Step | Hyp | Ref
| Expression |
1 | | rlimcn1.1 |
. . . 4
⊢ (𝜑 → 𝐺:𝐴⟶𝑋) |
2 | 1 | ffvelrnda 6267 |
. . 3
⊢ ((𝜑 ∧ 𝑤 ∈ 𝐴) → (𝐺‘𝑤) ∈ 𝑋) |
3 | 1 | feqmptd 6159 |
. . 3
⊢ (𝜑 → 𝐺 = (𝑤 ∈ 𝐴 ↦ (𝐺‘𝑤))) |
4 | | rlimcn1.4 |
. . . 4
⊢ (𝜑 → 𝐹:𝑋⟶ℂ) |
5 | 4 | feqmptd 6159 |
. . 3
⊢ (𝜑 → 𝐹 = (𝑣 ∈ 𝑋 ↦ (𝐹‘𝑣))) |
6 | | fveq2 6103 |
. . 3
⊢ (𝑣 = (𝐺‘𝑤) → (𝐹‘𝑣) = (𝐹‘(𝐺‘𝑤))) |
7 | 2, 3, 5, 6 | fmptco 6303 |
. 2
⊢ (𝜑 → (𝐹 ∘ 𝐺) = (𝑤 ∈ 𝐴 ↦ (𝐹‘(𝐺‘𝑤)))) |
8 | | rlimcn1.5 |
. . . . 5
⊢ ((𝜑 ∧ 𝑥 ∈ ℝ+) →
∃𝑦 ∈
ℝ+ ∀𝑧 ∈ 𝑋 ((abs‘(𝑧 − 𝐶)) < 𝑦 → (abs‘((𝐹‘𝑧) − (𝐹‘𝐶))) < 𝑥)) |
9 | | fvex 6113 |
. . . . . . . . . 10
⊢ (𝐺‘𝑤) ∈ V |
10 | 9 | a1i 11 |
. . . . . . . . 9
⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ 𝑦 ∈ ℝ+)
∧ 𝑤 ∈ 𝐴) → (𝐺‘𝑤) ∈ V) |
11 | 10 | ralrimiva 2949 |
. . . . . . . 8
⊢ (((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ 𝑦 ∈ ℝ+)
→ ∀𝑤 ∈
𝐴 (𝐺‘𝑤) ∈ V) |
12 | | simpr 476 |
. . . . . . . 8
⊢ (((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ 𝑦 ∈ ℝ+)
→ 𝑦 ∈
ℝ+) |
13 | | rlimcn1.3 |
. . . . . . . . . 10
⊢ (𝜑 → 𝐺 ⇝𝑟 𝐶) |
14 | 3, 13 | eqbrtrrd 4607 |
. . . . . . . . 9
⊢ (𝜑 → (𝑤 ∈ 𝐴 ↦ (𝐺‘𝑤)) ⇝𝑟 𝐶) |
15 | 14 | ad2antrr 758 |
. . . . . . . 8
⊢ (((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ 𝑦 ∈ ℝ+)
→ (𝑤 ∈ 𝐴 ↦ (𝐺‘𝑤)) ⇝𝑟 𝐶) |
16 | 11, 12, 15 | rlimi 14092 |
. . . . . . 7
⊢ (((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ 𝑦 ∈ ℝ+)
→ ∃𝑐 ∈
ℝ ∀𝑤 ∈
𝐴 (𝑐 ≤ 𝑤 → (abs‘((𝐺‘𝑤) − 𝐶)) < 𝑦)) |
17 | | simpll 786 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ (𝑦 ∈ ℝ+
