| Step |
Hyp |
Ref |
Expression |
| 1 |
|
dvne0.a |
⊢ ( 𝜑 → 𝐴 ∈ ℝ ) |
| 2 |
|
dvne0.b |
⊢ ( 𝜑 → 𝐵 ∈ ℝ ) |
| 3 |
|
dvne0.f |
⊢ ( 𝜑 → 𝐹 ∈ ( ( 𝐴 [,] 𝐵 ) –cn→ ℝ ) ) |
| 4 |
|
dvne0.d |
⊢ ( 𝜑 → dom ( ℝ D 𝐹 ) = ( 𝐴 (,) 𝐵 ) ) |
| 5 |
|
dvne0.z |
⊢ ( 𝜑 → ¬ 0 ∈ ran ( ℝ D 𝐹 ) ) |
| 6 |
|
eleq1 |
⊢ ( 𝑥 = 0 → ( 𝑥 ∈ ran ( ℝ D 𝐹 ) ↔ 0 ∈ ran ( ℝ D 𝐹 ) ) ) |
| 7 |
6
|
notbid |
⊢ ( 𝑥 = 0 → ( ¬ 𝑥 ∈ ran ( ℝ D 𝐹 ) ↔ ¬ 0 ∈ ran ( ℝ D 𝐹 ) ) ) |
| 8 |
5 7
|
syl5ibrcom |
⊢ ( 𝜑 → ( 𝑥 = 0 → ¬ 𝑥 ∈ ran ( ℝ D 𝐹 ) ) ) |
| 9 |
8
|
necon2ad |
⊢ ( 𝜑 → ( 𝑥 ∈ ran ( ℝ D 𝐹 ) → 𝑥 ≠ 0 ) ) |
| 10 |
9
|
imp |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ ran ( ℝ D 𝐹 ) ) → 𝑥 ≠ 0 ) |
| 11 |
|
cncff |
⊢ ( 𝐹 ∈ ( ( 𝐴 [,] 𝐵 ) –cn→ ℝ ) → 𝐹 : ( 𝐴 [,] 𝐵 ) ⟶ ℝ ) |
| 12 |
3 11
|
syl |
⊢ ( 𝜑 → 𝐹 : ( 𝐴 [,] 𝐵 ) ⟶ ℝ ) |
| 13 |
|
iccssre |
⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ) → ( 𝐴 [,] 𝐵 ) ⊆ ℝ ) |
| 14 |
1 2 13
|
syl2anc |
⊢ ( 𝜑 → ( 𝐴 [,] 𝐵 ) ⊆ ℝ ) |
| 15 |
|
dvfre |
⊢ ( ( 𝐹 : ( 𝐴 [,] 𝐵 ) ⟶ ℝ ∧ ( 𝐴 [,] 𝐵 ) ⊆ ℝ ) → ( ℝ D 𝐹 ) : dom ( ℝ D 𝐹 ) ⟶ ℝ ) |
| 16 |
12 14 15
|
syl2anc |
⊢ ( 𝜑 → ( ℝ D 𝐹 ) : dom ( ℝ D 𝐹 ) ⟶ ℝ ) |
| 17 |
16
|
frnd |
⊢ ( 𝜑 → ran ( ℝ D 𝐹 ) ⊆ ℝ ) |
| 18 |
17
|
sselda |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ ran ( ℝ D 𝐹 ) ) → 𝑥 ∈ ℝ ) |
| 19 |
|
0re |
⊢ 0 ∈ ℝ |
| 20 |
|
lttri2 |
⊢ ( ( 𝑥 ∈ ℝ ∧ 0 ∈ ℝ ) → ( 𝑥 ≠ 0 ↔ ( 𝑥 < 0 ∨ 0 < 𝑥 ) ) ) |
| 21 |
18 19 20
|
sylancl |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ ran ( ℝ D 𝐹 ) ) → ( 𝑥 ≠ 0 ↔ ( 𝑥 < 0 ∨ 0 < 𝑥 ) ) ) |
| 22 |
|
0xr |
⊢ 0 ∈ ℝ* |
| 23 |
|
elioomnf |
⊢ ( 0 ∈ ℝ* → ( 𝑥 ∈ ( -∞ (,) 0 ) ↔ ( 𝑥 ∈ ℝ ∧ 𝑥 < 0 ) ) ) |
| 24 |
22 23
|
ax-mp |
⊢ ( 𝑥 ∈ ( -∞ (,) 0 ) ↔ ( 𝑥 ∈ ℝ ∧ 𝑥 < 0 ) ) |
| 25 |
24
|
baib |
⊢ ( 𝑥 ∈ ℝ → ( 𝑥 ∈ ( -∞ (,) 0 ) ↔ 𝑥 < 0 ) ) |
| 26 |
|
elrp |
⊢ ( 𝑥 ∈ ℝ+ ↔ ( 𝑥 ∈ ℝ ∧ 0 < 𝑥 ) ) |
| 27 |
26
|
baib |
⊢ ( 𝑥 ∈ ℝ → ( 𝑥 ∈ ℝ+ ↔ 0 < 𝑥 ) ) |
| 28 |
25 27
|
orbi12d |
⊢ ( 𝑥 ∈ ℝ → ( ( 𝑥 ∈ ( -∞ (,) 0 ) ∨ 𝑥 ∈ ℝ+ ) ↔ ( 𝑥 < 0 ∨ 0 < 𝑥 ) ) ) |
| 29 |
18 28
|
syl |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ ran ( ℝ D 𝐹 ) ) → ( ( 𝑥 ∈ ( -∞ (,) 0 ) ∨ 𝑥 ∈ ℝ+ ) ↔ ( 𝑥 < 0 ∨ 0 < 𝑥 ) ) ) |
