Step | Hyp | Ref
| Expression |
1 | | imo72b2.2 |
. . . . 5
⊢ (𝜑 → 𝐺:ℝ⟶ℝ) |
2 | | imo72b2.4 |
. . . . 5
⊢ (𝜑 → 𝐵 ∈ ℝ) |
3 | 1, 2 | ffvelrnd 6862 |
. . . 4
⊢ (𝜑 → (𝐺‘𝐵) ∈ ℝ) |
4 | 3 | recnd 10747 |
. . 3
⊢ (𝜑 → (𝐺‘𝐵) ∈ ℂ) |
5 | 4 | abscld 14886 |
. 2
⊢ (𝜑 → (abs‘(𝐺‘𝐵)) ∈ ℝ) |
6 | | 1red 10720 |
. 2
⊢ (𝜑 → 1 ∈
ℝ) |
7 | | simpr 488 |
. . 3
⊢ ((𝜑 ∧ 1 < (abs‘(𝐺‘𝐵))) → 1 < (abs‘(𝐺‘𝐵))) |
8 | 1 | adantr 484 |
. . . . . . 7
⊢ ((𝜑 ∧ 1 < (abs‘(𝐺‘𝐵))) → 𝐺:ℝ⟶ℝ) |
9 | 2 | adantr 484 |
. . . . . . 7
⊢ ((𝜑 ∧ 1 < (abs‘(𝐺‘𝐵))) → 𝐵 ∈ ℝ) |
10 | 8, 9 | ffvelrnd 6862 |
. . . . . 6
⊢ ((𝜑 ∧ 1 < (abs‘(𝐺‘𝐵))) → (𝐺‘𝐵) ∈ ℝ) |
11 | 10 | recnd 10747 |
. . . . 5
⊢ ((𝜑 ∧ 1 < (abs‘(𝐺‘𝐵))) → (𝐺‘𝐵) ∈ ℂ) |
12 | 11 | abscld 14886 |
. . . 4
⊢ ((𝜑 ∧ 1 < (abs‘(𝐺‘𝐵))) → (abs‘(𝐺‘𝐵)) ∈ ℝ) |
13 | 6 | adantr 484 |
. . . 4
⊢ ((𝜑 ∧ 1 < (abs‘(𝐺‘𝐵))) → 1 ∈
ℝ) |
14 | | ax-resscn 10672 |
. . . . . . . . 9
⊢ ℝ
⊆ ℂ |
15 | | imaco 6084 |
. . . . . . . . . . . 12
⊢ ((abs
∘ 𝐹) “ ℝ)
= (abs “ (𝐹 “
ℝ)) |
16 | 15 | eqcomi 2747 |
. . . . . . . . . . 11
⊢ (abs
“ (𝐹 “
ℝ)) = ((abs ∘ 𝐹) “ ℝ) |
17 | | imassrn 5914 |
. . . . . . . . . . . . 13
⊢ ((abs
∘ 𝐹) “ ℝ)
⊆ ran (abs ∘ 𝐹) |
18 | 17 | a1i 11 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 1 < (abs‘(𝐺‘𝐵))) → ((abs ∘ 𝐹) “ ℝ) ⊆ ran (abs ∘
𝐹)) |
19 | | imo72b2.1 |
. . . . . . . . . . . . . . 15
⊢ (𝜑 → 𝐹:ℝ⟶ℝ) |
20 | 19 | adantr 484 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 1 < (abs‘(𝐺‘𝐵))) → 𝐹:ℝ⟶ℝ) |
21 | | absf 14787 |
. . . . . . . . . . . . . . . 16
⊢
abs:ℂ⟶ℝ |
22 | 21 | a1i 11 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 1 < (abs‘(𝐺‘𝐵))) →
abs:ℂ⟶ℝ) |
23 | 14 | a1i 11 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 1 < (abs‘(𝐺‘𝐵))) → ℝ ⊆
ℂ) |
24 | 22, 23 | fssresd 6545 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 1 < (abs‘(𝐺‘𝐵))) → (abs ↾
ℝ):ℝ⟶ℝ) |
25 | 20, 24 | fco2d 41319 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 1 < (abs‘(𝐺‘𝐵))) → (abs ∘ 𝐹):ℝ⟶ℝ) |
26 | 25 | frnd 6512 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 1 < (abs‘(𝐺‘𝐵))) → ran (abs ∘ 𝐹) ⊆
