Proof of Theorem eulerpartlemgf
| Step | Hyp | Ref
| Expression |
| 1 | | eulerpart.p |
. . . . . . 7
⊢ 𝑃 = {𝑓 ∈ (ℕ0
↑m ℕ) ∣ ((◡𝑓 “ ℕ) ∈ Fin ∧
Σ𝑘 ∈ ℕ
((𝑓‘𝑘) · 𝑘) = 𝑁)} |
| 2 | | eulerpart.o |
. . . . . . 7
⊢ 𝑂 = {𝑔 ∈ 𝑃 ∣ ∀𝑛 ∈ (◡𝑔 “ ℕ) ¬ 2 ∥ 𝑛} |
| 3 | | eulerpart.d |
. . . . . . 7
⊢ 𝐷 = {𝑔 ∈ 𝑃 ∣ ∀𝑛 ∈ ℕ (𝑔‘𝑛) ≤ 1} |
| 4 | | eulerpart.j |
. . . . . . 7
⊢ 𝐽 = {𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧} |
| 5 | | eulerpart.f |
. . . . . . 7
⊢ 𝐹 = (𝑥 ∈ 𝐽, 𝑦 ∈ ℕ0 ↦
((2↑𝑦) · 𝑥)) |
| 6 | | eulerpart.h |
. . . . . . 7
⊢ 𝐻 = {𝑟 ∈ ((𝒫 ℕ0 ∩
Fin) ↑m 𝐽)
∣ (𝑟 supp ∅)
∈ Fin} |
| 7 | | eulerpart.m |
. . . . . . 7
⊢ 𝑀 = (𝑟 ∈ 𝐻 ↦ {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ (𝑟‘𝑥))}) |
| 8 | | eulerpart.r |
. . . . . . 7
⊢ 𝑅 = {𝑓 ∣ (◡𝑓 “ ℕ) ∈
Fin} |
| 9 | | eulerpart.t |
. . . . . . 7
⊢ 𝑇 = {𝑓 ∈ (ℕ0
↑m ℕ) ∣ (◡𝑓 “ ℕ) ⊆ 𝐽} |
| 10 | | eulerpart.g |
. . . . . . 7
⊢ 𝐺 = (𝑜 ∈ (𝑇 ∩ 𝑅) ↦
((𝟭‘ℕ)‘(𝐹 “ (𝑀‘(bits ∘ (𝑜 ↾ 𝐽)))))) |
| 11 | 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 | eulerpartlemgv 34375 |
. . . . . 6
⊢ (𝐴 ∈ (𝑇 ∩ 𝑅) → (𝐺‘𝐴) = ((𝟭‘ℕ)‘(𝐹 “ (𝑀‘(bits ∘ (𝐴 ↾ 𝐽)))))) |
| 12 | 11 | cnveqd 5886 |
. . . . 5
⊢ (𝐴 ∈ (𝑇 ∩ 𝑅) → ◡(𝐺‘𝐴) = ◡((𝟭‘ℕ)‘(𝐹 “ (𝑀‘(bits ∘ (𝐴 ↾ 𝐽)))))) |
| 13 | 12 | imaeq1d 6077 |
. . . 4
⊢ (𝐴 ∈ (𝑇 ∩ 𝑅) → (◡(𝐺‘𝐴) “ {1}) = (◡((𝟭‘ℕ)‘(𝐹 “ (𝑀‘(bits ∘ (𝐴 ↾ 𝐽))))) “ {1})) |
| 14 | | nnex 12272 |
. . . . 5
⊢ ℕ
∈ V |
| 15 | | imassrn 6089 |
. . . . . 6
⊢ (𝐹 “ (𝑀‘(bits ∘ (𝐴 ↾ 𝐽)))) ⊆ ran 𝐹 |
| 16 | 4, 5 | oddpwdc 34356 |