∧ ∀𝑧 ∈
𝑋 ((abs‘(𝑧 − 𝐶)) < 𝑦 → (abs‘((𝐹‘𝑧) − (𝐹‘𝐶))) < 𝑥))) → 𝜑) |
18 | 17, 2 | sylan 487 |
. . . . . . . . . . . 12
⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ (𝑦 ∈ ℝ+
∧ ∀𝑧 ∈
𝑋 ((abs‘(𝑧 − 𝐶)) < 𝑦 → (abs‘((𝐹‘𝑧) − (𝐹‘𝐶))) < 𝑥))) ∧ 𝑤 ∈ 𝐴) → (𝐺‘𝑤) ∈ 𝑋) |
19 | | simplrr 797 |
. . . . . . . . . . . 12
⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ (𝑦 ∈ ℝ+
∧ ∀𝑧 ∈
𝑋 ((abs‘(𝑧 − 𝐶)) < 𝑦 → (abs‘((𝐹‘𝑧) − (𝐹‘𝐶))) < 𝑥))) ∧ 𝑤 ∈ 𝐴) → ∀𝑧 ∈ 𝑋 ((abs‘(𝑧 − 𝐶)) < 𝑦 → (abs‘((𝐹‘𝑧) − (𝐹‘𝐶))) < 𝑥)) |
20 | | oveq1 6556 |
. . . . . . . . . . . . . . . 16
⊢ (𝑧 = (𝐺‘𝑤) → (𝑧 − 𝐶) = ((𝐺‘𝑤) − 𝐶)) |
21 | 20 | fveq2d 6107 |
. . . . . . . . . . . . . . 15
⊢ (𝑧 = (𝐺‘𝑤) → (abs‘(𝑧 − 𝐶)) = (abs‘((𝐺‘𝑤) − 𝐶))) |
22 | 21 | breq1d 4593 |
. . . . . . . . . . . . . 14
⊢ (𝑧 = (𝐺‘𝑤) → ((abs‘(𝑧 − 𝐶)) < 𝑦 ↔ (abs‘((𝐺‘𝑤) − 𝐶)) < 𝑦)) |
23 | | fveq2 6103 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑧 = (𝐺‘𝑤) → (𝐹‘𝑧) = (𝐹‘(𝐺‘𝑤))) |
24 | 23 | oveq1d 6564 |
. . . . . . . . . . . . . . . 16
⊢ (𝑧 = (𝐺‘𝑤) → ((𝐹‘𝑧) − (𝐹‘𝐶)) = ((𝐹‘(𝐺‘𝑤)) − (𝐹‘𝐶))) |
25 | 24 | fveq2d 6107 |
. . . . . . . . . . . . . . 15
⊢ (𝑧 = (𝐺‘𝑤) → (abs‘((𝐹‘𝑧) − (𝐹‘𝐶))) = (abs‘((𝐹‘(𝐺‘𝑤)) − (𝐹‘𝐶)))) |
26 | 25 | breq1d 4593 |
. . . . . . . . . . . . . 14
⊢ (𝑧 = (𝐺‘𝑤) → ((abs‘((𝐹‘𝑧) − (𝐹‘𝐶))) < 𝑥 ↔ (abs‘((𝐹‘(𝐺‘𝑤)) − (𝐹‘𝐶))) < 𝑥)) |
27 | 22, 26 | imbi12d 333 |
. . . . . . . . . . . . 13
⊢ (𝑧 = (𝐺‘𝑤) → (((abs‘(𝑧 − 𝐶)) < 𝑦 → (abs‘((𝐹‘𝑧) − (𝐹‘𝐶))) < 𝑥) ↔ ((abs‘((𝐺‘𝑤) − 𝐶)) < 𝑦 → (abs‘((𝐹‘(𝐺‘𝑤)) − (𝐹‘𝐶))) < 𝑥))) |
