Step |
Hyp |
Ref |
Expression |
1 |
|
mvth.a |
⊢ ( 𝜑 → 𝐴 ∈ ℝ ) |
2 |
|
mvth.b |
⊢ ( 𝜑 → 𝐵 ∈ ℝ ) |
3 |
|
mvth.lt |
⊢ ( 𝜑 → 𝐴 < 𝐵 ) |
4 |
|
mvth.f |
⊢ ( 𝜑 → 𝐹 ∈ ( ( 𝐴 [,] 𝐵 ) –cn→ ℝ ) ) |
5 |
|
mvth.d |
⊢ ( 𝜑 → dom ( ℝ D 𝐹 ) = ( 𝐴 (,) 𝐵 ) ) |
6 |
|
mptresid |
⊢ ( I ↾ ( 𝐴 [,] 𝐵 ) ) = ( 𝑧 ∈ ( 𝐴 [,] 𝐵 ) ↦ 𝑧 ) |
7 |
|
iccssre |
⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ) → ( 𝐴 [,] 𝐵 ) ⊆ ℝ ) |
8 |
1 2 7
|
syl2anc |
⊢ ( 𝜑 → ( 𝐴 [,] 𝐵 ) ⊆ ℝ ) |
9 |
|
ax-resscn |
⊢ ℝ ⊆ ℂ |
10 |
|
cncfmptid |
⊢ ( ( ( 𝐴 [,] 𝐵 ) ⊆ ℝ ∧ ℝ ⊆ ℂ ) → ( 𝑧 ∈ ( 𝐴 [,] 𝐵 ) ↦ 𝑧 ) ∈ ( ( 𝐴 [,] 𝐵 ) –cn→ ℝ ) ) |
11 |
8 9 10
|
sylancl |
⊢ ( 𝜑 → ( 𝑧 ∈ ( 𝐴 [,] 𝐵 ) ↦ 𝑧 ) ∈ ( ( 𝐴 [,] 𝐵 ) –cn→ ℝ ) ) |
12 |
6 11
|
eqeltrid |
⊢ ( 𝜑 → ( I ↾ ( 𝐴 [,] 𝐵 ) ) ∈ ( ( 𝐴 [,] 𝐵 ) –cn→ ℝ ) ) |
13 |
6
|
eqcomi |
⊢ ( 𝑧 ∈ ( 𝐴 [,] 𝐵 ) ↦ 𝑧 ) = ( I ↾ ( 𝐴 [,] 𝐵 ) ) |
14 |
13
|
oveq2i |
⊢ ( ℝ D ( 𝑧 ∈ ( 𝐴 [,] 𝐵 ) ↦ 𝑧 ) ) = ( ℝ D ( I ↾ ( 𝐴 [,] 𝐵 ) ) ) |
15 |
|
reelprrecn |
⊢ ℝ ∈ { ℝ , ℂ } |
16 |
15
|
a1i |
⊢ ( 𝜑 → ℝ ∈ { ℝ , ℂ } ) |
17 |
|
simpr |
⊢ ( ( 𝜑 ∧ 𝑧 ∈ ℝ ) → 𝑧 ∈ ℝ ) |
18 |
17
|
recnd |
⊢ ( ( 𝜑 ∧ 𝑧 ∈ ℝ ) → 𝑧 ∈ ℂ ) |
19 |
|
1red |
⊢ ( ( 𝜑 ∧ 𝑧 ∈ ℝ ) → 1 ∈ ℝ ) |
20 |
16
|
dvmptid |
⊢ ( 𝜑 → ( ℝ D ( 𝑧 ∈ ℝ ↦ 𝑧 ) ) = ( 𝑧 ∈ ℝ ↦ 1 ) ) |
21 |
|
eqid |
⊢ ( TopOpen ‘ ℂfld ) = ( TopOpen ‘ ℂfld ) |
22 |
21
|
tgioo2 |
⊢ ( topGen ‘ ran (,) ) = ( ( TopOpen ‘ ℂfld ) ↾t ℝ ) |
23 |
|
iccntr |
⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ) → ( ( int ‘ ( topGen ‘ ran (,) ) ) ‘ ( 𝐴 [,] 𝐵 ) ) = ( 𝐴 (,) 𝐵 ) ) |
24 |
1 2 23
|
syl2anc |
⊢ ( 𝜑 → ( ( int ‘ ( topGen ‘ ran (,) ) ) ‘ ( 𝐴 [,] 𝐵 ) ) = ( 𝐴 (,) 𝐵 ) ) |
