| Step | Hyp | Ref
| Expression |
| 1 | | fveq2 6906 |
. . . . . 6
⊢ (𝑘 = 𝑛 → (𝐹‘𝑘) = (𝐹‘𝑛)) |
| 2 | 1 | cbvmptv 5255 |
. . . . 5
⊢ (𝑘 ∈ 𝑍 ↦ (𝐹‘𝑘)) = (𝑛 ∈ 𝑍 ↦ (𝐹‘𝑛)) |
| 3 | | caucvg.1 |
. . . . . . . . . 10
⊢ 𝑍 =
(ℤ≥‘𝑀) |
| 4 | | uzssz 12899 |
. . . . . . . . . 10
⊢
(ℤ≥‘𝑀) ⊆ ℤ |
| 5 | 3, 4 | eqsstri 4030 |
. . . . . . . . 9
⊢ 𝑍 ⊆
ℤ |
| 6 | | zssre 12620 |
. . . . . . . . 9
⊢ ℤ
⊆ ℝ |
| 7 | 5, 6 | sstri 3993 |
. . . . . . . 8
⊢ 𝑍 ⊆
ℝ |
| 8 | 7 | a1i 11 |
. . . . . . 7
⊢ (𝜑 → 𝑍 ⊆ ℝ) |
| 9 | | caucvg.2 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) ∈ ℂ) |
| 10 | 2 | eqcomi 2746 |
. . . . . . . 8
⊢ (𝑛 ∈ 𝑍 ↦ (𝐹‘𝑛)) = (𝑘 ∈ 𝑍 ↦ (𝐹‘𝑘)) |
| 11 | 9, 10 | fmptd 7134 |
. . . . . . 7
⊢ (𝜑 → (𝑛 ∈ 𝑍 ↦ (𝐹‘𝑛)):𝑍⟶ℂ) |
| 12 | | 1rp 13038 |
. . . . . . . . . . 11
⊢ 1 ∈
ℝ+ |
| 13 | 12 | ne0ii 4344 |
. . . . . . . . . 10
⊢
ℝ+ ≠ ∅ |
| 14 | | caucvg.3 |
. . . . . . . . . 10
⊢ (𝜑 → ∀𝑥 ∈ ℝ+ ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(abs‘((𝐹‘𝑘) − (𝐹‘𝑗))) < 𝑥) |
| 15 | | r19.2z 4495 |
. . . . . . . . . 10
⊢
((ℝ+ ≠ ∅ ∧ ∀𝑥 ∈ ℝ+ ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(abs‘((𝐹‘𝑘) − (𝐹‘𝑗))) < 𝑥) → ∃𝑥 ∈ ℝ+ ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(abs‘((𝐹‘𝑘) − (𝐹‘𝑗))) < 𝑥) |
| 16 | 13, 14, 15 | sylancr 587 |
. . . . . . . . 9
⊢ (𝜑 → ∃𝑥 ∈ ℝ+ ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(abs‘((𝐹‘𝑘) − (𝐹‘𝑗))) < 𝑥) |
| 17 | | eluzel2 12883 |
. . . . . . . . . . . . 13
⊢ (𝑗 ∈
(ℤ≥‘𝑀) → 𝑀 ∈ ℤ) |
| 18 | 17, 3 | eleq2s 2859 |
. . . . . . . . . . . 12
⊢ (𝑗 ∈ 𝑍 → 𝑀 ∈ ℤ) |
| 19 | 18 | a1d 25 |
. . . . . . . . . . 11
⊢ (𝑗 ∈ 𝑍 → (∀𝑘 ∈ (ℤ≥‘𝑗)(abs‘((𝐹‘𝑘) − (𝐹‘𝑗))) < 𝑥 → 𝑀 ∈ ℤ)) |
| 20 | 19 | rexlimiv 3148 |