28 | 27 | rspcv 3278 |
. . . . . . . . . . . 12
⊢ ((𝐺‘𝑤) ∈ 𝑋 → (∀𝑧 ∈ 𝑋 ((abs‘(𝑧 − 𝐶)) < 𝑦 → (abs‘((𝐹‘𝑧) − (𝐹‘𝐶))) < 𝑥) → ((abs‘((𝐺‘𝑤) − 𝐶)) < 𝑦 → (abs‘((𝐹‘(𝐺‘𝑤)) − (𝐹‘𝐶))) < 𝑥))) |
29 | 18, 19, 28 | sylc 63 |
. . . . . . . . . . 11
⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ (𝑦 ∈ ℝ+
∧ ∀𝑧 ∈
𝑋 ((abs‘(𝑧 − 𝐶)) < 𝑦 → (abs‘((𝐹‘𝑧) − (𝐹‘𝐶))) < 𝑥))) ∧ 𝑤 ∈ 𝐴) → ((abs‘((𝐺‘𝑤) − 𝐶)) < 𝑦 → (abs‘((𝐹‘(𝐺‘𝑤)) − (𝐹‘𝐶))) < 𝑥)) |
30 | 29 | imim2d 55 |
. . . . . . . . . 10
⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ (𝑦 ∈ ℝ+
∧ ∀𝑧 ∈
𝑋 ((abs‘(𝑧 − 𝐶)) < 𝑦 → (abs‘((𝐹‘𝑧) − (𝐹‘𝐶))) < 𝑥))) ∧ 𝑤 ∈ 𝐴) → ((𝑐 ≤ 𝑤 → (abs‘((𝐺‘𝑤) − 𝐶)) < 𝑦) → (𝑐 ≤ 𝑤 → (abs‘((𝐹‘(𝐺‘𝑤)) − (𝐹‘𝐶))) < 𝑥))) |
31 | 30 | ralimdva 2945 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ (𝑦 ∈ ℝ+
∧ ∀𝑧 ∈
𝑋 ((abs‘(𝑧 − 𝐶)) < 𝑦 → (abs‘((𝐹‘𝑧) − (𝐹‘𝐶))) < 𝑥))) → (∀𝑤 ∈ 𝐴 (𝑐 ≤ 𝑤 → (abs‘((𝐺‘𝑤) − 𝐶)) < 𝑦) → ∀𝑤 ∈ 𝐴 (𝑐 ≤ 𝑤 → (abs‘((𝐹‘(𝐺‘𝑤)) − (𝐹‘𝐶))) < 𝑥))) |
32 | 31 | reximdv 2999 |
. . . . . . . 8
⊢ (((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ (𝑦 ∈ ℝ+
∧ ∀𝑧 ∈
𝑋 ((abs‘(𝑧 − 𝐶)) < 𝑦 → (abs‘((𝐹‘𝑧) − (𝐹‘𝐶))) < 𝑥))) → (∃𝑐 ∈ ℝ ∀𝑤 ∈ 𝐴 (𝑐 ≤ 𝑤 → (abs‘((𝐺‘𝑤) − 𝐶)) < 𝑦) → ∃𝑐 ∈ ℝ ∀𝑤 ∈ 𝐴 (𝑐 ≤ 𝑤 → (abs‘((𝐹‘(𝐺‘𝑤)) − (𝐹‘𝐶))) < 𝑥))) |
33 | 32 | expr 641 |
. . . . . . 7
⊢ (((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ 𝑦 ∈ ℝ+)
→ (∀𝑧 ∈
𝑋 ((abs‘(𝑧 − 𝐶)) < 𝑦 → (abs‘((𝐹‘𝑧) − (𝐹‘𝐶))) < 𝑥) → (∃𝑐 ∈ ℝ ∀𝑤 ∈ 𝐴 (𝑐 ≤ 𝑤 → (abs‘((𝐺‘𝑤) − 𝐶)) < 𝑦) → ∃𝑐 ∈ ℝ ∀𝑤 ∈ 𝐴 (𝑐 ≤ 𝑤 → (abs‘((𝐹‘(𝐺‘𝑤)) − (𝐹‘𝐶))) < 𝑥)))) |