| 30 |
21 29
|
bitr4d |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ ran ( ℝ D 𝐹 ) ) → ( 𝑥 ≠ 0 ↔ ( 𝑥 ∈ ( -∞ (,) 0 ) ∨ 𝑥 ∈ ℝ+ ) ) ) |
| 31 |
10 30
|
mpbid |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ ran ( ℝ D 𝐹 ) ) → ( 𝑥 ∈ ( -∞ (,) 0 ) ∨ 𝑥 ∈ ℝ+ ) ) |
| 32 |
|
elun |
⊢ ( 𝑥 ∈ ( ( -∞ (,) 0 ) ∪ ℝ+ ) ↔ ( 𝑥 ∈ ( -∞ (,) 0 ) ∨ 𝑥 ∈ ℝ+ ) ) |
| 33 |
31 32
|
sylibr |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ ran ( ℝ D 𝐹 ) ) → 𝑥 ∈ ( ( -∞ (,) 0 ) ∪ ℝ+ ) ) |
| 34 |
33
|
ex |
⊢ ( 𝜑 → ( 𝑥 ∈ ran ( ℝ D 𝐹 ) → 𝑥 ∈ ( ( -∞ (,) 0 ) ∪ ℝ+ ) ) ) |
| 35 |
34
|
ssrdv |
⊢ ( 𝜑 → ran ( ℝ D 𝐹 ) ⊆ ( ( -∞ (,) 0 ) ∪ ℝ+ ) ) |
| 36 |
|
disjssun |
⊢ ( ( ran ( ℝ D 𝐹 ) ∩ ( -∞ (,) 0 ) ) = ∅ → ( ran ( ℝ D 𝐹 ) ⊆ ( ( -∞ (,) 0 ) ∪ ℝ+ ) ↔ ran ( ℝ D 𝐹 ) ⊆ ℝ+ ) ) |
| 37 |
35 36
|
syl5ibcom |
⊢ ( 𝜑 → ( ( ran ( ℝ D 𝐹 ) ∩ ( -∞ (,) 0 ) ) = ∅ → ran ( ℝ D 𝐹 ) ⊆ ℝ+ ) ) |
| 38 |
37
|
imp |
⊢ ( ( 𝜑 ∧ ( ran ( ℝ D 𝐹 ) ∩ ( -∞ (,) 0 ) ) = ∅ ) → ran ( ℝ D 𝐹 ) ⊆ ℝ+ ) |
| 39 |
1
|
adantr |
⊢ ( ( 𝜑 ∧ ran ( ℝ D 𝐹 ) ⊆ ℝ+ ) → 𝐴 ∈ ℝ ) |
| 40 |
2
|
adantr |
⊢ ( ( 𝜑 ∧ ran ( ℝ D 𝐹 ) ⊆ ℝ+ ) → 𝐵 ∈ ℝ ) |
| 41 |
3
|
adantr |
⊢ ( ( 𝜑 ∧ ran ( ℝ D 𝐹 ) ⊆ ℝ+ ) → 𝐹 ∈ ( ( 𝐴 [,] 𝐵 ) –cn→ ℝ ) ) |
| 42 |
4
|
feq2d |
⊢ ( 𝜑 → ( ( ℝ D 𝐹 ) : dom ( ℝ D 𝐹 ) ⟶ ℝ ↔ ( ℝ D 𝐹 ) : ( 𝐴 (,) 𝐵 ) ⟶ ℝ ) ) |
| 43 |
16 42
|
mpbid |
⊢ ( 𝜑 → ( ℝ D 𝐹 ) : ( 𝐴 (,) 𝐵 ) ⟶ ℝ ) |
| 44 |
43
|
ffnd |
⊢ ( 𝜑 → ( ℝ D 𝐹 ) Fn ( 𝐴 (,) 𝐵 ) ) |
| 45 |
44
|
anim1i |
⊢ ( ( 𝜑 ∧ ran ( ℝ D 𝐹 ) ⊆ ℝ+ ) → ( ( ℝ D 𝐹 ) Fn ( 𝐴 (,) 𝐵 ) ∧ ran ( ℝ D 𝐹 ) ⊆ ℝ+ ) ) |
| 46 |
|
df-f |
⊢ ( ( ℝ D 𝐹 ) : ( 𝐴 (,) 𝐵 ) ⟶ ℝ+ ↔ ( ( ℝ D 𝐹 ) Fn ( 𝐴 (,) 𝐵 ) ∧ ran ( ℝ D 𝐹 ) ⊆ ℝ+ ) ) |
| 47 |
45 46
|
sylibr |
⊢ ( ( 𝜑 ∧ ran ( ℝ D 𝐹 ) ⊆ ℝ+ ) → ( ℝ D 𝐹 ) : ( 𝐴 (,) 𝐵 ) ⟶ ℝ+ ) |
| 48 |
39 40 41 47
|
dvgt0 |
⊢ ( ( 𝜑 ∧ ran ( ℝ D 𝐹 ) ⊆ ℝ+ ) → 𝐹 Isom < , < ( ( 𝐴 [,] 𝐵 ) , ran 𝐹 ) ) |
| 49 |
48
|
orcd |
⊢ ( ( 𝜑 ∧ ran ( ℝ D 𝐹 ) ⊆ ℝ+ ) → ( 𝐹 Isom < , < ( ( 𝐴 [,] 𝐵 ) , ran 𝐹 ) ∨ 𝐹 Isom < , ◡ < ( ( 𝐴 [,] 𝐵 ) , ran 𝐹 ) ) ) |
| 50 |
38 49
|
syldan |
⊢ ( ( 𝜑 ∧ ( ran ( ℝ D 𝐹 ) ∩ ( -∞ (,) 0 ) ) = ∅ ) → ( 𝐹 Isom < , < ( ( 𝐴 [,] 𝐵 ) , ran 𝐹 ) ∨ 𝐹 Isom < , ◡ < ( ( 𝐴 [,] 𝐵 ) , ran 𝐹 ) ) ) |
