ℝ) ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) → (𝐹‘𝑥) ∈ ℂ) |
67 | 66 | subidd 10259 |
. . . . . . . . . . . 12
⊢ ((((𝜑 ∧ 𝐴 ≤ 𝐵) ∧ 𝑘 ∈ ℝ) ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) → ((𝐹‘𝑥) − (𝐹‘𝑥)) = 0) |
68 | 67 | abs00bd 13879 |
. . . . . . . . . . 11
⊢ ((((𝜑 ∧ 𝐴 ≤ 𝐵) ∧ 𝑘 ∈ ℝ) ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) → (abs‘((𝐹‘𝑥) − (𝐹‘𝑥))) = 0) |
69 | | iccssre 12126 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴[,]𝐵) ⊆ ℝ) |
70 | 4, 6, 69 | syl2anc 691 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝜑 → (𝐴[,]𝐵) ⊆ ℝ) |
71 | 70 | ad3antrrr 762 |
. . . . . . . . . . . . . . . . 17
⊢ ((((𝜑 ∧ 𝐴 ≤ 𝐵) ∧ 𝑘 ∈ ℝ) ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) → (𝐴[,]𝐵) ⊆ ℝ) |
72 | | simprl 790 |
. . . . . . . . . . . . . . . . 17
⊢ ((((𝜑 ∧ 𝐴 ≤ 𝐵) ∧ 𝑘 ∈ ℝ) ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) → 𝑥 ∈ (𝐴[,]𝐵)) |
73 | 71, 72 | sseldd 3569 |
. . . . . . . . . . . . . . . 16
⊢ ((((𝜑 ∧ 𝐴 ≤ 𝐵) ∧ 𝑘 ∈ ℝ) ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) → 𝑥 ∈ ℝ) |
74 | 73 | recnd 9947 |
. . . . . . . . . . . . . . 15
⊢ ((((𝜑 ∧ 𝐴 ≤ 𝐵) ∧ 𝑘 ∈ ℝ) ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) → 𝑥 ∈ ℂ) |
75 | 74 | subidd 10259 |
. . . . . . . . . . . . . 14
⊢ ((((𝜑 ∧ 𝐴 ≤ 𝐵) ∧ 𝑘 ∈ ℝ) ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) → (𝑥 − 𝑥) = 0) |
76 | 75 | abs00bd 13879 |
. . . . . . . . . . . . 13
⊢ ((((𝜑 ∧ 𝐴 ≤ 𝐵) ∧ 𝑘 ∈ ℝ) ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) → (abs‘(𝑥 − 𝑥)) = 0) |
77 | 76 | oveq2d 6565 |
. . . . . . . . . . . 12
⊢ ((((𝜑 ∧ 𝐴 ≤ 𝐵) ∧ 𝑘 ∈ ℝ) ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) → (𝑘 · (abs‘(𝑥 − 𝑥))) = (𝑘 · 0)) |
78 | | simplr 788 |
. . . . . . . . . . . . . 14
⊢ ((((𝜑 ∧ 𝐴 ≤ 𝐵) ∧ 𝑘 ∈ ℝ) ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) → 𝑘 ∈ ℝ) |
79 | 78 | recnd 9947 |
. . . . . . . . . . . . 13
⊢ ((((𝜑 ∧ 𝐴 ≤ 𝐵) ∧ 𝑘 ∈ ℝ) ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) → 𝑘 ∈ ℂ) |
80 | 79 | mul01d 10114 |
. . . . . . . . . . . 12
⊢ ((((𝜑 ∧ 𝐴 ≤ 𝐵) ∧ 𝑘 ∈ ℝ) ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) → (𝑘 · 0) = 0) |
