Step |
Hyp |
Ref |
Expression |
1 |
|
imo72b2.1 |
⊢ ( 𝜑 → 𝐹 : ℝ ⟶ ℝ ) |
2 |
|
imo72b2.2 |
⊢ ( 𝜑 → 𝐺 : ℝ ⟶ ℝ ) |
3 |
|
imo72b2.4 |
⊢ ( 𝜑 → 𝐵 ∈ ℝ ) |
4 |
|
imo72b2.5 |
⊢ ( 𝜑 → ∀ 𝑢 ∈ ℝ ∀ 𝑣 ∈ ℝ ( ( 𝐹 ‘ ( 𝑢 + 𝑣 ) ) + ( 𝐹 ‘ ( 𝑢 − 𝑣 ) ) ) = ( 2 · ( ( 𝐹 ‘ 𝑢 ) · ( 𝐺 ‘ 𝑣 ) ) ) ) |
5 |
|
imo72b2.6 |
⊢ ( 𝜑 → ∀ 𝑦 ∈ ℝ ( abs ‘ ( 𝐹 ‘ 𝑦 ) ) ≤ 1 ) |
6 |
|
imo72b2.7 |
⊢ ( 𝜑 → ∃ 𝑥 ∈ ℝ ( 𝐹 ‘ 𝑥 ) ≠ 0 ) |
7 |
2 3
|
ffvelrnd |
⊢ ( 𝜑 → ( 𝐺 ‘ 𝐵 ) ∈ ℝ ) |
8 |
7
|
recnd |
⊢ ( 𝜑 → ( 𝐺 ‘ 𝐵 ) ∈ ℂ ) |
9 |
8
|
abscld |
⊢ ( 𝜑 → ( abs ‘ ( 𝐺 ‘ 𝐵 ) ) ∈ ℝ ) |
10 |
|
1red |
⊢ ( 𝜑 → 1 ∈ ℝ ) |
11 |
|
simpr |
⊢ ( ( 𝜑 ∧ 1 < ( abs ‘ ( 𝐺 ‘ 𝐵 ) ) ) → 1 < ( abs ‘ ( 𝐺 ‘ 𝐵 ) ) ) |
12 |
2
|
adantr |
⊢ ( ( 𝜑 ∧ 1 < ( abs ‘ ( 𝐺 ‘ 𝐵 ) ) ) → 𝐺 : ℝ ⟶ ℝ ) |
13 |
3
|
adantr |
⊢ ( ( 𝜑 ∧ 1 < ( abs ‘ ( 𝐺 ‘ 𝐵 ) ) ) → 𝐵 ∈ ℝ ) |
14 |
12 13
|
ffvelrnd |
⊢ ( ( 𝜑 ∧ 1 < ( abs ‘ ( 𝐺 ‘ 𝐵 ) ) ) → ( 𝐺 ‘ 𝐵 ) ∈ ℝ ) |
15 |
14
|
recnd |
⊢ ( ( 𝜑 ∧ 1 < ( abs ‘ ( 𝐺 ‘ 𝐵 ) ) ) → ( 𝐺 ‘ 𝐵 ) ∈ ℂ ) |
16 |
15
|
abscld |
⊢ ( ( 𝜑 ∧ 1 < ( abs ‘ ( 𝐺 ‘ 𝐵 ) ) ) → ( abs ‘ ( 𝐺 ‘ 𝐵 ) ) ∈ ℝ ) |
17 |
10
|
adantr |
⊢ ( ( 𝜑 ∧ 1 < ( abs ‘ ( 𝐺 ‘ 𝐵 ) ) ) → 1 ∈ ℝ ) |
18 |
|
ax-resscn |
⊢ ℝ ⊆ ℂ |
19 |
|
imaco |
⊢ ( ( abs ∘ 𝐹 ) “ ℝ ) = ( abs “ ( 𝐹 “ ℝ ) ) |
20 |
19
|
eqcomi |
⊢ ( abs “ ( 𝐹 “ ℝ ) ) = ( ( abs ∘ 𝐹 ) “ ℝ ) |
21 |
|
imassrn |
⊢ ( ( abs ∘ 𝐹 ) “ ℝ ) ⊆ ran ( abs ∘ 𝐹 ) |
22 |
21
|
a1i |
