Step | Hyp | Ref
| Expression |
1 | | rlimcn1.1 |
. . . 4
⊢ (𝜑 → 𝐺:𝐴⟶𝑋) |
2 | 1 | ffvelrnda 6359 |
. . 3
⊢ ((𝜑 ∧ 𝑤 ∈ 𝐴) → (𝐺‘𝑤) ∈ 𝑋) |
3 | 1 | feqmptd 6249 |
. . 3
⊢ (𝜑 → 𝐺 = (𝑤 ∈ 𝐴 ↦ (𝐺‘𝑤))) |
4 | | rlimcn1.4 |
. . . 4
⊢ (𝜑 → 𝐹:𝑋⟶ℂ) |
5 | 4 | feqmptd 6249 |
. . 3
⊢ (𝜑 → 𝐹 = (𝑣 ∈ 𝑋 ↦ (𝐹‘𝑣))) |
6 | | fveq2 6191 |
. . 3
⊢ (𝑣 = (𝐺‘𝑤) → (𝐹‘𝑣) = (𝐹‘(𝐺‘𝑤))) |
7 | 2, 3, 5, 6 | fmptco 6396 |
. 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 | | fvexd 6203 |
. . . . . . . . 9
⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ 𝑦 ∈ ℝ+)
∧ 𝑤 ∈ 𝐴) → (𝐺‘𝑤) ∈ V) |
10 | 9 | ralrimiva 2966 |
. . . . . . . 8
⊢ (((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ 𝑦 ∈ ℝ+)
→ ∀𝑤 ∈
𝐴 (𝐺‘𝑤) ∈ V) |
11 | | simpr 477 |
. . . . . . . 8
⊢ (((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ 𝑦 ∈ ℝ+)
→ 𝑦 ∈
ℝ+) |
12 | | rlimcn1.3 |
. . . . . . . . . 10
⊢ (𝜑 → 𝐺 ⇝𝑟 𝐶) |
13 | 3, 12 | eqbrtrrd 4677 |
. . . . . . . . 9
⊢ (𝜑 → (𝑤 ∈ 𝐴 ↦ (𝐺‘𝑤)) ⇝𝑟 𝐶) |
14 | 13 | ad2antrr 762 |
. . . . . . . 8
⊢ (((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ 𝑦 ∈ ℝ+)
→ (𝑤 ∈ 𝐴 ↦ (𝐺‘𝑤)) ⇝𝑟 𝐶) |
15 | 10, 11, 14 | rlimi 14244 |
. . . . . . 7
⊢ (((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ 𝑦 ∈ ℝ+)
→ ∃𝑐 ∈
ℝ ∀𝑤 ∈
𝐴 (𝑐 ≤ 𝑤 → (abs‘((𝐺‘𝑤) − 𝐶)) < 𝑦)) |
16 | | simpll 790 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ (𝑦 ∈ ℝ+
∧ ∀𝑧 ∈
𝑋 ((abs‘(𝑧 − 𝐶)) < 𝑦 → (abs‘((𝐹‘𝑧) − (𝐹‘𝐶))) < 𝑥))) → 𝜑) |
17 | 16, 2 | sylan 488 |
. . . . . . . . . . . 12
⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ (𝑦 ∈ ℝ+
∧ ∀𝑧 ∈
𝑋 ((abs‘(𝑧 − 𝐶)) < 𝑦 → (abs‘((𝐹‘𝑧) − (𝐹‘𝐶))) < 𝑥))) ∧ 𝑤 ∈ 𝐴) → (𝐺‘𝑤) ∈ 𝑋) |
18 | | simplrr 801 |
. . . . . . . . . . . 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)) |
19 | | oveq1 6657 |
. . . . . . . . . . . . . . . 16
⊢ (𝑧 = (𝐺‘𝑤) → (𝑧 − 𝐶) = ((𝐺‘𝑤) − 𝐶)) |
20 | 19 | fveq2d 6195 |
. . . . . . . . . . . . . . 15
⊢ (𝑧 = (𝐺‘𝑤) → (abs‘(𝑧 − 𝐶)) = (abs‘((𝐺‘𝑤) − 𝐶))) |
21 | 20 | breq1d 4663 |
. . . . . . . . . . . . . 14
⊢ (𝑧 = (𝐺‘𝑤) → ((abs‘(𝑧 − 𝐶)) < 𝑦 ↔ (abs‘((𝐺‘𝑤) − 𝐶)) < 𝑦)) |
22 | | fveq2 6191 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑧 = (𝐺‘𝑤) → (𝐹‘𝑧) = (𝐹‘(𝐺‘𝑤))) |
23 | 22 | oveq1d 6665 |
. . . . . . . . . . . . . . . 16
⊢ (𝑧 = (𝐺‘𝑤) → ((𝐹‘𝑧) − (𝐹‘𝐶)) = ((𝐹‘(𝐺‘𝑤)) − (𝐹‘𝐶))) |
