Theorem eulerpartlemt0

Description: Lemma for eulerpart . (Contributed by Thierry Arnoux, 19-Sep-2017)

	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 }
Assertion	eulerpartlemt0	⊢ ( 𝐴 ∈ ( 𝑇 ∩ 𝑅 ) ↔ ( 𝐴 ∈ ( ℕ0 ↑m ℕ ) ∧ ( ◡ 𝐴 “ ℕ ) ∈ Fin ∧ ( ◡ 𝐴 “ ℕ ) ⊆ 𝐽 ) )

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		cnveq	⊢ ( 𝑓 = 𝐴 → ◡ 𝑓 = ◡ 𝐴 )
11	10	imaeq1d	⊢ ( 𝑓 = 𝐴 → ( ◡ 𝑓 “ ℕ ) = ( ◡ 𝐴 “ ℕ ) )
12	11	sseq1d	⊢ ( 𝑓 = 𝐴 → ( ( ◡ 𝑓 “ ℕ ) ⊆ 𝐽 ↔ ( ◡ 𝐴 “ ℕ ) ⊆ 𝐽 ) )
13	12 9	elrab2	⊢ ( 𝐴 ∈ 𝑇 ↔ ( 𝐴 ∈ ( ℕ0 ↑m ℕ ) ∧ ( ◡ 𝐴 “ ℕ ) ⊆ 𝐽 ) )
14	11	eleq1d	⊢ ( 𝑓 = 𝐴 → ( ( ◡ 𝑓 “ ℕ ) ∈ Fin ↔ ( ◡ 𝐴 “ ℕ ) ∈ Fin ) )
15	14 8	elab4g	⊢ ( 𝐴 ∈ 𝑅 ↔ ( 𝐴 ∈ V ∧ ( ◡ 𝐴 “ ℕ ) ∈ Fin ) )
16	13 15	anbi12i	⊢ ( ( 𝐴 ∈ 𝑇 ∧ 𝐴 ∈ 𝑅 ) ↔ ( ( 𝐴 ∈ ( ℕ0 ↑m ℕ ) ∧ ( ◡ 𝐴 “ ℕ ) ⊆ 𝐽 ) ∧ ( 𝐴 ∈ V ∧ ( ◡ 𝐴 “ ℕ ) ∈ Fin ) ) )
17		elin	⊢ ( 𝐴 ∈ ( 𝑇 ∩ 𝑅 ) ↔ ( 𝐴 ∈ 𝑇 ∧ 𝐴 ∈ 𝑅 ) )
18		elex	⊢ ( 𝐴 ∈ ( ℕ0 ↑m ℕ ) → 𝐴 ∈ V )
19	18	pm4.71i	⊢ ( 𝐴 ∈ ( ℕ0 ↑m ℕ ) ↔ ( 𝐴 ∈ ( ℕ0 ↑m ℕ ) ∧ 𝐴 ∈ V ) )
20	19	anbi1i	⊢ ( ( 𝐴 ∈ ( ℕ0 ↑m ℕ ) ∧ ( ( ◡ 𝐴 “ ℕ ) ∈ Fin ∧ ( ◡ 𝐴 “ ℕ ) ⊆ 𝐽 ) ) ↔ ( ( 𝐴 ∈ ( ℕ0 ↑m ℕ ) ∧ 𝐴 ∈ V ) ∧ ( ( ◡ 𝐴 “ ℕ ) ∈ Fin ∧ ( ◡ 𝐴 “ ℕ ) ⊆ 𝐽 ) ) )
21		3anass	⊢ ( ( 𝐴 ∈ ( ℕ0 ↑m ℕ ) ∧ ( ◡ 𝐴 “ ℕ ) ∈ Fin ∧ ( ◡ 𝐴 “ ℕ ) ⊆ 𝐽 ) ↔ ( 𝐴 ∈ ( ℕ0 ↑m ℕ ) ∧ ( ( ◡ 𝐴 “ ℕ ) ∈ Fin ∧ ( ◡ 𝐴 “ ℕ ) ⊆ 𝐽 ) ) )
22		an42	⊢ ( ( ( 𝐴 ∈ ( ℕ0 ↑m ℕ ) ∧ ( ◡ 𝐴 “ ℕ ) ⊆ 𝐽 ) ∧ ( 𝐴 ∈ V ∧ ( ◡ 𝐴 “ ℕ ) ∈ Fin ) ) ↔ ( ( 𝐴 ∈ ( ℕ0 ↑m ℕ ) ∧ 𝐴 ∈ V ) ∧ ( ( ◡ 𝐴 “ ℕ ) ∈ Fin ∧ ( ◡ 𝐴 “ ℕ ) ⊆ 𝐽 ) ) )
23	20 21 22	3bitr4i	⊢ ( ( 𝐴 ∈ ( ℕ0 ↑m ℕ ) ∧ ( ◡ 𝐴 “ ℕ ) ∈ Fin ∧ ( ◡ 𝐴 “ ℕ ) ⊆ 𝐽 ) ↔ ( ( 𝐴 ∈ ( ℕ0 ↑m ℕ ) ∧ ( ◡ 𝐴 “ ℕ ) ⊆ 𝐽 ) ∧ ( 𝐴 ∈ V ∧ ( ◡ 𝐴 “ ℕ ) ∈ Fin ) ) )
24	16 17 23	3bitr4i	⊢ ( 𝐴 ∈ ( 𝑇 ∩ 𝑅 ) ↔ ( 𝐴 ∈ ( ℕ0 ↑m ℕ ) ∧ ( ◡ 𝐴 “ ℕ ) ∈ Fin ∧ ( ◡ 𝐴 “ ℕ ) ⊆ 𝐽 ) )