. . . . . . 7
⊢ 𝐹:(𝐽 × ℕ0)–1-1-onto→ℕ |
| 17 | | f1of 6848 |
. . . . . . 7
⊢ (𝐹:(𝐽 × ℕ0)–1-1-onto→ℕ → 𝐹:(𝐽 ×
ℕ0)⟶ℕ) |
| 18 | | frn 6743 |
. . . . . . 7
⊢ (𝐹:(𝐽 × ℕ0)⟶ℕ
→ ran 𝐹 ⊆
ℕ) |
| 19 | 16, 17, 18 | mp2b 10 |
. . . . . 6
⊢ ran 𝐹 ⊆
ℕ |
| 20 | 15, 19 | sstri 3993 |
. . . . 5
⊢ (𝐹 “ (𝑀‘(bits ∘ (𝐴 ↾ 𝐽)))) ⊆ ℕ |
| 21 | | indpi1 32845 |
. . . . 5
⊢ ((ℕ
∈ V ∧ (𝐹 “
(𝑀‘(bits ∘
(𝐴 ↾ 𝐽)))) ⊆ ℕ) →
(◡((𝟭‘ℕ)‘(𝐹 “ (𝑀‘(bits ∘ (𝐴 ↾ 𝐽))))) “ {1}) = (𝐹 “ (𝑀‘(bits ∘ (𝐴 ↾ 𝐽))))) |
| 22 | 14, 20, 21 | mp2an 692 |
. . . 4
⊢ (◡((𝟭‘ℕ)‘(𝐹 “ (𝑀‘(bits ∘ (𝐴 ↾ 𝐽))))) “ {1}) = (𝐹 “ (𝑀‘(bits ∘ (𝐴 ↾ 𝐽)))) |
| 23 | 13, 22 | eqtrdi 2793 |
. . 3
⊢ (𝐴 ∈ (𝑇 ∩ 𝑅) → (◡(𝐺‘𝐴) “ {1}) = (𝐹 “ (𝑀‘(bits ∘ (𝐴 ↾ 𝐽))))) |
| 24 | | ffun 6739 |
. . . . 5
⊢ (𝐹:(𝐽 × ℕ0)⟶ℕ
→ Fun 𝐹) |
| 25 | 16, 17, 24 | mp2b 10 |
. . . 4
⊢ Fun 𝐹 |
| 26 | | inss2 4238 |
. . . . 5
⊢
(𝒫 (𝐽
× ℕ0) ∩ Fin) ⊆ Fin |
| 27 | 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 | eulerpartlemmf 34377 |
. . . . . 6
⊢ (𝐴 ∈ (𝑇 ∩ 𝑅) → (bits ∘ (𝐴 ↾ 𝐽)) ∈ 𝐻) |
| 28 | 1, 2, 3, 4, 5, 6, 7 | eulerpartlem1 34369 |
. . . . . . . 8
⊢ 𝑀:𝐻–1-1-onto→(𝒫 (𝐽 × ℕ0) ∩
Fin) |
| 29 | | f1of 6848 |
. . . . . . . 8
⊢ (𝑀:𝐻–1-1-onto→(𝒫 (𝐽 × ℕ0) ∩ Fin)
→ 𝑀:𝐻⟶(𝒫 (𝐽 × ℕ0) ∩
Fin)) |
| 30 | 28, 29 | ax-mp 5 |
. . . . . . 7
⊢ 𝑀:𝐻⟶(𝒫 (𝐽 × ℕ0) ∩
Fin) |
| 31 | 30 | ffvelcdmi 7103 |
. . . . . 6
⊢ ((bits
∘ (𝐴 ↾ 𝐽)) ∈ 𝐻 → (𝑀‘(bits ∘ (𝐴 ↾ 𝐽))) ∈ (𝒫 (𝐽 × ℕ0) ∩
Fin)) |
| 32 | 27, 31 | syl 17 |
. . . . 5
⊢ (𝐴 ∈ (𝑇 ∩ 𝑅) → (𝑀‘(bits ∘ (𝐴 ↾ 𝐽))) ∈ (𝒫 (𝐽 × ℕ0) ∩