24 | 23 | fveq2d 6195 |
. . . . . . . . . . . . . . 15
⊢ (𝑧 = (𝐺‘𝑤) → (abs‘((𝐹‘𝑧) − (𝐹‘𝐶))) = (abs‘((𝐹‘(𝐺‘𝑤)) − (𝐹‘𝐶)))) |
25 | 24 | breq1d 4663 |
. . . . . . . . . . . . . 14
⊢ (𝑧 = (𝐺‘𝑤) → ((abs‘((𝐹‘𝑧) − (𝐹‘𝐶))) < 𝑥 ↔ (abs‘((𝐹‘(𝐺‘𝑤)) − (𝐹‘𝐶))) < 𝑥)) |
26 | 21, 25 | imbi12d 334 |
. . . . . . . . . . . . 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))) |
27 | 26 | rspcv 3305 |
. . . . . . . . . . . 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))) |
28 | 17, 18, 27 | sylc 65 |
. . . . . . . . . . 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)) |
29 | 28 | imim2d 57 |
. . . . . . . . . 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))) |
30 | 29 | ralimdva 2962 |
. . . . . . . . 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))) |
31 | 30 | reximdv 3016 |
. . . . . . . 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))) |
32 | 31 | expr 643 |
. . . . . . 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)))) |
33 | 15, 32 | mpid 44 |
. . . . . 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))) |
34 | 33 | rexlimdva 3031 |
. . . . 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))) |
35 | 8, 34 | mpd 15 |
. . . 4
⊢ ((𝜑 ∧ 𝑥 ∈ ℝ+) →
∃𝑐 ∈ ℝ
∀𝑤 ∈ 𝐴 (𝑐 ≤ 𝑤 → (abs‘((𝐹‘(𝐺‘𝑤)) − (𝐹‘𝐶))) < 𝑥)) |
36 | 35 | ralrimiva 2966 |
. . 3
⊢ (𝜑 → ∀𝑥 ∈ ℝ+ ∃𝑐 ∈ ℝ ∀𝑤 ∈ 𝐴 (𝑐 ≤ 𝑤 → (abs‘((𝐹‘(𝐺‘𝑤)) − (𝐹‘𝐶))) < 𝑥)) |
37 | 4 | ffvelrnda 6359 |
. . . . . 6
⊢ ((𝜑 ∧ (𝐺‘𝑤) ∈ 𝑋) → (𝐹‘(𝐺‘𝑤)) ∈ ℂ) |
38 | 2, 37 | syldan 487 |
. . . . 5
⊢ ((𝜑 ∧ 𝑤 ∈ 𝐴) → (𝐹‘(𝐺‘𝑤)) ∈ ℂ) |
39 | 38 | ralrimiva 2966 |
. . . 4
⊢ (𝜑 → ∀𝑤 ∈ 𝐴 (𝐹‘(𝐺‘𝑤)) ∈ ℂ) |
40 | | fdm 6051 |
. . . . . 6
⊢ (𝐺:𝐴⟶𝑋 → dom 𝐺 = 𝐴) |
41 | 1, 40 | syl 17 |
. . . . 5
⊢ (𝜑 → dom 𝐺 = 𝐴) |
42 | | rlimss 14233 |
. . . . . 6
⊢ (𝐺 ⇝𝑟
𝐶 → dom 𝐺 ⊆
ℝ) |
43 | 12, 42 | syl 17 |
. . . . 5
⊢ (𝜑 → dom 𝐺 ⊆ ℝ) |
44 | 41, 43 | eqsstr3d 3640 |
. . . 4
⊢ (𝜑 → 𝐴 ⊆ ℝ) |
45 | | rlimcn1.2 |
. . . . 5
⊢ (𝜑 → 𝐶 ∈ 𝑋) |
46 | 4, 45 | ffvelrnd 6360 |
. . . 4
⊢ (𝜑 → (𝐹‘𝐶) ∈ ℂ) |
47 | 39, 44, 46 | rlim2 14227 |
. . 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))) |
48 | 36, 47 | mpbird 247 |
. 2
⊢ (𝜑 → (𝑤 ∈ 𝐴 ↦ (𝐹‘(𝐺‘𝑤))) ⇝𝑟 (𝐹‘𝐶)) |
49 | 7, 48 | eqbrtrd 4675 |
1
⊢ (𝜑 → (𝐹 ∘ 𝐺) ⇝𝑟 (𝐹‘𝐶)) |