Step | Hyp | Ref
| Expression |
1 | | iccssxr 12127 |
. . . . . . 7
⊢ (𝐴[,]𝐵) ⊆
ℝ* |
2 | | simplrl 796 |
. . . . . . 7
⊢ (((𝜑 ∧ (𝑋 ∈ (𝐴[,]𝐵) ∧ 𝑌 ∈ (𝐴[,]𝐵))) ∧ 𝑋 < 𝑌) → 𝑋 ∈ (𝐴[,]𝐵)) |
3 | 1, 2 | sseldi 3566 |
. . . . . 6
⊢ (((𝜑 ∧ (𝑋 ∈ (𝐴[,]𝐵) ∧ 𝑌 ∈ (𝐴[,]𝐵))) ∧ 𝑋 < 𝑌) → 𝑋 ∈
ℝ*) |
4 | | simplrr 797 |
. . . . . . 7
⊢ (((𝜑 ∧ (𝑋 ∈ (𝐴[,]𝐵) ∧ 𝑌 ∈ (𝐴[,]𝐵))) ∧ 𝑋 < 𝑌) → 𝑌 ∈ (𝐴[,]𝐵)) |
5 | 1, 4 | sseldi 3566 |
. . . . . 6
⊢ (((𝜑 ∧ (𝑋 ∈ (𝐴[,]𝐵) ∧ 𝑌 ∈ (𝐴[,]𝐵))) ∧ 𝑋 < 𝑌) → 𝑌 ∈
ℝ*) |
6 | | dvgt0.a |
. . . . . . . . . 10
⊢ (𝜑 → 𝐴 ∈ ℝ) |
7 | | dvgt0.b |
. . . . . . . . . 10
⊢ (𝜑 → 𝐵 ∈ ℝ) |
8 | | iccssre 12126 |
. . . . . . . . . 10
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴[,]𝐵) ⊆ ℝ) |
9 | 6, 7, 8 | syl2anc 691 |
. . . . . . . . 9
⊢ (𝜑 → (𝐴[,]𝐵) ⊆ ℝ) |
10 | 9 | ad2antrr 758 |
. . . . . . . 8
⊢ (((𝜑 ∧ (𝑋 ∈ (𝐴[,]𝐵) ∧ 𝑌 ∈ (𝐴[,]𝐵))) ∧ 𝑋 < 𝑌) → (𝐴[,]𝐵) ⊆ ℝ) |
11 | 10, 2 | sseldd 3569 |
. . . . . . 7
⊢ (((𝜑 ∧ (𝑋 ∈ (𝐴[,]𝐵) ∧ 𝑌 ∈ (𝐴[,]𝐵))) ∧ 𝑋 < 𝑌) → 𝑋 ∈ ℝ) |
12 | 10, 4 | sseldd 3569 |
. . . . . . 7
⊢ (((𝜑 ∧ (𝑋 ∈ (𝐴[,]𝐵) ∧ 𝑌 ∈ (𝐴[,]𝐵))) ∧ 𝑋 < 𝑌) → 𝑌 ∈ ℝ) |
13 | | simpr 476 |
. . . . . . 7
⊢ (((𝜑 ∧ (𝑋 ∈ (𝐴[,]𝐵) ∧ 𝑌 ∈ (𝐴[,]𝐵))) ∧ 𝑋 < 𝑌) → 𝑋 < 𝑌) |
14 | 11, 12, 13 | ltled 10064 |
. . . . . 6
⊢ (((𝜑 ∧ (𝑋 ∈ (𝐴[,]𝐵) ∧ 𝑌 ∈ (𝐴[,]𝐵))) ∧ 𝑋 < 𝑌) → 𝑋 ≤ 𝑌) |
15 | | ubicc2 12160 |
. . . . . 6
⊢ ((𝑋 ∈ ℝ*
∧ 𝑌 ∈
ℝ* ∧ 𝑋
≤ 𝑌) → 𝑌 ∈ (𝑋[,]𝑌)) |
16 | 3, 5, 14, 15 | syl3anc 1318 |
. . . . 5
⊢ (((𝜑 ∧ (𝑋 ∈ (𝐴[,]𝐵) ∧ 𝑌 ∈ (𝐴[,]𝐵))) ∧ 𝑋 < 𝑌) → 𝑌 ∈ (𝑋[,]𝑌)) |
17 | | fvres 6117 |
. . . . 5