81 | 77, 80 | eqtrd 2644 |
. . . . . . . . . . 11
⊢ ((((𝜑 ∧ 𝐴 ≤ 𝐵) ∧ 𝑘 ∈ ℝ) ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) → (𝑘 · (abs‘(𝑥 − 𝑥))) = 0) |
82 | 68, 81 | breq12d 4596 |
. . . . . . . . . 10
⊢ ((((𝜑 ∧ 𝐴 ≤ 𝐵) ∧ 𝑘 ∈ ℝ) ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) → ((abs‘((𝐹‘𝑥) − (𝐹‘𝑥))) ≤ (𝑘 · (abs‘(𝑥 − 𝑥))) ↔ 0 ≤ 0)) |
83 | 56, 82 | mpbiri 247 |
. . . . . . . . 9
⊢ ((((𝜑 ∧ 𝐴 ≤ 𝐵) ∧ 𝑘 ∈ ℝ) ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) → (abs‘((𝐹‘𝑥) − (𝐹‘𝑥))) ≤ (𝑘 · (abs‘(𝑥 − 𝑥)))) |
84 | | fveq2 6103 |
. . . . . . . . . . . 12
⊢ (𝑥 = 𝑦 → (𝐹‘𝑥) = (𝐹‘𝑦)) |
85 | 84 | oveq1d 6564 |
. . . . . . . . . . 11
⊢ (𝑥 = 𝑦 → ((𝐹‘𝑥) − (𝐹‘𝑥)) = ((𝐹‘𝑦) − (𝐹‘𝑥))) |
86 | 85 | fveq2d 6107 |
. . . . . . . . . 10
⊢ (𝑥 = 𝑦 → (abs‘((𝐹‘𝑥) − (𝐹‘𝑥))) = (abs‘((𝐹‘𝑦) − (𝐹‘𝑥)))) |
87 | | oveq1 6556 |
. . . . . . . . . . . 12
⊢ (𝑥 = 𝑦 → (𝑥 − 𝑥) = (𝑦 − 𝑥)) |
88 | 87 | fveq2d 6107 |
. . . . . . . . . . 11
⊢ (𝑥 = 𝑦 → (abs‘(𝑥 − 𝑥)) = (abs‘(𝑦 − 𝑥))) |
89 | 88 | oveq2d 6565 |
. . . . . . . . . 10
⊢ (𝑥 = 𝑦 → (𝑘 · (abs‘(𝑥 − 𝑥))) = (𝑘 · (abs‘(𝑦 − 𝑥)))) |
90 | 86, 89 | breq12d 4596 |
. . . . . . . . 9
⊢ (𝑥 = 𝑦 → ((abs‘((𝐹‘𝑥) − (𝐹‘𝑥))) ≤ (𝑘 · (abs‘(𝑥 − 𝑥))) ↔ (abs‘((𝐹‘𝑦) − (𝐹‘𝑥))) ≤ (𝑘 · (abs‘(𝑦 − 𝑥))))) |
91 | 83, 90 | syl5ibcom 234 |
. . . . . . . 8
⊢ ((((𝜑 ∧ 𝐴 ≤ 𝐵) ∧ 𝑘 ∈ ℝ) ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) → (𝑥 = 𝑦 → (abs‘((𝐹‘𝑦) − (𝐹‘𝑥))) ≤ (𝑘 · (abs‘(𝑦 − 𝑥))))) |
92 | 91 | imp 444 |
. . . . . . 7
⊢
(((((𝜑 ∧ 𝐴 ≤ 𝐵) ∧ 𝑘 ∈ ℝ) ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) ∧ 𝑥 = 𝑦) → (abs‘((𝐹‘𝑦) − (𝐹‘𝑥))) ≤ (𝑘 · (abs‘(𝑦 − 𝑥)))) |
93 | 92 | 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))))) |
94 | | breq1 4586 |
. . . . . . . . . . 11
⊢ (𝑎 = 𝑦 → (𝑎 < 𝑏 ↔ 𝑦 < 𝑏)) |
95 | | fveq2 6103 |
. . . . . . . . . . . . . 14
⊢ (𝑎 = 𝑦 → (𝐹‘𝑎) = (𝐹‘𝑦)) |
96 | 95 | oveq2d 6565 |
. . . . . . . . . . . . 13
⊢ (𝑎 = 𝑦 → ((𝐹‘𝑏) − (𝐹‘𝑎)) = ((𝐹‘𝑏) − (𝐹‘𝑦))) |
97 | 96 | fveq2d 6107 |
. . . . . . . . . . . 12
⊢ (𝑎 = 𝑦 → (abs‘((𝐹‘𝑏) − (𝐹‘𝑎))) = (abs‘((𝐹‘𝑏) − (𝐹‘𝑦)))) |
98 | | oveq2 6557 |