. . . . . . . . . 10
⊢
(∃𝑗 ∈
𝑍 ∀𝑘 ∈
(ℤ≥‘𝑗)(abs‘((𝐹‘𝑘) − (𝐹‘𝑗))) < 𝑥 → 𝑀 ∈ ℤ) |
| 21 | 20 | rexlimivw 3151 |
. . . . . . . . 9
⊢
(∃𝑥 ∈
ℝ+ ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(abs‘((𝐹‘𝑘) − (𝐹‘𝑗))) < 𝑥 → 𝑀 ∈ ℤ) |
| 22 | 16, 21 | syl 17 |
. . . . . . . 8
⊢ (𝜑 → 𝑀 ∈ ℤ) |
| 23 | 3 | uzsup 13903 |
. . . . . . . 8
⊢ (𝑀 ∈ ℤ → sup(𝑍, ℝ*, < ) =
+∞) |
| 24 | 22, 23 | syl 17 |
. . . . . . 7
⊢ (𝜑 → sup(𝑍, ℝ*, < ) =
+∞) |
| 25 | 5 | sseli 3979 |
. . . . . . . . . . . . . . . 16
⊢ (𝑗 ∈ 𝑍 → 𝑗 ∈ ℤ) |
| 26 | 5 | sseli 3979 |
. . . . . . . . . . . . . . . 16
⊢ (𝑘 ∈ 𝑍 → 𝑘 ∈ ℤ) |
| 27 | | eluz 12892 |
. . . . . . . . . . . . . . . 16
⊢ ((𝑗 ∈ ℤ ∧ 𝑘 ∈ ℤ) → (𝑘 ∈
(ℤ≥‘𝑗) ↔ 𝑗 ≤ 𝑘)) |
| 28 | 25, 26, 27 | syl2an 596 |
. . . . . . . . . . . . . . 15
⊢ ((𝑗 ∈ 𝑍 ∧ 𝑘 ∈ 𝑍) → (𝑘 ∈ (ℤ≥‘𝑗) ↔ 𝑗 ≤ 𝑘)) |
| 29 | 28 | biimprd 248 |
. . . . . . . . . . . . . 14
⊢ ((𝑗 ∈ 𝑍 ∧ 𝑘 ∈ 𝑍) → (𝑗 ≤ 𝑘 → 𝑘 ∈ (ℤ≥‘𝑗))) |
| 30 | | fveq2 6906 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑛 = 𝑘 → (𝐹‘𝑛) = (𝐹‘𝑘)) |
| 31 | | eqid 2737 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑛 ∈ 𝑍 ↦ (𝐹‘𝑛)) = (𝑛 ∈ 𝑍 ↦ (𝐹‘𝑛)) |
| 32 | | fvex 6919 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝐹‘𝑛) ∈ V |
| 33 | 30, 31, 32 | fvmpt3i 7021 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑘 ∈ 𝑍 → ((𝑛 ∈ 𝑍 ↦ (𝐹‘𝑛))‘𝑘) = (𝐹‘𝑘)) |
| 34 | | fveq2 6906 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑛 = 𝑗 → (𝐹‘𝑛) = (𝐹‘𝑗)) |
| 35 | 34, 31, 32 | fvmpt3i 7021 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑗 ∈ 𝑍 → ((𝑛 ∈ 𝑍 ↦ (𝐹‘𝑛))‘𝑗) = (𝐹‘𝑗)) |
