Proof of Theorem caucvgrlem2
| Step | Hyp | Ref | Expression | 
|---|
| 1 |  | caucvgrlem2.5 | . . 3
⊢ 𝐻:ℂ⟶ℝ | 
| 2 |  | caucvgr.2 | . . 3
⊢ (𝜑 → 𝐹:𝐴⟶ℂ) | 
| 3 |  | fcompt 7152 | . . 3
⊢ ((𝐻:ℂ⟶ℝ ∧
𝐹:𝐴⟶ℂ) → (𝐻 ∘ 𝐹) = (𝑛 ∈ 𝐴 ↦ (𝐻‘(𝐹‘𝑛)))) | 
| 4 | 1, 2, 3 | sylancr 587 | . 2
⊢ (𝜑 → (𝐻 ∘ 𝐹) = (𝑛 ∈ 𝐴 ↦ (𝐻‘(𝐹‘𝑛)))) | 
| 5 |  | caucvgr.1 | . . . . 5
⊢ (𝜑 → 𝐴 ⊆ ℝ) | 
| 6 |  | fco 6759 | . . . . . 6
⊢ ((𝐻:ℂ⟶ℝ ∧
𝐹:𝐴⟶ℂ) → (𝐻 ∘ 𝐹):𝐴⟶ℝ) | 
| 7 | 1, 2, 6 | sylancr 587 | . . . . 5
⊢ (𝜑 → (𝐻 ∘ 𝐹):𝐴⟶ℝ) | 
| 8 |  | caucvgr.3 | . . . . 5
⊢ (𝜑 → sup(𝐴, ℝ*, < ) =
+∞) | 
| 9 |  | caucvgr.4 | . . . . . 6
⊢ (𝜑 → ∀𝑥 ∈ ℝ+ ∃𝑗 ∈ 𝐴 ∀𝑘 ∈ 𝐴 (𝑗 ≤ 𝑘 → (abs‘((𝐹‘𝑘) − (𝐹‘𝑗))) < 𝑥)) | 
| 10 | 2 | ad2antrr 726 | . . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ (𝑗 ∈ 𝐴 ∧ 𝑘 ∈ 𝐴)) → 𝐹:𝐴⟶ℂ) | 
| 11 |  | simprr 772 | . . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ (𝑗 ∈ 𝐴 ∧ 𝑘 ∈ 𝐴)) → 𝑘 ∈ 𝐴) | 
| 12 | 10, 11 | ffvelcdmd 7104 | . . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ (𝑗 ∈ 𝐴 ∧ 𝑘 ∈ 𝐴)) → (𝐹‘𝑘) ∈ ℂ) | 
| 13 |  | simprl 770 | . . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ (𝑗 ∈ 𝐴 ∧ 𝑘 ∈ 𝐴)) → 𝑗 ∈ 𝐴) | 
| 14 | 10, 13 | ffvelcdmd 7104 | . . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ (𝑗 ∈ 𝐴 ∧ 𝑘 ∈ 𝐴)) → (𝐹‘𝑗) ∈ ℂ) | 
| 15 |  | caucvgrlem2.6 | . . . . . . . . . . . . . 14
⊢ (((𝐹‘𝑘) ∈ ℂ ∧ (𝐹‘𝑗) ∈ ℂ) → (abs‘((𝐻‘(𝐹‘𝑘)) − (𝐻‘(𝐹‘𝑗)))) ≤ (abs‘((𝐹‘𝑘) − (𝐹‘𝑗)))) | 
| 16 | 12, 14, 15 | syl2anc 584 | . . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ (𝑗 ∈ 𝐴 ∧ 𝑘 ∈ 𝐴)) → (abs‘((𝐻‘(𝐹‘𝑘)) − (𝐻‘(𝐹‘𝑗)))) ≤ (abs‘((𝐹‘𝑘) − (𝐹‘𝑗)))) | 
| 17 | 1 | ffvelcdmi 7102 | . . . . . . . . . . . . . . . . . 18
⊢ ((𝐹‘𝑘) ∈ ℂ → (𝐻‘(𝐹‘𝑘)) ∈ ℝ) | 