⊢ ( ( 𝜑 ∧ 1 < ( abs ‘ ( 𝐺 ‘ 𝐵 ) ) ) → ( ( abs ∘ 𝐹 ) “ ℝ ) ⊆ ran ( abs ∘ 𝐹 ) ) |
23 |
1
|
adantr |
⊢ ( ( 𝜑 ∧ 1 < ( abs ‘ ( 𝐺 ‘ 𝐵 ) ) ) → 𝐹 : ℝ ⟶ ℝ ) |
24 |
|
absf |
⊢ abs : ℂ ⟶ ℝ |
25 |
24
|
a1i |
⊢ ( ( 𝜑 ∧ 1 < ( abs ‘ ( 𝐺 ‘ 𝐵 ) ) ) → abs : ℂ ⟶ ℝ ) |
26 |
18
|
a1i |
⊢ ( ( 𝜑 ∧ 1 < ( abs ‘ ( 𝐺 ‘ 𝐵 ) ) ) → ℝ ⊆ ℂ ) |
27 |
25 26
|
fssresd |
⊢ ( ( 𝜑 ∧ 1 < ( abs ‘ ( 𝐺 ‘ 𝐵 ) ) ) → ( abs ↾ ℝ ) : ℝ ⟶ ℝ ) |
28 |
23 27
|
fco2d |
⊢ ( ( 𝜑 ∧ 1 < ( abs ‘ ( 𝐺 ‘ 𝐵 ) ) ) → ( abs ∘ 𝐹 ) : ℝ ⟶ ℝ ) |
29 |
28
|
frnd |
⊢ ( ( 𝜑 ∧ 1 < ( abs ‘ ( 𝐺 ‘ 𝐵 ) ) ) → ran ( abs ∘ 𝐹 ) ⊆ ℝ ) |
30 |
22 29
|
sstrd |
⊢ ( ( 𝜑 ∧ 1 < ( abs ‘ ( 𝐺 ‘ 𝐵 ) ) ) → ( ( abs ∘ 𝐹 ) “ ℝ ) ⊆ ℝ ) |
31 |
20 30
|
eqsstrid |
⊢ ( ( 𝜑 ∧ 1 < ( abs ‘ ( 𝐺 ‘ 𝐵 ) ) ) → ( abs “ ( 𝐹 “ ℝ ) ) ⊆ ℝ ) |
32 |
|
0re |
⊢ 0 ∈ ℝ |
33 |
32
|
ne0ii |
⊢ ℝ ≠ ∅ |
34 |
33
|
a1i |
⊢ ( ( 𝜑 ∧ 1 < ( abs ‘ ( 𝐺 ‘ 𝐵 ) ) ) → ℝ ≠ ∅ ) |
35 |
34 28
|
wnefimgd |
⊢ ( ( 𝜑 ∧ 1 < ( abs ‘ ( 𝐺 ‘ 𝐵 ) ) ) → ( ( abs ∘ 𝐹 ) “ ℝ ) ≠ ∅ ) |
36 |
35
|
necomd |
⊢ ( ( 𝜑 ∧ 1 < ( abs ‘ ( 𝐺 ‘ 𝐵 ) ) ) → ∅ ≠ ( ( abs ∘ 𝐹 ) “ ℝ ) ) |
37 |
20
|
a1i |
⊢ ( ( 𝜑 ∧ 1 < ( abs ‘ ( 𝐺 ‘ 𝐵 ) ) ) → ( abs “ ( 𝐹 “ ℝ ) ) = ( ( abs ∘ 𝐹 ) “ ℝ ) ) |
38 |
36 37
|
neeqtrrd |
⊢ ( ( 𝜑 ∧ 1 < ( abs ‘ ( 𝐺 ‘ 𝐵 ) ) ) → ∅ ≠ ( abs “ ( 𝐹 “ ℝ ) ) ) |
39 |
38
|
necomd |
⊢ ( ( 𝜑 ∧ 1 < ( abs ‘ ( 𝐺 ‘ 𝐵 ) ) ) → ( abs “ ( 𝐹 “ ℝ ) ) ≠ ∅ ) |
40 |
|
simpr |
