Step |
Hyp |
Ref |
Expression |
1 |
|
dvrelog2.1 |
⊢ ( 𝜑 → 𝐴 ∈ ℝ ) |
2 |
|
dvrelog2.2 |
⊢ ( 𝜑 → 𝐵 ∈ ℝ ) |
3 |
|
dvrelog2.3 |
⊢ ( 𝜑 → 0 < 𝐴 ) |
4 |
|
dvrelog2.4 |
⊢ ( 𝜑 → 𝐴 ≤ 𝐵 ) |
5 |
|
dvrelog2.5 |
⊢ 𝐹 = ( 𝑥 ∈ ( 𝐴 [,] 𝐵 ) ↦ ( log ‘ 𝑥 ) ) |
6 |
|
dvrelog2.6 |
⊢ 𝐺 = ( 𝑥 ∈ ( 𝐴 (,) 𝐵 ) ↦ ( 1 / 𝑥 ) ) |
7 |
5
|
a1i |
⊢ ( 𝜑 → 𝐹 = ( 𝑥 ∈ ( 𝐴 [,] 𝐵 ) ↦ ( log ‘ 𝑥 ) ) ) |
8 |
7
|
oveq2d |
⊢ ( 𝜑 → ( ℝ D 𝐹 ) = ( ℝ D ( 𝑥 ∈ ( 𝐴 [,] 𝐵 ) ↦ ( log ‘ 𝑥 ) ) ) ) |
9 |
|
reelprrecn |
⊢ ℝ ∈ { ℝ , ℂ } |
10 |
9
|
a1i |
⊢ ( 𝜑 → ℝ ∈ { ℝ , ℂ } ) |
11 |
|
rpssre |
⊢ ℝ+ ⊆ ℝ |
12 |
|
ax-resscn |
⊢ ℝ ⊆ ℂ |
13 |
11 12
|
sstri |
⊢ ℝ+ ⊆ ℂ |
14 |
13
|
sseli |
⊢ ( 𝑥 ∈ ℝ+ → 𝑥 ∈ ℂ ) |
15 |
14
|
adantl |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) → 𝑥 ∈ ℂ ) |
16 |
|
rpne0 |
⊢ ( 𝑥 ∈ ℝ+ → 𝑥 ≠ 0 ) |
17 |
16
|
adantl |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) → 𝑥 ≠ 0 ) |
18 |
15 17
|
logcld |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) → ( log ‘ 𝑥 ) ∈ ℂ ) |
19 |
|
1red |
⊢ ( 𝑥 ∈ ℝ+ → 1 ∈ ℝ ) |
20 |
11
|
sseli |
⊢ ( 𝑥 ∈ ℝ+ → 𝑥 ∈ ℝ ) |
21 |
19 20 16
|
redivcld |
⊢ ( 𝑥 ∈ ℝ+ → ( 1 / 𝑥 ) ∈ ℝ ) |
22 |
21
|
adantl |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) → ( 1 / 𝑥 ) ∈ ℝ ) |
23 |
|
logf1o |
⊢ log : ( ℂ ∖ { 0 } ) –1-1-onto→ ran log |
24 |
|
f1of |
⊢ ( log : ( ℂ ∖ { 0 } ) –1-1-onto→ ran log → log : ( ℂ ∖ { 0 } ) ⟶ ran log ) |
25 |
23 24
|
ax-mp |
⊢ log : ( ℂ ∖ { 0 } ) ⟶ ran log |
26 |
25
|
a1i |
⊢ ( 𝜑 → log : ( ℂ ∖ { 0 } ) ⟶ ran log ) |
27 |
|
0nrp |
⊢ ¬ 0 ∈ ℝ+ |
28 |
|
disjsn |
⊢ ( ( ℝ+ ∩ { 0 } ) = ∅ ↔ ¬ 0 ∈ ℝ+ ) |
29 |
27 28
|
mpbir |
⊢ ( ℝ+ ∩ { 0 } ) = ∅ |
30 |
|
disjdif2 |
⊢ ( ( ℝ+ ∩ { 0 } ) = ∅ → ( ℝ+ ∖ { 0 } ) = ℝ+ ) |
31 |
29 30
|
ax-mp |
⊢ ( ℝ+ ∖ { 0 } ) = ℝ+ |
32 |
|
ssdif |
⊢ ( ℝ+ ⊆ ℂ → ( ℝ+ ∖ { 0 } ) ⊆ ( ℂ ∖ { 0 } ) ) |
33 |
13 32
|
ax-mp |
⊢ ( ℝ+ ∖ { 0 } ) ⊆ ( ℂ ∖ { 0 } ) |
34 |
31 33
|
eqsstrri |
⊢ ℝ+ ⊆ ( ℂ ∖ { 0 } ) |
35 |
34
|
a1i |
⊢ ( 𝜑 → ℝ+ ⊆ ( ℂ ∖ { 0 } ) ) |
36 |
26 35
|
feqresmpt |
⊢ ( 𝜑 → ( log ↾ ℝ+ ) = ( 𝑥 ∈ ℝ+ ↦ ( log ‘ 𝑥 ) ) ) |
37 |
36
|
eqcomd |
⊢ ( 𝜑 → ( 𝑥 ∈ ℝ+ ↦ ( log ‘ 𝑥 ) ) = ( log ↾ ℝ+ ) ) |
38 |
37
|
oveq2d |
⊢ ( 𝜑 → ( ℝ D ( 𝑥 ∈ ℝ+ ↦ ( log ‘ 𝑥 ) ) ) = ( ℝ D ( log ↾ ℝ+ ) ) ) |
39 |
|
dvrelog |
⊢ ( ℝ D ( log ↾ ℝ+ ) ) = ( 𝑥 ∈ ℝ+ ↦ ( 1 / 𝑥 ) ) |
