| Step | Hyp | Ref
| Expression |
| 1 | | climrlim2.5 |
. 2
⊢ (𝜑 → (𝑛 ∈ 𝑍 ↦ 𝐵) ⇝ 𝐷) |
| 2 | | eluzelz 12888 |
. . . . . . . . . . . . . . . 16
⊢ (𝑗 ∈
(ℤ≥‘𝑀) → 𝑗 ∈ ℤ) |
| 3 | | climrlim2.1 |
. . . . . . . . . . . . . . . 16
⊢ 𝑍 =
(ℤ≥‘𝑀) |
| 4 | 2, 3 | eleq2s 2859 |
. . . . . . . . . . . . . . 15
⊢ (𝑗 ∈ 𝑍 → 𝑗 ∈ ℤ) |
| 5 | 4 | ad2antlr 727 |
. . . . . . . . . . . . . 14
⊢ ((((𝜑 ∧ 𝑦 ∈ ℝ+) ∧ 𝑗 ∈ 𝑍) ∧ (𝑥 ∈ 𝐴 ∧ 𝑗 ≤ 𝑥)) → 𝑗 ∈ ℤ) |
| 6 | | climrlim2.3 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝜑 → 𝐴 ⊆ ℝ) |
| 7 | 6 | sselda 3983 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ ℝ) |
| 8 | 7 | flcld 13838 |
. . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (⌊‘𝑥) ∈ ℤ) |
| 9 | 8 | adantlr 715 |
. . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ 𝑦 ∈ ℝ+) ∧ 𝑥 ∈ 𝐴) → (⌊‘𝑥) ∈ ℤ) |
| 10 | 9 | ad2ant2r 747 |
. . . . . . . . . . . . . 14
⊢ ((((𝜑 ∧ 𝑦 ∈ ℝ+) ∧ 𝑗 ∈ 𝑍) ∧ (𝑥 ∈ 𝐴 ∧ 𝑗 ≤ 𝑥)) → (⌊‘𝑥) ∈ ℤ) |
| 11 | | simprr 773 |
. . . . . . . . . . . . . . 15
⊢ ((((𝜑 ∧ 𝑦 ∈ ℝ+) ∧ 𝑗 ∈ 𝑍) ∧ (𝑥 ∈ 𝐴 ∧ 𝑗 ≤ 𝑥)) → 𝑗 ≤ 𝑥) |
| 12 | 7 | adantlr 715 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝜑 ∧ 𝑦 ∈ ℝ+) ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ ℝ) |
| 13 | 12 | ad2ant2r 747 |
. . . . . . . . . . . . . . . 16
⊢ ((((𝜑 ∧ 𝑦 ∈ ℝ+) ∧ 𝑗 ∈ 𝑍) ∧ (𝑥 ∈ 𝐴 ∧ 𝑗 ≤ 𝑥)) → 𝑥 ∈ ℝ) |
| 14 | | flge 13845 |
. . . . . . . . . . . . . . . 16
⊢ ((𝑥 ∈ ℝ ∧ 𝑗 ∈ ℤ) → (𝑗 ≤ 𝑥 ↔ 𝑗 ≤ (⌊‘𝑥))) |
| 15 | 13, 5, 14 | syl2anc 584 |
. . . . . . . . . . . . . . 15
⊢ ((((𝜑 ∧ 𝑦 ∈ ℝ+) ∧ 𝑗 ∈ 𝑍) ∧ (𝑥 ∈ 𝐴 ∧ 𝑗 ≤ 𝑥)) → (𝑗 ≤ 𝑥 ↔ 𝑗 ≤ (⌊‘𝑥))) |
| 16 | 11, 15 | mpbid 232 |
. . . . . . . . . . . . . 14
⊢ ((((𝜑 ∧ 𝑦 ∈ ℝ+) ∧ 𝑗 ∈ 𝑍) ∧ (𝑥 ∈ 𝐴 ∧ 𝑗 ≤ 𝑥)) → 𝑗 ≤ (⌊‘𝑥)) |