ℝ) |
27 | 18, 26 | sstrd 3887 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 1 < (abs‘(𝐺‘𝐵))) → ((abs ∘ 𝐹) “ ℝ) ⊆
ℝ) |
28 | 16, 27 | eqsstrid 3925 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 1 < (abs‘(𝐺‘𝐵))) → (abs “ (𝐹 “ ℝ)) ⊆
ℝ) |
29 | | 0re 10721 |
. . . . . . . . . . . . . . . 16
⊢ 0 ∈
ℝ |
30 | 29 | ne0ii 4226 |
. . . . . . . . . . . . . . 15
⊢ ℝ
≠ ∅ |
31 | 30 | a1i 11 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 1 < (abs‘(𝐺‘𝐵))) → ℝ ≠
∅) |
32 | 31, 25 | wnefimgd 41318 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 1 < (abs‘(𝐺‘𝐵))) → ((abs ∘ 𝐹) “ ℝ) ≠
∅) |
33 | 32 | necomd 2989 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 1 < (abs‘(𝐺‘𝐵))) → ∅ ≠ ((abs ∘ 𝐹) “
ℝ)) |
34 | 16 | a1i 11 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 1 < (abs‘(𝐺‘𝐵))) → (abs “ (𝐹 “ ℝ)) = ((abs ∘ 𝐹) “
ℝ)) |
35 | 33, 34 | neeqtrrd 3008 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 1 < (abs‘(𝐺‘𝐵))) → ∅ ≠ (abs “ (𝐹 “
ℝ))) |
36 | 35 | necomd 2989 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 1 < (abs‘(𝐺‘𝐵))) → (abs “ (𝐹 “ ℝ)) ≠
∅) |
37 | | simpr 488 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 1 < (abs‘(𝐺‘𝐵))) ∧ 𝑐 = 1) → 𝑐 = 1) |
38 | 37 | breq2d 5042 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ 1 < (abs‘(𝐺‘𝐵))) ∧ 𝑐 = 1) → (𝑡 ≤ 𝑐 ↔ 𝑡 ≤ 1)) |
39 | 38 | ralbidv 3109 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ 1 < (abs‘(𝐺‘𝐵))) ∧ 𝑐 = 1) → (∀𝑡 ∈ (abs “ (𝐹 “ ℝ))𝑡 ≤ 𝑐 ↔ ∀𝑡 ∈ (abs “ (𝐹 “ ℝ))𝑡 ≤ 1)) |
40 | | imo72b2.6 |
. . . . . . . . . . . . 13
⊢ (𝜑 → ∀𝑦 ∈ ℝ (abs‘(𝐹‘𝑦)) ≤ 1) |
41 | 19, 40 | extoimad 41321 |
. . . . . . . . . . . 12
⊢ (𝜑 → ∀𝑡 ∈ (abs “ (𝐹 “ ℝ))𝑡 ≤ 1) |
42 | 41 | adantr 484 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 1 < (abs‘(𝐺‘𝐵))) → ∀𝑡 ∈ (abs “ (𝐹 “ ℝ))𝑡 ≤ 1) |
43 | 13, 39, 42 | rspcedvd 3529 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 1 < (abs‘(𝐺‘𝐵))) → ∃𝑐 ∈ ℝ ∀𝑡 ∈ (abs “ (𝐹 “ ℝ))𝑡 ≤ 𝑐) |
