| Step | Hyp | Ref
| Expression |
| 1 | | eqid 2737 |
. . . . . 6
⊢ (𝑘 ∈ ℕ0
↦ (bits‘𝑘)) =
(𝑘 ∈
ℕ0 ↦ (bits‘𝑘)) |
| 2 | | bitsss 16463 |
. . . . . . . . 9
⊢
(bits‘𝑘)
⊆ ℕ0 |
| 3 | 2 | a1i 11 |
. . . . . . . 8
⊢ (𝑘 ∈ ℕ0
→ (bits‘𝑘)
⊆ ℕ0) |
| 4 | | bitsfi 16474 |
. . . . . . . 8
⊢ (𝑘 ∈ ℕ0
→ (bits‘𝑘)
∈ Fin) |
| 5 | | elfpw 9394 |
. . . . . . . 8
⊢
((bits‘𝑘)
∈ (𝒫 ℕ0 ∩ Fin) ↔ ((bits‘𝑘) ⊆ ℕ0
∧ (bits‘𝑘) ∈
Fin)) |
| 6 | 3, 4, 5 | sylanbrc 583 |
. . . . . . 7
⊢ (𝑘 ∈ ℕ0
→ (bits‘𝑘)
∈ (𝒫 ℕ0 ∩ Fin)) |
| 7 | 6 | adantl 481 |
. . . . . 6
⊢
((⊤ ∧ 𝑘
∈ ℕ0) → (bits‘𝑘) ∈ (𝒫 ℕ0 ∩
Fin)) |
| 8 | | elinel2 4202 |
. . . . . . . 8
⊢ (𝑥 ∈ (𝒫
ℕ0 ∩ Fin) → 𝑥 ∈ Fin) |
| 9 | | 2nn0 12543 |
. . . . . . . . . 10
⊢ 2 ∈
ℕ0 |
| 10 | 9 | a1i 11 |
. . . . . . . . 9
⊢ ((𝑥 ∈ (𝒫
ℕ0 ∩ Fin) ∧ 𝑛 ∈ 𝑥) → 2 ∈
ℕ0) |
| 11 | | elfpw 9394 |
. . . . . . . . . . 11
⊢ (𝑥 ∈ (𝒫
ℕ0 ∩ Fin) ↔ (𝑥 ⊆ ℕ0 ∧ 𝑥 ∈ Fin)) |
| 12 | 11 | simplbi 497 |
. . . . . . . . . 10
⊢ (𝑥 ∈ (𝒫
ℕ0 ∩ Fin) → 𝑥 ⊆
ℕ0) |
| 13 | 12 | sselda 3983 |
. . . . . . . . 9
⊢ ((𝑥 ∈ (𝒫
ℕ0 ∩ Fin) ∧ 𝑛 ∈ 𝑥) → 𝑛 ∈ ℕ0) |
| 14 | 10, 13 | nn0expcld 14285 |
. . . . . . . 8
⊢ ((𝑥 ∈ (𝒫
ℕ0 ∩ Fin) ∧ 𝑛 ∈ 𝑥) → (2↑𝑛) ∈
ℕ0) |
| 15 | 8, 14 | fsumnn0cl 15772 |
. . . . . . 7
⊢ (𝑥 ∈ (𝒫
ℕ0 ∩ Fin) → Σ𝑛 ∈ 𝑥 (2↑𝑛) ∈
ℕ0) |
| 16 | 15 | adantl 481 |
. . . . . 6
⊢
((⊤ ∧ 𝑥
∈ (𝒫 ℕ0 ∩ Fin)) → Σ𝑛 ∈ 𝑥 (2↑𝑛) ∈
ℕ0) |
| 17 | | bitsinv2 16480 |
. . . . . . . . . 10
⊢ (𝑥 ∈ (𝒫
ℕ0 ∩ Fin) → (bits‘Σ𝑛 ∈ 𝑥 (2↑𝑛)) = 𝑥) |
| 18 | 17 | eqcomd 2743 |