25 |
16 18 19 20 8 22 21 24
|
dvmptres2 |
⊢ ( 𝜑 → ( ℝ D ( 𝑧 ∈ ( 𝐴 [,] 𝐵 ) ↦ 𝑧 ) ) = ( 𝑧 ∈ ( 𝐴 (,) 𝐵 ) ↦ 1 ) ) |
26 |
14 25
|
syl5eqr |
⊢ ( 𝜑 → ( ℝ D ( I ↾ ( 𝐴 [,] 𝐵 ) ) ) = ( 𝑧 ∈ ( 𝐴 (,) 𝐵 ) ↦ 1 ) ) |
27 |
26
|
dmeqd |
⊢ ( 𝜑 → dom ( ℝ D ( I ↾ ( 𝐴 [,] 𝐵 ) ) ) = dom ( 𝑧 ∈ ( 𝐴 (,) 𝐵 ) ↦ 1 ) ) |
28 |
|
1ex |
⊢ 1 ∈ V |
29 |
|
eqid |
⊢ ( 𝑧 ∈ ( 𝐴 (,) 𝐵 ) ↦ 1 ) = ( 𝑧 ∈ ( 𝐴 (,) 𝐵 ) ↦ 1 ) |
30 |
28 29
|
dmmpti |
⊢ dom ( 𝑧 ∈ ( 𝐴 (,) 𝐵 ) ↦ 1 ) = ( 𝐴 (,) 𝐵 ) |
31 |
27 30
|
eqtrdi |
⊢ ( 𝜑 → dom ( ℝ D ( I ↾ ( 𝐴 [,] 𝐵 ) ) ) = ( 𝐴 (,) 𝐵 ) ) |
32 |
1 2 3 4 12 5 31
|
cmvth |
⊢ ( 𝜑 → ∃ 𝑥 ∈ ( 𝐴 (,) 𝐵 ) ( ( ( 𝐹 ‘ 𝐵 ) − ( 𝐹 ‘ 𝐴 ) ) · ( ( ℝ D ( I ↾ ( 𝐴 [,] 𝐵 ) ) ) ‘ 𝑥 ) ) = ( ( ( ( I ↾ ( 𝐴 [,] 𝐵 ) ) ‘ 𝐵 ) − ( ( I ↾ ( 𝐴 [,] 𝐵 ) ) ‘ 𝐴 ) ) · ( ( ℝ D 𝐹 ) ‘ 𝑥 ) ) ) |
33 |
1
|
rexrd |
⊢ ( 𝜑 → 𝐴 ∈ ℝ* ) |
34 |
2
|
rexrd |
⊢ ( 𝜑 → 𝐵 ∈ ℝ* ) |
35 |
1 2 3
|
ltled |
⊢ ( 𝜑 → 𝐴 ≤ 𝐵 ) |
36 |
|
ubicc2 |
⊢ ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐴 ≤ 𝐵 ) → 𝐵 ∈ ( 𝐴 [,] 𝐵 ) ) |
37 |
33 34 35 36
|
syl3anc |
⊢ ( 𝜑 → 𝐵 ∈ ( 𝐴 [,] 𝐵 ) ) |
38 |
|
fvresi |
⊢ ( 𝐵 ∈ ( 𝐴 [,] 𝐵 ) → ( ( I ↾ ( 𝐴 [,] 𝐵 ) ) ‘ 𝐵 ) = 𝐵 ) |
39 |
37 38
|
syl |
⊢ ( 𝜑 → ( ( I ↾ ( 𝐴 [,] 𝐵 ) ) ‘ 𝐵 ) = 𝐵 ) |
40 |
|
lbicc2 |
⊢ ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐴 ≤ 𝐵 ) → 𝐴 ∈ ( 𝐴 [,] 𝐵 ) ) |
41 |
33 34 35 40
|
syl3anc |
⊢ ( 𝜑 → 𝐴 ∈ ( 𝐴 [,] 𝐵 ) ) |
42 |
|
fvresi |
⊢ ( 𝐴 ∈ ( 𝐴 [,] 𝐵 ) → ( ( I ↾ ( 𝐴 [,] 𝐵 ) ) ‘ 𝐴 ) = 𝐴 ) |
43 |
41 42
|
syl |
⊢ ( 𝜑 → ( ( I ↾ ( 𝐴 [,] 𝐵 ) ) ‘ 𝐴 ) = 𝐴 ) |
44 |
39 43
|
oveq12d |