| 36 | 33, 35 | oveqan12rd 7451 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝑗 ∈ 𝑍 ∧ 𝑘 ∈ 𝑍) → (((𝑛 ∈ 𝑍 ↦ (𝐹‘𝑛))‘𝑘) − ((𝑛 ∈ 𝑍 ↦ (𝐹‘𝑛))‘𝑗)) = ((𝐹‘𝑘) − (𝐹‘𝑗))) |
| 37 | 36 | fveq2d 6910 |
. . . . . . . . . . . . . . . 16
⊢ ((𝑗 ∈ 𝑍 ∧ 𝑘 ∈ 𝑍) → (abs‘(((𝑛 ∈ 𝑍 ↦ (𝐹‘𝑛))‘𝑘) − ((𝑛 ∈ 𝑍 ↦ (𝐹‘𝑛))‘𝑗))) = (abs‘((𝐹‘𝑘) − (𝐹‘𝑗)))) |
| 38 | 37 | breq1d 5153 |
. . . . . . . . . . . . . . 15
⊢ ((𝑗 ∈ 𝑍 ∧ 𝑘 ∈ 𝑍) → ((abs‘(((𝑛 ∈ 𝑍 ↦ (𝐹‘𝑛))‘𝑘) − ((𝑛 ∈ 𝑍 ↦ (𝐹‘𝑛))‘𝑗))) < 𝑥 ↔ (abs‘((𝐹‘𝑘) − (𝐹‘𝑗))) < 𝑥)) |
| 39 | 38 | biimprd 248 |
. . . . . . . . . . . . . 14
⊢ ((𝑗 ∈ 𝑍 ∧ 𝑘 ∈ 𝑍) → ((abs‘((𝐹‘𝑘) − (𝐹‘𝑗))) < 𝑥 → (abs‘(((𝑛 ∈ 𝑍 ↦ (𝐹‘𝑛))‘𝑘) − ((𝑛 ∈ 𝑍 ↦ (𝐹‘𝑛))‘𝑗))) < 𝑥)) |
| 40 | 29, 39 | imim12d 81 |
. . . . . . . . . . . . 13
⊢ ((𝑗 ∈ 𝑍 ∧ 𝑘 ∈ 𝑍) → ((𝑘 ∈ (ℤ≥‘𝑗) → (abs‘((𝐹‘𝑘) − (𝐹‘𝑗))) < 𝑥) → (𝑗 ≤ 𝑘 → (abs‘(((𝑛 ∈ 𝑍 ↦ (𝐹‘𝑛))‘𝑘) − ((𝑛 ∈ 𝑍 ↦ (𝐹‘𝑛))‘𝑗))) < 𝑥))) |
| 41 | 40 | ex 412 |
. . . . . . . . . . . 12
⊢ (𝑗 ∈ 𝑍 → (𝑘 ∈ 𝑍 → ((𝑘 ∈ (ℤ≥‘𝑗) → (abs‘((𝐹‘𝑘) − (𝐹‘𝑗))) < 𝑥) → (𝑗 ≤ 𝑘 → (abs‘(((𝑛 ∈ 𝑍 ↦ (𝐹‘𝑛))‘𝑘) − ((𝑛 ∈ 𝑍 ↦ (𝐹‘𝑛))‘𝑗))) < 𝑥)))) |
| 42 | 41 | com23 86 |
. . . . . . . . . . 11
⊢ (𝑗 ∈ 𝑍 → ((𝑘 ∈ (ℤ≥‘𝑗) → (abs‘((𝐹‘𝑘) − (𝐹‘𝑗))) < 𝑥) → (𝑘 ∈ 𝑍 → (𝑗 ≤ 𝑘 → (abs‘(((𝑛 ∈ 𝑍 ↦ (𝐹‘𝑛))‘𝑘) − ((𝑛 ∈ 𝑍 ↦ (𝐹‘𝑛))‘𝑗))) < 𝑥)))) |
| 43 | 42 | ralimdv2 3163 |
. . . . . . . . . 10
⊢ (𝑗 ∈ 𝑍 → (∀𝑘 ∈ (ℤ≥‘𝑗)(abs‘((𝐹‘𝑘) − (𝐹‘𝑗))) < 𝑥 → ∀𝑘 ∈ 𝑍 (𝑗 ≤ 𝑘 → (abs‘(((𝑛 ∈ 𝑍 ↦ (𝐹‘𝑛))‘𝑘) − ((𝑛 ∈ 𝑍 ↦ (𝐹‘𝑛))‘𝑗))) < 𝑥))) |
| 44 | 43 | reximia 3081 |
. . . . . . . . 9
⊢
(∃𝑗 ∈
𝑍 ∀𝑘 ∈
(ℤ≥‘𝑗)(abs‘((𝐹‘𝑘) − (𝐹‘𝑗))) < 𝑥 → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ 𝑍 (𝑗 ≤ 𝑘 → (abs‘(((𝑛 ∈ 𝑍 ↦ (𝐹‘𝑛))‘𝑘) − ((𝑛 ∈ 𝑍 ↦ (𝐹‘𝑛))‘𝑗))) < 𝑥)) |
| 45 | 44 | ralimi 3083 |
. . . . . . . 8
⊢
(∀𝑥 ∈