| 17 | | eluz2 12884 |
. . . . . . . . . . . . . 14
⊢
((⌊‘𝑥)
∈ (ℤ≥‘𝑗) ↔ (𝑗 ∈ ℤ ∧ (⌊‘𝑥) ∈ ℤ ∧ 𝑗 ≤ (⌊‘𝑥))) |
| 18 | 5, 10, 16, 17 | syl3anbrc 1344 |
. . . . . . . . . . . . 13
⊢ ((((𝜑 ∧ 𝑦 ∈ ℝ+) ∧ 𝑗 ∈ 𝑍) ∧ (𝑥 ∈ 𝐴 ∧ 𝑗 ≤ 𝑥)) → (⌊‘𝑥) ∈ (ℤ≥‘𝑗)) |
| 19 | | simpr 484 |
. . . . . . . . . . . . . 14
⊢ ((((𝑛 ∈ 𝑍 ↦ 𝐵)‘𝑘) ∈ ℂ ∧ (abs‘(((𝑛 ∈ 𝑍 ↦ 𝐵)‘𝑘) − 𝐷)) < 𝑦) → (abs‘(((𝑛 ∈ 𝑍 ↦ 𝐵)‘𝑘) − 𝐷)) < 𝑦) |
| 20 | 19 | ralimi 3083 |
. . . . . . . . . . . . 13
⊢
(∀𝑘 ∈
(ℤ≥‘𝑗)(((𝑛 ∈ 𝑍 ↦ 𝐵)‘𝑘) ∈ ℂ ∧ (abs‘(((𝑛 ∈ 𝑍 ↦ 𝐵)‘𝑘) − 𝐷)) < 𝑦) → ∀𝑘 ∈ (ℤ≥‘𝑗)(abs‘(((𝑛 ∈ 𝑍 ↦ 𝐵)‘𝑘) − 𝐷)) < 𝑦) |
| 21 | | fveq2 6906 |
. . . . . . . . . . . . . . . 16
⊢ (𝑘 = (⌊‘𝑥) → ((𝑛 ∈ 𝑍 ↦ 𝐵)‘𝑘) = ((𝑛 ∈ 𝑍 ↦ 𝐵)‘(⌊‘𝑥))) |
| 22 | 21 | fvoveq1d 7453 |
. . . . . . . . . . . . . . 15
⊢ (𝑘 = (⌊‘𝑥) → (abs‘(((𝑛 ∈ 𝑍 ↦ 𝐵)‘𝑘) − 𝐷)) = (abs‘(((𝑛 ∈ 𝑍 ↦ 𝐵)‘(⌊‘𝑥)) − 𝐷))) |
| 23 | 22 | breq1d 5153 |
. . . . . . . . . . . . . 14
⊢ (𝑘 = (⌊‘𝑥) → ((abs‘(((𝑛 ∈ 𝑍 ↦ 𝐵)‘𝑘) − 𝐷)) < 𝑦 ↔ (abs‘(((𝑛 ∈ 𝑍 ↦ 𝐵)‘(⌊‘𝑥)) − 𝐷)) < 𝑦)) |
| 24 | 23 | rspcv 3618 |
. . . . . . . . . . . . 13
⊢
((⌊‘𝑥)
∈ (ℤ≥‘𝑗) → (∀𝑘 ∈ (ℤ≥‘𝑗)(abs‘(((𝑛 ∈ 𝑍 ↦ 𝐵)‘𝑘) − 𝐷)) < 𝑦 → (abs‘(((𝑛 ∈ 𝑍 ↦ 𝐵)‘(⌊‘𝑥)) − 𝐷)) < 𝑦)) |
| 25 | 18, 20, 24 | syl2im 40 |
. . . . . . . . . . . 12
⊢ ((((𝜑 ∧ 𝑦 ∈ ℝ+) ∧ 𝑗 ∈ 𝑍) ∧ (𝑥 ∈ 𝐴 ∧ 𝑗 ≤ 𝑥)) → (∀𝑘 ∈ (ℤ≥‘𝑗)(((𝑛 ∈ 𝑍 ↦ 𝐵)‘𝑘) ∈ ℂ ∧ (abs‘(((𝑛 ∈ 𝑍 ↦ 𝐵)‘𝑘) − 𝐷)) < 𝑦) → (abs‘(((𝑛 ∈ 𝑍 ↦ 𝐵)‘(⌊‘𝑥)) − 𝐷)) < 𝑦)) |
| 26 | | eqid 2737 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑛 ∈ 𝑍 ↦ 𝐵) = (𝑛 ∈ 𝑍 ↦ 𝐵) |
| 27 | | climrlim2.2 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑛 = (⌊‘𝑥) → 𝐵 = 𝐶) |
| 28 | | climrlim2.4 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝜑 → 𝑀 ∈ ℤ) |