⊢ ( ( ( 𝜑 ∧ 1 < ( abs ‘ ( 𝐺 ‘ 𝐵 ) ) ) ∧ 𝑐 = 1 ) → 𝑐 = 1 ) |
41 |
40
|
breq2d |
⊢ ( ( ( 𝜑 ∧ 1 < ( abs ‘ ( 𝐺 ‘ 𝐵 ) ) ) ∧ 𝑐 = 1 ) → ( 𝑡 ≤ 𝑐 ↔ 𝑡 ≤ 1 ) ) |
42 |
41
|
ralbidv |
⊢ ( ( ( 𝜑 ∧ 1 < ( abs ‘ ( 𝐺 ‘ 𝐵 ) ) ) ∧ 𝑐 = 1 ) → ( ∀ 𝑡 ∈ ( abs “ ( 𝐹 “ ℝ ) ) 𝑡 ≤ 𝑐 ↔ ∀ 𝑡 ∈ ( abs “ ( 𝐹 “ ℝ ) ) 𝑡 ≤ 1 ) ) |
43 |
1 5
|
extoimad |
⊢ ( 𝜑 → ∀ 𝑡 ∈ ( abs “ ( 𝐹 “ ℝ ) ) 𝑡 ≤ 1 ) |
44 |
43
|
adantr |
⊢ ( ( 𝜑 ∧ 1 < ( abs ‘ ( 𝐺 ‘ 𝐵 ) ) ) → ∀ 𝑡 ∈ ( abs “ ( 𝐹 “ ℝ ) ) 𝑡 ≤ 1 ) |
45 |
17 42 44
|
rspcedvd |
⊢ ( ( 𝜑 ∧ 1 < ( abs ‘ ( 𝐺 ‘ 𝐵 ) ) ) → ∃ 𝑐 ∈ ℝ ∀ 𝑡 ∈ ( abs “ ( 𝐹 “ ℝ ) ) 𝑡 ≤ 𝑐 ) |
46 |
31 39 45
|
suprcld |
⊢ ( ( 𝜑 ∧ 1 < ( abs ‘ ( 𝐺 ‘ 𝐵 ) ) ) → sup ( ( abs “ ( 𝐹 “ ℝ ) ) , ℝ , < ) ∈ ℝ ) |
47 |
18 46
|
sselid |
⊢ ( ( 𝜑 ∧ 1 < ( abs ‘ ( 𝐺 ‘ 𝐵 ) ) ) → sup ( ( abs “ ( 𝐹 “ ℝ ) ) , ℝ , < ) ∈ ℂ ) |
48 |
18 16
|
sselid |
⊢ ( ( 𝜑 ∧ 1 < ( abs ‘ ( 𝐺 ‘ 𝐵 ) ) ) → ( abs ‘ ( 𝐺 ‘ 𝐵 ) ) ∈ ℂ ) |
49 |
47 48
|
mulcomd |
⊢ ( ( 𝜑 ∧ 1 < ( abs ‘ ( 𝐺 ‘ 𝐵 ) ) ) → ( sup ( ( abs “ ( 𝐹 “ ℝ ) ) , ℝ , < ) · ( abs ‘ ( 𝐺 ‘ 𝐵 ) ) ) = ( ( abs ‘ ( 𝐺 ‘ 𝐵 ) ) · sup ( ( abs “ ( 𝐹 “ ℝ ) ) , ℝ , < ) ) ) |
50 |
32
|
a1i |
⊢ ( ( 𝜑 ∧ 1 < ( abs ‘ ( 𝐺 ‘ 𝐵 ) ) ) → 0 ∈ ℝ ) |
51 |
|
0lt1 |
⊢ 0 < 1 |
52 |
51
|
a1i |
⊢ ( ( 𝜑 ∧ 1 < ( abs ‘ ( 𝐺 ‘ 𝐵 ) ) ) → 0 < 1 ) |
53 |
50 17 16 52 11
|
lttrd |
⊢ ( ( 𝜑 ∧ 1 < ( abs ‘ ( 𝐺 ‘ 𝐵 ) ) ) → 0 < ( abs ‘ ( 𝐺 ‘ 𝐵 ) ) ) |
54 |
53
|
gt0ne0d |