. . . . . . . . . . . . . 14
⊢ (𝑎 = 𝑦 → (𝑏 − 𝑎) = (𝑏 − 𝑦)) |
99 | 98 | fveq2d 6107 |
. . . . . . . . . . . . 13
⊢ (𝑎 = 𝑦 → (abs‘(𝑏 − 𝑎)) = (abs‘(𝑏 − 𝑦))) |
100 | 99 | oveq2d 6565 |
. . . . . . . . . . . 12
⊢ (𝑎 = 𝑦 → (𝑘 · (abs‘(𝑏 − 𝑎))) = (𝑘 · (abs‘(𝑏 − 𝑦)))) |
101 | 97, 100 | breq12d 4596 |
. . . . . . . . . . 11
⊢ (𝑎 = 𝑦 → ((abs‘((𝐹‘𝑏) − (𝐹‘𝑎))) ≤ (𝑘 · (abs‘(𝑏 − 𝑎))) ↔ (abs‘((𝐹‘𝑏) − (𝐹‘𝑦))) ≤ (𝑘 · (abs‘(𝑏 − 𝑦))))) |
102 | 94, 101 | imbi12d 333 |
. . . . . . . . . 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)))))) |
103 | | breq2 4587 |
. . . . . . . . . . 11
⊢ (𝑏 = 𝑥 → (𝑦 < 𝑏 ↔ 𝑦 < 𝑥)) |
104 | | fveq2 6103 |
. . . . . . . . . . . . . 14
⊢ (𝑏 = 𝑥 → (𝐹‘𝑏) = (𝐹‘𝑥)) |
105 | 104 | oveq1d 6564 |
. . . . . . . . . . . . 13
⊢ (𝑏 = 𝑥 → ((𝐹‘𝑏) − (𝐹‘𝑦)) = ((𝐹‘𝑥) − (𝐹‘𝑦))) |
106 | 105 | fveq2d 6107 |
. . . . . . . . . . . 12
⊢ (𝑏 = 𝑥 → (abs‘((𝐹‘𝑏) − (𝐹‘𝑦))) = (abs‘((𝐹‘𝑥) − (𝐹‘𝑦)))) |
107 | | oveq1 6556 |
. . . . . . . . . . . . . 14
⊢ (𝑏 = 𝑥 → (𝑏 − 𝑦) = (𝑥 − 𝑦)) |
108 | 107 | fveq2d 6107 |
. . . . . . . . . . . . 13
⊢ (𝑏 = 𝑥 → (abs‘(𝑏 − 𝑦)) = (abs‘(𝑥 − 𝑦))) |
109 | 108 | oveq2d 6565 |
. . . . . . . . . . . 12
⊢ (𝑏 = 𝑥 → (𝑘 · (abs‘(𝑏 − 𝑦))) = (𝑘 · (abs‘(𝑥 − 𝑦)))) |
110 | 106, 109 | breq12d 4596 |
. . . . . . . . . . 11
⊢ (𝑏 = 𝑥 → ((abs‘((𝐹‘𝑏) − (𝐹‘𝑦))) ≤ (𝑘 · (abs‘(𝑏 − 𝑦))) ↔ (abs‘((𝐹‘𝑥) − (𝐹‘𝑦))) ≤ (𝑘 · (abs‘(𝑥 − 𝑦))))) |
111 | 103, 110 | imbi12d 333 |
. . . . . . . . . 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)))))) |
112 | 102, 111 | rspc2v 3293 |
. . . . . . . . 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)))))) |
113 | 112 | ancoms 468 |
. . . . . . . 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)))))) |
114 | 113 | ad2antlr 759 |
. . . . . . 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)))))) |
115 | | simpr 476 |
. . . . . . . 8
⊢
(((((𝜑 ∧ 𝐴 ≤ 𝐵) ∧ 𝑘 ∈ ℝ) ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) ∧ 𝑦 < 𝑥) → 𝑦 < 𝑥) |
116 | | fvres 6117 |
. . . . . . . . . . . . . . 15
⊢ (𝑦 ∈ (𝐴[,]𝐵) → ((𝐹 ↾ (𝐴[,]𝐵))‘𝑦) = (𝐹‘𝑦)) |
117 | 116 | ad2antll 761 |
. . . . . . . . . . . . . 14