| 18 | 12, 17 | syl 17 | . . . . . . . . . . . . . . . . 17
⊢ (((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ (𝑗 ∈ 𝐴 ∧ 𝑘 ∈ 𝐴)) → (𝐻‘(𝐹‘𝑘)) ∈ ℝ) | 
| 19 | 1 | ffvelcdmi 7102 | . . . . . . . . . . . . . . . . . 18
⊢ ((𝐹‘𝑗) ∈ ℂ → (𝐻‘(𝐹‘𝑗)) ∈ ℝ) | 
| 20 | 14, 19 | syl 17 | . . . . . . . . . . . . . . . . 17
⊢ (((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ (𝑗 ∈ 𝐴 ∧ 𝑘 ∈ 𝐴)) → (𝐻‘(𝐹‘𝑗)) ∈ ℝ) | 
| 21 | 18, 20 | resubcld 11692 | . . . . . . . . . . . . . . . 16
⊢ (((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ (𝑗 ∈ 𝐴 ∧ 𝑘 ∈ 𝐴)) → ((𝐻‘(𝐹‘𝑘)) − (𝐻‘(𝐹‘𝑗))) ∈ ℝ) | 
| 22 | 21 | recnd 11290 | . . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ (𝑗 ∈ 𝐴 ∧ 𝑘 ∈ 𝐴)) → ((𝐻‘(𝐹‘𝑘)) − (𝐻‘(𝐹‘𝑗))) ∈ ℂ) | 
| 23 | 22 | abscld 15476 | . . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ (𝑗 ∈ 𝐴 ∧ 𝑘 ∈ 𝐴)) → (abs‘((𝐻‘(𝐹‘𝑘)) − (𝐻‘(𝐹‘𝑗)))) ∈ ℝ) | 
| 24 | 12, 14 | subcld 11621 | . . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ (𝑗 ∈ 𝐴 ∧ 𝑘 ∈ 𝐴)) → ((𝐹‘𝑘) − (𝐹‘𝑗)) ∈ ℂ) | 
| 25 | 24 | abscld 15476 | . . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ (𝑗 ∈ 𝐴 ∧ 𝑘 ∈ 𝐴)) → (abs‘((𝐹‘𝑘) − (𝐹‘𝑗))) ∈ ℝ) | 
| 26 |  | rpre 13044 | . . . . . . . . . . . . . . 15
⊢ (𝑥 ∈ ℝ+
→ 𝑥 ∈
ℝ) | 
| 27 | 26 | ad2antlr 727 | . . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ (𝑗 ∈ 𝐴 ∧ 𝑘 ∈ 𝐴)) → 𝑥 ∈ ℝ) | 
| 28 |  | lelttr 11352 | . . . . . . . . . . . . . 14
⊢
(((abs‘((𝐻‘(𝐹‘𝑘)) − (𝐻‘(𝐹‘𝑗)))) ∈ ℝ ∧ (abs‘((𝐹‘𝑘) − (𝐹‘𝑗))) ∈ ℝ ∧ 𝑥 ∈ ℝ) → (((abs‘((𝐻‘(𝐹‘𝑘)) − (𝐻‘(𝐹‘𝑗)))) ≤ (abs‘((𝐹‘𝑘) − (𝐹‘𝑗))) ∧ (abs‘((𝐹‘𝑘) − (𝐹‘𝑗))) < 𝑥) → (abs‘((𝐻‘(𝐹‘𝑘)) − (𝐻‘(𝐹‘𝑗)))) < 𝑥)) | 
| 29 | 23, 25, 27, 28 | syl3anc 1372 | . . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ (𝑗 ∈ 𝐴 ∧ 𝑘 ∈ 𝐴)) → (((abs‘((𝐻‘(𝐹‘𝑘)) − (𝐻‘(𝐹‘𝑗)))) ≤ (abs‘((𝐹‘𝑘) − (𝐹‘𝑗))) ∧ (abs‘((𝐹‘𝑘) − (𝐹‘𝑗))) < 𝑥) → (abs‘((𝐻‘(𝐹‘𝑘)) − (𝐻‘(𝐹‘𝑗)))) < 𝑥)) | 
