Proof of Theorem cvgcaule
| Step | Hyp | Ref | Expression | 
|---|
| 1 |  | cvgcaule.7 | . 2
⊢ (𝜑 → 𝑋 ∈
ℝ+) | 
| 2 |  | cvgcaule.1 | . . 3
⊢
Ⅎ𝑗𝐹 | 
| 3 |  | cvgcaule.2 | . . 3
⊢
Ⅎ𝑘𝐹 | 
| 4 |  | cvgcaule.3 | . . 3
⊢ (𝜑 → 𝑀 ∈ 𝑍) | 
| 5 |  | cvgcaule.4 | . . 3
⊢ (𝜑 → 𝐹 ∈ 𝑉) | 
| 6 |  | cvgcaule.5 | . . 3
⊢ 𝑍 =
(ℤ≥‘𝑀) | 
| 7 |  | cvgcaule.6 | . . 3
⊢ (𝜑 → 𝐹 ∈ dom ⇝ ) | 
| 8 | 2, 3, 4, 5, 6, 7, 1 | cvgcau 45501 | . 2
⊢ (𝜑 → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)((𝐹‘𝑘) ∈ ℂ ∧ (abs‘((𝐹‘𝑘) − (𝐹‘𝑗))) < 𝑋)) | 
| 9 |  | nfv 1914 | . . . . . 6
⊢
Ⅎ𝑘(𝑋 ∈ ℝ+
∧ 𝑗 ∈ 𝑍) | 
| 10 |  | nfra1 3284 | . . . . . 6
⊢
Ⅎ𝑘∀𝑘 ∈ (ℤ≥‘𝑗)((𝐹‘𝑘) ∈ ℂ ∧ (abs‘((𝐹‘𝑘) − (𝐹‘𝑗))) < 𝑋) | 
| 11 | 9, 10 | nfan 1899 | . . . . 5
⊢
Ⅎ𝑘((𝑋 ∈ ℝ+
∧ 𝑗 ∈ 𝑍) ∧ ∀𝑘 ∈
(ℤ≥‘𝑗)((𝐹‘𝑘) ∈ ℂ ∧ (abs‘((𝐹‘𝑘) − (𝐹‘𝑗))) < 𝑋)) | 
| 12 |  | rspa 3248 | . . . . . . . 8
⊢
((∀𝑘 ∈
(ℤ≥‘𝑗)((𝐹‘𝑘) ∈ ℂ ∧ (abs‘((𝐹‘𝑘) − (𝐹‘𝑗))) < 𝑋) ∧ 𝑘 ∈ (ℤ≥‘𝑗)) → ((𝐹‘𝑘) ∈ ℂ ∧ (abs‘((𝐹‘𝑘) − (𝐹‘𝑗))) < 𝑋)) | 
| 13 | 12 | simpld 494 | . . . . . . 7
⊢
((∀𝑘 ∈
(ℤ≥‘𝑗)((𝐹‘𝑘) ∈ ℂ ∧ (abs‘((𝐹‘𝑘) − (𝐹‘𝑗))) < 𝑋) ∧ 𝑘 ∈ (ℤ≥‘𝑗)) → (𝐹‘𝑘) ∈ ℂ) | 
| 14 | 13 | adantll 714 | . . . . . 6
⊢ ((((𝑋 ∈ ℝ+
∧ 𝑗 ∈ 𝑍) ∧ ∀𝑘 ∈
(ℤ≥‘𝑗)((𝐹‘𝑘) ∈ ℂ ∧ (abs‘((𝐹‘𝑘) − (𝐹‘𝑗))) < 𝑋)) ∧ 𝑘 ∈ (ℤ≥‘𝑗)) → (𝐹‘𝑘) ∈ ℂ) | 
| 15 | 13 | adantll 714 | . . . . . . . . . 10
⊢ (((𝑗 ∈ 𝑍 ∧ ∀𝑘 ∈ (ℤ≥‘𝑗)((𝐹‘𝑘) ∈ ℂ ∧ (abs‘((𝐹‘𝑘) − (𝐹‘𝑗))) < 𝑋)) ∧ 𝑘 ∈ (ℤ≥‘𝑗)) → (𝐹‘𝑘) ∈ ℂ) | 
| 16 | 6 | uzid3 45446 | . . . . . . . . . . . . . 14
⊢ (𝑗 ∈ 𝑍 → 𝑗 ∈ (ℤ≥‘𝑗)) | 
