Step | Hyp | Ref
| Expression |
1 | | cncficcgt0.fcn |
. . . . . . . 8
⊢ (𝜑 → 𝐹 ∈ ((𝐴[,]𝐵)–cn→(ℝ ∖ {0}))) |
2 | | cncff 23645 |
. . . . . . . 8
⊢ (𝐹 ∈ ((𝐴[,]𝐵)–cn→(ℝ ∖ {0})) → 𝐹:(𝐴[,]𝐵)⟶(ℝ ∖
{0})) |
3 | | ffun 6507 |
. . . . . . . 8
⊢ (𝐹:(𝐴[,]𝐵)⟶(ℝ ∖ {0}) → Fun
𝐹) |
4 | 1, 2, 3 | 3syl 18 |
. . . . . . 7
⊢ (𝜑 → Fun 𝐹) |
5 | 4 | adantr 484 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑐 ∈ (𝐴[,]𝐵)) → Fun 𝐹) |
6 | | simpr 488 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑐 ∈ (𝐴[,]𝐵)) → 𝑐 ∈ (𝐴[,]𝐵)) |
7 | 1, 2 | syl 17 |
. . . . . . . . . 10
⊢ (𝜑 → 𝐹:(𝐴[,]𝐵)⟶(ℝ ∖
{0})) |
8 | 7 | fdmd 6515 |
. . . . . . . . 9
⊢ (𝜑 → dom 𝐹 = (𝐴[,]𝐵)) |
9 | 8 | eqcomd 2744 |
. . . . . . . 8
⊢ (𝜑 → (𝐴[,]𝐵) = dom 𝐹) |
10 | 9 | adantr 484 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑐 ∈ (𝐴[,]𝐵)) → (𝐴[,]𝐵) = dom 𝐹) |
11 | 6, 10 | eleqtrd 2835 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑐 ∈ (𝐴[,]𝐵)) → 𝑐 ∈ dom 𝐹) |
12 | | fvco 6766 |
. . . . . 6
⊢ ((Fun
𝐹 ∧ 𝑐 ∈ dom 𝐹) → ((abs ∘ 𝐹)‘𝑐) = (abs‘(𝐹‘𝑐))) |
13 | 5, 11, 12 | syl2anc 587 |
. . . . 5
⊢ ((𝜑 ∧ 𝑐 ∈ (𝐴[,]𝐵)) → ((abs ∘ 𝐹)‘𝑐) = (abs‘(𝐹‘𝑐))) |
14 | 7 | ffvelrnda 6861 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑐 ∈ (𝐴[,]𝐵)) → (𝐹‘𝑐) ∈ (ℝ ∖
{0})) |
15 | 14 | eldifad 3855 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑐 ∈ (𝐴[,]𝐵)) → (𝐹‘𝑐) ∈ ℝ) |
16 | 15 | recnd 10747 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑐 ∈ (𝐴[,]𝐵)) → (𝐹‘𝑐) ∈ ℂ) |
17 | | eldifsni 4678 |
. . . . . . 7
⊢ ((𝐹‘𝑐) ∈ (ℝ ∖ {0}) → (𝐹‘𝑐) ≠ 0) |