⊢ (𝑌 ∈ (𝑋[,]𝑌) → ((𝐹 ↾ (𝑋[,]𝑌))‘𝑌) = (𝐹‘𝑌)) |
18 | 16, 17 | syl 17 |
. . . 4
⊢ (((𝜑 ∧ (𝑋 ∈ (𝐴[,]𝐵) ∧ 𝑌 ∈ (𝐴[,]𝐵))) ∧ 𝑋 < 𝑌) → ((𝐹 ↾ (𝑋[,]𝑌))‘𝑌) = (𝐹‘𝑌)) |
19 | | lbicc2 12159 |
. . . . . 6
⊢ ((𝑋 ∈ ℝ*
∧ 𝑌 ∈
ℝ* ∧ 𝑋
≤ 𝑌) → 𝑋 ∈ (𝑋[,]𝑌)) |
20 | 3, 5, 14, 19 | syl3anc 1318 |
. . . . 5
⊢ (((𝜑 ∧ (𝑋 ∈ (𝐴[,]𝐵) ∧ 𝑌 ∈ (𝐴[,]𝐵))) ∧ 𝑋 < 𝑌) → 𝑋 ∈ (𝑋[,]𝑌)) |
21 | | fvres 6117 |
. . . . 5
⊢ (𝑋 ∈ (𝑋[,]𝑌) → ((𝐹 ↾ (𝑋[,]𝑌))‘𝑋) = (𝐹‘𝑋)) |
22 | 20, 21 | syl 17 |
. . . 4
⊢ (((𝜑 ∧ (𝑋 ∈ (𝐴[,]𝐵) ∧ 𝑌 ∈ (𝐴[,]𝐵))) ∧ 𝑋 < 𝑌) → ((𝐹 ↾ (𝑋[,]𝑌))‘𝑋) = (𝐹‘𝑋)) |
23 | 18, 22 | oveq12d 6567 |
. . 3
⊢ (((𝜑 ∧ (𝑋 ∈ (𝐴[,]𝐵) ∧ 𝑌 ∈ (𝐴[,]𝐵))) ∧ 𝑋 < 𝑌) → (((𝐹 ↾ (𝑋[,]𝑌))‘𝑌) − ((𝐹 ↾ (𝑋[,]𝑌))‘𝑋)) = ((𝐹‘𝑌) − (𝐹‘𝑋))) |
24 | 23 | oveq1d 6564 |
. 2
⊢ (((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))) |
25 | | iccss2 12115 |
. . . . . 6
⊢ ((𝑋 ∈ (𝐴[,]𝐵) ∧ 𝑌 ∈ (𝐴[,]𝐵)) → (𝑋[,]𝑌) ⊆ (𝐴[,]𝐵)) |
26 | 25 | ad2antlr 759 |
. . . . 5
⊢ (((𝜑 ∧ (𝑋 ∈ (𝐴[,]𝐵) ∧ 𝑌 ∈ (𝐴[,]𝐵))) ∧ 𝑋 < 𝑌) → (𝑋[,]𝑌) ⊆ (𝐴[,]𝐵)) |
27 | | dvgt0.f |
. . . . . 6
⊢ (𝜑 → 𝐹 ∈ ((𝐴[,]𝐵)–cn→ℝ)) |
28 | 27 | ad2antrr 758 |
. . . . 5
⊢ (((𝜑 ∧ (𝑋 ∈ (𝐴[,]𝐵) ∧ 𝑌 ∈ (𝐴[,]𝐵))) ∧ 𝑋 < 𝑌) → 𝐹 ∈ ((𝐴[,]𝐵)–cn→ℝ)) |
29 | | rescncf 22508 |
. . . . 5
⊢ ((𝑋[,]𝑌) ⊆ (𝐴[,]𝐵) → (𝐹 ∈ ((𝐴[,]𝐵)–cn→ℝ) → (𝐹 ↾ (𝑋[,]𝑌)) ∈ ((𝑋[,]𝑌)–cn→ℝ))) |
30 | 26, 28, 29 | sylc 63 |
. . . 4
⊢ (((𝜑 ∧ (𝑋 ∈ (𝐴[,]𝐵) ∧ 𝑌 ∈ (𝐴[,]𝐵))) ∧ 𝑋 < 𝑌) → (𝐹 ↾ (𝑋[,]𝑌)) ∈ ((𝑋[,]𝑌)–cn→ℝ)) |
31 | | dvgt0lem.d |
. . . . . . . 8
⊢ (𝜑 → (ℝ D 𝐹):(𝐴(,)𝐵)⟶𝑆) |
32 | 31 | ad2antrr 758 |
. . . . . . 7