| 29 | 28 | adantr 480 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑀 ∈ ℤ) |
| 30 | | climrlim2.7 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑀 ≤ 𝑥) |
| 31 | | flge 13845 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((𝑥 ∈ ℝ ∧ 𝑀 ∈ ℤ) → (𝑀 ≤ 𝑥 ↔ 𝑀 ≤ (⌊‘𝑥))) |
| 32 | 7, 29, 31 | syl2anc 584 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝑀 ≤ 𝑥 ↔ 𝑀 ≤ (⌊‘𝑥))) |
| 33 | 30, 32 | mpbid 232 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑀 ≤ (⌊‘𝑥)) |
| 34 | | eluz2 12884 |
. . . . . . . . . . . . . . . . . . 19
⊢
((⌊‘𝑥)
∈ (ℤ≥‘𝑀) ↔ (𝑀 ∈ ℤ ∧ (⌊‘𝑥) ∈ ℤ ∧ 𝑀 ≤ (⌊‘𝑥))) |
| 35 | 29, 8, 33, 34 | syl3anbrc 1344 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (⌊‘𝑥) ∈ (ℤ≥‘𝑀)) |
| 36 | 35, 3 | eleqtrrdi 2852 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (⌊‘𝑥) ∈ 𝑍) |
| 37 | 27 | eleq1d 2826 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑛 = (⌊‘𝑥) → (𝐵 ∈ ℂ ↔ 𝐶 ∈ ℂ)) |
| 38 | | climrlim2.6 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → 𝐵 ∈ ℂ) |
| 39 | 38 | ralrimiva 3146 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝜑 → ∀𝑛 ∈ 𝑍 𝐵 ∈ ℂ) |
| 40 | 39 | adantr 480 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ∀𝑛 ∈ 𝑍 𝐵 ∈ ℂ) |
| 41 | 37, 40, 36 | rspcdva 3623 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐶 ∈ ℂ) |
| 42 | 26, 27, 36, 41 | fvmptd3 7039 |
. . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ((𝑛 ∈ 𝑍 ↦ 𝐵)‘(⌊‘𝑥)) = 𝐶) |
| 43 | 42 | adantlr 715 |
. . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ 𝑦 ∈ ℝ+) ∧ 𝑥 ∈ 𝐴) → ((𝑛 ∈ 𝑍 ↦ 𝐵)‘(⌊‘𝑥)) = 𝐶) |
| 44 | 43 | ad2ant2r 747 |
. . . . . . . . . . . . . 14
⊢ ((((𝜑 ∧ 𝑦 ∈ ℝ+) ∧ 𝑗 ∈ 𝑍) ∧ (𝑥 ∈ 𝐴 ∧ 𝑗 ≤ 𝑥)) → ((𝑛 ∈ 𝑍 ↦ 𝐵)‘(⌊‘𝑥)) = 𝐶) |