| 51 |
|
n0 |
⊢ ( ( ran ( ℝ D 𝐹 ) ∩ ( -∞ (,) 0 ) ) ≠ ∅ ↔ ∃ 𝑥 𝑥 ∈ ( ran ( ℝ D 𝐹 ) ∩ ( -∞ (,) 0 ) ) ) |
| 52 |
|
elin |
⊢ ( 𝑥 ∈ ( ran ( ℝ D 𝐹 ) ∩ ( -∞ (,) 0 ) ) ↔ ( 𝑥 ∈ ran ( ℝ D 𝐹 ) ∧ 𝑥 ∈ ( -∞ (,) 0 ) ) ) |
| 53 |
|
fvelrnb |
⊢ ( ( ℝ D 𝐹 ) Fn ( 𝐴 (,) 𝐵 ) → ( 𝑥 ∈ ran ( ℝ D 𝐹 ) ↔ ∃ 𝑦 ∈ ( 𝐴 (,) 𝐵 ) ( ( ℝ D 𝐹 ) ‘ 𝑦 ) = 𝑥 ) ) |
| 54 |
44 53
|
syl |
⊢ ( 𝜑 → ( 𝑥 ∈ ran ( ℝ D 𝐹 ) ↔ ∃ 𝑦 ∈ ( 𝐴 (,) 𝐵 ) ( ( ℝ D 𝐹 ) ‘ 𝑦 ) = 𝑥 ) ) |
| 55 |
1
|
adantr |
⊢ ( ( 𝜑 ∧ ( 𝑦 ∈ ( 𝐴 (,) 𝐵 ) ∧ ( ( ℝ D 𝐹 ) ‘ 𝑦 ) ∈ ( -∞ (,) 0 ) ) ) → 𝐴 ∈ ℝ ) |
| 56 |
2
|
adantr |
⊢ ( ( 𝜑 ∧ ( 𝑦 ∈ ( 𝐴 (,) 𝐵 ) ∧ ( ( ℝ D 𝐹 ) ‘ 𝑦 ) ∈ ( -∞ (,) 0 ) ) ) → 𝐵 ∈ ℝ ) |
| 57 |
3
|
adantr |
⊢ ( ( 𝜑 ∧ ( 𝑦 ∈ ( 𝐴 (,) 𝐵 ) ∧ ( ( ℝ D 𝐹 ) ‘ 𝑦 ) ∈ ( -∞ (,) 0 ) ) ) → 𝐹 ∈ ( ( 𝐴 [,] 𝐵 ) –cn→ ℝ ) ) |
| 58 |
44
|
adantr |
⊢ ( ( 𝜑 ∧ ( 𝑦 ∈ ( 𝐴 (,) 𝐵 ) ∧ ( ( ℝ D 𝐹 ) ‘ 𝑦 ) ∈ ( -∞ (,) 0 ) ) ) → ( ℝ D 𝐹 ) Fn ( 𝐴 (,) 𝐵 ) ) |
| 59 |
43
|
adantr |
⊢ ( ( 𝜑 ∧ ( 𝑦 ∈ ( 𝐴 (,) 𝐵 ) ∧ ( ( ℝ D 𝐹 ) ‘ 𝑦 ) ∈ ( -∞ (,) 0 ) ) ) → ( ℝ D 𝐹 ) : ( 𝐴 (,) 𝐵 ) ⟶ ℝ ) |
| 60 |
59
|
ffvelrnda |
⊢ ( ( ( 𝜑 ∧ ( 𝑦 ∈ ( 𝐴 (,) 𝐵 ) ∧ ( ( ℝ D 𝐹 ) ‘ 𝑦 ) ∈ ( -∞ (,) 0 ) ) ) ∧ 𝑧 ∈ ( 𝐴 (,) 𝐵 ) ) → ( ( ℝ D 𝐹 ) ‘ 𝑧 ) ∈ ℝ ) |
| 61 |
5
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ ( 𝑦 ∈ ( 𝐴 (,) 𝐵 ) ∧ ( ( ℝ D 𝐹 ) ‘ 𝑦 ) ∈ ( -∞ (,) 0 ) ) ) ∧ 𝑧 ∈ ( 𝐴 (,) 𝐵 ) ) → ¬ 0 ∈ ran ( ℝ D 𝐹 ) ) |
| 62 |
|
simplrl |
⊢ ( ( ( 𝜑 ∧ ( 𝑦 ∈ ( 𝐴 (,) 𝐵 ) ∧ ( ( ℝ D 𝐹 ) ‘ 𝑦 ) ∈ ( -∞ (,) 0 ) ) ) ∧ ( 𝑧 ∈ ( 𝐴 (,) 𝐵 ) ∧ 0 ≤ ( ( ℝ D 𝐹 ) ‘ 𝑧 ) ) ) → 𝑦 ∈ ( 𝐴 (,) 𝐵 ) ) |
| 63 |
|
simprl |
⊢ ( ( ( 𝜑 ∧ ( 𝑦 ∈ ( 𝐴 (,) 𝐵 ) ∧ ( ( ℝ D 𝐹 ) ‘ 𝑦 ) ∈ ( -∞ (,) 0 ) ) ) ∧ ( 𝑧 ∈ ( 𝐴 (,) 𝐵 ) ∧ 0 ≤ ( ( ℝ D 𝐹 ) ‘ 𝑧 ) ) ) → 𝑧 ∈ ( 𝐴 (,) 𝐵 ) ) |
| 64 |
|
ioossicc |
⊢ ( 𝐴 (,) 𝐵 ) ⊆ ( 𝐴 [,] 𝐵 ) |
| 65 |
|
rescncf |
⊢ ( ( 𝐴 (,) 𝐵 ) ⊆ ( 𝐴 [,] 𝐵 ) → ( 𝐹 ∈ ( ( 𝐴 [,] 𝐵 ) –cn→ ℝ ) → ( 𝐹 ↾ ( 𝐴 (,) 𝐵 ) ) ∈ ( ( 𝐴 (,) 𝐵 ) –cn→ ℝ ) ) ) |
| 66 |
64 3 65
|
mpsyl |
⊢ ( 𝜑 → ( 𝐹 ↾ ( 𝐴 (,) 𝐵 ) ) ∈ ( ( 𝐴 (,) 𝐵 ) –cn→ ℝ ) ) |
| 67 |
66
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ ( 𝑦 ∈ ( 𝐴 (,) 𝐵 ) ∧ ( ( ℝ D 𝐹 ) ‘ 𝑦 ) ∈ ( -∞ (,) 0 ) ) ) ∧ ( 𝑧 ∈ ( 𝐴 (,) 𝐵 ) ∧ 0 ≤ ( ( ℝ D 𝐹 ) ‘ 𝑧 ) ) ) → ( 𝐹 ↾ ( 𝐴 (,) 𝐵 ) ) ∈ ( ( 𝐴 (,) 𝐵 ) –cn→ ℝ ) ) |