⊢ (((𝜑 ∧ (𝑋 ∈ (𝐴[,]𝐵) ∧ 𝑌 ∈ (𝐴[,]𝐵))) ∧ 𝑋 < 𝑌) → (ℝ D 𝐹):(𝐴(,)𝐵)⟶𝑆) |
33 | 6 | ad2antrr 758 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ (𝑋 ∈ (𝐴[,]𝐵) ∧ 𝑌 ∈ (𝐴[,]𝐵))) ∧ 𝑋 < 𝑌) → 𝐴 ∈ ℝ) |
34 | 33 | rexrd 9968 |
. . . . . . . . 9
⊢ (((𝜑 ∧ (𝑋 ∈ (𝐴[,]𝐵) ∧ 𝑌 ∈ (𝐴[,]𝐵))) ∧ 𝑋 < 𝑌) → 𝐴 ∈
ℝ*) |
35 | 7 | ad2antrr 758 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ (𝑋 ∈ (𝐴[,]𝐵) ∧ 𝑌 ∈ (𝐴[,]𝐵))) ∧ 𝑋 < 𝑌) → 𝐵 ∈ ℝ) |
36 | | elicc2 12109 |
. . . . . . . . . . . 12
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝑋 ∈ (𝐴[,]𝐵) ↔ (𝑋 ∈ ℝ ∧ 𝐴 ≤ 𝑋 ∧ 𝑋 ≤ 𝐵))) |
37 | 33, 35, 36 | syl2anc 691 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ (𝑋 ∈ (𝐴[,]𝐵) ∧ 𝑌 ∈ (𝐴[,]𝐵))) ∧ 𝑋 < 𝑌) → (𝑋 ∈ (𝐴[,]𝐵) ↔ (𝑋 ∈ ℝ ∧ 𝐴 ≤ 𝑋 ∧ 𝑋 ≤ 𝐵))) |
38 | 2, 37 | mpbid 221 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ (𝑋 ∈ (𝐴[,]𝐵) ∧ 𝑌 ∈ (𝐴[,]𝐵))) ∧ 𝑋 < 𝑌) → (𝑋 ∈ ℝ ∧ 𝐴 ≤ 𝑋 ∧ 𝑋 ≤ 𝐵)) |
39 | 38 | simp2d 1067 |
. . . . . . . . 9
⊢ (((𝜑 ∧ (𝑋 ∈ (𝐴[,]𝐵) ∧ 𝑌 ∈ (𝐴[,]𝐵))) ∧ 𝑋 < 𝑌) → 𝐴 ≤ 𝑋) |
40 | | iooss1 12081 |
. . . . . . . . 9
⊢ ((𝐴 ∈ ℝ*
∧ 𝐴 ≤ 𝑋) → (𝑋(,)𝑌) ⊆ (𝐴(,)𝑌)) |
41 | 34, 39, 40 | syl2anc 691 |
. . . . . . . 8
⊢ (((𝜑 ∧ (𝑋 ∈ (𝐴[,]𝐵) ∧ 𝑌 ∈ (𝐴[,]𝐵))) ∧ 𝑋 < 𝑌) → (𝑋(,)𝑌) ⊆ (𝐴(,)𝑌)) |
42 | 35 | rexrd 9968 |
. . . . . . . . 9
⊢ (((𝜑 ∧ (𝑋 ∈ (𝐴[,]𝐵) ∧ 𝑌 ∈ (𝐴[,]𝐵))) ∧ 𝑋 < 𝑌) → 𝐵 ∈
ℝ*) |
43 | | elicc2 12109 |
. . . . . . . . . . . 12
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝑌 ∈ (𝐴[,]𝐵) ↔ (𝑌 ∈ ℝ ∧ 𝐴 ≤ 𝑌 ∧ 𝑌 ≤ 𝐵))) |