18 | 14, 17 | syl 17 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑐 ∈ (𝐴[,]𝐵)) → (𝐹‘𝑐) ≠ 0) |
19 | 16, 18 | absrpcld 14898 |
. . . . 5
⊢ ((𝜑 ∧ 𝑐 ∈ (𝐴[,]𝐵)) → (abs‘(𝐹‘𝑐)) ∈
ℝ+) |
20 | 13, 19 | eqeltrd 2833 |
. . . 4
⊢ ((𝜑 ∧ 𝑐 ∈ (𝐴[,]𝐵)) → ((abs ∘ 𝐹)‘𝑐) ∈
ℝ+) |
21 | 20 | adantr 484 |
. . 3
⊢ (((𝜑 ∧ 𝑐 ∈ (𝐴[,]𝐵)) ∧ ∀𝑑 ∈ (𝐴[,]𝐵)((abs ∘ 𝐹)‘𝑐) ≤ ((abs ∘ 𝐹)‘𝑑)) → ((abs ∘ 𝐹)‘𝑐) ∈
ℝ+) |
22 | | nfv 1921 |
. . . . 5
⊢
Ⅎ𝑥(𝜑 ∧ 𝑐 ∈ (𝐴[,]𝐵)) |
23 | | nfcv 2899 |
. . . . . 6
⊢
Ⅎ𝑥(𝐴[,]𝐵) |
24 | | nfcv 2899 |
. . . . . . . . 9
⊢
Ⅎ𝑥abs |
25 | | cncficcgt0.f |
. . . . . . . . . 10
⊢ 𝐹 = (𝑥 ∈ (𝐴[,]𝐵) ↦ 𝐶) |
26 | | nfmpt1 5128 |
. . . . . . . . . 10
⊢
Ⅎ𝑥(𝑥 ∈ (𝐴[,]𝐵) ↦ 𝐶) |
27 | 25, 26 | nfcxfr 2897 |
. . . . . . . . 9
⊢
Ⅎ𝑥𝐹 |
28 | 24, 27 | nfco 5708 |
. . . . . . . 8
⊢
Ⅎ𝑥(abs
∘ 𝐹) |
29 | | nfcv 2899 |
. . . . . . . 8
⊢
Ⅎ𝑥𝑐 |
30 | 28, 29 | nffv 6684 |
. . . . . . 7
⊢
Ⅎ𝑥((abs
∘ 𝐹)‘𝑐) |
31 | | nfcv 2899 |
. . . . . . 7
⊢
Ⅎ𝑥
≤ |
32 | | nfcv 2899 |
. . . . . . . 8
⊢
Ⅎ𝑥𝑑 |
33 | 28, 32 | nffv 6684 |
. . . . . . 7
⊢
Ⅎ𝑥((abs
∘ 𝐹)‘𝑑) |
34 | 30, 31, 33 | nfbr 5077 |
. . . . . 6
⊢
Ⅎ𝑥((abs
∘ 𝐹)‘𝑐) ≤ ((abs ∘ 𝐹)‘𝑑) |
35 | 23, 34 | nfralw 3138 |
. . . . 5
⊢
Ⅎ𝑥∀𝑑 ∈ (𝐴[,]𝐵)((abs ∘ 𝐹)‘𝑐) ≤ ((abs ∘ 𝐹)‘𝑑) |
36 | 22, 35 | nfan 1906 |
. . . 4
⊢
Ⅎ𝑥((𝜑 ∧ 𝑐 ∈ (𝐴[,]𝐵)) ∧ ∀𝑑 ∈ (𝐴[,]𝐵)((abs ∘ 𝐹)‘𝑐) ≤ ((abs ∘ 𝐹)‘𝑑)) |
37 | | fveq2 6674 |
. . . . . . . . 9
⊢ (𝑑 = 𝑥 → ((abs ∘ 𝐹)‘𝑑) = ((abs ∘ 𝐹)‘𝑥)) |
38 | 37 | breq2d 5042 |
. . . . . . . 8
⊢ (𝑑 = 𝑥 → (((abs ∘ 𝐹)‘𝑐) ≤ ((abs ∘ 𝐹)‘𝑑) ↔ ((abs ∘ 𝐹)‘𝑐) ≤ ((abs ∘ 𝐹)‘𝑥))) |
39 | 38 | rspccva 3525 |
. . . . . . 7
⊢
((∀𝑑 ∈