| 68 |
|
ax-resscn |
⊢ ℝ ⊆ ℂ |
| 69 |
68
|
a1i |
⊢ ( 𝜑 → ℝ ⊆ ℂ ) |
| 70 |
|
fss |
⊢ ( ( 𝐹 : ( 𝐴 [,] 𝐵 ) ⟶ ℝ ∧ ℝ ⊆ ℂ ) → 𝐹 : ( 𝐴 [,] 𝐵 ) ⟶ ℂ ) |
| 71 |
12 68 70
|
sylancl |
⊢ ( 𝜑 → 𝐹 : ( 𝐴 [,] 𝐵 ) ⟶ ℂ ) |
| 72 |
64 14
|
sstrid |
⊢ ( 𝜑 → ( 𝐴 (,) 𝐵 ) ⊆ ℝ ) |
| 73 |
|
eqid |
⊢ ( TopOpen ‘ ℂfld ) = ( TopOpen ‘ ℂfld ) |
| 74 |
73
|
tgioo2 |
⊢ ( topGen ‘ ran (,) ) = ( ( TopOpen ‘ ℂfld ) ↾t ℝ ) |
| 75 |
73 74
|
dvres |
⊢ ( ( ( ℝ ⊆ ℂ ∧ 𝐹 : ( 𝐴 [,] 𝐵 ) ⟶ ℂ ) ∧ ( ( 𝐴 [,] 𝐵 ) ⊆ ℝ ∧ ( 𝐴 (,) 𝐵 ) ⊆ ℝ ) ) → ( ℝ D ( 𝐹 ↾ ( 𝐴 (,) 𝐵 ) ) ) = ( ( ℝ D 𝐹 ) ↾ ( ( int ‘ ( topGen ‘ ran (,) ) ) ‘ ( 𝐴 (,) 𝐵 ) ) ) ) |
| 76 |
69 71 14 72 75
|
syl22anc |
⊢ ( 𝜑 → ( ℝ D ( 𝐹 ↾ ( 𝐴 (,) 𝐵 ) ) ) = ( ( ℝ D 𝐹 ) ↾ ( ( int ‘ ( topGen ‘ ran (,) ) ) ‘ ( 𝐴 (,) 𝐵 ) ) ) ) |
| 77 |
|
retop |
⊢ ( topGen ‘ ran (,) ) ∈ Top |
| 78 |
|
iooretop |
⊢ ( 𝐴 (,) 𝐵 ) ∈ ( topGen ‘ ran (,) ) |
| 79 |
|
isopn3i |
⊢ ( ( ( topGen ‘ ran (,) ) ∈ Top ∧ ( 𝐴 (,) 𝐵 ) ∈ ( topGen ‘ ran (,) ) ) → ( ( int ‘ ( topGen ‘ ran (,) ) ) ‘ ( 𝐴 (,) 𝐵 ) ) = ( 𝐴 (,) 𝐵 ) ) |
| 80 |
77 78 79
|
mp2an |
⊢ ( ( int ‘ ( topGen ‘ ran (,) ) ) ‘ ( 𝐴 (,) 𝐵 ) ) = ( 𝐴 (,) 𝐵 ) |
| 81 |
80
|
reseq2i |
⊢ ( ( ℝ D 𝐹 ) ↾ ( ( int ‘ ( topGen ‘ ran (,) ) ) ‘ ( 𝐴 (,) 𝐵 ) ) ) = ( ( ℝ D 𝐹 ) ↾ ( 𝐴 (,) 𝐵 ) ) |
| 82 |
|
fnresdm |
⊢ ( ( ℝ D 𝐹 ) Fn ( 𝐴 (,) 𝐵 ) → ( ( ℝ D 𝐹 ) ↾ ( 𝐴 (,) 𝐵 ) ) = ( ℝ D 𝐹 ) ) |
| 83 |
44 82
|
syl |
⊢ ( 𝜑 → ( ( ℝ D 𝐹 ) ↾ ( 𝐴 (,) 𝐵 ) ) = ( ℝ D 𝐹 ) ) |
| 84 |
81 83
|
eqtrid |
⊢ ( 𝜑 → ( ( ℝ D 𝐹 ) ↾ ( ( int ‘ ( topGen ‘ ran (,) ) ) ‘ ( 𝐴 (,) 𝐵 ) ) ) = ( ℝ D 𝐹 ) ) |
| 85 |
76 84
|
eqtrd |
⊢ ( 𝜑 → ( ℝ D ( 𝐹 ↾ ( 𝐴 (,) 𝐵 ) ) ) = ( ℝ D 𝐹 ) ) |
| 86 |
85
|
dmeqd |
⊢ ( 𝜑 → dom ( ℝ D ( 𝐹 ↾ ( 𝐴 (,) 𝐵 ) ) ) = dom ( ℝ D 𝐹 ) ) |
| 87 |
86 4
|
eqtrd |
⊢ ( 𝜑 → dom ( ℝ D ( 𝐹 ↾ ( 𝐴 (,) 𝐵 ) ) ) = ( 𝐴 (,) 𝐵 ) ) |
| 88 |
87
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ ( 𝑦 ∈ ( 𝐴 (,) 𝐵 ) ∧ ( ( ℝ D 𝐹 ) ‘ 𝑦 ) ∈ ( -∞ (,) 0 ) ) ) ∧ ( 𝑧 ∈ ( 𝐴 (,) 𝐵 ) ∧ 0 ≤ ( ( ℝ D 𝐹 ) ‘ 𝑧 ) ) ) → dom ( ℝ D ( 𝐹 ↾ ( 𝐴 (,) 𝐵 ) ) ) = ( 𝐴 (,) 𝐵 ) ) |
| 89 |
62 63 67 88
|
dvivth |