⊢ ((((𝜑 ∧ 𝐴 ≤ 𝐵) ∧ 𝑘 ∈ ℝ) ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) → ((𝐹 ↾ (𝐴[,]𝐵))‘𝑦) = (𝐹‘𝑦)) |
118 | | simpr 476 |
. . . . . . . . . . . . . . 15
⊢ ((𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵)) → 𝑦 ∈ (𝐴[,]𝐵)) |
119 | | ffvelrn 6265 |
. . . . . . . . . . . . . . 15
⊢ (((𝐹 ↾ (𝐴[,]𝐵)):(𝐴[,]𝐵)⟶ℝ ∧ 𝑦 ∈ (𝐴[,]𝐵)) → ((𝐹 ↾ (𝐴[,]𝐵))‘𝑦) ∈ ℝ) |
120 | 61, 118, 119 | syl2an 493 |
. . . . . . . . . . . . . 14
⊢ ((((𝜑 ∧ 𝐴 ≤ 𝐵) ∧ 𝑘 ∈ ℝ) ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) → ((𝐹 ↾ (𝐴[,]𝐵))‘𝑦) ∈ ℝ) |
121 | 117, 120 | eqeltrrd 2689 |
. . . . . . . . . . . . 13
⊢ ((((𝜑 ∧ 𝐴 ≤ 𝐵) ∧ 𝑘 ∈ ℝ) ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) → (𝐹‘𝑦) ∈ ℝ) |
122 | 121 | recnd 9947 |
. . . . . . . . . . . 12
⊢ ((((𝜑 ∧ 𝐴 ≤ 𝐵) ∧ 𝑘 ∈ ℝ) ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) → (𝐹‘𝑦) ∈ ℂ) |
123 | 66, 122 | abssubd 14040 |
. . . . . . . . . . 11
⊢ ((((𝜑 ∧ 𝐴 ≤ 𝐵) ∧ 𝑘 ∈ ℝ) ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) → (abs‘((𝐹‘𝑥) − (𝐹‘𝑦))) = (abs‘((𝐹‘𝑦) − (𝐹‘𝑥)))) |
124 | 123 | 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)))) |
125 | 70 | ad2antrr 758 |
. . . . . . . . . . . . . . . 16
⊢ (((𝜑 ∧ 𝐴 ≤ 𝐵) ∧ 𝑘 ∈ ℝ) → (𝐴[,]𝐵) ⊆ ℝ) |
126 | 125 | sseld 3567 |
. . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ 𝐴 ≤ 𝐵) ∧ 𝑘 ∈ ℝ) → (𝑥 ∈ (𝐴[,]𝐵) → 𝑥 ∈ ℝ)) |
127 | 125 | sseld 3567 |
. . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ 𝐴 ≤ 𝐵) ∧ 𝑘 ∈ ℝ) → (𝑦 ∈ (𝐴[,]𝐵) → 𝑦 ∈ ℝ)) |
128 | 126, 127 | anim12d 584 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ 𝐴 ≤ 𝐵) ∧ 𝑘 ∈ ℝ) → ((𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵)) → (𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ))) |
129 | 128 | imp 444 |
. . . . . . . . . . . . 13
⊢ ((((𝜑 ∧ 𝐴 ≤ 𝐵) ∧ 𝑘 ∈ ℝ) ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) → (𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ)) |
130 | | recn 9905 |
. . . . . . . . . . . . . 14
⊢ (𝑥 ∈ ℝ → 𝑥 ∈
ℂ) |
131 | | recn 9905 |
. . . . . . . . . . . . . 14
⊢ (𝑦 ∈ ℝ → 𝑦 ∈
ℂ) |
132 | | abssub 13914 |
. . . . . . . . . . . . . 14
⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ) →
(abs‘(𝑥 − 𝑦)) = (abs‘(𝑦 − 𝑥))) |
133 | 130, 131,
132 | syl2an 493 |
. . . . . . . . . . . . 13
⊢ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) →
(abs‘(𝑥 − 𝑦)) = (abs‘(𝑦 − 𝑥))) |
134 | 129, 133 | syl 17 |
. . . . . . . . . . . 12
⊢ ((((𝜑 ∧ 𝐴 ≤ 𝐵) ∧ 𝑘 ∈ ℝ) ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) → (abs‘(𝑥 − 𝑦)) = (abs‘(𝑦 − 𝑥))) |
135 | 134 | adantr 480 |
. . . . . . . . . . 11
⊢
(((((𝜑 ∧ 𝐴 ≤ 𝐵) ∧ 𝑘 ∈ ℝ) ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) ∧ 𝑦 < 𝑥) → (abs‘(𝑥 − 𝑦)) = (abs‘(𝑦 − 𝑥))) |
136 | 135 | oveq2d 6565 |
. . . . . . . . . 10
⊢
(((((𝜑 ∧ 𝐴 ≤ 𝐵) ∧ 𝑘 ∈ ℝ) ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) ∧ 𝑦 < 𝑥) → (𝑘 · (abs‘(𝑥 − 𝑦))) = (𝑘 · (abs‘(𝑦 − 𝑥)))) |
137 | 124, 136 | breq12d 4596 |
. . . . . . . . 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))))) |
138 | 137 | biimpd 218 |
. . . . . . . 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))))) |
139 | 115, 138 | embantd 57 |
. . . . . . 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))))) |
140 | 114, 139 | syld 46 |
. . . . . 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))))) |
141 | | lttri4 10001 |
. . . . . . 7
⊢ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) → (𝑥 < 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 < 𝑥)) |
142 | 129, 141 | syl 17 |
. . . . . 6
⊢ ((((𝜑 ∧ 𝐴 ≤ 𝐵) ∧ 𝑘 ∈ ℝ) ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) → (𝑥 < 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 < 𝑥)) |
143 | 55, 93, 140, 142 | mpjao3dan 1387 |
. . . . 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))))) |
144 | 143 | ralrimdvva 2957 |
. . . 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))))) |
145 | 144 | reximdva 3000 |
. . 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))))) |
146 | 32, 145 | mpd 15 |
. 2
⊢ ((𝜑 ∧ 𝐴 ≤ 𝐵) → ∃𝑘 ∈ ℝ ∀𝑥 ∈ (𝐴[,]𝐵)∀𝑦 ∈ (𝐴[,]𝐵)(abs‘((𝐹‘𝑦) − (𝐹‘𝑥))) ≤ (𝑘 · (abs‘(𝑦 − 𝑥)))) |
147 | 15, 146, 6, 4 | ltlecasei 10024 |
1
⊢ (𝜑 → ∃𝑘 ∈ ℝ ∀𝑥 ∈ (𝐴[,]𝐵)∀𝑦 ∈ (𝐴[,]𝐵)(abs‘((𝐹‘𝑦) − (𝐹‘𝑥))) ≤ (𝑘 · (abs‘(𝑦 − 𝑥)))) |