44 | 33, 35, 43 | syl2anc 691 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ (𝑋 ∈ (𝐴[,]𝐵) ∧ 𝑌 ∈ (𝐴[,]𝐵))) ∧ 𝑋 < 𝑌) → (𝑌 ∈ (𝐴[,]𝐵) ↔ (𝑌 ∈ ℝ ∧ 𝐴 ≤ 𝑌 ∧ 𝑌 ≤ 𝐵))) |
45 | 4, 44 | mpbid 221 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ (𝑋 ∈ (𝐴[,]𝐵) ∧ 𝑌 ∈ (𝐴[,]𝐵))) ∧ 𝑋 < 𝑌) → (𝑌 ∈ ℝ ∧ 𝐴 ≤ 𝑌 ∧ 𝑌 ≤ 𝐵)) |
46 | 45 | simp3d 1068 |
. . . . . . . . 9
⊢ (((𝜑 ∧ (𝑋 ∈ (𝐴[,]𝐵) ∧ 𝑌 ∈ (𝐴[,]𝐵))) ∧ 𝑋 < 𝑌) → 𝑌 ≤ 𝐵) |
47 | | iooss2 12082 |
. . . . . . . . 9
⊢ ((𝐵 ∈ ℝ*
∧ 𝑌 ≤ 𝐵) → (𝐴(,)𝑌) ⊆ (𝐴(,)𝐵)) |
48 | 42, 46, 47 | syl2anc 691 |
. . . . . . . 8
⊢ (((𝜑 ∧ (𝑋 ∈ (𝐴[,]𝐵) ∧ 𝑌 ∈ (𝐴[,]𝐵))) ∧ 𝑋 < 𝑌) → (𝐴(,)𝑌) ⊆ (𝐴(,)𝐵)) |
49 | 41, 48 | sstrd 3578 |
. . . . . . 7
⊢ (((𝜑 ∧ (𝑋 ∈ (𝐴[,]𝐵) ∧ 𝑌 ∈ (𝐴[,]𝐵))) ∧ 𝑋 < 𝑌) → (𝑋(,)𝑌) ⊆ (𝐴(,)𝐵)) |
50 | 32, 49 | fssresd 5984 |
. . . . . 6
⊢ (((𝜑 ∧ (𝑋 ∈ (𝐴[,]𝐵) ∧ 𝑌 ∈ (𝐴[,]𝐵))) ∧ 𝑋 < 𝑌) → ((ℝ D 𝐹) ↾ (𝑋(,)𝑌)):(𝑋(,)𝑌)⟶𝑆) |
51 | | ax-resscn 9872 |
. . . . . . . . . 10
⊢ ℝ
⊆ ℂ |
52 | 51 | a1i 11 |
. . . . . . . . 9
⊢ (((𝜑 ∧ (𝑋 ∈ (𝐴[,]𝐵) ∧ 𝑌 ∈ (𝐴[,]𝐵))) ∧ 𝑋 < 𝑌) → ℝ ⊆
ℂ) |
53 | | cncff 22504 |
. . . . . . . . . . . 12
⊢ (𝐹 ∈ ((𝐴[,]𝐵)–cn→ℝ) → 𝐹:(𝐴[,]𝐵)⟶ℝ) |
54 | 27, 53 | syl 17 |
. . . . . . . . . . 11
⊢ (𝜑 → 𝐹:(𝐴[,]𝐵)⟶ℝ) |
55 | 54 | ad2antrr 758 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ (𝑋 ∈ (𝐴[,]𝐵) ∧ 𝑌 ∈ (𝐴[,]𝐵))) ∧ 𝑋 < 𝑌) → 𝐹:(𝐴[,]𝐵)⟶ℝ) |
56 | | fss 5969 |
. . . . . . . . . 10
⊢ ((𝐹:(𝐴[,]𝐵)⟶ℝ ∧ ℝ ⊆
ℂ) → 𝐹:(𝐴[,]𝐵)⟶ℂ) |
57 | 55, 51, 56 | sylancl 693 |
. . . . . . . . 9
⊢ (((𝜑 ∧ (𝑋 ∈ (𝐴[,]𝐵) ∧ 𝑌 ∈ (𝐴[,]𝐵))) ∧ 𝑋 < 𝑌) → 𝐹:(𝐴[,]𝐵)⟶ℂ) |
58 | | iccssre 12126 |
. . . . . . . . . 10