⊢ ( ( ( 𝜑 ∧ ( 𝑦 ∈ ( 𝐴 (,) 𝐵 ) ∧ ( ( ℝ D 𝐹 ) ‘ 𝑦 ) ∈ ( -∞ (,) 0 ) ) ) ∧ ( 𝑧 ∈ ( 𝐴 (,) 𝐵 ) ∧ 0 ≤ ( ( ℝ D 𝐹 ) ‘ 𝑧 ) ) ) → ( ( ( ℝ D ( 𝐹 ↾ ( 𝐴 (,) 𝐵 ) ) ) ‘ 𝑦 ) [,] ( ( ℝ D ( 𝐹 ↾ ( 𝐴 (,) 𝐵 ) ) ) ‘ 𝑧 ) ) ⊆ ran ( ℝ D ( 𝐹 ↾ ( 𝐴 (,) 𝐵 ) ) ) ) |
| 90 |
85
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ ( 𝑦 ∈ ( 𝐴 (,) 𝐵 ) ∧ ( ( ℝ D 𝐹 ) ‘ 𝑦 ) ∈ ( -∞ (,) 0 ) ) ) ∧ ( 𝑧 ∈ ( 𝐴 (,) 𝐵 ) ∧ 0 ≤ ( ( ℝ D 𝐹 ) ‘ 𝑧 ) ) ) → ( ℝ D ( 𝐹 ↾ ( 𝐴 (,) 𝐵 ) ) ) = ( ℝ D 𝐹 ) ) |
| 91 |
90
|
fveq1d |
⊢ ( ( ( 𝜑 ∧ ( 𝑦 ∈ ( 𝐴 (,) 𝐵 ) ∧ ( ( ℝ D 𝐹 ) ‘ 𝑦 ) ∈ ( -∞ (,) 0 ) ) ) ∧ ( 𝑧 ∈ ( 𝐴 (,) 𝐵 ) ∧ 0 ≤ ( ( ℝ D 𝐹 ) ‘ 𝑧 ) ) ) → ( ( ℝ D ( 𝐹 ↾ ( 𝐴 (,) 𝐵 ) ) ) ‘ 𝑦 ) = ( ( ℝ D 𝐹 ) ‘ 𝑦 ) ) |
| 92 |
90
|
fveq1d |
⊢ ( ( ( 𝜑 ∧ ( 𝑦 ∈ ( 𝐴 (,) 𝐵 ) ∧ ( ( ℝ D 𝐹 ) ‘ 𝑦 ) ∈ ( -∞ (,) 0 ) ) ) ∧ ( 𝑧 ∈ ( 𝐴 (,) 𝐵 ) ∧ 0 ≤ ( ( ℝ D 𝐹 ) ‘ 𝑧 ) ) ) → ( ( ℝ D ( 𝐹 ↾ ( 𝐴 (,) 𝐵 ) ) ) ‘ 𝑧 ) = ( ( ℝ D 𝐹 ) ‘ 𝑧 ) ) |
| 93 |
91 92
|
oveq12d |
⊢ ( ( ( 𝜑 ∧ ( 𝑦 ∈ ( 𝐴 (,) 𝐵 ) ∧ ( ( ℝ D 𝐹 ) ‘ 𝑦 ) ∈ ( -∞ (,) 0 ) ) ) ∧ ( 𝑧 ∈ ( 𝐴 (,) 𝐵 ) ∧ 0 ≤ ( ( ℝ D 𝐹 ) ‘ 𝑧 ) ) ) → ( ( ( ℝ D ( 𝐹 ↾ ( 𝐴 (,) 𝐵 ) ) ) ‘ 𝑦 ) [,] ( ( ℝ D ( 𝐹 ↾ ( 𝐴 (,) 𝐵 ) ) ) ‘ 𝑧 ) ) = ( ( ( ℝ D 𝐹 ) ‘ 𝑦 ) [,] ( ( ℝ D 𝐹 ) ‘ 𝑧 ) ) ) |
| 94 |
90
|
rneqd |
⊢ ( ( ( 𝜑 ∧ ( 𝑦 ∈ ( 𝐴 (,) 𝐵 ) ∧ ( ( ℝ D 𝐹 ) ‘ 𝑦 ) ∈ ( -∞ (,) 0 ) ) ) ∧ ( 𝑧 ∈ ( 𝐴 (,) 𝐵 ) ∧ 0 ≤ ( ( ℝ D 𝐹 ) ‘ 𝑧 ) ) ) → ran ( ℝ D ( 𝐹 ↾ ( 𝐴 (,) 𝐵 ) ) ) = ran ( ℝ D 𝐹 ) ) |
| 95 |
89 93 94
|
3sstr3d |
⊢ ( ( ( 𝜑 ∧ ( 𝑦 ∈ ( 𝐴 (,) 𝐵 ) ∧ ( ( ℝ D 𝐹 ) ‘ 𝑦 ) ∈ ( -∞ (,) 0 ) ) ) ∧ ( 𝑧 ∈ ( 𝐴 (,) 𝐵 ) ∧ 0 ≤ ( ( ℝ D 𝐹 ) ‘ 𝑧 ) ) ) → ( ( ( ℝ D 𝐹 ) ‘ 𝑦 ) [,] ( ( ℝ D 𝐹 ) ‘ 𝑧 ) ) ⊆ ran ( ℝ D 𝐹 ) ) |
| 96 |
19
|
a1i |
⊢ ( ( ( 𝜑 ∧ ( 𝑦 ∈ ( 𝐴 (,) 𝐵 ) ∧ ( ( ℝ D 𝐹 ) ‘ 𝑦 ) ∈ ( -∞ (,) 0 ) ) ) ∧ ( 𝑧 ∈ ( 𝐴 (,) 𝐵 ) ∧ 0 ≤ ( ( ℝ D 𝐹 ) ‘ 𝑧 ) ) ) → 0 ∈ ℝ ) |
| 97 |
|
simplrr |
⊢ ( ( ( 𝜑 ∧ ( 𝑦 ∈ ( 𝐴 (,) 𝐵 ) ∧ ( ( ℝ D 𝐹 ) ‘ 𝑦 ) ∈ ( -∞ (,) 0 ) ) ) ∧ ( 𝑧 ∈ ( 𝐴 (,) 𝐵 ) ∧ 0 ≤ ( ( ℝ D 𝐹 ) ‘ 𝑧 ) ) ) → ( ( ℝ D 𝐹 ) ‘ 𝑦 ) ∈ ( -∞ (,) 0 ) ) |
| 98 |
|
elioomnf |
⊢ ( 0 ∈ ℝ* → ( ( ( ℝ D 𝐹 ) ‘ 𝑦 ) ∈ ( -∞ (,) 0 ) ↔ ( ( ( ℝ D 𝐹 ) ‘ 𝑦 ) ∈ ℝ ∧ ( ( ℝ D 𝐹 ) ‘ 𝑦 ) < 0 ) ) ) |
| 99 |
22 98
|
ax-mp |
⊢ ( ( ( ℝ D 𝐹 ) ‘ 𝑦 ) ∈ ( -∞ (,) 0 ) ↔ ( ( ( ℝ D 𝐹 ) ‘ 𝑦 ) ∈ ℝ ∧ ( ( ℝ D 𝐹 ) ‘ 𝑦 ) < 0 ) ) |
| 100 |
97 99
|
