Metamath Proof Explorer

Theorem eulerpartlemgu

Description: Lemma for eulerpart : Rewriting the U set for an odd partition Note that interestingly, this proof reuses marypha2lem2 . (Contributed by Thierry Arnoux, 10-Aug-2018)

Ref Expression
Hypotheses eulerpart.p ⊢ 𝑃 = { 𝑓 ∈ ( ℕ0 ↑m ℕ ) ∣ ( ( ◡ 𝑓 “ ℕ ) ∈ Fin ∧ Σ 𝑘 ∈ ℕ ( ( 𝑓 ‘ 𝑘 ) · 𝑘 ) = 𝑁 ) }
eulerpart.o ⊢ 𝑂 = { 𝑔 ∈ 𝑃 ∣ ∀ 𝑛 ∈ ( ◡ 𝑔 “ ℕ ) ¬ 2 ∥ 𝑛 }
eulerpart.d ⊢ 𝐷 = { 𝑔 ∈ 𝑃 ∣ ∀ 𝑛 ∈ ℕ ( 𝑔 ‘ 𝑛 ) ≤ 1 }
eulerpart.j ⊢ 𝐽 = { 𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧 }
eulerpart.f ⊢ 𝐹 = ( 𝑥 ∈ 𝐽 , 𝑦 ∈ ℕ0 ↦ ( ( 2 ↑ 𝑦 ) · 𝑥 ) )
eulerpart.h ⊢ 𝐻 = { 𝑟 ∈ ( ( 𝒫 ℕ0 ∩ Fin ) ↑m 𝐽 ) ∣ ( 𝑟 supp ∅ ) ∈ Fin }
eulerpart.m ⊢ 𝑀 = ( 𝑟 ∈ 𝐻 ↦ { ⟨ 𝑥 , 𝑦 ⟩ ∣ ( 𝑥 ∈ 𝐽 ∧ 𝑦 ∈ ( 𝑟 ‘ 𝑥 ) ) } )
eulerpart.r ⊢ 𝑅 = { 𝑓 ∣ ( ◡ 𝑓 “ ℕ ) ∈ Fin }
eulerpart.t ⊢ 𝑇 = { 𝑓 ∈ ( ℕ0 ↑m ℕ ) ∣ ( ◡ 𝑓 “ ℕ ) ⊆ 𝐽 }
eulerpart.g ⊢ 𝐺 = ( 𝑜 ∈ ( 𝑇 ∩ 𝑅 ) ↦ ( ( 𝟭 ‘ ℕ ) ‘ ( 𝐹 “ ( 𝑀 ‘ ( bits ∘ ( 𝑜 ↾ 𝐽 ) ) ) ) ) )
eulerpartlemgh.1 ⊢ 𝑈 = ∪ 𝑡 ∈ ( ( ◡ 𝐴 “ ℕ ) ∩ 𝐽 ) ( { 𝑡 } × ( bits ‘ ( 𝐴 ‘ 𝑡 ) ) )
Assertion eulerpartlemgu ( 𝐴 ∈ ( 𝑇 ∩ 𝑅 ) → 𝑈 = { ⟨ 𝑡 , 𝑛 ⟩ ∣ ( 𝑡 ∈ ( ( ◡ 𝐴 “ ℕ ) ∩ 𝐽 ) ∧ 𝑛 ∈ ( ( bits ∘ 𝐴 ) ‘ 𝑡 ) ) } )