Fin)) |
| 33 | 26, 32 | sselid 3981 |
. . . 4
⊢ (𝐴 ∈ (𝑇 ∩ 𝑅) → (𝑀‘(bits ∘ (𝐴 ↾ 𝐽))) ∈ Fin) |
| 34 | | imafi 9353 |
. . . 4
⊢ ((Fun
𝐹 ∧ (𝑀‘(bits ∘ (𝐴 ↾ 𝐽))) ∈ Fin) → (𝐹 “ (𝑀‘(bits ∘ (𝐴 ↾ 𝐽)))) ∈ Fin) |
| 35 | 25, 33, 34 | sylancr 587 |
. . 3
⊢ (𝐴 ∈ (𝑇 ∩ 𝑅) → (𝐹 “ (𝑀‘(bits ∘ (𝐴 ↾ 𝐽)))) ∈ Fin) |
| 36 | 23, 35 | eqeltrd 2841 |
. 2
⊢ (𝐴 ∈ (𝑇 ∩ 𝑅) → (◡(𝐺‘𝐴) “ {1}) ∈ Fin) |
| 37 | 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 | eulerpartgbij 34374 |
. . . . . . . 8
⊢ 𝐺:(𝑇 ∩ 𝑅)–1-1-onto→(({0,
1} ↑m ℕ) ∩ 𝑅) |
| 38 | | f1of 6848 |
. . . . . . . 8
⊢ (𝐺:(𝑇 ∩ 𝑅)–1-1-onto→(({0,
1} ↑m ℕ) ∩ 𝑅) → 𝐺:(𝑇 ∩ 𝑅)⟶(({0, 1} ↑m
ℕ) ∩ 𝑅)) |
| 39 | 37, 38 | ax-mp 5 |
. . . . . . 7
⊢ 𝐺:(𝑇 ∩ 𝑅)⟶(({0, 1} ↑m
ℕ) ∩ 𝑅) |
| 40 | 39 | ffvelcdmi 7103 |
. . . . . 6
⊢ (𝐴 ∈ (𝑇 ∩ 𝑅) → (𝐺‘𝐴) ∈ (({0, 1} ↑m
ℕ) ∩ 𝑅)) |
| 41 | | elin 3967 |
. . . . . . 7
⊢ ((𝐺‘𝐴) ∈ (({0, 1} ↑m
ℕ) ∩ 𝑅) ↔
((𝐺‘𝐴) ∈ ({0, 1} ↑m ℕ)
∧ (𝐺‘𝐴) ∈ 𝑅)) |
| 42 | 41 | simplbi 497 |
. . . . . 6
⊢ ((𝐺‘𝐴) ∈ (({0, 1} ↑m
ℕ) ∩ 𝑅) →
(𝐺‘𝐴) ∈ ({0, 1} ↑m
ℕ)) |
| 43 | | elmapi 8889 |
. . . . . 6
⊢ ((𝐺‘𝐴) ∈ ({0, 1} ↑m ℕ)
→ (𝐺‘𝐴):ℕ⟶{0,
1}) |
| 44 | 40, 42, 43 | 3syl 18 |
. . . . 5
⊢ (𝐴 ∈ (𝑇 ∩ 𝑅) → (𝐺‘𝐴):ℕ⟶{0, 1}) |
| 45 | 44 | ffund 6740 |
. . . 4
⊢ (𝐴 ∈ (𝑇 ∩ 𝑅) → Fun (𝐺‘𝐴)) |
| 46 | | ssv 4008 |
. . . . 5
⊢
ℕ0 ⊆ V |
| 47 | | dfn2 12539 |
. . . . . 6
⊢ ℕ =
(ℕ0 ∖ {0}) |
| 48 | | ssdif 4144 |
. . . . . 6
⊢
(ℕ0 ⊆ V → (ℕ0 ∖ {0})
⊆ (V ∖ {0})) |
| 49 | 47, 48 | eqsstrid 4022 |
. . . . 5
⊢
(ℕ0 ⊆ V → ℕ ⊆ (V ∖