⊢ ( ( 𝜑 ∧ 1 < ( abs ‘ ( 𝐺 ‘ 𝐵 ) ) ) → ( abs ‘ ( 𝐺 ‘ 𝐵 ) ) ≠ 0 ) |
55 |
46 16 54
|
redivcld |
⊢ ( ( 𝜑 ∧ 1 < ( abs ‘ ( 𝐺 ‘ 𝐵 ) ) ) → ( sup ( ( abs “ ( 𝐹 “ ℝ ) ) , ℝ , < ) / ( abs ‘ ( 𝐺 ‘ 𝐵 ) ) ) ∈ ℝ ) |
56 |
23
|
adantr |
⊢ ( ( ( 𝜑 ∧ 1 < ( abs ‘ ( 𝐺 ‘ 𝐵 ) ) ) ∧ 𝑢 ∈ ℝ ) → 𝐹 : ℝ ⟶ ℝ ) |
57 |
12
|
adantr |
⊢ ( ( ( 𝜑 ∧ 1 < ( abs ‘ ( 𝐺 ‘ 𝐵 ) ) ) ∧ 𝑢 ∈ ℝ ) → 𝐺 : ℝ ⟶ ℝ ) |
58 |
|
simpr |
⊢ ( ( ( 𝜑 ∧ 1 < ( abs ‘ ( 𝐺 ‘ 𝐵 ) ) ) ∧ 𝑢 ∈ ℝ ) → 𝑢 ∈ ℝ ) |
59 |
13
|
adantr |
⊢ ( ( ( 𝜑 ∧ 1 < ( abs ‘ ( 𝐺 ‘ 𝐵 ) ) ) ∧ 𝑢 ∈ ℝ ) → 𝐵 ∈ ℝ ) |
60 |
|
simpr |
⊢ ( ( 𝜑 ∧ 𝑣 = 𝐵 ) → 𝑣 = 𝐵 ) |
61 |
60
|
oveq2d |
⊢ ( ( 𝜑 ∧ 𝑣 = 𝐵 ) → ( 𝑢 + 𝑣 ) = ( 𝑢 + 𝐵 ) ) |
62 |
61
|
fveq2d |
⊢ ( ( 𝜑 ∧ 𝑣 = 𝐵 ) → ( 𝐹 ‘ ( 𝑢 + 𝑣 ) ) = ( 𝐹 ‘ ( 𝑢 + 𝐵 ) ) ) |
63 |
60
|
oveq2d |
⊢ ( ( 𝜑 ∧ 𝑣 = 𝐵 ) → ( 𝑢 − 𝑣 ) = ( 𝑢 − 𝐵 ) ) |
64 |
63
|
fveq2d |
⊢ ( ( 𝜑 ∧ 𝑣 = 𝐵 ) → ( 𝐹 ‘ ( 𝑢 − 𝑣 ) ) = ( 𝐹 ‘ ( 𝑢 − 𝐵 ) ) ) |
65 |
62 64
|
oveq12d |
⊢ ( ( 𝜑 ∧ 𝑣 = 𝐵 ) → ( ( 𝐹 ‘ ( 𝑢 + 𝑣 ) ) + ( 𝐹 ‘ ( 𝑢 − 𝑣 ) ) ) = ( ( 𝐹 ‘ ( 𝑢 + 𝐵 ) ) + ( 𝐹 ‘ ( 𝑢 − 𝐵 ) ) ) ) |
66 |
60
|
fveq2d |
⊢ ( ( 𝜑 ∧ 𝑣 = 𝐵 ) → ( 𝐺 ‘ 𝑣 ) = ( 𝐺 ‘ 𝐵 ) ) |
67 |
66
|
oveq2d |
⊢ ( ( 𝜑 ∧ 𝑣 = 𝐵 ) → ( ( 𝐹 ‘ 𝑢 ) · ( 𝐺 ‘ 𝑣 ) ) = ( ( 𝐹 ‘ 𝑢 ) · ( 𝐺 ‘ 𝐵 ) ) ) |
68 |
67
|
oveq2d |
⊢ ( ( 𝜑 ∧ 𝑣 = 𝐵 ) → ( 2 · ( ( 𝐹 ‘ 𝑢 ) · ( 𝐺 ‘ 𝑣 ) ) ) = ( 2 · ( ( 𝐹 ‘ 𝑢 ) · ( 𝐺 ‘ 𝐵 ) ) ) ) |
69 |
65 68
|
eqeq12d |