sylib |
⊢ ( ( ( 𝜑 ∧ ( 𝑦 ∈ ( 𝐴 (,) 𝐵 ) ∧ ( ( ℝ D 𝐹 ) ‘ 𝑦 ) ∈ ( -∞ (,) 0 ) ) ) ∧ ( 𝑧 ∈ ( 𝐴 (,) 𝐵 ) ∧ 0 ≤ ( ( ℝ D 𝐹 ) ‘ 𝑧 ) ) ) → ( ( ( ℝ D 𝐹 ) ‘ 𝑦 ) ∈ ℝ ∧ ( ( ℝ D 𝐹 ) ‘ 𝑦 ) < 0 ) ) |
| 101 |
100
|
simprd |
⊢ ( ( ( 𝜑 ∧ ( 𝑦 ∈ ( 𝐴 (,) 𝐵 ) ∧ ( ( ℝ D 𝐹 ) ‘ 𝑦 ) ∈ ( -∞ (,) 0 ) ) ) ∧ ( 𝑧 ∈ ( 𝐴 (,) 𝐵 ) ∧ 0 ≤ ( ( ℝ D 𝐹 ) ‘ 𝑧 ) ) ) → ( ( ℝ D 𝐹 ) ‘ 𝑦 ) < 0 ) |
| 102 |
100
|
simpld |
⊢ ( ( ( 𝜑 ∧ ( 𝑦 ∈ ( 𝐴 (,) 𝐵 ) ∧ ( ( ℝ D 𝐹 ) ‘ 𝑦 ) ∈ ( -∞ (,) 0 ) ) ) ∧ ( 𝑧 ∈ ( 𝐴 (,) 𝐵 ) ∧ 0 ≤ ( ( ℝ D 𝐹 ) ‘ 𝑧 ) ) ) → ( ( ℝ D 𝐹 ) ‘ 𝑦 ) ∈ ℝ ) |
| 103 |
|
ltle |
⊢ ( ( ( ( ℝ D 𝐹 ) ‘ 𝑦 ) ∈ ℝ ∧ 0 ∈ ℝ ) → ( ( ( ℝ D 𝐹 ) ‘ 𝑦 ) < 0 → ( ( ℝ D 𝐹 ) ‘ 𝑦 ) ≤ 0 ) ) |
| 104 |
102 19 103
|
sylancl |
⊢ ( ( ( 𝜑 ∧ ( 𝑦 ∈ ( 𝐴 (,) 𝐵 ) ∧ ( ( ℝ D 𝐹 ) ‘ 𝑦 ) ∈ ( -∞ (,) 0 ) ) ) ∧ ( 𝑧 ∈ ( 𝐴 (,) 𝐵 ) ∧ 0 ≤ ( ( ℝ D 𝐹 ) ‘ 𝑧 ) ) ) → ( ( ( ℝ D 𝐹 ) ‘ 𝑦 ) < 0 → ( ( ℝ D 𝐹 ) ‘ 𝑦 ) ≤ 0 ) ) |
| 105 |
101 104
|
mpd |
⊢ ( ( ( 𝜑 ∧ ( 𝑦 ∈ ( 𝐴 (,) 𝐵 ) ∧ ( ( ℝ D 𝐹 ) ‘ 𝑦 ) ∈ ( -∞ (,) 0 ) ) ) ∧ ( 𝑧 ∈ ( 𝐴 (,) 𝐵 ) ∧ 0 ≤ ( ( ℝ D 𝐹 ) ‘ 𝑧 ) ) ) → ( ( ℝ D 𝐹 ) ‘ 𝑦 ) ≤ 0 ) |
| 106 |
|
simprr |
⊢ ( ( ( 𝜑 ∧ ( 𝑦 ∈ ( 𝐴 (,) 𝐵 ) ∧ ( ( ℝ D 𝐹 ) ‘ 𝑦 ) ∈ ( -∞ (,) 0 ) ) ) ∧ ( 𝑧 ∈ ( 𝐴 (,) 𝐵 ) ∧ 0 ≤ ( ( ℝ D 𝐹 ) ‘ 𝑧 ) ) ) → 0 ≤ ( ( ℝ D 𝐹 ) ‘ 𝑧 ) ) |
| 107 |
63 60
|
syldan |
⊢ ( ( ( 𝜑 ∧ ( 𝑦 ∈ ( 𝐴 (,) 𝐵 ) ∧ ( ( ℝ D 𝐹 ) ‘ 𝑦 ) ∈ ( -∞ (,) 0 ) ) ) ∧ ( 𝑧 ∈ ( 𝐴 (,) 𝐵 ) ∧ 0 ≤ ( ( ℝ D 𝐹 ) ‘ 𝑧 ) ) ) → ( ( ℝ D 𝐹 ) ‘ 𝑧 ) ∈ ℝ ) |
| 108 |
|
elicc2 |
⊢ ( ( ( ( ℝ D 𝐹 ) ‘ 𝑦 ) ∈ ℝ ∧ ( ( ℝ D 𝐹 ) ‘ 𝑧 ) ∈ ℝ ) → ( 0 ∈ ( ( ( ℝ D 𝐹 ) ‘ 𝑦 ) [,] ( ( ℝ D 𝐹 ) ‘ 𝑧 ) ) ↔ ( 0 ∈ ℝ ∧ ( ( ℝ D 𝐹 ) ‘ 𝑦 ) ≤ 0 ∧ 0 ≤ ( ( ℝ D 𝐹 ) ‘ 𝑧 ) ) ) ) |
| 109 |
102 107 108
|
syl2anc |
⊢ ( ( ( 𝜑 ∧ ( 𝑦 ∈ ( 𝐴 (,) 𝐵 ) ∧ ( ( ℝ D 𝐹 ) ‘ 𝑦 ) ∈ ( -∞ (,) 0 ) ) ) ∧ ( 𝑧 ∈ ( 𝐴 (,) 𝐵 ) ∧ 0 ≤ ( ( ℝ D 𝐹 ) ‘ 𝑧 ) ) ) → ( 0 ∈ ( ( ( ℝ D 𝐹 ) ‘ 𝑦 ) [,] ( ( ℝ D 𝐹 ) ‘ 𝑧 ) ) ↔ ( 0 ∈ ℝ ∧ ( ( ℝ D 𝐹 ) ‘ 𝑦 ) ≤ 0 ∧ 0 ≤ ( ( ℝ D 𝐹 ) ‘ 𝑧 ) ) ) ) |
| 110 |
96 105 106 109
|
mpbir3and |
⊢ ( ( ( 𝜑 ∧ ( 𝑦 ∈ ( 𝐴 (,) 𝐵 ) ∧ ( ( ℝ D 𝐹 ) ‘ 𝑦 ) ∈ ( -∞ (,) 0 ) ) ) ∧ ( 𝑧 ∈ ( 𝐴 (,) 𝐵 ) ∧ 0 ≤ ( ( ℝ D 𝐹 ) ‘ 𝑧 ) ) ) → 0 ∈ ( ( ( ℝ D 𝐹 ) ‘ 𝑦 ) [,] ( ( ℝ D 𝐹 ) ‘ 𝑧 ) ) ) |
| 111 |
95 110
|
sseldd |