40 |
39
|
a1i |
⊢ ( 𝜑 → ( ℝ D ( log ↾ ℝ+ ) ) = ( 𝑥 ∈ ℝ+ ↦ ( 1 / 𝑥 ) ) ) |
41 |
38 40
|
eqtrd |
⊢ ( 𝜑 → ( ℝ D ( 𝑥 ∈ ℝ+ ↦ ( log ‘ 𝑥 ) ) ) = ( 𝑥 ∈ ℝ+ ↦ ( 1 / 𝑥 ) ) ) |
42 |
|
elicc2 |
⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ) → ( 𝑦 ∈ ( 𝐴 [,] 𝐵 ) ↔ ( 𝑦 ∈ ℝ ∧ 𝐴 ≤ 𝑦 ∧ 𝑦 ≤ 𝐵 ) ) ) |
43 |
1 2 42
|
syl2anc |
⊢ ( 𝜑 → ( 𝑦 ∈ ( 𝐴 [,] 𝐵 ) ↔ ( 𝑦 ∈ ℝ ∧ 𝐴 ≤ 𝑦 ∧ 𝑦 ≤ 𝐵 ) ) ) |
44 |
43
|
biimpa |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 𝐴 [,] 𝐵 ) ) → ( 𝑦 ∈ ℝ ∧ 𝐴 ≤ 𝑦 ∧ 𝑦 ≤ 𝐵 ) ) |
45 |
44
|
simp1d |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 𝐴 [,] 𝐵 ) ) → 𝑦 ∈ ℝ ) |
46 |
|
0red |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 𝐴 [,] 𝐵 ) ) → 0 ∈ ℝ ) |
47 |
1
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 𝐴 [,] 𝐵 ) ) → 𝐴 ∈ ℝ ) |
48 |
3
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 𝐴 [,] 𝐵 ) ) → 0 < 𝐴 ) |
49 |
44
|
simp2d |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 𝐴 [,] 𝐵 ) ) → 𝐴 ≤ 𝑦 ) |
50 |
46 47 45 48 49
|
ltletrd |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 𝐴 [,] 𝐵 ) ) → 0 < 𝑦 ) |
51 |
45 50
|
jca |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 𝐴 [,] 𝐵 ) ) → ( 𝑦 ∈ ℝ ∧ 0 < 𝑦 ) ) |
52 |
|
elrp |
⊢ ( 𝑦 ∈ ℝ+ ↔ ( 𝑦 ∈ ℝ ∧ 0 < 𝑦 ) ) |
53 |
51 52
|
sylibr |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 𝐴 [,] 𝐵 ) ) → 𝑦 ∈ ℝ+ ) |
54 |
53
|
ex |
⊢ ( 𝜑 → ( 𝑦 ∈ ( 𝐴 [,] 𝐵 ) → 𝑦 ∈ ℝ+ ) ) |
55 |
54
|
ssrdv |
⊢ ( 𝜑 → ( 𝐴 [,] 𝐵 ) ⊆ ℝ+ ) |
56 |
|
eqid |
⊢ ( TopOpen ‘ ℂfld ) = ( TopOpen ‘ ℂfld ) |
57 |
56
|
tgioo2 |
⊢ ( topGen ‘ ran (,) ) = ( ( TopOpen ‘ ℂfld ) ↾t ℝ ) |
58 |
|
iccntr |
⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ) → ( ( int ‘ ( topGen ‘ ran (,) ) ) ‘ ( 𝐴 [,] 𝐵 ) ) = ( 𝐴 (,) 𝐵 ) ) |
59 |
1 2 58
|
syl2anc |
⊢ ( 𝜑 → ( ( int ‘ ( topGen ‘ ran (,) ) ) ‘ ( 𝐴 [,] 𝐵 ) ) = ( 𝐴 (,) 𝐵 ) ) |
60 |
10 18 22 41 55 57 56 59
|
dvmptres2 |
⊢ ( 𝜑 → ( ℝ D ( 𝑥 ∈ ( 𝐴 [,] 𝐵 ) ↦ ( log ‘ 𝑥 ) ) ) = ( 𝑥 ∈ ( 𝐴 (,) 𝐵 ) ↦ ( 1 / 𝑥 ) ) ) |
61 |
8 60
|
eqtrd |
⊢ ( 𝜑 → ( ℝ D 𝐹 ) = ( 𝑥 ∈ ( 𝐴 (,) 𝐵 ) ↦ ( 1 / 𝑥 ) ) ) |
62 |
6
|
a1i |
⊢ ( 𝜑 → 𝐺 = ( 𝑥 ∈ ( 𝐴 (,) 𝐵 ) ↦ ( 1 / 𝑥 ) ) ) |
63 |
62
|
eqcomd |
⊢ ( 𝜑 → ( 𝑥 ∈ ( 𝐴 (,) 𝐵 ) ↦ ( 1 / 𝑥 ) ) = 𝐺 ) |
64 |
61 63
|
eqtrd |
⊢ ( 𝜑 → ( ℝ D 𝐹 ) = 𝐺 ) |