44 | 28, 36, 43 | suprcld 11681 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 1 < (abs‘(𝐺‘𝐵))) → sup((abs “ (𝐹 “ ℝ)), ℝ,
< ) ∈ ℝ) |
45 | 14, 44 | sseldi 3875 |
. . . . . . . 8
⊢ ((𝜑 ∧ 1 < (abs‘(𝐺‘𝐵))) → sup((abs “ (𝐹 “ ℝ)), ℝ,
< ) ∈ ℂ) |
46 | 14, 12 | sseldi 3875 |
. . . . . . . 8
⊢ ((𝜑 ∧ 1 < (abs‘(𝐺‘𝐵))) → (abs‘(𝐺‘𝐵)) ∈ ℂ) |
47 | 45, 46 | mulcomd 10740 |
. . . . . . 7
⊢ ((𝜑 ∧ 1 < (abs‘(𝐺‘𝐵))) → (sup((abs “ (𝐹 “ ℝ)), ℝ,
< ) · (abs‘(𝐺‘𝐵))) = ((abs‘(𝐺‘𝐵)) · sup((abs “ (𝐹 “ ℝ)), ℝ,
< ))) |
48 | 29 | a1i 11 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 1 < (abs‘(𝐺‘𝐵))) → 0 ∈
ℝ) |
49 | | 0lt1 11240 |
. . . . . . . . . . . . 13
⊢ 0 <
1 |
50 | 49 | a1i 11 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 1 < (abs‘(𝐺‘𝐵))) → 0 < 1) |
51 | 48, 13, 12, 50, 7 | lttrd 10879 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 1 < (abs‘(𝐺‘𝐵))) → 0 < (abs‘(𝐺‘𝐵))) |
52 | 51 | gt0ne0d 11282 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 1 < (abs‘(𝐺‘𝐵))) → (abs‘(𝐺‘𝐵)) ≠ 0) |
53 | 44, 12, 52 | redivcld 11546 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 1 < (abs‘(𝐺‘𝐵))) → (sup((abs “ (𝐹 “ ℝ)), ℝ,
< ) / (abs‘(𝐺‘𝐵))) ∈ ℝ) |
54 | 20 | adantr 484 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ 1 < (abs‘(𝐺‘𝐵))) ∧ 𝑢 ∈ ℝ) → 𝐹:ℝ⟶ℝ) |
55 | 8 | adantr 484 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ 1 < (abs‘(𝐺‘𝐵))) ∧ 𝑢 ∈ ℝ) → 𝐺:ℝ⟶ℝ) |
56 | | simpr 488 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ 1 < (abs‘(𝐺‘𝐵))) ∧ 𝑢 ∈ ℝ) → 𝑢 ∈ ℝ) |
57 | 9 | adantr 484 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ 1 < (abs‘(𝐺‘𝐵))) ∧ 𝑢 ∈ ℝ) → 𝐵 ∈ ℝ) |
58 | | simpr 488 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝜑 ∧ 𝑣 = 𝐵) → 𝑣 = 𝐵) |
59 | 58 | oveq2d 7186 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝜑 ∧ 𝑣 = 𝐵) → (𝑢 + 𝑣) = (𝑢 + 𝐵)) |
60 | 59 | fveq2d 6678 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝜑 ∧ 𝑣 = 𝐵) → (𝐹‘(𝑢 + 𝑣)) = (𝐹‘(𝑢 + 𝐵))) |
61 | 58 | oveq2d 7186 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝜑 ∧ 𝑣 = 𝐵) → (𝑢 − 𝑣) = (𝑢 − 𝐵)) |