| 17 |  | nfcv 2905 | . . . . . . . . . . . . . . . . . 18
⊢
Ⅎ𝑘𝑗 | 
| 18 | 3, 17 | nffv 6916 | . . . . . . . . . . . . . . . . 17
⊢
Ⅎ𝑘(𝐹‘𝑗) | 
| 19 | 18 | nfel1 2922 | . . . . . . . . . . . . . . . 16
⊢
Ⅎ𝑘(𝐹‘𝑗) ∈ ℂ | 
| 20 |  | nfcv 2905 | . . . . . . . . . . . . . . . . . 18
⊢
Ⅎ𝑘abs | 
| 21 |  | nfcv 2905 | . . . . . . . . . . . . . . . . . . 19
⊢
Ⅎ𝑘
− | 
| 22 | 18, 21, 18 | nfov 7461 | . . . . . . . . . . . . . . . . . 18
⊢
Ⅎ𝑘((𝐹‘𝑗) − (𝐹‘𝑗)) | 
| 23 | 20, 22 | nffv 6916 | . . . . . . . . . . . . . . . . 17
⊢
Ⅎ𝑘(abs‘((𝐹‘𝑗) − (𝐹‘𝑗))) | 
| 24 |  | nfcv 2905 | . . . . . . . . . . . . . . . . 17
⊢
Ⅎ𝑘
< | 
| 25 |  | nfcv 2905 | . . . . . . . . . . . . . . . . 17
⊢
Ⅎ𝑘𝑋 | 
| 26 | 23, 24, 25 | nfbr 5190 | . . . . . . . . . . . . . . . 16
⊢
Ⅎ𝑘(abs‘((𝐹‘𝑗) − (𝐹‘𝑗))) < 𝑋 | 
| 27 | 19, 26 | nfan 1899 | . . . . . . . . . . . . . . 15
⊢
Ⅎ𝑘((𝐹‘𝑗) ∈ ℂ ∧ (abs‘((𝐹‘𝑗) − (𝐹‘𝑗))) < 𝑋) | 
| 28 |  | fveq2 6906 | . . . . . . . . . . . . . . . . 17
⊢ (𝑘 = 𝑗 → (𝐹‘𝑘) = (𝐹‘𝑗)) | 
| 29 | 28 | eleq1d 2826 | . . . . . . . . . . . . . . . 16
⊢ (𝑘 = 𝑗 → ((𝐹‘𝑘) ∈ ℂ ↔ (𝐹‘𝑗) ∈ ℂ)) | 
| 30 | 28 | fvoveq1d 7453 | . . . . . . . . . . . . . . . . 17
⊢ (𝑘 = 𝑗 → (abs‘((𝐹‘𝑘) − (𝐹‘𝑗))) = (abs‘((𝐹‘𝑗) − (𝐹‘𝑗)))) | 
| 31 | 30 | breq1d 5153 | . . . . . . . . . . . . . . . 16
⊢ (𝑘 = 𝑗 → ((abs‘((𝐹‘𝑘) − (𝐹‘𝑗))) < 𝑋 ↔ (abs‘((𝐹‘𝑗) − (𝐹‘𝑗))) < 𝑋)) | 
| 32 | 29, 31 | anbi12d 632 | . . . . . . . . . . . . . . 15
⊢ (𝑘 = 𝑗 → (((𝐹‘𝑘) ∈ ℂ ∧ (abs‘((𝐹‘𝑘) − (𝐹‘𝑗))) < 𝑋) ↔ ((𝐹‘𝑗) ∈ ℂ ∧ (abs‘((𝐹‘𝑗) − (𝐹‘𝑗))) < 𝑋))) | 