⊢ ( ( 𝜑 ∧ 𝑣 = 𝐵 ) → ( ( ( 𝐹 ‘ ( 𝑢 + 𝑣 ) ) + ( 𝐹 ‘ ( 𝑢 − 𝑣 ) ) ) = ( 2 · ( ( 𝐹 ‘ 𝑢 ) · ( 𝐺 ‘ 𝑣 ) ) ) ↔ ( ( 𝐹 ‘ ( 𝑢 + 𝐵 ) ) + ( 𝐹 ‘ ( 𝑢 − 𝐵 ) ) ) = ( 2 · ( ( 𝐹 ‘ 𝑢 ) · ( 𝐺 ‘ 𝐵 ) ) ) ) ) |
70 |
69
|
ralbidv |
⊢ ( ( 𝜑 ∧ 𝑣 = 𝐵 ) → ( ∀ 𝑢 ∈ ℝ ( ( 𝐹 ‘ ( 𝑢 + 𝑣 ) ) + ( 𝐹 ‘ ( 𝑢 − 𝑣 ) ) ) = ( 2 · ( ( 𝐹 ‘ 𝑢 ) · ( 𝐺 ‘ 𝑣 ) ) ) ↔ ∀ 𝑢 ∈ ℝ ( ( 𝐹 ‘ ( 𝑢 + 𝐵 ) ) + ( 𝐹 ‘ ( 𝑢 − 𝐵 ) ) ) = ( 2 · ( ( 𝐹 ‘ 𝑢 ) · ( 𝐺 ‘ 𝐵 ) ) ) ) ) |
71 |
|
ralcom |
⊢ ( ∀ 𝑢 ∈ ℝ ∀ 𝑣 ∈ ℝ ( ( 𝐹 ‘ ( 𝑢 + 𝑣 ) ) + ( 𝐹 ‘ ( 𝑢 − 𝑣 ) ) ) = ( 2 · ( ( 𝐹 ‘ 𝑢 ) · ( 𝐺 ‘ 𝑣 ) ) ) ↔ ∀ 𝑣 ∈ ℝ ∀ 𝑢 ∈ ℝ ( ( 𝐹 ‘ ( 𝑢 + 𝑣 ) ) + ( 𝐹 ‘ ( 𝑢 − 𝑣 ) ) ) = ( 2 · ( ( 𝐹 ‘ 𝑢 ) · ( 𝐺 ‘ 𝑣 ) ) ) ) |
72 |
71
|
biimpi |
⊢ ( ∀ 𝑢 ∈ ℝ ∀ 𝑣 ∈ ℝ ( ( 𝐹 ‘ ( 𝑢 + 𝑣 ) ) + ( 𝐹 ‘ ( 𝑢 − 𝑣 ) ) ) = ( 2 · ( ( 𝐹 ‘ 𝑢 ) · ( 𝐺 ‘ 𝑣 ) ) ) → ∀ 𝑣 ∈ ℝ ∀ 𝑢 ∈ ℝ ( ( 𝐹 ‘ ( 𝑢 + 𝑣 ) ) + ( 𝐹 ‘ ( 𝑢 − 𝑣 ) ) ) = ( 2 · ( ( 𝐹 ‘ 𝑢 ) · ( 𝐺 ‘ 𝑣 ) ) ) ) |
73 |
72
|
a1i |
⊢ ( 𝜑 → ( ∀ 𝑢 ∈ ℝ ∀ 𝑣 ∈ ℝ ( ( 𝐹 ‘ ( 𝑢 + 𝑣 ) ) + ( 𝐹 ‘ ( 𝑢 − 𝑣 ) ) ) = ( 2 · ( ( 𝐹 ‘ 𝑢 ) · ( 𝐺 ‘ 𝑣 ) ) ) → ∀ 𝑣 ∈ ℝ ∀ 𝑢 ∈ ℝ ( ( 𝐹 ‘ ( 𝑢 + 𝑣 ) ) + ( 𝐹 ‘ ( 𝑢 − 𝑣 ) ) ) = ( 2 · ( ( 𝐹 ‘ 𝑢 ) · ( 𝐺 ‘ 𝑣 ) ) ) ) ) |
74 |
73
|
imp |
⊢ ( ( 𝜑 ∧ ∀ 𝑢 ∈ ℝ ∀ 𝑣 ∈ ℝ ( ( 𝐹 ‘ ( 𝑢 + 𝑣 ) ) + ( 𝐹 ‘ ( 𝑢 − 𝑣 ) ) ) = ( 2 · ( ( 𝐹 ‘ 𝑢 ) · ( 𝐺 ‘ 𝑣 ) ) ) ) → ∀ 𝑣 ∈ ℝ ∀ 𝑢 ∈ ℝ ( ( 𝐹 ‘ ( 𝑢 + 𝑣 ) ) + ( 𝐹 ‘ ( 𝑢 − 𝑣 ) ) ) = ( 2 · ( ( 𝐹 ‘ 𝑢 ) · ( 𝐺 ‘ 𝑣 ) ) ) ) |
75 |
4 74
|
mpdan |