. . . . . . . . 9
⊢ (𝑥 ∈ (𝒫
ℕ0 ∩ Fin) → 𝑥 = (bits‘Σ𝑛 ∈ 𝑥 (2↑𝑛))) |
| 19 | 18 | ad2antll 729 |
. . . . . . . 8
⊢
((⊤ ∧ (𝑘
∈ ℕ0 ∧ 𝑥 ∈ (𝒫 ℕ0 ∩
Fin))) → 𝑥 =
(bits‘Σ𝑛 ∈
𝑥 (2↑𝑛))) |
| 20 | | fveq2 6906 |
. . . . . . . . 9
⊢ (𝑘 = Σ𝑛 ∈ 𝑥 (2↑𝑛) → (bits‘𝑘) = (bits‘Σ𝑛 ∈ 𝑥 (2↑𝑛))) |
| 21 | 20 | eqeq2d 2748 |
. . . . . . . 8
⊢ (𝑘 = Σ𝑛 ∈ 𝑥 (2↑𝑛) → (𝑥 = (bits‘𝑘) ↔ 𝑥 = (bits‘Σ𝑛 ∈ 𝑥 (2↑𝑛)))) |
| 22 | 19, 21 | syl5ibrcom 247 |
. . . . . . 7
⊢
((⊤ ∧ (𝑘
∈ ℕ0 ∧ 𝑥 ∈ (𝒫 ℕ0 ∩
Fin))) → (𝑘 =
Σ𝑛 ∈ 𝑥 (2↑𝑛) → 𝑥 = (bits‘𝑘))) |
| 23 | | bitsinv1 16479 |
. . . . . . . . . 10
⊢ (𝑘 ∈ ℕ0
→ Σ𝑛 ∈
(bits‘𝑘)(2↑𝑛) = 𝑘) |
| 24 | 23 | eqcomd 2743 |
. . . . . . . . 9
⊢ (𝑘 ∈ ℕ0
→ 𝑘 = Σ𝑛 ∈ (bits‘𝑘)(2↑𝑛)) |
| 25 | 24 | ad2antrl 728 |
. . . . . . . 8
⊢
((⊤ ∧ (𝑘
∈ ℕ0 ∧ 𝑥 ∈ (𝒫 ℕ0 ∩
Fin))) → 𝑘 =
Σ𝑛 ∈
(bits‘𝑘)(2↑𝑛)) |
| 26 | | sumeq1 15725 |
. . . . . . . . 9
⊢ (𝑥 = (bits‘𝑘) → Σ𝑛 ∈ 𝑥 (2↑𝑛) = Σ𝑛 ∈ (bits‘𝑘)(2↑𝑛)) |
| 27 | 26 | eqeq2d 2748 |
. . . . . . . 8
⊢ (𝑥 = (bits‘𝑘) → (𝑘 = Σ𝑛 ∈ 𝑥 (2↑𝑛) ↔ 𝑘 = Σ𝑛 ∈ (bits‘𝑘)(2↑𝑛))) |
| 28 | 25, 27 | syl5ibrcom 247 |
. . . . . . 7
⊢
((⊤ ∧ (𝑘
∈ ℕ0 ∧ 𝑥 ∈ (𝒫 ℕ0 ∩
Fin))) → (𝑥 =
(bits‘𝑘) → 𝑘 = Σ𝑛 ∈ 𝑥 (2↑𝑛))) |
| 29 | 22, 28 | impbid 212 |
. . . . . 6
⊢
((⊤ ∧ (𝑘
∈ ℕ0 ∧ 𝑥 ∈ (𝒫 ℕ0 ∩
Fin))) → (𝑘 =
Σ𝑛 ∈ 𝑥 (2↑𝑛) ↔ 𝑥 = (bits‘𝑘))) |
| 30 | 1, 7, 16, 29 | f1ocnv2d 7686 |
. . . . 5
⊢ (⊤
→ ((𝑘 ∈
ℕ0 ↦ (bits‘𝑘)):ℕ0–1-1-onto→(𝒫 ℕ0 ∩ Fin)
∧ ◡(𝑘 ∈ ℕ0 ↦
(bits‘𝑘)) = (𝑥 ∈ (𝒫
ℕ0 ∩ Fin) ↦ Σ𝑛 ∈ 𝑥 (2↑𝑛)))) |
| 31 | 30 | simpld 494 |