62 | 61 | fveq2d 6678 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝜑 ∧ 𝑣 = 𝐵) → (𝐹‘(𝑢 − 𝑣)) = (𝐹‘(𝑢 − 𝐵))) |
63 | 60, 62 | oveq12d 7188 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝜑 ∧ 𝑣 = 𝐵) → ((𝐹‘(𝑢 + 𝑣)) + (𝐹‘(𝑢 − 𝑣))) = ((𝐹‘(𝑢 + 𝐵)) + (𝐹‘(𝑢 − 𝐵)))) |
64 | 58 | fveq2d 6678 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝜑 ∧ 𝑣 = 𝐵) → (𝐺‘𝑣) = (𝐺‘𝐵)) |
65 | 64 | oveq2d 7186 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝜑 ∧ 𝑣 = 𝐵) → ((𝐹‘𝑢) · (𝐺‘𝑣)) = ((𝐹‘𝑢) · (𝐺‘𝐵))) |
66 | 65 | oveq2d 7186 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝜑 ∧ 𝑣 = 𝐵) → (2 · ((𝐹‘𝑢) · (𝐺‘𝑣))) = (2 · ((𝐹‘𝑢) · (𝐺‘𝐵)))) |
67 | 63, 66 | eqeq12d 2754 |
. . . . . . . . . . . . . . . 16
⊢ ((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))))) |
68 | 67 | ralbidv 3109 |
. . . . . . . . . . . . . . 15
⊢ ((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))))) |
69 | | imo72b2.5 |
. . . . . . . . . . . . . . . 16
⊢ (𝜑 → ∀𝑢 ∈ ℝ ∀𝑣 ∈ ℝ ((𝐹‘(𝑢 + 𝑣)) + (𝐹‘(𝑢 − 𝑣))) = (2 · ((𝐹‘𝑢) · (𝐺‘𝑣)))) |
70 | | ralcom 3258 |
. . . . . . . . . . . . . . . . . . 19
⊢
(∀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)))) |
71 | 70 | biimpi 219 |
. . . . . . . . . . . . . . . . . 18
⊢
(∀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 | a1i 11 |
. . . . . . . . . . . . . . . . 17
⊢ (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))))) |
73 | 72 | imp 410 |
. . . . . . . . . . . . . . . 16
⊢ ((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 | 69, 73 | mpdan 687 |
. . . . . . . . . . . . . . 15
⊢ (𝜑 → ∀𝑣 ∈ ℝ ∀𝑢 ∈ ℝ ((𝐹‘(𝑢 + 𝑣)) + (𝐹‘(𝑢 − 𝑣))) = (2 · ((𝐹‘𝑢) · (𝐺‘𝑣)))) |
75 | 68, 2, 74 | rspcdvinvd 41329 |
. . . . . . . . . . . . . 14
⊢ (𝜑 → ∀𝑢 ∈ ℝ ((𝐹‘(𝑢 + 𝐵)) + (𝐹‘(𝑢 − 𝐵))) = (2 · ((𝐹‘𝑢) · (𝐺‘𝐵)))) |
76 | 75 | r19.21bi 3121 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑢 ∈ ℝ) → ((𝐹‘(𝑢 + 𝐵)) + (𝐹‘(𝑢 − 𝐵))) = (2 · ((𝐹‘𝑢) · (𝐺‘𝐵)))) |