⊢ ( 𝜑 → ( ( ( I ↾ ( 𝐴 [,] 𝐵 ) ) ‘ 𝐵 ) − ( ( I ↾ ( 𝐴 [,] 𝐵 ) ) ‘ 𝐴 ) ) = ( 𝐵 − 𝐴 ) ) |
45 |
44
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ ( 𝐴 (,) 𝐵 ) ) → ( ( ( I ↾ ( 𝐴 [,] 𝐵 ) ) ‘ 𝐵 ) − ( ( I ↾ ( 𝐴 [,] 𝐵 ) ) ‘ 𝐴 ) ) = ( 𝐵 − 𝐴 ) ) |
46 |
45
|
oveq1d |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ ( 𝐴 (,) 𝐵 ) ) → ( ( ( ( I ↾ ( 𝐴 [,] 𝐵 ) ) ‘ 𝐵 ) − ( ( I ↾ ( 𝐴 [,] 𝐵 ) ) ‘ 𝐴 ) ) · ( ( ℝ D 𝐹 ) ‘ 𝑥 ) ) = ( ( 𝐵 − 𝐴 ) · ( ( ℝ D 𝐹 ) ‘ 𝑥 ) ) ) |
47 |
26
|
fveq1d |
⊢ ( 𝜑 → ( ( ℝ D ( I ↾ ( 𝐴 [,] 𝐵 ) ) ) ‘ 𝑥 ) = ( ( 𝑧 ∈ ( 𝐴 (,) 𝐵 ) ↦ 1 ) ‘ 𝑥 ) ) |
48 |
|
eqidd |
⊢ ( 𝑧 = 𝑥 → 1 = 1 ) |
49 |
48 29 28
|
fvmpt3i |
⊢ ( 𝑥 ∈ ( 𝐴 (,) 𝐵 ) → ( ( 𝑧 ∈ ( 𝐴 (,) 𝐵 ) ↦ 1 ) ‘ 𝑥 ) = 1 ) |
50 |
47 49
|
sylan9eq |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ ( 𝐴 (,) 𝐵 ) ) → ( ( ℝ D ( I ↾ ( 𝐴 [,] 𝐵 ) ) ) ‘ 𝑥 ) = 1 ) |
51 |
50
|
oveq2d |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ ( 𝐴 (,) 𝐵 ) ) → ( ( ( 𝐹 ‘ 𝐵 ) − ( 𝐹 ‘ 𝐴 ) ) · ( ( ℝ D ( I ↾ ( 𝐴 [,] 𝐵 ) ) ) ‘ 𝑥 ) ) = ( ( ( 𝐹 ‘ 𝐵 ) − ( 𝐹 ‘ 𝐴 ) ) · 1 ) ) |
52 |
|
cncff |
⊢ ( 𝐹 ∈ ( ( 𝐴 [,] 𝐵 ) –cn→ ℝ ) → 𝐹 : ( 𝐴 [,] 𝐵 ) ⟶ ℝ ) |
53 |
4 52
|
syl |
⊢ ( 𝜑 → 𝐹 : ( 𝐴 [,] 𝐵 ) ⟶ ℝ ) |
54 |
53 37
|
ffvelrnd |
⊢ ( 𝜑 → ( 𝐹 ‘ 𝐵 ) ∈ ℝ ) |
55 |
53 41
|
ffvelrnd |
⊢ ( 𝜑 → ( 𝐹 ‘ 𝐴 ) ∈ ℝ ) |
56 |
54 55
|
resubcld |
⊢ ( 𝜑 → ( ( 𝐹 ‘ 𝐵 ) − ( 𝐹 ‘ 𝐴 ) ) ∈ ℝ ) |
57 |
56
|
recnd |
⊢ ( 𝜑 → ( ( 𝐹 ‘ 𝐵 ) − ( 𝐹 ‘ 𝐴 ) ) ∈ ℂ ) |
58 |
57
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ ( 𝐴 (,) 𝐵 ) ) → ( ( 𝐹 ‘ 𝐵 ) − ( 𝐹 ‘ 𝐴 ) ) ∈ ℂ ) |
59 |
58
|
mulid1d |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ ( 𝐴 (,) 𝐵 ) ) → ( ( ( 𝐹 ‘ 𝐵 ) − ( 𝐹 ‘ 𝐴 ) ) · 1 ) = ( ( 𝐹 ‘ 𝐵 ) − ( 𝐹 ‘ 𝐴 ) ) ) |