| 45 | 44 | fvoveq1d 7453 |
. . . . . . . . . . . . 13
⊢ ((((𝜑 ∧ 𝑦 ∈ ℝ+) ∧ 𝑗 ∈ 𝑍) ∧ (𝑥 ∈ 𝐴 ∧ 𝑗 ≤ 𝑥)) → (abs‘(((𝑛 ∈ 𝑍 ↦ 𝐵)‘(⌊‘𝑥)) − 𝐷)) = (abs‘(𝐶 − 𝐷))) |
| 46 | 45 | breq1d 5153 |
. . . . . . . . . . . 12
⊢ ((((𝜑 ∧ 𝑦 ∈ ℝ+) ∧ 𝑗 ∈ 𝑍) ∧ (𝑥 ∈ 𝐴 ∧ 𝑗 ≤ 𝑥)) → ((abs‘(((𝑛 ∈ 𝑍 ↦ 𝐵)‘(⌊‘𝑥)) − 𝐷)) < 𝑦 ↔ (abs‘(𝐶 − 𝐷)) < 𝑦)) |
| 47 | 25, 46 | sylibd 239 |
. . . . . . . . . . 11
⊢ ((((𝜑 ∧ 𝑦 ∈ ℝ+) ∧ 𝑗 ∈ 𝑍) ∧ (𝑥 ∈ 𝐴 ∧ 𝑗 ≤ 𝑥)) → (∀𝑘 ∈ (ℤ≥‘𝑗)(((𝑛 ∈ 𝑍 ↦ 𝐵)‘𝑘) ∈ ℂ ∧ (abs‘(((𝑛 ∈ 𝑍 ↦ 𝐵)‘𝑘) − 𝐷)) < 𝑦) → (abs‘(𝐶 − 𝐷)) < 𝑦)) |
| 48 | 47 | expr 456 |
. . . . . . . . . 10
⊢ ((((𝜑 ∧ 𝑦 ∈ ℝ+) ∧ 𝑗 ∈ 𝑍) ∧ 𝑥 ∈ 𝐴) → (𝑗 ≤ 𝑥 → (∀𝑘 ∈ (ℤ≥‘𝑗)(((𝑛 ∈ 𝑍 ↦ 𝐵)‘𝑘) ∈ ℂ ∧ (abs‘(((𝑛 ∈ 𝑍 ↦ 𝐵)‘𝑘) − 𝐷)) < 𝑦) → (abs‘(𝐶 − 𝐷)) < 𝑦))) |
| 49 | 48 | com23 86 |
. . . . . . . . 9
⊢ ((((𝜑 ∧ 𝑦 ∈ ℝ+) ∧ 𝑗 ∈ 𝑍) ∧ 𝑥 ∈ 𝐴) → (∀𝑘 ∈ (ℤ≥‘𝑗)(((𝑛 ∈ 𝑍 ↦ 𝐵)‘𝑘) ∈ ℂ ∧ (abs‘(((𝑛 ∈ 𝑍 ↦ 𝐵)‘𝑘) − 𝐷)) < 𝑦) → (𝑗 ≤ 𝑥 → (abs‘(𝐶 − 𝐷)) < 𝑦))) |
| 50 | 49 | ralrimdva 3154 |
. . . . . . . 8
⊢ (((𝜑 ∧ 𝑦 ∈ ℝ+) ∧ 𝑗 ∈ 𝑍) → (∀𝑘 ∈ (ℤ≥‘𝑗)(((𝑛 ∈ 𝑍 ↦ 𝐵)‘𝑘) ∈ ℂ ∧ (abs‘(((𝑛 ∈ 𝑍 ↦ 𝐵)‘𝑘) − 𝐷)) < 𝑦) → ∀𝑥 ∈ 𝐴 (𝑗 ≤ 𝑥 → (abs‘(𝐶 − 𝐷)) < 𝑦))) |
| 51 | | eluzelre 12889 |
. . . . . . . . . 10
⊢ (𝑗 ∈
(ℤ≥‘𝑀) → 𝑗 ∈ ℝ) |
| 52 | 51, 3 | eleq2s 2859 |
. . . . . . . . 9
⊢ (𝑗 ∈ 𝑍 → 𝑗 ∈ ℝ) |
| 53 | 52 | adantl 481 |
. . . . . . . 8
⊢ (((𝜑 ∧ 𝑦 ∈ ℝ+) ∧ 𝑗 ∈ 𝑍) → 𝑗 ∈ ℝ) |
| 54 | 50, 53 | jctild 525 |
. . . . . . 7
⊢ (((𝜑 ∧ 𝑦 ∈ ℝ+) ∧ 𝑗 ∈ 𝑍) → (∀𝑘 ∈ (ℤ≥‘𝑗)(((𝑛 ∈ 𝑍 ↦ 𝐵)‘𝑘) ∈ ℂ ∧ (abs‘(((𝑛 ∈ 𝑍 ↦ 𝐵)‘𝑘) − 𝐷)) < 𝑦) → (𝑗 ∈ ℝ ∧ ∀𝑥 ∈ 𝐴 (𝑗 ≤ 𝑥 → (abs‘(𝐶 − 𝐷)) < 𝑦)))) |
| 55 | 54 | expimpd 453 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑦 ∈ ℝ+) → ((𝑗 ∈ 𝑍 ∧ ∀𝑘 ∈ (ℤ≥‘𝑗)(((𝑛 ∈ 𝑍 ↦ 𝐵)‘𝑘) ∈ ℂ ∧ (abs‘(((𝑛 ∈ 𝑍 ↦ 𝐵)‘𝑘) − 𝐷)) < 𝑦)) → (𝑗 ∈ ℝ ∧ ∀𝑥 ∈ 𝐴 (𝑗 ≤ 𝑥 → (abs‘(𝐶 − 𝐷)) < 𝑦)))) |