⊢ ((𝑋 ∈ ℝ ∧ 𝑌 ∈ ℝ) → (𝑋[,]𝑌) ⊆ ℝ) |
59 | 11, 12, 58 | syl2anc 691 |
. . . . . . . . 9
⊢ (((𝜑 ∧ (𝑋 ∈ (𝐴[,]𝐵) ∧ 𝑌 ∈ (𝐴[,]𝐵))) ∧ 𝑋 < 𝑌) → (𝑋[,]𝑌) ⊆ ℝ) |
60 | | eqid 2610 |
. . . . . . . . . 10
⊢
(TopOpen‘ℂfld) =
(TopOpen‘ℂfld) |
61 | 60 | tgioo2 22414 |
. . . . . . . . . 10
⊢
(topGen‘ran (,)) = ((TopOpen‘ℂfld)
↾t ℝ) |
62 | 60, 61 | dvres 23481 |
. . . . . . . . 9
⊢
(((ℝ ⊆ ℂ ∧ 𝐹:(𝐴[,]𝐵)⟶ℂ) ∧ ((𝐴[,]𝐵) ⊆ ℝ ∧ (𝑋[,]𝑌) ⊆ ℝ)) → (ℝ D (𝐹 ↾ (𝑋[,]𝑌))) = ((ℝ D 𝐹) ↾ ((int‘(topGen‘ran
(,)))‘(𝑋[,]𝑌)))) |
63 | 52, 57, 10, 59, 62 | syl22anc 1319 |
. . . . . . . 8
⊢ (((𝜑 ∧ (𝑋 ∈ (𝐴[,]𝐵) ∧ 𝑌 ∈ (𝐴[,]𝐵))) ∧ 𝑋 < 𝑌) → (ℝ D (𝐹 ↾ (𝑋[,]𝑌))) = ((ℝ D 𝐹) ↾ ((int‘(topGen‘ran
(,)))‘(𝑋[,]𝑌)))) |
64 | | iccntr 22432 |
. . . . . . . . . 10
⊢ ((𝑋 ∈ ℝ ∧ 𝑌 ∈ ℝ) →
((int‘(topGen‘ran (,)))‘(𝑋[,]𝑌)) = (𝑋(,)𝑌)) |
65 | 11, 12, 64 | syl2anc 691 |
. . . . . . . . 9
⊢ (((𝜑 ∧ (𝑋 ∈ (𝐴[,]𝐵) ∧ 𝑌 ∈ (𝐴[,]𝐵))) ∧ 𝑋 < 𝑌) → ((int‘(topGen‘ran
(,)))‘(𝑋[,]𝑌)) = (𝑋(,)𝑌)) |
66 | 65 | reseq2d 5317 |
. . . . . . . 8
⊢ (((𝜑 ∧ (𝑋 ∈ (𝐴[,]𝐵) ∧ 𝑌 ∈ (𝐴[,]𝐵))) ∧ 𝑋 < 𝑌) → ((ℝ D 𝐹) ↾ ((int‘(topGen‘ran
(,)))‘(𝑋[,]𝑌))) = ((ℝ D 𝐹) ↾ (𝑋(,)𝑌))) |
67 | 63, 66 | eqtrd 2644 |
. . . . . . 7
⊢ (((𝜑 ∧ (𝑋 ∈ (𝐴[,]𝐵) ∧ 𝑌 ∈ (𝐴[,]𝐵))) ∧ 𝑋 < 𝑌) → (ℝ D (𝐹 ↾ (𝑋[,]𝑌))) = ((ℝ D 𝐹) ↾ (𝑋(,)𝑌))) |
68 | 67 | feq1d 5943 |
. . . . . 6
⊢ (((𝜑 ∧ (𝑋 ∈ (𝐴[,]𝐵) ∧ 𝑌 ∈ (𝐴[,]𝐵))) ∧ 𝑋 < 𝑌) → ((ℝ D (𝐹 ↾ (𝑋[,]𝑌))):(𝑋(,)𝑌)⟶𝑆 ↔ ((ℝ D 𝐹) ↾ (𝑋(,)𝑌)):(𝑋(,)𝑌)⟶𝑆)) |
69 | 50, 68 | mpbird 246 |
. . . . 5
⊢ (((𝜑 ∧ (𝑋 ∈ (𝐴[,]𝐵) ∧ 𝑌 ∈ (𝐴[,]𝐵))) ∧ 𝑋 < 𝑌) → (ℝ D (𝐹 ↾ (𝑋[,]𝑌))):(𝑋(,)𝑌)⟶𝑆) |