| 33 | 27, 32 | rspc 3610 | . . . . . . . . . . . . . 14
⊢ (𝑗 ∈
(ℤ≥‘𝑗) → (∀𝑘 ∈ (ℤ≥‘𝑗)((𝐹‘𝑘) ∈ ℂ ∧ (abs‘((𝐹‘𝑘) − (𝐹‘𝑗))) < 𝑋) → ((𝐹‘𝑗) ∈ ℂ ∧ (abs‘((𝐹‘𝑗) − (𝐹‘𝑗))) < 𝑋))) | 
| 34 | 16, 33 | syl 17 | . . . . . . . . . . . . 13
⊢ (𝑗 ∈ 𝑍 → (∀𝑘 ∈ (ℤ≥‘𝑗)((𝐹‘𝑘) ∈ ℂ ∧ (abs‘((𝐹‘𝑘) − (𝐹‘𝑗))) < 𝑋) → ((𝐹‘𝑗) ∈ ℂ ∧ (abs‘((𝐹‘𝑗) − (𝐹‘𝑗))) < 𝑋))) | 
| 35 | 34 | imp 406 | . . . . . . . . . . . 12
⊢ ((𝑗 ∈ 𝑍 ∧ ∀𝑘 ∈ (ℤ≥‘𝑗)((𝐹‘𝑘) ∈ ℂ ∧ (abs‘((𝐹‘𝑘) − (𝐹‘𝑗))) < 𝑋)) → ((𝐹‘𝑗) ∈ ℂ ∧ (abs‘((𝐹‘𝑗) − (𝐹‘𝑗))) < 𝑋)) | 
| 36 | 35 | simpld 494 | . . . . . . . . . . 11
⊢ ((𝑗 ∈ 𝑍 ∧ ∀𝑘 ∈ (ℤ≥‘𝑗)((𝐹‘𝑘) ∈ ℂ ∧ (abs‘((𝐹‘𝑘) − (𝐹‘𝑗))) < 𝑋)) → (𝐹‘𝑗) ∈ ℂ) | 
| 37 | 36 | adantr 480 | . . . . . . . . . 10
⊢ (((𝑗 ∈ 𝑍 ∧ ∀𝑘 ∈ (ℤ≥‘𝑗)((𝐹‘𝑘) ∈ ℂ ∧ (abs‘((𝐹‘𝑘) − (𝐹‘𝑗))) < 𝑋)) ∧ 𝑘 ∈ (ℤ≥‘𝑗)) → (𝐹‘𝑗) ∈ ℂ) | 
| 38 | 15, 37 | subcld 11620 | . . . . . . . . 9
⊢ (((𝑗 ∈ 𝑍 ∧ ∀𝑘 ∈ (ℤ≥‘𝑗)((𝐹‘𝑘) ∈ ℂ ∧ (abs‘((𝐹‘𝑘) − (𝐹‘𝑗))) < 𝑋)) ∧ 𝑘 ∈ (ℤ≥‘𝑗)) → ((𝐹‘𝑘) − (𝐹‘𝑗)) ∈ ℂ) | 
| 39 | 38 | abscld 15475 | . . . . . . . 8
⊢ (((𝑗 ∈ 𝑍 ∧ ∀𝑘 ∈ (ℤ≥‘𝑗)((𝐹‘𝑘) ∈ ℂ ∧ (abs‘((𝐹‘𝑘) − (𝐹‘𝑗))) < 𝑋)) ∧ 𝑘 ∈ (ℤ≥‘𝑗)) → (abs‘((𝐹‘𝑘) − (𝐹‘𝑗))) ∈ ℝ) | 
| 40 | 39 | adantlll 718 | . . . . . . 7
⊢ ((((𝑋 ∈ ℝ+
∧ 𝑗 ∈ 𝑍) ∧ ∀𝑘 ∈
(ℤ≥‘𝑗)((𝐹‘𝑘) ∈ ℂ ∧ (abs‘((𝐹‘𝑘) − (𝐹‘𝑗))) < 𝑋)) ∧ 𝑘 ∈ (ℤ≥‘𝑗)) → (abs‘((𝐹‘𝑘) − (𝐹‘𝑗))) ∈ ℝ) | 
| 41 |  | simplll 775 | . . . . . . . 8
⊢ ((((𝑋 ∈ ℝ+
∧ 𝑗 ∈ 𝑍) ∧ ∀𝑘 ∈
(ℤ≥‘𝑗)((𝐹‘𝑘) ∈ ℂ ∧ (abs‘((𝐹‘𝑘) − (𝐹‘𝑗))) < 𝑋)) ∧ 𝑘 ∈ (ℤ≥‘𝑗)) → 𝑋 ∈
ℝ+) | 
| 42 | 41 | rpred 13077 | . . . . . . 7