| 56 | 55 | reximdv2 3164 |
. . . . 5
⊢ ((𝜑 ∧ 𝑦 ∈ ℝ+) →
(∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(((𝑛 ∈ 𝑍 ↦ 𝐵)‘𝑘) ∈ ℂ ∧ (abs‘(((𝑛 ∈ 𝑍 ↦ 𝐵)‘𝑘) − 𝐷)) < 𝑦) → ∃𝑗 ∈ ℝ ∀𝑥 ∈ 𝐴 (𝑗 ≤ 𝑥 → (abs‘(𝐶 − 𝐷)) < 𝑦))) |
| 57 | 56 | ralimdva 3167 |
. . . 4
⊢ (𝜑 → (∀𝑦 ∈ ℝ+
∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(((𝑛 ∈ 𝑍 ↦ 𝐵)‘𝑘) ∈ ℂ ∧ (abs‘(((𝑛 ∈ 𝑍 ↦ 𝐵)‘𝑘) − 𝐷)) < 𝑦) → ∀𝑦 ∈ ℝ+ ∃𝑗 ∈ ℝ ∀𝑥 ∈ 𝐴 (𝑗 ≤ 𝑥 → (abs‘(𝐶 − 𝐷)) < 𝑦))) |
| 58 | 57 | adantld 490 |
. . 3
⊢ (𝜑 → ((𝐷 ∈ ℂ ∧ ∀𝑦 ∈ ℝ+
∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(((𝑛 ∈ 𝑍 ↦ 𝐵)‘𝑘) ∈ ℂ ∧ (abs‘(((𝑛 ∈ 𝑍 ↦ 𝐵)‘𝑘) − 𝐷)) < 𝑦)) → ∀𝑦 ∈ ℝ+ ∃𝑗 ∈ ℝ ∀𝑥 ∈ 𝐴 (𝑗 ≤ 𝑥 → (abs‘(𝐶 − 𝐷)) < 𝑦))) |
| 59 | | climrel 15528 |
. . . . . 6
⊢ Rel
⇝ |
| 60 | 59 | brrelex1i 5741 |
. . . . 5
⊢ ((𝑛 ∈ 𝑍 ↦ 𝐵) ⇝ 𝐷 → (𝑛 ∈ 𝑍 ↦ 𝐵) ∈ V) |
| 61 | 1, 60 | syl 17 |
. . . 4
⊢ (𝜑 → (𝑛 ∈ 𝑍 ↦ 𝐵) ∈ V) |
| 62 | | eqidd 2738 |
. . . 4
⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → ((𝑛 ∈ 𝑍 ↦ 𝐵)‘𝑘) = ((𝑛 ∈ 𝑍 ↦ 𝐵)‘𝑘)) |
| 63 | 3, 28, 61, 62 | clim2 15540 |
. . 3
⊢ (𝜑 → ((𝑛 ∈ 𝑍 ↦ 𝐵) ⇝ 𝐷 ↔ (𝐷 ∈ ℂ ∧ ∀𝑦 ∈ ℝ+
∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(((𝑛 ∈ 𝑍 ↦ 𝐵)‘𝑘) ∈ ℂ ∧ (abs‘(((𝑛 ∈ 𝑍 ↦ 𝐵)‘𝑘) − 𝐷)) < 𝑦)))) |
| 64 | 41 | ralrimiva 3146 |
. . . 4
⊢ (𝜑 → ∀𝑥 ∈ 𝐴 𝐶 ∈ ℂ) |
| 65 | | climcl 15535 |
. . . . 5
⊢ ((𝑛 ∈ 𝑍 ↦ 𝐵) ⇝ 𝐷 → 𝐷 ∈ ℂ) |
| 66 | 1, 65 | syl 17 |
. . . 4
⊢ (𝜑 → 𝐷 ∈ ℂ) |
| 67 | 64, 6, 66 | rlim2 15532 |
. . 3
⊢ (𝜑 → ((𝑥 ∈ 𝐴 ↦ 𝐶) ⇝𝑟 𝐷 ↔ ∀𝑦 ∈ ℝ+
∃𝑗 ∈ ℝ
∀𝑥 ∈ 𝐴 (𝑗 ≤ 𝑥 → (abs‘(𝐶 − 𝐷)) < 𝑦))) |
| 68 | 58, 63, 67 | 3imtr4d 294 |
. 2
⊢ (𝜑 → ((𝑛 ∈ 𝑍 ↦ 𝐵) ⇝ 𝐷 → (𝑥 ∈ 𝐴 ↦ 𝐶) ⇝𝑟 𝐷)) |
| 69 | 1, 68 | mpd 15 |
1
⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐶) ⇝𝑟 𝐷) |