{0})) |
| 50 | 46, 49 | ax-mp 5 |
. . . 4
⊢ ℕ
⊆ (V ∖ {0}) |
| 51 | | sspreima 7088 |
. . . 4
⊢ ((Fun
(𝐺‘𝐴) ∧ ℕ ⊆ (V ∖ {0}))
→ (◡(𝐺‘𝐴) “ ℕ) ⊆ (◡(𝐺‘𝐴) “ (V ∖ {0}))) |
| 52 | 45, 50, 51 | sylancl 586 |
. . 3
⊢ (𝐴 ∈ (𝑇 ∩ 𝑅) → (◡(𝐺‘𝐴) “ ℕ) ⊆ (◡(𝐺‘𝐴) “ (V ∖ {0}))) |
| 53 | | fvex 6919 |
. . . . 5
⊢ (𝐺‘𝐴) ∈ V |
| 54 | | 0nn0 12541 |
. . . . 5
⊢ 0 ∈
ℕ0 |
| 55 | | suppimacnv 8199 |
. . . . 5
⊢ (((𝐺‘𝐴) ∈ V ∧ 0 ∈
ℕ0) → ((𝐺‘𝐴) supp 0) = (◡(𝐺‘𝐴) “ (V ∖ {0}))) |
| 56 | 53, 54, 55 | mp2an 692 |
. . . 4
⊢ ((𝐺‘𝐴) supp 0) = (◡(𝐺‘𝐴) “ (V ∖ {0})) |
| 57 | | 0ne1 12337 |
. . . . . . . . 9
⊢ 0 ≠
1 |
| 58 | | difprsn1 4800 |
. . . . . . . . 9
⊢ (0 ≠ 1
→ ({0, 1} ∖ {0}) = {1}) |
| 59 | 57, 58 | ax-mp 5 |
. . . . . . . 8
⊢ ({0, 1}
∖ {0}) = {1} |
| 60 | 59 | eqcomi 2746 |
. . . . . . 7
⊢ {1} =
({0, 1} ∖ {0}) |
| 61 | 60 | ffs2 32739 |
. . . . . 6
⊢ ((ℕ
∈ V ∧ 0 ∈ ℕ0 ∧ (𝐺‘𝐴):ℕ⟶{0, 1}) → ((𝐺‘𝐴) supp 0) = (◡(𝐺‘𝐴) “ {1})) |
| 62 | 14, 54, 61 | mp3an12 1453 |
. . . . 5
⊢ ((𝐺‘𝐴):ℕ⟶{0, 1} → ((𝐺‘𝐴) supp 0) = (◡(𝐺‘𝐴) “ {1})) |
| 63 | 44, 62 | syl 17 |
. . . 4
⊢ (𝐴 ∈ (𝑇 ∩ 𝑅) → ((𝐺‘𝐴) supp 0) = (◡(𝐺‘𝐴) “ {1})) |
| 64 | 56, 63 | eqtr3id 2791 |
. . 3
⊢ (𝐴 ∈ (𝑇 ∩ 𝑅) → (◡(𝐺‘𝐴) “ (V ∖ {0})) = (◡(𝐺‘𝐴) “ {1})) |
| 65 | 52, 64 | sseqtrd 4020 |
. 2
⊢ (𝐴 ∈ (𝑇 ∩ 𝑅) → (◡(𝐺‘𝐴) “ ℕ) ⊆ (◡(𝐺‘𝐴) “ {1})) |
| 66 | | ssfi 9213 |
. 2
⊢ (((◡(𝐺‘𝐴) “ {1}) ∈ Fin ∧ (◡(𝐺‘𝐴) “ ℕ) ⊆ (◡(𝐺‘𝐴) “ {1})) → (◡(𝐺‘𝐴) “ ℕ) ∈
Fin) |
| 67 | 36, 65, 66 | syl2anc 584 |
1
⊢ (𝐴 ∈ (𝑇 ∩ 𝑅) → (◡(𝐺‘𝐴) “ ℕ) ∈
Fin) |