| 30 | 16, 29 | mpand 695 | . . . . . . . . . . . 12
⊢ (((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ (𝑗 ∈ 𝐴 ∧ 𝑘 ∈ 𝐴)) → ((abs‘((𝐹‘𝑘) − (𝐹‘𝑗))) < 𝑥 → (abs‘((𝐻‘(𝐹‘𝑘)) − (𝐻‘(𝐹‘𝑗)))) < 𝑥)) | 
| 31 |  | fvco3 7007 | . . . . . . . . . . . . . . . 16
⊢ ((𝐹:𝐴⟶ℂ ∧ 𝑘 ∈ 𝐴) → ((𝐻 ∘ 𝐹)‘𝑘) = (𝐻‘(𝐹‘𝑘))) | 
| 32 | 10, 11, 31 | syl2anc 584 | . . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ (𝑗 ∈ 𝐴 ∧ 𝑘 ∈ 𝐴)) → ((𝐻 ∘ 𝐹)‘𝑘) = (𝐻‘(𝐹‘𝑘))) | 
| 33 |  | fvco3 7007 | . . . . . . . . . . . . . . . 16
⊢ ((𝐹:𝐴⟶ℂ ∧ 𝑗 ∈ 𝐴) → ((𝐻 ∘ 𝐹)‘𝑗) = (𝐻‘(𝐹‘𝑗))) | 
| 34 | 10, 13, 33 | syl2anc 584 | . . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ (𝑗 ∈ 𝐴 ∧ 𝑘 ∈ 𝐴)) → ((𝐻 ∘ 𝐹)‘𝑗) = (𝐻‘(𝐹‘𝑗))) | 
| 35 | 32, 34 | oveq12d 7450 | . . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ (𝑗 ∈ 𝐴 ∧ 𝑘 ∈ 𝐴)) → (((𝐻 ∘ 𝐹)‘𝑘) − ((𝐻 ∘ 𝐹)‘𝑗)) = ((𝐻‘(𝐹‘𝑘)) − (𝐻‘(𝐹‘𝑗)))) | 
| 36 | 35 | fveq2d 6909 | . . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ (𝑗 ∈ 𝐴 ∧ 𝑘 ∈ 𝐴)) → (abs‘(((𝐻 ∘ 𝐹)‘𝑘) − ((𝐻 ∘ 𝐹)‘𝑗))) = (abs‘((𝐻‘(𝐹‘𝑘)) − (𝐻‘(𝐹‘𝑗))))) | 
| 37 | 36 | breq1d 5152 | . . . . . . . . . . . 12
⊢ (((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ (𝑗 ∈ 𝐴 ∧ 𝑘 ∈ 𝐴)) → ((abs‘(((𝐻 ∘ 𝐹)‘𝑘) − ((𝐻 ∘ 𝐹)‘𝑗))) < 𝑥 ↔ (abs‘((𝐻‘(𝐹‘𝑘)) − (𝐻‘(𝐹‘𝑗)))) < 𝑥)) | 
| 38 | 30, 37 | sylibrd 259 | . . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ (𝑗 ∈ 𝐴 ∧ 𝑘 ∈ 𝐴)) → ((abs‘((𝐹‘𝑘) − (𝐹‘𝑗))) < 𝑥 → (abs‘(((𝐻 ∘ 𝐹)‘𝑘) − ((𝐻 ∘ 𝐹)‘𝑗))) < 𝑥)) | 
| 39 | 38 | imim2d 57 | . . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ (𝑗 ∈ 𝐴 ∧ 𝑘 ∈ 𝐴)) → ((𝑗 ≤ 𝑘 → (abs‘((𝐹‘𝑘) − (𝐹‘𝑗))) < 𝑥) → (𝑗 ≤ 𝑘 → (abs‘(((𝐻 ∘ 𝐹)‘𝑘) − ((𝐻 ∘ 𝐹)‘𝑗))) < 𝑥))) | 