Proof

Step Hyp Ref Expression
1 eulerpart.p ⊢ 𝑃 = { 𝑓 ∈ ( ℕ0 ↑m ℕ ) ∣ ( ( ◡ 𝑓 “ ℕ ) ∈ Fin ∧ Σ 𝑘 ∈ ℕ ( ( 𝑓 ‘ 𝑘 ) · 𝑘 ) = 𝑁 ) }
2 eulerpart.o ⊢ 𝑂 = { 𝑔 ∈ 𝑃 ∣ ∀ 𝑛 ∈ ( ◡ 𝑔 “ ℕ ) ¬ 2 ∥ 𝑛 }
3 eulerpart.d ⊢ 𝐷 = { 𝑔 ∈ 𝑃 ∣ ∀ 𝑛 ∈ ℕ ( 𝑔 ‘ 𝑛 ) ≤ 1 }
4 eulerpart.j ⊢ 𝐽 = { 𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧 }
5 eulerpart.f ⊢ 𝐹 = ( 𝑥 ∈ 𝐽 , 𝑦 ∈ ℕ0 ↦ ( ( 2 ↑ 𝑦 ) · 𝑥 ) )
6 eulerpart.h ⊢ 𝐻 = { 𝑟 ∈ ( ( 𝒫 ℕ0 ∩ Fin ) ↑m 𝐽 ) ∣ ( 𝑟 supp ∅ ) ∈ Fin }
7 eulerpart.m ⊢ 𝑀 = ( 𝑟 ∈ 𝐻 ↦ { ⟨ 𝑥 , 𝑦 ⟩ ∣ ( 𝑥 ∈ 𝐽 ∧ 𝑦 ∈ ( 𝑟 ‘ 𝑥 ) ) } )
8 eulerpart.r ⊢ 𝑅 = { 𝑓 ∣ ( ◡ 𝑓 “ ℕ ) ∈ Fin }
9 eulerpart.t ⊢ 𝑇 = { 𝑓 ∈ ( ℕ0 ↑m ℕ ) ∣ ( ◡ 𝑓 “ ℕ ) ⊆ 𝐽 }
10 eulerpart.g ⊢ 𝐺 = ( 𝑜 ∈ ( 𝑇 ∩ 𝑅 ) ↦ ( ( 𝟭 ‘ ℕ ) ‘ ( 𝐹 “ ( 𝑀 ‘ ( bits ∘ ( 𝑜 ↾ 𝐽 ) ) ) ) ) )
11 eulerpartlemgh.1 ⊢ 𝑈 = ∪ 𝑡 ∈ ( ( ◡ 𝐴 “ ℕ ) ∩ 𝐽 ) ( { 𝑡 } × ( bits ‘ ( 𝐴 ‘ 𝑡 ) ) )
12 1 2 3 4 5 6 7 8 9 eulerpartlemt0 ⊢ ( 𝐴 ∈ ( 𝑇 ∩ 𝑅 ) ↔ ( 𝐴 ∈ ( ℕ0 ↑m ℕ ) ∧ ( ◡ 𝐴 “ ℕ ) ∈ Fin ∧ ( ◡ 𝐴 “ ℕ ) ⊆ 𝐽 ) )
13 12 simp1bi ⊢ ( 𝐴 ∈ ( 𝑇 ∩ 𝑅 ) → 𝐴 ∈ ( ℕ0 ↑m ℕ ) )
14 elmapi ⊢ ( 𝐴 ∈ ( ℕ0 ↑m ℕ ) → 𝐴 : ℕ ⟶ ℕ0 )
15 13 14 syl ⊢ ( 𝐴 ∈ ( 𝑇 ∩ 𝑅 ) → 𝐴 : ℕ ⟶ ℕ0 )
16 15 adantr ⊢ ( ( 𝐴 ∈ ( 𝑇 ∩ 𝑅 ) ∧ 𝑡 ∈ ( ( ◡ 𝐴 “ ℕ ) ∩ 𝐽 ) ) → 𝐴 : ℕ ⟶ ℕ0 )
17 16 ffund ⊢ ( ( 𝐴 ∈ ( 𝑇 ∩ 𝑅 ) ∧ 𝑡 ∈ ( ( ◡ 𝐴 “ ℕ ) ∩ 𝐽 ) ) → Fun 𝐴 )
18 inss1 ⊢ ( ( ◡ 𝐴 “ ℕ ) ∩ 𝐽 ) ⊆ ( ◡ 𝐴 “ ℕ )
19 cnvimass ⊢ ( ◡ 𝐴 “ ℕ ) ⊆ dom 𝐴
20 19 15 fssdm ⊢ ( 𝐴 ∈ ( 𝑇 ∩ 𝑅 ) → ( ◡ 𝐴 “ ℕ ) ⊆ ℕ )
21 18 20 sstrid ⊢ ( 𝐴 ∈ ( 𝑇 ∩ 𝑅 ) → ( ( ◡ 𝐴 “ ℕ ) ∩ 𝐽 ) ⊆ ℕ )