77 | 76 | adantlr 715 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ 1 < (abs‘(𝐺‘𝐵))) ∧ 𝑢 ∈ ℝ) → ((𝐹‘(𝑢 + 𝐵)) + (𝐹‘(𝑢 − 𝐵))) = (2 · ((𝐹‘𝑢) · (𝐺‘𝐵)))) |
78 | 40 | ad2antrr 726 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ 1 < (abs‘(𝐺‘𝐵))) ∧ 𝑢 ∈ ℝ) → ∀𝑦 ∈ ℝ
(abs‘(𝐹‘𝑦)) ≤ 1) |
79 | 54, 55, 56, 57, 77, 78 | imo72b2lem0 41322 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ 1 < (abs‘(𝐺‘𝐵))) ∧ 𝑢 ∈ ℝ) → ((abs‘(𝐹‘𝑢)) · (abs‘(𝐺‘𝐵))) ≤ sup((abs “ (𝐹 “ ℝ)), ℝ, <
)) |
80 | | 0xr 10766 |
. . . . . . . . . . . . 13
⊢ 0 ∈
ℝ* |
81 | 80 | a1i 11 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ 1 < (abs‘(𝐺‘𝐵))) ∧ 𝑢 ∈ ℝ) → 0 ∈
ℝ*) |
82 | | 1xr 10778 |
. . . . . . . . . . . . 13
⊢ 1 ∈
ℝ* |
83 | 82 | a1i 11 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ 1 < (abs‘(𝐺‘𝐵))) ∧ 𝑢 ∈ ℝ) → 1 ∈
ℝ*) |
84 | 12 | adantr 484 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 1 < (abs‘(𝐺‘𝐵))) ∧ 𝑢 ∈ ℝ) → (abs‘(𝐺‘𝐵)) ∈ ℝ) |
85 | 84 | rexrd 10769 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ 1 < (abs‘(𝐺‘𝐵))) ∧ 𝑢 ∈ ℝ) → (abs‘(𝐺‘𝐵)) ∈
ℝ*) |
86 | 49 | a1i 11 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ 1 < (abs‘(𝐺‘𝐵))) ∧ 𝑢 ∈ ℝ) → 0 <
1) |
87 | | simplr 769 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ 1 < (abs‘(𝐺‘𝐵))) ∧ 𝑢 ∈ ℝ) → 1 <
(abs‘(𝐺‘𝐵))) |
88 | 81, 83, 85, 86, 87 | xrlttrd 12635 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ 1 < (abs‘(𝐺‘𝐵))) ∧ 𝑢 ∈ ℝ) → 0 <
(abs‘(𝐺‘𝐵))) |
89 | 20 | ffvelrnda 6861 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 1 < (abs‘(𝐺‘𝐵))) ∧ 𝑢 ∈ ℝ) → (𝐹‘𝑢) ∈ ℝ) |
90 | 89 | recnd 10747 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ 1 < (abs‘(𝐺‘𝐵))) ∧ 𝑢 ∈ ℝ) → (𝐹‘𝑢) ∈ ℂ) |
91 | 90 | abscld 14886 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ 1 < (abs‘(𝐺‘𝐵))) ∧ 𝑢 ∈ ℝ) → (abs‘(𝐹‘𝑢)) ∈ ℝ) |
92 | 44 | adantr 484 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ 1 < (abs‘(𝐺‘𝐵))) ∧ 𝑢 ∈ ℝ) → sup((abs “
(𝐹 “ ℝ)),
ℝ, < ) ∈ ℝ) |
93 | 79, 88, 84, 91, 92 | lemuldiv3d 41327 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 1 < (abs‘(𝐺‘𝐵))) ∧ 𝑢 ∈ ℝ) → (abs‘(𝐹‘𝑢)) ≤ (sup((abs “ (𝐹 “ ℝ)), ℝ, < ) /