(𝐴[,]𝐵)((abs ∘ 𝐹)‘𝑐) ≤ ((abs ∘ 𝐹)‘𝑑) ∧ 𝑥 ∈ (𝐴[,]𝐵)) → ((abs ∘ 𝐹)‘𝑐) ≤ ((abs ∘ 𝐹)‘𝑥)) |
40 | 39 | adantll 714 |
. . . . . 6
⊢ ((((𝜑 ∧ 𝑐 ∈ (𝐴[,]𝐵)) ∧ ∀𝑑 ∈ (𝐴[,]𝐵)((abs ∘ 𝐹)‘𝑐) ≤ ((abs ∘ 𝐹)‘𝑑)) ∧ 𝑥 ∈ (𝐴[,]𝐵)) → ((abs ∘ 𝐹)‘𝑐) ≤ ((abs ∘ 𝐹)‘𝑥)) |
41 | | absf 14787 |
. . . . . . . . . . 11
⊢
abs:ℂ⟶ℝ |
42 | 41 | a1i 11 |
. . . . . . . . . 10
⊢ (𝜑 →
abs:ℂ⟶ℝ) |
43 | | difss 4022 |
. . . . . . . . . . . . 13
⊢ (ℝ
∖ {0}) ⊆ ℝ |
44 | | ax-resscn 10672 |
. . . . . . . . . . . . 13
⊢ ℝ
⊆ ℂ |
45 | 43, 44 | sstri 3886 |
. . . . . . . . . . . 12
⊢ (ℝ
∖ {0}) ⊆ ℂ |
46 | 45 | a1i 11 |
. . . . . . . . . . 11
⊢ (𝜑 → (ℝ ∖ {0})
⊆ ℂ) |
47 | 7, 46 | fssd 6522 |
. . . . . . . . . 10
⊢ (𝜑 → 𝐹:(𝐴[,]𝐵)⟶ℂ) |
48 | | fcompt 6905 |
. . . . . . . . . 10
⊢
((abs:ℂ⟶ℝ ∧ 𝐹:(𝐴[,]𝐵)⟶ℂ) → (abs ∘ 𝐹) = (𝑧 ∈ (𝐴[,]𝐵) ↦ (abs‘(𝐹‘𝑧)))) |
49 | 42, 47, 48 | syl2anc 587 |
. . . . . . . . 9
⊢ (𝜑 → (abs ∘ 𝐹) = (𝑧 ∈ (𝐴[,]𝐵) ↦ (abs‘(𝐹‘𝑧)))) |
50 | | nfcv 2899 |
. . . . . . . . . . . . 13
⊢
Ⅎ𝑥𝑧 |
51 | 27, 50 | nffv 6684 |
. . . . . . . . . . . 12
⊢
Ⅎ𝑥(𝐹‘𝑧) |
52 | 24, 51 | nffv 6684 |
. . . . . . . . . . 11
⊢
Ⅎ𝑥(abs‘(𝐹‘𝑧)) |
53 | | nfcv 2899 |
. . . . . . . . . . 11
⊢
Ⅎ𝑧(abs‘(𝐹‘𝑥)) |
54 | | fveq2 6674 |
. . . . . . . . . . . 12
⊢ (𝑧 = 𝑥 → (𝐹‘𝑧) = (𝐹‘𝑥)) |
55 | 54 | fveq2d 6678 |
. . . . . . . . . . 11
⊢ (𝑧 = 𝑥 → (abs‘(𝐹‘𝑧)) = (abs‘(𝐹‘𝑥))) |
56 | 52, 53, 55 | cbvmpt 5131 |
. . . . . . . . . 10
⊢ (𝑧 ∈ (𝐴[,]𝐵) ↦ (abs‘(𝐹‘𝑧))) = (𝑥 ∈ (𝐴[,]𝐵) ↦ (abs‘(𝐹‘𝑥))) |
57 | 56 | a1i 11 |
. . . . . . . . 9
⊢ (𝜑 → (𝑧 ∈ (𝐴[,]𝐵) ↦ (abs‘(𝐹‘𝑧))) = (𝑥 ∈ (𝐴[,]𝐵) ↦ (abs‘(𝐹‘𝑥)))) |
58 | 25 | a1i 11 |