| 40 | 39 | anassrs 467 | . . . . . . . . 9
⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ 𝑗 ∈ 𝐴) ∧ 𝑘 ∈ 𝐴) → ((𝑗 ≤ 𝑘 → (abs‘((𝐹‘𝑘) − (𝐹‘𝑗))) < 𝑥) → (𝑗 ≤ 𝑘 → (abs‘(((𝐻 ∘ 𝐹)‘𝑘) − ((𝐻 ∘ 𝐹)‘𝑗))) < 𝑥))) | 
| 41 | 40 | ralimdva 3166 | . . . . . . . 8
⊢ (((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ 𝑗 ∈ 𝐴) → (∀𝑘 ∈ 𝐴 (𝑗 ≤ 𝑘 → (abs‘((𝐹‘𝑘) − (𝐹‘𝑗))) < 𝑥) → ∀𝑘 ∈ 𝐴 (𝑗 ≤ 𝑘 → (abs‘(((𝐻 ∘ 𝐹)‘𝑘) − ((𝐻 ∘ 𝐹)‘𝑗))) < 𝑥))) | 
| 42 | 41 | reximdva 3167 | . . . . . . 7
⊢ ((𝜑 ∧ 𝑥 ∈ ℝ+) →
(∃𝑗 ∈ 𝐴 ∀𝑘 ∈ 𝐴 (𝑗 ≤ 𝑘 → (abs‘((𝐹‘𝑘) − (𝐹‘𝑗))) < 𝑥) → ∃𝑗 ∈ 𝐴 ∀𝑘 ∈ 𝐴 (𝑗 ≤ 𝑘 → (abs‘(((𝐻 ∘ 𝐹)‘𝑘) − ((𝐻 ∘ 𝐹)‘𝑗))) < 𝑥))) | 
| 43 | 42 | ralimdva 3166 | . . . . . 6
⊢ (𝜑 → (∀𝑥 ∈ ℝ+
∃𝑗 ∈ 𝐴 ∀𝑘 ∈ 𝐴 (𝑗 ≤ 𝑘 → (abs‘((𝐹‘𝑘) − (𝐹‘𝑗))) < 𝑥) → ∀𝑥 ∈ ℝ+ ∃𝑗 ∈ 𝐴 ∀𝑘 ∈ 𝐴 (𝑗 ≤ 𝑘 → (abs‘(((𝐻 ∘ 𝐹)‘𝑘) − ((𝐻 ∘ 𝐹)‘𝑗))) < 𝑥))) | 
| 44 | 9, 43 | mpd 15 | . . . . 5
⊢ (𝜑 → ∀𝑥 ∈ ℝ+ ∃𝑗 ∈ 𝐴 ∀𝑘 ∈ 𝐴 (𝑗 ≤ 𝑘 → (abs‘(((𝐻 ∘ 𝐹)‘𝑘) − ((𝐻 ∘ 𝐹)‘𝑗))) < 𝑥)) | 
| 45 | 5, 7, 8, 44 | caurcvgr 15711 | . . . 4
⊢ (𝜑 → (𝐻 ∘ 𝐹) ⇝𝑟 (lim
sup‘(𝐻 ∘ 𝐹))) | 
| 46 |  | rlimrel 15530 | . . . . 5
⊢ Rel
⇝𝑟 | 
| 47 | 46 | releldmi 5958 | . . . 4
⊢ ((𝐻 ∘ 𝐹) ⇝𝑟 (lim
sup‘(𝐻 ∘ 𝐹)) → (𝐻 ∘ 𝐹) ∈ dom ⇝𝑟
) | 
| 48 | 45, 47 | syl 17 | . . 3
⊢ (𝜑 → (𝐻 ∘ 𝐹) ∈ dom ⇝𝑟
) | 
| 49 |  | ax-resscn 11213 | . . . . 5
⊢ ℝ
⊆ ℂ | 
| 50 |  | fss 6751 | . . . . 5
⊢ (((𝐻 ∘ 𝐹):𝐴⟶ℝ ∧ ℝ ⊆
ℂ) → (𝐻 ∘
𝐹):𝐴⟶ℂ) | 
| 51 | 7, 49, 50 | sylancl 586 | . . . 4
⊢ (𝜑 → (𝐻 ∘ 𝐹):𝐴⟶ℂ) | 
| 52 | 51, 8 | rlimdm 15588 | . . 3
⊢ (𝜑 → ((𝐻 ∘ 𝐹) ∈ dom ⇝𝑟
↔ (𝐻 ∘ 𝐹) ⇝𝑟 (
⇝𝑟 ‘(𝐻 ∘ 𝐹)))) | 
| 53 | 48, 52 | mpbid 232 | . 2
⊢ (𝜑 → (𝐻 ∘ 𝐹) ⇝𝑟 (
⇝𝑟 ‘(𝐻 ∘ 𝐹))) | 
| 54 | 4, 53 | eqbrtrrd 5166 | 1
⊢ (𝜑 → (𝑛 ∈ 𝐴 ↦ (𝐻‘(𝐹‘𝑛))) ⇝𝑟 (
⇝𝑟 ‘(𝐻 ∘ 𝐹))) |