⊢ ( ( ( 𝜑 ∧ ( 𝑦 ∈ ( 𝐴 (,) 𝐵 ) ∧ ( ( ℝ D 𝐹 ) ‘ 𝑦 ) ∈ ( -∞ (,) 0 ) ) ) ∧ ( 𝑧 ∈ ( 𝐴 (,) 𝐵 ) ∧ 0 ≤ ( ( ℝ D 𝐹 ) ‘ 𝑧 ) ) ) → 0 ∈ ran ( ℝ D 𝐹 ) ) |
| 112 |
111
|
expr |
⊢ ( ( ( 𝜑 ∧ ( 𝑦 ∈ ( 𝐴 (,) 𝐵 ) ∧ ( ( ℝ D 𝐹 ) ‘ 𝑦 ) ∈ ( -∞ (,) 0 ) ) ) ∧ 𝑧 ∈ ( 𝐴 (,) 𝐵 ) ) → ( 0 ≤ ( ( ℝ D 𝐹 ) ‘ 𝑧 ) → 0 ∈ ran ( ℝ D 𝐹 ) ) ) |
| 113 |
61 112
|
mtod |
⊢ ( ( ( 𝜑 ∧ ( 𝑦 ∈ ( 𝐴 (,) 𝐵 ) ∧ ( ( ℝ D 𝐹 ) ‘ 𝑦 ) ∈ ( -∞ (,) 0 ) ) ) ∧ 𝑧 ∈ ( 𝐴 (,) 𝐵 ) ) → ¬ 0 ≤ ( ( ℝ D 𝐹 ) ‘ 𝑧 ) ) |
| 114 |
|
ltnle |
⊢ ( ( ( ( ℝ D 𝐹 ) ‘ 𝑧 ) ∈ ℝ ∧ 0 ∈ ℝ ) → ( ( ( ℝ D 𝐹 ) ‘ 𝑧 ) < 0 ↔ ¬ 0 ≤ ( ( ℝ D 𝐹 ) ‘ 𝑧 ) ) ) |
| 115 |
60 19 114
|
sylancl |
⊢ ( ( ( 𝜑 ∧ ( 𝑦 ∈ ( 𝐴 (,) 𝐵 ) ∧ ( ( ℝ D 𝐹 ) ‘ 𝑦 ) ∈ ( -∞ (,) 0 ) ) ) ∧ 𝑧 ∈ ( 𝐴 (,) 𝐵 ) ) → ( ( ( ℝ D 𝐹 ) ‘ 𝑧 ) < 0 ↔ ¬ 0 ≤ ( ( ℝ D 𝐹 ) ‘ 𝑧 ) ) ) |
| 116 |
113 115
|
mpbird |
⊢ ( ( ( 𝜑 ∧ ( 𝑦 ∈ ( 𝐴 (,) 𝐵 ) ∧ ( ( ℝ D 𝐹 ) ‘ 𝑦 ) ∈ ( -∞ (,) 0 ) ) ) ∧ 𝑧 ∈ ( 𝐴 (,) 𝐵 ) ) → ( ( ℝ D 𝐹 ) ‘ 𝑧 ) < 0 ) |
| 117 |
|
elioomnf |
⊢ ( 0 ∈ ℝ* → ( ( ( ℝ D 𝐹 ) ‘ 𝑧 ) ∈ ( -∞ (,) 0 ) ↔ ( ( ( ℝ D 𝐹 ) ‘ 𝑧 ) ∈ ℝ ∧ ( ( ℝ D 𝐹 ) ‘ 𝑧 ) < 0 ) ) ) |
| 118 |
22 117
|
ax-mp |
⊢ ( ( ( ℝ D 𝐹 ) ‘ 𝑧 ) ∈ ( -∞ (,) 0 ) ↔ ( ( ( ℝ D 𝐹 ) ‘ 𝑧 ) ∈ ℝ ∧ ( ( ℝ D 𝐹 ) ‘ 𝑧 ) < 0 ) ) |
| 119 |
60 116 118
|
sylanbrc |
⊢ ( ( ( 𝜑 ∧ ( 𝑦 ∈ ( 𝐴 (,) 𝐵 ) ∧ ( ( ℝ D 𝐹 ) ‘ 𝑦 ) ∈ ( -∞ (,) 0 ) ) ) ∧ 𝑧 ∈ ( 𝐴 (,) 𝐵 ) ) → ( ( ℝ D 𝐹 ) ‘ 𝑧 ) ∈ ( -∞ (,) 0 ) ) |
| 120 |
119
|
ralrimiva |
⊢ ( ( 𝜑 ∧ ( 𝑦 ∈ ( 𝐴 (,) 𝐵 ) ∧ ( ( ℝ D 𝐹 ) ‘ 𝑦 ) ∈ ( -∞ (,) 0 ) ) ) → ∀ 𝑧 ∈ ( 𝐴 (,) 𝐵 ) ( ( ℝ D 𝐹 ) ‘ 𝑧 ) ∈ ( -∞ (,) 0 ) ) |
| 121 |
|
ffnfv |
⊢ ( ( ℝ D 𝐹 ) : ( 𝐴 (,) 𝐵 ) ⟶ ( -∞ (,) 0 ) ↔ ( ( ℝ D 𝐹 ) Fn ( 𝐴 (,) 𝐵 ) ∧ ∀ 𝑧 ∈ ( 𝐴 (,) 𝐵 ) ( ( ℝ D 𝐹 ) ‘ 𝑧 ) ∈ ( -∞ (,) 0 ) ) ) |
| 122 |
58 120 121
|
sylanbrc |
⊢ ( ( 𝜑 ∧ ( 𝑦 ∈ ( 𝐴 (,) 𝐵 ) ∧ ( ( ℝ D 𝐹 ) ‘ 𝑦 ) ∈ ( -∞ (,) 0 ) ) ) → ( ℝ D 𝐹 ) : ( 𝐴 (,) 𝐵 ) ⟶ ( -∞ (,) 0 ) ) |
| 123 |
55 56 57 122
|
dvlt0 |
⊢ ( ( 𝜑 ∧ ( 𝑦 ∈ ( 𝐴 (,) 𝐵 ) ∧ ( ( ℝ D 𝐹 ) ‘ 𝑦 ) ∈ ( -∞ (,) 0 ) ) ) → 𝐹 Isom < , ◡ < ( ( 𝐴 [,] 𝐵 ) , ran 𝐹 ) ) |
| 124 |
123
|
olcd |
⊢ ( ( 𝜑 ∧ ( 𝑦 ∈ ( 𝐴 (,) 𝐵 ) ∧ ( ( ℝ D 𝐹 ) ‘ 𝑦 ) ∈ ( -∞ (,) 0 ) ) ) → ( 𝐹 Isom < , < ( ( 𝐴 [,] 𝐵 ) , ran 𝐹 ) ∨ 𝐹 Isom < , ◡ < ( ( 𝐴 [,] 𝐵 ) , ran 𝐹 ) ) ) |
| 125 |
124
|