. . . . . . . . . . . 12
⊢ (𝜑 → 𝐹 = (𝑥 ∈ (𝐴[,]𝐵) ↦ 𝐶)) |
59 | 58, 7 | feq1dd 42241 |
. . . . . . . . . . . . 13
⊢ (𝜑 → (𝑥 ∈ (𝐴[,]𝐵) ↦ 𝐶):(𝐴[,]𝐵)⟶(ℝ ∖
{0})) |
60 | 59 | fvmptelrn 6887 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) → 𝐶 ∈ (ℝ ∖
{0})) |
61 | 58, 60 | fvmpt2d 6788 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) → (𝐹‘𝑥) = 𝐶) |
62 | 61 | fveq2d 6678 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) → (abs‘(𝐹‘𝑥)) = (abs‘𝐶)) |
63 | 62 | mpteq2dva 5125 |
. . . . . . . . 9
⊢ (𝜑 → (𝑥 ∈ (𝐴[,]𝐵) ↦ (abs‘(𝐹‘𝑥))) = (𝑥 ∈ (𝐴[,]𝐵) ↦ (abs‘𝐶))) |
64 | 49, 57, 63 | 3eqtrd 2777 |
. . . . . . . 8
⊢ (𝜑 → (abs ∘ 𝐹) = (𝑥 ∈ (𝐴[,]𝐵) ↦ (abs‘𝐶))) |
65 | 45, 60 | sseldi 3875 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) → 𝐶 ∈ ℂ) |
66 | 65 | abscld 14886 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) → (abs‘𝐶) ∈ ℝ) |
67 | 64, 66 | fvmpt2d 6788 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) → ((abs ∘ 𝐹)‘𝑥) = (abs‘𝐶)) |
68 | 67 | ad4ant14 752 |
. . . . . 6
⊢ ((((𝜑 ∧ 𝑐 ∈ (𝐴[,]𝐵)) ∧ ∀𝑑 ∈ (𝐴[,]𝐵)((abs ∘ 𝐹)‘𝑐) ≤ ((abs ∘ 𝐹)‘𝑑)) ∧ 𝑥 ∈ (𝐴[,]𝐵)) → ((abs ∘ 𝐹)‘𝑥) = (abs‘𝐶)) |
69 | 40, 68 | breqtrd 5056 |
. . . . 5
⊢ ((((𝜑 ∧ 𝑐 ∈ (𝐴[,]𝐵)) ∧ ∀𝑑 ∈ (𝐴[,]𝐵)((abs ∘ 𝐹)‘𝑐) ≤ ((abs ∘ 𝐹)‘𝑑)) ∧ 𝑥 ∈ (𝐴[,]𝐵)) → ((abs ∘ 𝐹)‘𝑐) ≤ (abs‘𝐶)) |
70 | 69 | ex 416 |
. . . 4
⊢ (((𝜑 ∧ 𝑐 ∈ (𝐴[,]𝐵)) ∧ ∀𝑑 ∈ (𝐴[,]𝐵)((abs ∘ 𝐹)‘𝑐) ≤ ((abs ∘ 𝐹)‘𝑑)) → (𝑥 ∈ (𝐴[,]𝐵) → ((abs ∘ 𝐹)‘𝑐) ≤ (abs‘𝐶))) |
71 | 36, 70 | ralrimi 3128 |
. . 3
⊢ (((𝜑 ∧ 𝑐 ∈ (𝐴[,]𝐵)) ∧ ∀𝑑 ∈ (𝐴[,]𝐵)((abs ∘ 𝐹)‘𝑐) ≤ ((abs ∘ 𝐹)‘𝑑)) → ∀𝑥 ∈ (𝐴[,]𝐵)((abs ∘ 𝐹)‘𝑐) ≤ (abs‘𝐶)) |