34 | 16, 33 | mpid 43 |
. . . . . 6
⊢ (((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ 𝑦 ∈ ℝ+)
→ (∀𝑧 ∈
𝑋 ((abs‘(𝑧 − 𝐶)) < 𝑦 → (abs‘((𝐹‘𝑧) − (𝐹‘𝐶))) < 𝑥) → ∃𝑐 ∈ ℝ ∀𝑤 ∈ 𝐴 (𝑐 ≤ 𝑤 → (abs‘((𝐹‘(𝐺‘𝑤)) − (𝐹‘𝐶))) < 𝑥))) |
35 | 34 | rexlimdva 3013 |
. . . . 5
⊢ ((𝜑 ∧ 𝑥 ∈ ℝ+) →
(∃𝑦 ∈
ℝ+ ∀𝑧 ∈ 𝑋 ((abs‘(𝑧 − 𝐶)) < 𝑦 → (abs‘((𝐹‘𝑧) − (𝐹‘𝐶))) < 𝑥) → ∃𝑐 ∈ ℝ ∀𝑤 ∈ 𝐴 (𝑐 ≤ 𝑤 → (abs‘((𝐹‘(𝐺‘𝑤)) − (𝐹‘𝐶))) < 𝑥))) |
36 | 8, 35 | mpd 15 |
. . . 4
⊢ ((𝜑 ∧ 𝑥 ∈ ℝ+) →
∃𝑐 ∈ ℝ
∀𝑤 ∈ 𝐴 (𝑐 ≤ 𝑤 → (abs‘((𝐹‘(𝐺‘𝑤)) − (𝐹‘𝐶))) < 𝑥)) |
37 | 36 | ralrimiva 2949 |
. . 3
⊢ (𝜑 → ∀𝑥 ∈ ℝ+ ∃𝑐 ∈ ℝ ∀𝑤 ∈ 𝐴 (𝑐 ≤ 𝑤 → (abs‘((𝐹‘(𝐺‘𝑤)) − (𝐹‘𝐶))) < 𝑥)) |
38 | 4 | ffvelrnda 6267 |
. . . . . 6
⊢ ((𝜑 ∧ (𝐺‘𝑤) ∈ 𝑋) → (𝐹‘(𝐺‘𝑤)) ∈ ℂ) |
39 | 2, 38 | syldan 486 |
. . . . 5
⊢ ((𝜑 ∧ 𝑤 ∈ 𝐴) → (𝐹‘(𝐺‘𝑤)) ∈ ℂ) |
40 | 39 | ralrimiva 2949 |
. . . 4
⊢ (𝜑 → ∀𝑤 ∈ 𝐴 (𝐹‘(𝐺‘𝑤)) ∈ ℂ) |
41 | | fdm 5964 |
. . . . . 6
⊢ (𝐺:𝐴⟶𝑋 → dom 𝐺 = 𝐴) |
42 | 1, 41 | syl 17 |
. . . . 5
⊢ (𝜑 → dom 𝐺 = 𝐴) |
43 | | rlimss 14081 |
. . . . . 6
⊢ (𝐺 ⇝𝑟
𝐶 → dom 𝐺 ⊆
ℝ) |
44 | 13, 43 | syl 17 |
. . . . 5
⊢ (𝜑 → dom 𝐺 ⊆ ℝ) |
45 | 42, 44 | eqsstr3d 3603 |
. . . 4
⊢ (𝜑 → 𝐴 ⊆ ℝ) |
46 | | rlimcn1.2 |
. . . . 5
⊢ (𝜑 → 𝐶 ∈ 𝑋) |
47 | 4, 46 | ffvelrnd 6268 |
. . . 4
⊢ (𝜑 → (𝐹‘𝐶) ∈ ℂ) |
48 | 40, 45, 47 | rlim2 14075 |
. . 3
⊢ (𝜑 → ((𝑤 ∈ 𝐴 ↦ (𝐹‘(𝐺‘𝑤))) ⇝𝑟 (𝐹‘𝐶) ↔ ∀𝑥 ∈ ℝ+ ∃𝑐 ∈ ℝ ∀𝑤 ∈ 𝐴 (𝑐 ≤ 𝑤 → (abs‘((𝐹‘(𝐺‘𝑤)) − (𝐹‘𝐶))) < 𝑥))) |
49 | 37, 48 | mpbird 246 |
. 2
⊢ (𝜑 → (𝑤 ∈ 𝐴 ↦ (𝐹‘(𝐺‘𝑤))) ⇝𝑟 (𝐹‘𝐶)) |
50 | 7, 49 | eqbrtrd 4605 |
1
⊢ (𝜑 → (𝐹 ∘ 𝐺) ⇝𝑟 (𝐹‘𝐶)) |