. . . 4
⊢ (⊤
→ (𝑘 ∈
ℕ0 ↦ (bits‘𝑘)):ℕ0–1-1-onto→(𝒫 ℕ0 ∩
Fin)) |
| 32 | | bitsf 16464 |
. . . . . . . . 9
⊢
bits:ℤ⟶𝒫 ℕ0 |
| 33 | 32 | a1i 11 |
. . . . . . . 8
⊢ (⊤
→ bits:ℤ⟶𝒫 ℕ0) |
| 34 | 33 | feqmptd 6977 |
. . . . . . 7
⊢ (⊤
→ bits = (𝑘 ∈
ℤ ↦ (bits‘𝑘))) |
| 35 | 34 | reseq1d 5996 |
. . . . . 6
⊢ (⊤
→ (bits ↾ ℕ0) = ((𝑘 ∈ ℤ ↦ (bits‘𝑘)) ↾
ℕ0)) |
| 36 | | nn0ssz 12636 |
. . . . . . 7
⊢
ℕ0 ⊆ ℤ |
| 37 | | resmpt 6055 |
. . . . . . 7
⊢
(ℕ0 ⊆ ℤ → ((𝑘 ∈ ℤ ↦ (bits‘𝑘)) ↾ ℕ0)
= (𝑘 ∈
ℕ0 ↦ (bits‘𝑘))) |
| 38 | 36, 37 | ax-mp 5 |
. . . . . 6
⊢ ((𝑘 ∈ ℤ ↦
(bits‘𝑘)) ↾
ℕ0) = (𝑘
∈ ℕ0 ↦ (bits‘𝑘)) |
| 39 | 35, 38 | eqtrdi 2793 |
. . . . 5
⊢ (⊤
→ (bits ↾ ℕ0) = (𝑘 ∈ ℕ0 ↦
(bits‘𝑘))) |
| 40 | 39 | f1oeq1d 6843 |
. . . 4
⊢ (⊤
→ ((bits ↾ ℕ0):ℕ0–1-1-onto→(𝒫 ℕ0 ∩ Fin)
↔ (𝑘 ∈
ℕ0 ↦ (bits‘𝑘)):ℕ0–1-1-onto→(𝒫 ℕ0 ∩
Fin))) |
| 41 | 31, 40 | mpbird 257 |
. . 3
⊢ (⊤
→ (bits ↾ ℕ0):ℕ0–1-1-onto→(𝒫 ℕ0 ∩
Fin)) |
| 42 | 39 | cnveqd 5886 |
. . . 4
⊢ (⊤
→ ◡(bits ↾
ℕ0) = ◡(𝑘 ∈ ℕ0
↦ (bits‘𝑘))) |
| 43 | 30 | simprd 495 |
. . . 4
⊢ (⊤
→ ◡(𝑘 ∈ ℕ0 ↦
(bits‘𝑘)) = (𝑥 ∈ (𝒫
ℕ0 ∩ Fin) ↦ Σ𝑛 ∈ 𝑥 (2↑𝑛))) |
| 44 | 42, 43 | eqtrd 2777 |
. . 3
⊢ (⊤
→ ◡(bits ↾
ℕ0) = (𝑥
∈ (𝒫 ℕ0 ∩ Fin) ↦ Σ𝑛 ∈ 𝑥 (2↑𝑛))) |
| 45 | 41, 44 | jca 511 |
. 2
⊢ (⊤
→ ((bits ↾ ℕ0):ℕ0–1-1-onto→(𝒫 ℕ0 ∩ Fin)
∧ ◡(bits ↾
ℕ0) = (𝑥
∈ (𝒫 ℕ0 ∩ Fin) ↦ Σ𝑛 ∈ 𝑥 (2↑𝑛)))) |
| 46 | 45 | mptru 1547 |
1
⊢ ((bits
↾ ℕ0):ℕ0–1-1-onto→(𝒫 ℕ0 ∩ Fin)
∧ ◡(bits ↾
ℕ0) = (𝑥
∈ (𝒫 ℕ0 ∩ Fin) ↦ Σ𝑛 ∈ 𝑥 (2↑𝑛))) |