72 | 30 | nfeq2 2916 |
. . . . 5
⊢
Ⅎ𝑥 𝑦 = ((abs ∘ 𝐹)‘𝑐) |
73 | | breq1 5033 |
. . . . 5
⊢ (𝑦 = ((abs ∘ 𝐹)‘𝑐) → (𝑦 ≤ (abs‘𝐶) ↔ ((abs ∘ 𝐹)‘𝑐) ≤ (abs‘𝐶))) |
74 | 72, 73 | ralbid 3145 |
. . . 4
⊢ (𝑦 = ((abs ∘ 𝐹)‘𝑐) → (∀𝑥 ∈ (𝐴[,]𝐵)𝑦 ≤ (abs‘𝐶) ↔ ∀𝑥 ∈ (𝐴[,]𝐵)((abs ∘ 𝐹)‘𝑐) ≤ (abs‘𝐶))) |
75 | 74 | rspcev 3526 |
. . 3
⊢ ((((abs
∘ 𝐹)‘𝑐) ∈ ℝ+
∧ ∀𝑥 ∈
(𝐴[,]𝐵)((abs ∘ 𝐹)‘𝑐) ≤ (abs‘𝐶)) → ∃𝑦 ∈ ℝ+ ∀𝑥 ∈ (𝐴[,]𝐵)𝑦 ≤ (abs‘𝐶)) |
76 | 21, 71, 75 | syl2anc 587 |
. 2
⊢ (((𝜑 ∧ 𝑐 ∈ (𝐴[,]𝐵)) ∧ ∀𝑑 ∈ (𝐴[,]𝐵)((abs ∘ 𝐹)‘𝑐) ≤ ((abs ∘ 𝐹)‘𝑑)) → ∃𝑦 ∈ ℝ+ ∀𝑥 ∈ (𝐴[,]𝐵)𝑦 ≤ (abs‘𝐶)) |
77 | | cncficcgt0.a |
. . . 4
⊢ (𝜑 → 𝐴 ∈ ℝ) |
78 | | cncficcgt0.b |
. . . 4
⊢ (𝜑 → 𝐵 ∈ ℝ) |
79 | | cncficcgt0.aleb |
. . . 4
⊢ (𝜑 → 𝐴 ≤ 𝐵) |
80 | | ssidd 3900 |
. . . . . . 7
⊢ (𝜑 → ℂ ⊆
ℂ) |
81 | | cncfss 23651 |
. . . . . . 7
⊢
(((ℝ ∖ {0}) ⊆ ℂ ∧ ℂ ⊆ ℂ)
→ ((𝐴[,]𝐵)–cn→(ℝ ∖ {0})) ⊆ ((𝐴[,]𝐵)–cn→ℂ)) |
82 | 46, 80, 81 | syl2anc 587 |
. . . . . 6
⊢ (𝜑 → ((𝐴[,]𝐵)–cn→(ℝ ∖ {0})) ⊆ ((𝐴[,]𝐵)–cn→ℂ)) |
83 | 82, 1 | sseldd 3878 |
. . . . 5
⊢ (𝜑 → 𝐹 ∈ ((𝐴[,]𝐵)–cn→ℂ)) |
84 | | abscncf 23653 |
. . . . . 6
⊢ abs
∈ (ℂ–cn→ℝ) |
85 | 84 | a1i 11 |
. . . . 5
⊢ (𝜑 → abs ∈
(ℂ–cn→ℝ)) |
86 | 83, 85 | cncfco 23659 |
. . . 4
⊢ (𝜑 → (abs ∘ 𝐹) ∈ ((𝐴[,]𝐵)–cn→ℝ)) |
87 | 77, 78, 79, 86 | evthicc 24211 |
. . 3
⊢ (𝜑 → (∃𝑎 ∈ (𝐴[,]𝐵)∀𝑏 ∈ (𝐴[,]𝐵)((abs ∘ 𝐹)‘𝑏) ≤ ((abs ∘ 𝐹)‘𝑎) ∧ ∃𝑐 ∈ (𝐴[,]𝐵)∀𝑑 ∈ (𝐴[,]𝐵)((abs ∘ 𝐹)‘𝑐) ≤ ((abs ∘ 𝐹)‘𝑑))) |
88 | 87 | simprd 499 |
. 2
⊢ (𝜑 → ∃𝑐 ∈ (𝐴[,]𝐵)∀𝑑 ∈ (𝐴[,]𝐵)((abs ∘ 𝐹)‘𝑐) ≤ ((abs ∘ 𝐹)‘𝑑)) |
89 | 76, 88 | r19.29a 3199 |
1
⊢ (𝜑 → ∃𝑦 ∈ ℝ+ ∀𝑥 ∈ (𝐴[,]𝐵)𝑦 ≤ (abs‘𝐶)) |