70 | | fdm 5964 |
. . . . 5
⊢ ((ℝ
D (𝐹 ↾ (𝑋[,]𝑌))):(𝑋(,)𝑌)⟶𝑆 → dom (ℝ D (𝐹 ↾ (𝑋[,]𝑌))) = (𝑋(,)𝑌)) |
71 | 69, 70 | syl 17 |
. . . 4
⊢ (((𝜑 ∧ (𝑋 ∈ (𝐴[,]𝐵) ∧ 𝑌 ∈ (𝐴[,]𝐵))) ∧ 𝑋 < 𝑌) → dom (ℝ D (𝐹 ↾ (𝑋[,]𝑌))) = (𝑋(,)𝑌)) |
72 | 11, 12, 13, 30, 71 | mvth 23559 |
. . 3
⊢ (((𝜑 ∧ (𝑋 ∈ (𝐴[,]𝐵) ∧ 𝑌 ∈ (𝐴[,]𝐵))) ∧ 𝑋 < 𝑌) → ∃𝑧 ∈ (𝑋(,)𝑌)((ℝ D (𝐹 ↾ (𝑋[,]𝑌)))‘𝑧) = ((((𝐹 ↾ (𝑋[,]𝑌))‘𝑌) − ((𝐹 ↾ (𝑋[,]𝑌))‘𝑋)) / (𝑌 − 𝑋))) |
73 | 69 | ffvelrnda 6267 |
. . . . 5
⊢ ((((𝜑 ∧ (𝑋 ∈ (𝐴[,]𝐵) ∧ 𝑌 ∈ (𝐴[,]𝐵))) ∧ 𝑋 < 𝑌) ∧ 𝑧 ∈ (𝑋(,)𝑌)) → ((ℝ D (𝐹 ↾ (𝑋[,]𝑌)))‘𝑧) ∈ 𝑆) |
74 | | eleq1 2676 |
. . . . 5
⊢
(((ℝ D (𝐹
↾ (𝑋[,]𝑌)))‘𝑧) = ((((𝐹 ↾ (𝑋[,]𝑌))‘𝑌) − ((𝐹 ↾ (𝑋[,]𝑌))‘𝑋)) / (𝑌 − 𝑋)) → (((ℝ D (𝐹 ↾ (𝑋[,]𝑌)))‘𝑧) ∈ 𝑆 ↔ ((((𝐹 ↾ (𝑋[,]𝑌))‘𝑌) − ((𝐹 ↾ (𝑋[,]𝑌))‘𝑋)) / (𝑌 − 𝑋)) ∈ 𝑆)) |
75 | 73, 74 | syl5ibcom 234 |
. . . 4
⊢ ((((𝜑 ∧ (𝑋 ∈ (𝐴[,]𝐵) ∧ 𝑌 ∈ (𝐴[,]𝐵))) ∧ 𝑋 < 𝑌) ∧ 𝑧 ∈ (𝑋(,)𝑌)) → (((ℝ D (𝐹 ↾ (𝑋[,]𝑌)))‘𝑧) = ((((𝐹 ↾ (𝑋[,]𝑌))‘𝑌) − ((𝐹 ↾ (𝑋[,]𝑌))‘𝑋)) / (𝑌 − 𝑋)) → ((((𝐹 ↾ (𝑋[,]𝑌))‘𝑌) − ((𝐹 ↾ (𝑋[,]𝑌))‘𝑋)) / (𝑌 − 𝑋)) ∈ 𝑆)) |
76 | 75 | rexlimdva 3013 |
. . 3
⊢ (((𝜑 ∧ (𝑋 ∈ (𝐴[,]𝐵) ∧ 𝑌 ∈ (𝐴[,]𝐵))) ∧ 𝑋 < 𝑌) → (∃𝑧 ∈ (𝑋(,)𝑌)((ℝ D (𝐹 ↾ (𝑋[,]𝑌)))‘𝑧) = ((((𝐹 ↾ (𝑋[,]𝑌))‘𝑌) − ((𝐹 ↾ (𝑋[,]𝑌))‘𝑋)) / (𝑌 − 𝑋)) → ((((𝐹 ↾ (𝑋[,]𝑌))‘𝑌) − ((𝐹 ↾ (𝑋[,]𝑌))‘𝑋)) / (𝑌 − 𝑋)) ∈ 𝑆)) |
77 | 72, 76 | mpd 15 |
. 2
⊢ (((𝜑 ∧ (𝑋 ∈ (𝐴[,]𝐵) ∧ 𝑌 ∈ (𝐴[,]𝐵))) ∧ 𝑋 < 𝑌) → ((((𝐹 ↾ (𝑋[,]𝑌))‘𝑌) − ((𝐹 ↾ (𝑋[,]𝑌))‘𝑋)) / (𝑌 − 𝑋)) ∈ 𝑆) |
78 | 24, 77 | eqeltrrd 2689 |
1
⊢ (((𝜑 ∧ (𝑋 ∈ (𝐴[,]𝐵) ∧ 𝑌 ∈ (𝐴[,]𝐵))) ∧ 𝑋 < 𝑌) → (((𝐹‘𝑌) − (𝐹‘𝑋)) / (𝑌 − 𝑋)) ∈ 𝑆) |