| Step | Hyp | Ref
| Expression |
| 1 | | rlimcn1.1 |
. . . 4
⊢ (𝜑 → 𝐺:𝐴⟶𝑋) |
| 2 | 1 | ffvelcdmda 7104 |
. . 3
⊢ ((𝜑 ∧ 𝑤 ∈ 𝐴) → (𝐺‘𝑤) ∈ 𝑋) |
| 3 | 1 | feqmptd 6977 |
. . 3
⊢ (𝜑 → 𝐺 = (𝑤 ∈ 𝐴 ↦ (𝐺‘𝑤))) |
| 4 | | rlimcn1.4 |
. . . 4
⊢ (𝜑 → 𝐹:𝑋⟶ℂ) |
| 5 | 4 | feqmptd 6977 |
. . 3
⊢ (𝜑 → 𝐹 = (𝑣 ∈ 𝑋 ↦ (𝐹‘𝑣))) |
| 6 | | fveq2 6906 |
. . 3
⊢ (𝑣 = (𝐺‘𝑤) → (𝐹‘𝑣) = (𝐹‘(𝐺‘𝑤))) |
| 7 | 2, 3, 5, 6 | fmptco 7149 |
. 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 6921 |
. . . . . . . . 9
⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ 𝑦 ∈ ℝ+)
∧ 𝑤 ∈ 𝐴) → (𝐺‘𝑤) ∈ V) |
| 10 | 9 | ralrimiva 3146 |
. . . . . . . 8
⊢ (((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ 𝑦 ∈ ℝ+)
→ ∀𝑤 ∈
𝐴 (𝐺‘𝑤) ∈ V) |
| 11 | | simpr 484 |
. . . . . . . 8
⊢ (((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ 𝑦 ∈ ℝ+)
→ 𝑦 ∈
ℝ+) |
| 12 | | rlimcn1.3 |
. . . . . . . . . 10
⊢ (𝜑 → 𝐺 ⇝𝑟 𝐶) |
| 13 | 3, 12 | eqbrtrrd 5167 |
. . . . . . . . 9
⊢ (𝜑 → (𝑤 ∈ 𝐴 ↦ (𝐺‘𝑤)) ⇝𝑟 𝐶) |
| 14 | 13 | ad2antrr 726 |
. . . . . . . 8
⊢ (((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ 𝑦 ∈ ℝ+)
→ (𝑤 ∈ 𝐴 ↦ (𝐺‘𝑤)) ⇝𝑟 𝐶) |
| 15 | 10, 11, 14 | rlimi 15549 |
. . . . . . 7
⊢ (((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ 𝑦 ∈ ℝ+)
→ ∃𝑐 ∈
ℝ ∀𝑤 ∈
𝐴 (𝑐 ≤ 𝑤 → (abs‘((𝐺‘𝑤) − 𝐶)) < 𝑦)) |
| 16 | | fvoveq1 7454 |
. . . . . . . . . . . . . 14
⊢ (𝑧 = (𝐺‘𝑤) → (abs‘(𝑧 − 𝐶)) = (abs‘((𝐺‘𝑤) − 𝐶))) |
| 17 | 16 | breq1d 5153 |
. . . . . . . . . . . . 13
⊢ (𝑧 = (𝐺‘𝑤) → ((abs‘(𝑧 − 𝐶)) < 𝑦 ↔ (abs‘((𝐺‘𝑤) − 𝐶)) < 𝑦)) |
| 18 | 17 | imbrov2fvoveq 7456 |
. . . . . . . . . . . 12
⊢ (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))) |
| 19 | | simplrr 778 |
. . . . . . . . . . . 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 | 2 | ad4ant14 752 |
. . . . . . . . . . . 12
⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ (𝑦 ∈ ℝ+
∧ ∀𝑧 ∈
𝑋 ((abs‘(𝑧 − 𝐶)) < 𝑦 → (abs‘((𝐹‘𝑧) − (𝐹‘𝐶))) < 𝑥))) ∧ 𝑤 ∈ 𝐴) → (𝐺‘𝑤) ∈ 𝑋) |
| 21 | 18, 19, 20 | rspcdva 3623 |
. . . . . . . . . . 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)) |
| 22 | 21 | 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))) |
| 23 | 22 | ralimdva 3167 |
. . . . . . . . 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))) |
| 24 | 23 | reximdv 3170 |
. . . . . . . 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))) |
| 25 | 24 | expr 456 |
. . . . . . 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)))) |
| 26 | 15, 25 | 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))) |
| 27 | 26 | rexlimdva 3155 |
. . . . 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))) |
| 28 | 8, 27 | mpd 15 |
. . . 4
⊢ ((𝜑 ∧ 𝑥 ∈ ℝ+) →
∃𝑐 ∈ ℝ
∀𝑤 ∈ 𝐴 (𝑐 ≤ 𝑤 → (abs‘((𝐹‘(𝐺‘𝑤)) − (𝐹‘𝐶))) < 𝑥)) |
| 29 | 28 | ralrimiva 3146 |
. . 3
⊢ (𝜑 → ∀𝑥 ∈ ℝ+ ∃𝑐 ∈ ℝ ∀𝑤 ∈ 𝐴 (𝑐 ≤ 𝑤 → (abs‘((𝐹‘(𝐺‘𝑤)) − (𝐹‘𝐶))) < 𝑥)) |
| 30 | 4 | ffvelcdmda 7104 |
. . . . . 6
⊢ ((𝜑 ∧ (𝐺‘𝑤) ∈ 𝑋) → (𝐹‘(𝐺‘𝑤)) ∈ ℂ) |
| 31 | 2, 30 | syldan 591 |
. . . . 5
⊢ ((𝜑 ∧ 𝑤 ∈ 𝐴) → (𝐹‘(𝐺‘𝑤)) ∈ ℂ) |
| 32 | 31 | ralrimiva 3146 |
. . . 4
⊢ (𝜑 → ∀𝑤 ∈ 𝐴 (𝐹‘(𝐺‘𝑤)) ∈ ℂ) |
| 33 | 1 | fdmd 6746 |
. . . . 5
⊢ (𝜑 → dom 𝐺 = 𝐴) |
| 34 | | rlimss 15538 |
. . . . . 6
⊢ (𝐺 ⇝𝑟
𝐶 → dom 𝐺 ⊆
ℝ) |
| 35 | 12, 34 | syl 17 |
. . . . 5
⊢ (𝜑 → dom 𝐺 ⊆ ℝ) |
| 36 | 33, 35 | eqsstrrd 4019 |
. . . 4
⊢ (𝜑 → 𝐴 ⊆ ℝ) |
| 37 | | rlimcn1.2 |
. . . . 5
⊢ (𝜑 → 𝐶 ∈ 𝑋) |
| 38 | 4, 37 | ffvelcdmd 7105 |
. . . 4
⊢ (𝜑 → (𝐹‘𝐶) ∈ ℂ) |
| 39 | 32, 36, 38 | rlim2 15532 |
. . 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))) |
| 40 | 29, 39 | mpbird 257 |
. 2
⊢ (𝜑 → (𝑤 ∈ 𝐴 ↦ (𝐹‘(𝐺‘𝑤))) ⇝𝑟 (𝐹‘𝐶)) |
| 41 | 7, 40 | eqbrtrd 5165 |
1
⊢ (𝜑 → (𝐹 ∘ 𝐺) ⇝𝑟 (𝐹‘𝐶)) |