⊢ ( 𝜑 → ∀ 𝑣 ∈ ℝ ∀ 𝑢 ∈ ℝ ( ( 𝐹 ‘ ( 𝑢 + 𝑣 ) ) + ( 𝐹 ‘ ( 𝑢 − 𝑣 ) ) ) = ( 2 · ( ( 𝐹 ‘ 𝑢 ) · ( 𝐺 ‘ 𝑣 ) ) ) ) |
76 |
70 3 75
|
rspcdv2 |
⊢ ( 𝜑 → ∀ 𝑢 ∈ ℝ ( ( 𝐹 ‘ ( 𝑢 + 𝐵 ) ) + ( 𝐹 ‘ ( 𝑢 − 𝐵 ) ) ) = ( 2 · ( ( 𝐹 ‘ 𝑢 ) · ( 𝐺 ‘ 𝐵 ) ) ) ) |
77 |
76
|
r19.21bi |
⊢ ( ( 𝜑 ∧ 𝑢 ∈ ℝ ) → ( ( 𝐹 ‘ ( 𝑢 + 𝐵 ) ) + ( 𝐹 ‘ ( 𝑢 − 𝐵 ) ) ) = ( 2 · ( ( 𝐹 ‘ 𝑢 ) · ( 𝐺 ‘ 𝐵 ) ) ) ) |
78 |
77
|
adantlr |
⊢ ( ( ( 𝜑 ∧ 1 < ( abs ‘ ( 𝐺 ‘ 𝐵 ) ) ) ∧ 𝑢 ∈ ℝ ) → ( ( 𝐹 ‘ ( 𝑢 + 𝐵 ) ) + ( 𝐹 ‘ ( 𝑢 − 𝐵 ) ) ) = ( 2 · ( ( 𝐹 ‘ 𝑢 ) · ( 𝐺 ‘ 𝐵 ) ) ) ) |
79 |
5
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ 1 < ( abs ‘ ( 𝐺 ‘ 𝐵 ) ) ) ∧ 𝑢 ∈ ℝ ) → ∀ 𝑦 ∈ ℝ ( abs ‘ ( 𝐹 ‘ 𝑦 ) ) ≤ 1 ) |
80 |
56 57 58 59 78 79
|
imo72b2lem0 |
⊢ ( ( ( 𝜑 ∧ 1 < ( abs ‘ ( 𝐺 ‘ 𝐵 ) ) ) ∧ 𝑢 ∈ ℝ ) → ( ( abs ‘ ( 𝐹 ‘ 𝑢 ) ) · ( abs ‘ ( 𝐺 ‘ 𝐵 ) ) ) ≤ sup ( ( abs “ ( 𝐹 “ ℝ ) ) , ℝ , < ) ) |
81 |
|
0xr |
⊢ 0 ∈ ℝ* |
82 |
81
|
a1i |
⊢ ( ( ( 𝜑 ∧ 1 < ( abs ‘ ( 𝐺 ‘ 𝐵 ) ) ) ∧ 𝑢 ∈ ℝ ) → 0 ∈ ℝ* ) |
83 |
|
1xr |
⊢ 1 ∈ ℝ* |
84 |
83
|
a1i |
⊢ ( ( ( 𝜑 ∧ 1 < ( abs ‘ ( 𝐺 ‘ 𝐵 ) ) ) ∧ 𝑢 ∈ ℝ ) → 1 ∈ ℝ* ) |
85 |
16
|
adantr |
⊢ ( ( ( 𝜑 ∧ 1 < ( abs ‘ ( 𝐺 ‘ 𝐵 ) ) ) ∧ 𝑢 ∈ ℝ ) → ( abs ‘ ( 𝐺 ‘ 𝐵 ) ) ∈ ℝ ) |
86 |
85
|
rexrd |
⊢ ( ( ( 𝜑 ∧ 1 < ( abs ‘ ( 𝐺 ‘ 𝐵 ) ) ) ∧ 𝑢 ∈ ℝ ) → ( abs ‘ ( 𝐺 ‘ 𝐵 ) ) ∈ ℝ* ) |
87 |
51
|
a1i |
⊢ ( ( ( 𝜑 ∧ 1 < ( abs ‘ ( 𝐺 ‘ 𝐵 ) ) ) ∧ 𝑢 ∈ ℝ ) → 0 < 1 ) |
88 |
|
simplr |
⊢ ( ( ( 𝜑 ∧ 1 < ( abs ‘ ( 𝐺 ‘ 𝐵 ) ) ) ∧ 𝑢 ∈ ℝ ) → 1 < ( abs ‘ ( 𝐺 ‘ 𝐵 ) ) ) |