expr |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 𝐴 (,) 𝐵 ) ) → ( ( ( ℝ D 𝐹 ) ‘ 𝑦 ) ∈ ( -∞ (,) 0 ) → ( 𝐹 Isom < , < ( ( 𝐴 [,] 𝐵 ) , ran 𝐹 ) ∨ 𝐹 Isom < , ◡ < ( ( 𝐴 [,] 𝐵 ) , ran 𝐹 ) ) ) ) |
| 126 |
|
eleq1 |
⊢ ( ( ( ℝ D 𝐹 ) ‘ 𝑦 ) = 𝑥 → ( ( ( ℝ D 𝐹 ) ‘ 𝑦 ) ∈ ( -∞ (,) 0 ) ↔ 𝑥 ∈ ( -∞ (,) 0 ) ) ) |
| 127 |
126
|
imbi1d |
⊢ ( ( ( ℝ D 𝐹 ) ‘ 𝑦 ) = 𝑥 → ( ( ( ( ℝ D 𝐹 ) ‘ 𝑦 ) ∈ ( -∞ (,) 0 ) → ( 𝐹 Isom < , < ( ( 𝐴 [,] 𝐵 ) , ran 𝐹 ) ∨ 𝐹 Isom < , ◡ < ( ( 𝐴 [,] 𝐵 ) , ran 𝐹 ) ) ) ↔ ( 𝑥 ∈ ( -∞ (,) 0 ) → ( 𝐹 Isom < , < ( ( 𝐴 [,] 𝐵 ) , ran 𝐹 ) ∨ 𝐹 Isom < , ◡ < ( ( 𝐴 [,] 𝐵 ) , ran 𝐹 ) ) ) ) ) |
| 128 |
125 127
|
syl5ibcom |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 𝐴 (,) 𝐵 ) ) → ( ( ( ℝ D 𝐹 ) ‘ 𝑦 ) = 𝑥 → ( 𝑥 ∈ ( -∞ (,) 0 ) → ( 𝐹 Isom < , < ( ( 𝐴 [,] 𝐵 ) , ran 𝐹 ) ∨ 𝐹 Isom < , ◡ < ( ( 𝐴 [,] 𝐵 ) , ran 𝐹 ) ) ) ) ) |
| 129 |
128
|
rexlimdva |
⊢ ( 𝜑 → ( ∃ 𝑦 ∈ ( 𝐴 (,) 𝐵 ) ( ( ℝ D 𝐹 ) ‘ 𝑦 ) = 𝑥 → ( 𝑥 ∈ ( -∞ (,) 0 ) → ( 𝐹 Isom < , < ( ( 𝐴 [,] 𝐵 ) , ran 𝐹 ) ∨ 𝐹 Isom < , ◡ < ( ( 𝐴 [,] 𝐵 ) , ran 𝐹 ) ) ) ) ) |
| 130 |
54 129
|
sylbid |
⊢ ( 𝜑 → ( 𝑥 ∈ ran ( ℝ D 𝐹 ) → ( 𝑥 ∈ ( -∞ (,) 0 ) → ( 𝐹 Isom < , < ( ( 𝐴 [,] 𝐵 ) , ran 𝐹 ) ∨ 𝐹 Isom < , ◡ < ( ( 𝐴 [,] 𝐵 ) , ran 𝐹 ) ) ) ) ) |
| 131 |
130
|
impd |
⊢ ( 𝜑 → ( ( 𝑥 ∈ ran ( ℝ D 𝐹 ) ∧ 𝑥 ∈ ( -∞ (,) 0 ) ) → ( 𝐹 Isom < , < ( ( 𝐴 [,] 𝐵 ) , ran 𝐹 ) ∨ 𝐹 Isom < , ◡ < ( ( 𝐴 [,] 𝐵 ) , ran 𝐹 ) ) ) ) |
| 132 |
52 131
|
syl5bi |
⊢ ( 𝜑 → ( 𝑥 ∈ ( ran ( ℝ D 𝐹 ) ∩ ( -∞ (,) 0 ) ) → ( 𝐹 Isom < , < ( ( 𝐴 [,] 𝐵 ) , ran 𝐹 ) ∨ 𝐹 Isom < , ◡ < ( ( 𝐴 [,] 𝐵 ) , ran 𝐹 ) ) ) ) |
| 133 |
132
|
exlimdv |
⊢ ( 𝜑 → ( ∃ 𝑥 𝑥 ∈ ( ran ( ℝ D 𝐹 ) ∩ ( -∞ (,) 0 ) ) → ( 𝐹 Isom < , < ( ( 𝐴 [,] 𝐵 ) , ran 𝐹 ) ∨ 𝐹 Isom < , ◡ < ( ( 𝐴 [,] 𝐵 ) , ran 𝐹 ) ) ) ) |
| 134 |
51 133
|
syl5bi |
⊢ ( 𝜑 → ( ( ran ( ℝ D 𝐹 ) ∩ ( -∞ (,) 0 ) ) ≠ ∅ → ( 𝐹 Isom < , < ( ( 𝐴 [,] 𝐵 ) , ran 𝐹 ) ∨ 𝐹 Isom < , ◡ < ( ( 𝐴 [,] 𝐵 ) , ran 𝐹 ) ) ) ) |
| 135 |
134
|
imp |
⊢ ( ( 𝜑 ∧ ( ran ( ℝ D 𝐹 ) ∩ ( -∞ (,) 0 ) ) ≠ ∅ ) → ( 𝐹 Isom < , < ( ( 𝐴 [,] 𝐵 ) , ran 𝐹 ) ∨ 𝐹 Isom < , ◡ < ( ( 𝐴 [,] 𝐵 ) , ran 𝐹 ) ) ) |
| 136 |
50 135
|
pm2.61dane |
⊢ ( 𝜑 → ( 𝐹 Isom < , < ( ( 𝐴 [,] 𝐵 ) , ran 𝐹 ) ∨ 𝐹 Isom < , ◡ < ( ( 𝐴 [,] 𝐵 ) , ran 𝐹 ) ) ) |