60 |
51 59
|
eqtrd |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ ( 𝐴 (,) 𝐵 ) ) → ( ( ( 𝐹 ‘ 𝐵 ) − ( 𝐹 ‘ 𝐴 ) ) · ( ( ℝ D ( I ↾ ( 𝐴 [,] 𝐵 ) ) ) ‘ 𝑥 ) ) = ( ( 𝐹 ‘ 𝐵 ) − ( 𝐹 ‘ 𝐴 ) ) ) |
61 |
46 60
|
eqeq12d |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ ( 𝐴 (,) 𝐵 ) ) → ( ( ( ( ( I ↾ ( 𝐴 [,] 𝐵 ) ) ‘ 𝐵 ) − ( ( I ↾ ( 𝐴 [,] 𝐵 ) ) ‘ 𝐴 ) ) · ( ( ℝ D 𝐹 ) ‘ 𝑥 ) ) = ( ( ( 𝐹 ‘ 𝐵 ) − ( 𝐹 ‘ 𝐴 ) ) · ( ( ℝ D ( I ↾ ( 𝐴 [,] 𝐵 ) ) ) ‘ 𝑥 ) ) ↔ ( ( 𝐵 − 𝐴 ) · ( ( ℝ D 𝐹 ) ‘ 𝑥 ) ) = ( ( 𝐹 ‘ 𝐵 ) − ( 𝐹 ‘ 𝐴 ) ) ) ) |
62 |
2 1
|
resubcld |
⊢ ( 𝜑 → ( 𝐵 − 𝐴 ) ∈ ℝ ) |
63 |
62
|
recnd |
⊢ ( 𝜑 → ( 𝐵 − 𝐴 ) ∈ ℂ ) |
64 |
63
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ ( 𝐴 (,) 𝐵 ) ) → ( 𝐵 − 𝐴 ) ∈ ℂ ) |
65 |
|
dvf |
⊢ ( ℝ D 𝐹 ) : dom ( ℝ D 𝐹 ) ⟶ ℂ |
66 |
5
|
feq2d |
⊢ ( 𝜑 → ( ( ℝ D 𝐹 ) : dom ( ℝ D 𝐹 ) ⟶ ℂ ↔ ( ℝ D 𝐹 ) : ( 𝐴 (,) 𝐵 ) ⟶ ℂ ) ) |
67 |
65 66
|
mpbii |
⊢ ( 𝜑 → ( ℝ D 𝐹 ) : ( 𝐴 (,) 𝐵 ) ⟶ ℂ ) |
68 |
67
|
ffvelrnda |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ ( 𝐴 (,) 𝐵 ) ) → ( ( ℝ D 𝐹 ) ‘ 𝑥 ) ∈ ℂ ) |
69 |
1 2
|
posdifd |
⊢ ( 𝜑 → ( 𝐴 < 𝐵 ↔ 0 < ( 𝐵 − 𝐴 ) ) ) |
70 |
3 69
|
mpbid |
⊢ ( 𝜑 → 0 < ( 𝐵 − 𝐴 ) ) |
71 |
70
|
gt0ne0d |
⊢ ( 𝜑 → ( 𝐵 − 𝐴 ) ≠ 0 ) |
72 |
71
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ ( 𝐴 (,) 𝐵 ) ) → ( 𝐵 − 𝐴 ) ≠ 0 ) |
73 |
58 64 68 72
|
divmuld |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ ( 𝐴 (,) 𝐵 ) ) → ( ( ( ( 𝐹 ‘ 𝐵 ) − ( 𝐹 ‘ 𝐴 ) ) / ( 𝐵 − 𝐴 ) ) = ( ( ℝ D 𝐹 ) ‘ 𝑥 ) ↔ ( ( 𝐵 − 𝐴 ) · ( ( ℝ D 𝐹 ) ‘ 𝑥 ) ) = ( ( 𝐹 ‘ 𝐵 ) − ( 𝐹 ‘ 𝐴 ) ) ) ) |