(abs‘(𝐺‘𝐵)))) |
94 | 93 | ralrimiva 3096 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 1 < (abs‘(𝐺‘𝐵))) → ∀𝑢 ∈ ℝ (abs‘(𝐹‘𝑢)) ≤ (sup((abs “ (𝐹 “ ℝ)), ℝ, < ) /
(abs‘(𝐺‘𝐵)))) |
95 | 20, 53, 94 | imo72b2lem2 41324 |
. . . . . . . 8
⊢ ((𝜑 ∧ 1 < (abs‘(𝐺‘𝐵))) → sup((abs “ (𝐹 “ ℝ)), ℝ,
< ) ≤ (sup((abs “ (𝐹 “ ℝ)), ℝ, < ) /
(abs‘(𝐺‘𝐵)))) |
96 | 95, 51, 12, 44, 44 | lemuldiv4d 41328 |
. . . . . . 7
⊢ ((𝜑 ∧ 1 < (abs‘(𝐺‘𝐵))) → (sup((abs “ (𝐹 “ ℝ)), ℝ,
< ) · (abs‘(𝐺‘𝐵))) ≤ sup((abs “ (𝐹 “ ℝ)), ℝ, <
)) |
97 | 47, 96 | eqbrtrrd 5054 |
. . . . . 6
⊢ ((𝜑 ∧ 1 < (abs‘(𝐺‘𝐵))) → ((abs‘(𝐺‘𝐵)) · sup((abs “ (𝐹 “ ℝ)), ℝ,
< )) ≤ sup((abs “ (𝐹 “ ℝ)), ℝ, <
)) |
98 | | imo72b2.7 |
. . . . . . . 8
⊢ (𝜑 → ∃𝑥 ∈ ℝ (𝐹‘𝑥) ≠ 0) |
99 | 98 | adantr 484 |
. . . . . . 7
⊢ ((𝜑 ∧ 1 < (abs‘(𝐺‘𝐵))) → ∃𝑥 ∈ ℝ (𝐹‘𝑥) ≠ 0) |
100 | 40 | adantr 484 |
. . . . . . 7
⊢ ((𝜑 ∧ 1 < (abs‘(𝐺‘𝐵))) → ∀𝑦 ∈ ℝ (abs‘(𝐹‘𝑦)) ≤ 1) |
101 | 20, 99, 100 | imo72b2lem1 41326 |
. . . . . 6
⊢ ((𝜑 ∧ 1 < (abs‘(𝐺‘𝐵))) → 0 < sup((abs “ (𝐹 “ ℝ)), ℝ,
< )) |
102 | 97, 101, 44, 12, 44 | lemuldiv3d 41327 |
. . . . 5
⊢ ((𝜑 ∧ 1 < (abs‘(𝐺‘𝐵))) → (abs‘(𝐺‘𝐵)) ≤ (sup((abs “ (𝐹 “ ℝ)), ℝ, < ) /
sup((abs “ (𝐹 “
ℝ)), ℝ, < ))) |
103 | 23, 44 | sseldd 3878 |
. . . . . . 7
⊢ ((𝜑 ∧ 1 < (abs‘(𝐺‘𝐵))) → sup((abs “ (𝐹 “ ℝ)), ℝ,
< ) ∈ ℂ) |
104 | 101 | gt0ne0d 11282 |
. . . . . . 7
⊢ ((𝜑 ∧ 1 < (abs‘(𝐺‘𝐵))) → sup((abs “ (𝐹 “ ℝ)), ℝ,
< ) ≠ 0) |
105 | 103, 104 | dividd 11492 |
. . . . . 6
⊢ ((𝜑 ∧ 1 < (abs‘(𝐺‘𝐵))) → (sup((abs “ (𝐹 “ ℝ)), ℝ,
< ) / sup((abs “ (𝐹 “ ℝ)), ℝ, < )) =
1) |
106 | 105 | eqcomd 2744 |
. . . . 5
⊢ ((𝜑 ∧ 1 < (abs‘(𝐺‘𝐵))) → 1 = (sup((abs “ (𝐹 “ ℝ)), ℝ,
< ) / sup((abs “ (𝐹 “ ℝ)), ℝ, <
))) |
107 | 102, 106 | breqtrrd 5058 |
. . . 4
⊢ ((𝜑 ∧ 1 < (abs‘(𝐺‘𝐵))) → (abs‘(𝐺‘𝐵)) ≤ 1) |
108 | 12, 13, 107 | lensymd 10869 |
. . 3
⊢ ((𝜑 ∧ 1 < (abs‘(𝐺‘𝐵))) → ¬ 1 < (abs‘(𝐺‘𝐵))) |
109 | 7, 108 | pm2.65da 817 |
. 2
⊢ (𝜑 → ¬ 1 <
(abs‘(𝐺‘𝐵))) |
110 | 5, 6, 109 | nltled 10868 |
1
⊢ (𝜑 → (abs‘(𝐺‘𝐵)) ≤ 1) |