⊢ ((((𝑋 ∈ ℝ+
∧ 𝑗 ∈ 𝑍) ∧ ∀𝑘 ∈
(ℤ≥‘𝑗)((𝐹‘𝑘) ∈ ℂ ∧ (abs‘((𝐹‘𝑘) − (𝐹‘𝑗))) < 𝑋)) ∧ 𝑘 ∈ (ℤ≥‘𝑗)) → 𝑋 ∈ ℝ) | 
| 43 | 12 | adantll 714 | . . . . . . . 8
⊢ ((((𝑋 ∈ ℝ+
∧ 𝑗 ∈ 𝑍) ∧ ∀𝑘 ∈
(ℤ≥‘𝑗)((𝐹‘𝑘) ∈ ℂ ∧ (abs‘((𝐹‘𝑘) − (𝐹‘𝑗))) < 𝑋)) ∧ 𝑘 ∈ (ℤ≥‘𝑗)) → ((𝐹‘𝑘) ∈ ℂ ∧ (abs‘((𝐹‘𝑘) − (𝐹‘𝑗))) < 𝑋)) | 
| 44 | 43 | simprd 495 | . . . . . . 7
⊢ ((((𝑋 ∈ ℝ+
∧ 𝑗 ∈ 𝑍) ∧ ∀𝑘 ∈
(ℤ≥‘𝑗)((𝐹‘𝑘) ∈ ℂ ∧ (abs‘((𝐹‘𝑘) − (𝐹‘𝑗))) < 𝑋)) ∧ 𝑘 ∈ (ℤ≥‘𝑗)) → (abs‘((𝐹‘𝑘) − (𝐹‘𝑗))) < 𝑋) | 
| 45 | 40, 42, 44 | ltled 11409 | . . . . . 6
⊢ ((((𝑋 ∈ ℝ+
∧ 𝑗 ∈ 𝑍) ∧ ∀𝑘 ∈
(ℤ≥‘𝑗)((𝐹‘𝑘) ∈ ℂ ∧ (abs‘((𝐹‘𝑘) − (𝐹‘𝑗))) < 𝑋)) ∧ 𝑘 ∈ (ℤ≥‘𝑗)) → (abs‘((𝐹‘𝑘) − (𝐹‘𝑗))) ≤ 𝑋) | 
| 46 | 14, 45 | jca 511 | . . . . 5
⊢ ((((𝑋 ∈ ℝ+
∧ 𝑗 ∈ 𝑍) ∧ ∀𝑘 ∈
(ℤ≥‘𝑗)((𝐹‘𝑘) ∈ ℂ ∧ (abs‘((𝐹‘𝑘) − (𝐹‘𝑗))) < 𝑋)) ∧ 𝑘 ∈ (ℤ≥‘𝑗)) → ((𝐹‘𝑘) ∈ ℂ ∧ (abs‘((𝐹‘𝑘) − (𝐹‘𝑗))) ≤ 𝑋)) | 
| 47 | 11, 46 | ralrimia 3258 | . . . 4
⊢ (((𝑋 ∈ ℝ+
∧ 𝑗 ∈ 𝑍) ∧ ∀𝑘 ∈
(ℤ≥‘𝑗)((𝐹‘𝑘) ∈ ℂ ∧ (abs‘((𝐹‘𝑘) − (𝐹‘𝑗))) < 𝑋)) → ∀𝑘 ∈ (ℤ≥‘𝑗)((𝐹‘𝑘) ∈ ℂ ∧ (abs‘((𝐹‘𝑘) − (𝐹‘𝑗))) ≤ 𝑋)) | 
| 48 | 47 | ex 412 | . . 3
⊢ ((𝑋 ∈ ℝ+
∧ 𝑗 ∈ 𝑍) → (∀𝑘 ∈
(ℤ≥‘𝑗)((𝐹‘𝑘) ∈ ℂ ∧ (abs‘((𝐹‘𝑘) − (𝐹‘𝑗))) < 𝑋) → ∀𝑘 ∈ (ℤ≥‘𝑗)((𝐹‘𝑘) ∈ ℂ ∧ (abs‘((𝐹‘𝑘) − (𝐹‘𝑗))) ≤ 𝑋))) | 
| 49 | 48 | reximdva 3168 | . 2
⊢ (𝑋 ∈ ℝ+
→ (∃𝑗 ∈
𝑍 ∀𝑘 ∈
(ℤ≥‘𝑗)((𝐹‘𝑘) ∈ ℂ ∧ (abs‘((𝐹‘𝑘) − (𝐹‘𝑗))) < 𝑋) → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)((𝐹‘𝑘) ∈ ℂ ∧ (abs‘((𝐹‘𝑘) − (𝐹‘𝑗))) ≤ 𝑋))) | 
| 50 | 1, 8, 49 | sylc 65 | 1
⊢ (𝜑 → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)((𝐹‘𝑘) ∈ ℂ ∧ (abs‘((𝐹‘𝑘) − (𝐹‘𝑗))) ≤ 𝑋)) |