ℝ+ ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(abs‘((𝐹‘𝑘) − (𝐹‘𝑗))) < 𝑥 → ∀𝑥 ∈ ℝ+ ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ 𝑍 (𝑗 ≤ 𝑘 → (abs‘(((𝑛 ∈ 𝑍 ↦ (𝐹‘𝑛))‘𝑘) − ((𝑛 ∈ 𝑍 ↦ (𝐹‘𝑛))‘𝑗))) < 𝑥)) |
| 46 | 14, 45 | syl 17 |
. . . . . . 7
⊢ (𝜑 → ∀𝑥 ∈ ℝ+ ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ 𝑍 (𝑗 ≤ 𝑘 → (abs‘(((𝑛 ∈ 𝑍 ↦ (𝐹‘𝑛))‘𝑘) − ((𝑛 ∈ 𝑍 ↦ (𝐹‘𝑛))‘𝑗))) < 𝑥)) |
| 47 | 8, 11, 24, 46 | caucvgr 15712 |
. . . . . 6
⊢ (𝜑 → (𝑛 ∈ 𝑍 ↦ (𝐹‘𝑛)) ∈ dom ⇝𝑟
) |
| 48 | 11, 24 | rlimdm 15587 |
. . . . . 6
⊢ (𝜑 → ((𝑛 ∈ 𝑍 ↦ (𝐹‘𝑛)) ∈ dom ⇝𝑟
↔ (𝑛 ∈ 𝑍 ↦ (𝐹‘𝑛)) ⇝𝑟 (
⇝𝑟 ‘(𝑛 ∈ 𝑍 ↦ (𝐹‘𝑛))))) |
| 49 | 47, 48 | mpbid 232 |
. . . . 5
⊢ (𝜑 → (𝑛 ∈ 𝑍 ↦ (𝐹‘𝑛)) ⇝𝑟 (
⇝𝑟 ‘(𝑛 ∈ 𝑍 ↦ (𝐹‘𝑛)))) |
| 50 | 2, 49 | eqbrtrid 5178 |
. . . 4
⊢ (𝜑 → (𝑘 ∈ 𝑍 ↦ (𝐹‘𝑘)) ⇝𝑟 (
⇝𝑟 ‘(𝑛 ∈ 𝑍 ↦ (𝐹‘𝑛)))) |
| 51 | | eqid 2737 |
. . . . . 6
⊢ (𝑘 ∈ 𝑍 ↦ (𝐹‘𝑘)) = (𝑘 ∈ 𝑍 ↦ (𝐹‘𝑘)) |
| 52 | 9, 51 | fmptd 7134 |
. . . . 5
⊢ (𝜑 → (𝑘 ∈ 𝑍 ↦ (𝐹‘𝑘)):𝑍⟶ℂ) |
| 53 | 3, 22, 52 | rlimclim 15582 |
. . . 4
⊢ (𝜑 → ((𝑘 ∈ 𝑍 ↦ (𝐹‘𝑘)) ⇝𝑟 (
⇝𝑟 ‘(𝑛 ∈ 𝑍 ↦ (𝐹‘𝑛))) ↔ (𝑘 ∈ 𝑍 ↦ (𝐹‘𝑘)) ⇝ ( ⇝𝑟
‘(𝑛 ∈ 𝑍 ↦ (𝐹‘𝑛))))) |
| 54 | 50, 53 | mpbid 232 |
. . 3
⊢ (𝜑 → (𝑘 ∈ 𝑍 ↦ (𝐹‘𝑘)) ⇝ ( ⇝𝑟
‘(𝑛 ∈ 𝑍 ↦ (𝐹‘𝑛)))) |
| 55 | | caucvg.4 |
. . . 4
⊢ (𝜑 → 𝐹 ∈ 𝑉) |
| 56 | 3, 51 | climmpt 15607 |
. . . 4
⊢ ((𝑀 ∈ ℤ ∧ 𝐹 ∈ 𝑉) → (𝐹 ⇝ ( ⇝𝑟
‘(𝑛 ∈ 𝑍 ↦ (𝐹‘𝑛))) ↔ (𝑘 ∈ 𝑍 ↦ (𝐹‘𝑘)) ⇝ ( ⇝𝑟
‘(𝑛 ∈ 𝑍 ↦ (𝐹‘𝑛))))) |
| 57 | 22, 55, 56 | syl2anc 584 |
. . 3
⊢ (𝜑 → (𝐹 ⇝ ( ⇝𝑟
‘(𝑛 ∈ 𝑍 ↦ (𝐹‘𝑛))) ↔ (𝑘 ∈ 𝑍 ↦ (𝐹‘𝑘)) ⇝ ( ⇝𝑟
‘(𝑛 ∈ 𝑍 ↦ (𝐹‘𝑛))))) |
| 58 | 54, 57 | mpbird 257 |
. 2
⊢ (𝜑 → 𝐹 ⇝ ( ⇝𝑟
‘(𝑛 ∈ 𝑍 ↦ (𝐹‘𝑛)))) |
| 59 | | climrel 15528 |
. . 3
⊢ Rel
⇝ |
| 60 | 59 | releldmi 5959 |
. 2
⊢ (𝐹 ⇝ (
⇝𝑟 ‘(𝑛 ∈ 𝑍 ↦ (𝐹‘𝑛))) → 𝐹 ∈ dom ⇝ ) |
| 61 | 58, 60 | syl 17 |
1
⊢ (𝜑 → 𝐹 ∈ dom ⇝ ) |