74 |
61 73
|
bitr4d |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ ( 𝐴 (,) 𝐵 ) ) → ( ( ( ( ( I ↾ ( 𝐴 [,] 𝐵 ) ) ‘ 𝐵 ) − ( ( I ↾ ( 𝐴 [,] 𝐵 ) ) ‘ 𝐴 ) ) · ( ( ℝ D 𝐹 ) ‘ 𝑥 ) ) = ( ( ( 𝐹 ‘ 𝐵 ) − ( 𝐹 ‘ 𝐴 ) ) · ( ( ℝ D ( I ↾ ( 𝐴 [,] 𝐵 ) ) ) ‘ 𝑥 ) ) ↔ ( ( ( 𝐹 ‘ 𝐵 ) − ( 𝐹 ‘ 𝐴 ) ) / ( 𝐵 − 𝐴 ) ) = ( ( ℝ D 𝐹 ) ‘ 𝑥 ) ) ) |
75 |
|
eqcom |
⊢ ( ( ( ( 𝐹 ‘ 𝐵 ) − ( 𝐹 ‘ 𝐴 ) ) · ( ( ℝ D ( I ↾ ( 𝐴 [,] 𝐵 ) ) ) ‘ 𝑥 ) ) = ( ( ( ( I ↾ ( 𝐴 [,] 𝐵 ) ) ‘ 𝐵 ) − ( ( I ↾ ( 𝐴 [,] 𝐵 ) ) ‘ 𝐴 ) ) · ( ( ℝ D 𝐹 ) ‘ 𝑥 ) ) ↔ ( ( ( ( I ↾ ( 𝐴 [,] 𝐵 ) ) ‘ 𝐵 ) − ( ( I ↾ ( 𝐴 [,] 𝐵 ) ) ‘ 𝐴 ) ) · ( ( ℝ D 𝐹 ) ‘ 𝑥 ) ) = ( ( ( 𝐹 ‘ 𝐵 ) − ( 𝐹 ‘ 𝐴 ) ) · ( ( ℝ D ( I ↾ ( 𝐴 [,] 𝐵 ) ) ) ‘ 𝑥 ) ) ) |
76 |
|
eqcom |
⊢ ( ( ( ℝ D 𝐹 ) ‘ 𝑥 ) = ( ( ( 𝐹 ‘ 𝐵 ) − ( 𝐹 ‘ 𝐴 ) ) / ( 𝐵 − 𝐴 ) ) ↔ ( ( ( 𝐹 ‘ 𝐵 ) − ( 𝐹 ‘ 𝐴 ) ) / ( 𝐵 − 𝐴 ) ) = ( ( ℝ D 𝐹 ) ‘ 𝑥 ) ) |
77 |
74 75 76
|
3bitr4g |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ ( 𝐴 (,) 𝐵 ) ) → ( ( ( ( 𝐹 ‘ 𝐵 ) − ( 𝐹 ‘ 𝐴 ) ) · ( ( ℝ D ( I ↾ ( 𝐴 [,] 𝐵 ) ) ) ‘ 𝑥 ) ) = ( ( ( ( I ↾ ( 𝐴 [,] 𝐵 ) ) ‘ 𝐵 ) − ( ( I ↾ ( 𝐴 [,] 𝐵 ) ) ‘ 𝐴 ) ) · ( ( ℝ D 𝐹 ) ‘ 𝑥 ) ) ↔ ( ( ℝ D 𝐹 ) ‘ 𝑥 ) = ( ( ( 𝐹 ‘ 𝐵 ) − ( 𝐹 ‘ 𝐴 ) ) / ( 𝐵 − 𝐴 ) ) ) ) |
78 |
77
|
rexbidva |
⊢ ( 𝜑 → ( ∃ 𝑥 ∈ ( 𝐴 (,) 𝐵 ) ( ( ( 𝐹 ‘ 𝐵 ) − ( 𝐹 ‘ 𝐴 ) ) · ( ( ℝ D ( I ↾ ( 𝐴 [,] 𝐵 ) ) ) ‘ 𝑥 ) ) = ( ( ( ( I ↾ ( 𝐴 [,] 𝐵 ) ) ‘ 𝐵 ) − ( ( I ↾ ( 𝐴 [,] 𝐵 ) ) ‘ 𝐴 ) ) · ( ( ℝ D 𝐹 ) ‘ 𝑥 ) ) ↔ ∃ 𝑥 ∈ ( 𝐴 (,) 𝐵 ) ( ( ℝ D 𝐹 ) ‘ 𝑥 ) = ( ( ( 𝐹 ‘ 𝐵 ) − ( 𝐹 ‘ 𝐴 ) ) / ( 𝐵 − 𝐴 ) ) ) ) |
79 |
32 78
|
mpbid |
⊢ ( 𝜑 → ∃ 𝑥 ∈ ( 𝐴 (,) 𝐵 ) ( ( ℝ D 𝐹 ) ‘ 𝑥 ) = ( ( ( 𝐹 ‘ 𝐵 ) − ( 𝐹 ‘ 𝐴 ) ) / ( 𝐵 − 𝐴 ) ) ) |