89 |
82 84 86 87 88
|
xrlttrd |
⊢ ( ( ( 𝜑 ∧ 1 < ( abs ‘ ( 𝐺 ‘ 𝐵 ) ) ) ∧ 𝑢 ∈ ℝ ) → 0 < ( abs ‘ ( 𝐺 ‘ 𝐵 ) ) ) |
90 |
23
|
ffvelrnda |
⊢ ( ( ( 𝜑 ∧ 1 < ( abs ‘ ( 𝐺 ‘ 𝐵 ) ) ) ∧ 𝑢 ∈ ℝ ) → ( 𝐹 ‘ 𝑢 ) ∈ ℝ ) |
91 |
90
|
recnd |
⊢ ( ( ( 𝜑 ∧ 1 < ( abs ‘ ( 𝐺 ‘ 𝐵 ) ) ) ∧ 𝑢 ∈ ℝ ) → ( 𝐹 ‘ 𝑢 ) ∈ ℂ ) |
92 |
91
|
abscld |
⊢ ( ( ( 𝜑 ∧ 1 < ( abs ‘ ( 𝐺 ‘ 𝐵 ) ) ) ∧ 𝑢 ∈ ℝ ) → ( abs ‘ ( 𝐹 ‘ 𝑢 ) ) ∈ ℝ ) |
93 |
46
|
adantr |
⊢ ( ( ( 𝜑 ∧ 1 < ( abs ‘ ( 𝐺 ‘ 𝐵 ) ) ) ∧ 𝑢 ∈ ℝ ) → sup ( ( abs “ ( 𝐹 “ ℝ ) ) , ℝ , < ) ∈ ℝ ) |
94 |
80 89 85 92 93
|
lemuldiv3d |
⊢ ( ( ( 𝜑 ∧ 1 < ( abs ‘ ( 𝐺 ‘ 𝐵 ) ) ) ∧ 𝑢 ∈ ℝ ) → ( abs ‘ ( 𝐹 ‘ 𝑢 ) ) ≤ ( sup ( ( abs “ ( 𝐹 “ ℝ ) ) , ℝ , < ) / ( abs ‘ ( 𝐺 ‘ 𝐵 ) ) ) ) |
95 |
94
|
ralrimiva |
⊢ ( ( 𝜑 ∧ 1 < ( abs ‘ ( 𝐺 ‘ 𝐵 ) ) ) → ∀ 𝑢 ∈ ℝ ( abs ‘ ( 𝐹 ‘ 𝑢 ) ) ≤ ( sup ( ( abs “ ( 𝐹 “ ℝ ) ) , ℝ , < ) / ( abs ‘ ( 𝐺 ‘ 𝐵 ) ) ) ) |
96 |
23 55 95
|
imo72b2lem2 |
⊢ ( ( 𝜑 ∧ 1 < ( abs ‘ ( 𝐺 ‘ 𝐵 ) ) ) → sup ( ( abs “ ( 𝐹 “ ℝ ) ) , ℝ , < ) ≤ ( sup ( ( abs “ ( 𝐹 “ ℝ ) ) , ℝ , < ) / ( abs ‘ ( 𝐺 ‘ 𝐵 ) ) ) ) |
97 |
96 53 16 46 46
|
lemuldiv4d |
⊢ ( ( 𝜑 ∧ 1 < ( abs ‘ ( 𝐺 ‘ 𝐵 ) ) ) → ( sup ( ( abs “ ( 𝐹 “ ℝ ) ) , ℝ , < ) · ( abs ‘ ( 𝐺 ‘ 𝐵 ) ) ) ≤ sup ( ( abs “ ( 𝐹 “ ℝ ) ) , ℝ , < ) ) |
98 |
49 97
|
eqbrtrrd |
⊢ ( ( 𝜑 ∧ 1 < ( abs ‘ ( 𝐺 ‘ 𝐵 ) ) ) → ( ( abs ‘ ( 𝐺 ‘ 𝐵 ) ) · sup ( ( abs “ ( 𝐹 “ ℝ ) ) , ℝ , < ) ) ≤ sup ( ( abs “ ( 𝐹 “ ℝ ) ) , ℝ , < ) ) |