22 21 sselda ⊢ ( ( 𝐴 ∈ ( 𝑇 ∩ 𝑅 ) ∧ 𝑡 ∈ ( ( ◡ 𝐴 “ ℕ ) ∩ 𝐽 ) ) → 𝑡 ∈ ℕ )
23 15 fdmd ⊢ ( 𝐴 ∈ ( 𝑇 ∩ 𝑅 ) → dom 𝐴 = ℕ )
24 23 eleq2d ⊢ ( 𝐴 ∈ ( 𝑇 ∩ 𝑅 ) → ( 𝑡 ∈ dom 𝐴 ↔ 𝑡 ∈ ℕ ) )
25 24 adantr ⊢ ( ( 𝐴 ∈ ( 𝑇 ∩ 𝑅 ) ∧ 𝑡 ∈ ( ( ◡ 𝐴 “ ℕ ) ∩ 𝐽 ) ) → ( 𝑡 ∈ dom 𝐴 ↔ 𝑡 ∈ ℕ ) )
26 22 25 mpbird ⊢ ( ( 𝐴 ∈ ( 𝑇 ∩ 𝑅 ) ∧ 𝑡 ∈ ( ( ◡ 𝐴 “ ℕ ) ∩ 𝐽 ) ) → 𝑡 ∈ dom 𝐴 )
27 fvco ⊢ ( ( Fun 𝐴 ∧ 𝑡 ∈ dom 𝐴 ) → ( ( bits ∘ 𝐴 ) ‘ 𝑡 ) = ( bits ‘ ( 𝐴 ‘ 𝑡 ) ) )
28 17 26 27 syl2anc ⊢ ( ( 𝐴 ∈ ( 𝑇 ∩ 𝑅 ) ∧ 𝑡 ∈ ( ( ◡ 𝐴 “ ℕ ) ∩ 𝐽 ) ) → ( ( bits ∘ 𝐴 ) ‘ 𝑡 ) = ( bits ‘ ( 𝐴 ‘ 𝑡 ) ) )
29 28 xpeq2d ⊢ ( ( 𝐴 ∈ ( 𝑇 ∩ 𝑅 ) ∧ 𝑡 ∈ ( ( ◡ 𝐴 “ ℕ ) ∩ 𝐽 ) ) → ( { 𝑡 } × ( ( bits ∘ 𝐴 ) ‘ 𝑡 ) ) = ( { 𝑡 } × ( bits ‘ ( 𝐴 ‘ 𝑡 ) ) ) )
30 29 iuneq2dv ⊢ ( 𝐴 ∈ ( 𝑇 ∩ 𝑅 ) → ∪ 𝑡 ∈ ( ( ◡ 𝐴 “ ℕ ) ∩ 𝐽 ) ( { 𝑡 } × ( ( bits ∘ 𝐴 ) ‘ 𝑡 ) ) = ∪ 𝑡 ∈ ( ( ◡ 𝐴 “ ℕ ) ∩ 𝐽 ) ( { 𝑡 } × ( bits ‘ ( 𝐴 ‘ 𝑡 ) ) ) )
31 eqid ⊢ ∪ 𝑡 ∈ ( ( ◡ 𝐴 “ ℕ ) ∩ 𝐽 ) ( { 𝑡 } × ( ( bits ∘ 𝐴 ) ‘ 𝑡 ) ) = ∪ 𝑡 ∈ ( ( ◡ 𝐴 “ ℕ ) ∩ 𝐽 ) ( { 𝑡 } × ( ( bits ∘ 𝐴 ) ‘ 𝑡 ) )
32 31 marypha2lem2 ⊢ ∪ 𝑡 ∈ ( ( ◡ 𝐴 “ ℕ ) ∩ 𝐽 ) ( { 𝑡 } × ( ( bits ∘ 𝐴 ) ‘ 𝑡 ) ) = { ⟨ 𝑡 , 𝑛 ⟩ ∣ ( 𝑡 ∈ ( ( ◡ 𝐴 “ ℕ ) ∩ 𝐽 ) ∧ 𝑛 ∈ ( ( bits ∘ 𝐴 ) ‘ 𝑡 ) ) }
33 30 32 eqtr3di ⊢ ( 𝐴 ∈ ( 𝑇 ∩ 𝑅 ) → ∪ 𝑡 ∈ ( ( ◡ 𝐴 “ ℕ ) ∩ 𝐽 ) ( { 𝑡 } × ( bits ‘ ( 𝐴 ‘ 𝑡 ) ) ) = { ⟨ 𝑡 , 𝑛 ⟩ ∣ ( 𝑡 ∈ ( ( ◡ 𝐴 “ ℕ ) ∩ 𝐽 ) ∧ 𝑛 ∈ ( ( bits ∘ 𝐴 ) ‘ 𝑡 ) ) } )
34 11 33 syl5eq ⊢ ( 𝐴 ∈ ( 𝑇 ∩ 𝑅 ) → 𝑈 = { ⟨ 𝑡 , 𝑛 ⟩ ∣ ( 𝑡 ∈ ( ( ◡ 𝐴 “ ℕ ) ∩ 𝐽 ) ∧ 𝑛 ∈ ( ( bits ∘ 𝐴 ) ‘ 𝑡 ) ) } )