99 |
6
|
adantr |
⊢ ( ( 𝜑 ∧ 1 < ( abs ‘ ( 𝐺 ‘ 𝐵 ) ) ) → ∃ 𝑥 ∈ ℝ ( 𝐹 ‘ 𝑥 ) ≠ 0 ) |
100 |
5
|
adantr |
⊢ ( ( 𝜑 ∧ 1 < ( abs ‘ ( 𝐺 ‘ 𝐵 ) ) ) → ∀ 𝑦 ∈ ℝ ( abs ‘ ( 𝐹 ‘ 𝑦 ) ) ≤ 1 ) |
101 |
23 99 100
|
imo72b2lem1 |
⊢ ( ( 𝜑 ∧ 1 < ( abs ‘ ( 𝐺 ‘ 𝐵 ) ) ) → 0 < sup ( ( abs “ ( 𝐹 “ ℝ ) ) , ℝ , < ) ) |
102 |
98 101 46 16 46
|
lemuldiv3d |
⊢ ( ( 𝜑 ∧ 1 < ( abs ‘ ( 𝐺 ‘ 𝐵 ) ) ) → ( abs ‘ ( 𝐺 ‘ 𝐵 ) ) ≤ ( sup ( ( abs “ ( 𝐹 “ ℝ ) ) , ℝ , < ) / sup ( ( abs “ ( 𝐹 “ ℝ ) ) , ℝ , < ) ) ) |
103 |
26 46
|
sseldd |
⊢ ( ( 𝜑 ∧ 1 < ( abs ‘ ( 𝐺 ‘ 𝐵 ) ) ) → sup ( ( abs “ ( 𝐹 “ ℝ ) ) , ℝ , < ) ∈ ℂ ) |
104 |
101
|
gt0ne0d |
⊢ ( ( 𝜑 ∧ 1 < ( abs ‘ ( 𝐺 ‘ 𝐵 ) ) ) → sup ( ( abs “ ( 𝐹 “ ℝ ) ) , ℝ , < ) ≠ 0 ) |
105 |
103 104
|
dividd |
⊢ ( ( 𝜑 ∧ 1 < ( abs ‘ ( 𝐺 ‘ 𝐵 ) ) ) → ( sup ( ( abs “ ( 𝐹 “ ℝ ) ) , ℝ , < ) / sup ( ( abs “ ( 𝐹 “ ℝ ) ) , ℝ , < ) ) = 1 ) |
106 |
105
|
eqcomd |
⊢ ( ( 𝜑 ∧ 1 < ( abs ‘ ( 𝐺 ‘ 𝐵 ) ) ) → 1 = ( sup ( ( abs “ ( 𝐹 “ ℝ ) ) , ℝ , < ) / sup ( ( abs “ ( 𝐹 “ ℝ ) ) , ℝ , < ) ) ) |
107 |
102 106
|
breqtrrd |
⊢ ( ( 𝜑 ∧ 1 < ( abs ‘ ( 𝐺 ‘ 𝐵 ) ) ) → ( abs ‘ ( 𝐺 ‘ 𝐵 ) ) ≤ 1 ) |
108 |
16 17 107
|
lensymd |
⊢ ( ( 𝜑 ∧ 1 < ( abs ‘ ( 𝐺 ‘ 𝐵 ) ) ) → ¬ 1 < ( abs ‘ ( 𝐺 ‘ 𝐵 ) ) ) |
109 |
11 108
|
pm2.65da |
⊢ ( 𝜑 → ¬ 1 < ( abs ‘ ( 𝐺 ‘ 𝐵 ) ) ) |
110 |
9 10 109
|
nltled |
⊢ ( 𝜑 → ( abs ‘ ( 𝐺 ‘ 𝐵 ) ) ≤ 1 ) |