Theorem eulerpartlemsv1

Description: Lemma for eulerpart . Value of the sum of a partition A . (Contributed by Thierry Arnoux, 26-Aug-2018)

	Ref	Expression
Hypotheses	eulerpartlems.r	⊢ 𝑅 = { 𝑓 ∣ ( ◡ 𝑓 “ ℕ ) ∈ Fin }
	eulerpartlems.s	⊢ 𝑆 = ( 𝑓 ∈ ( ( ℕ0 ↑m ℕ ) ∩ 𝑅 ) ↦ Σ 𝑘 ∈ ℕ ( ( 𝑓 ‘ 𝑘 ) · 𝑘 ) )
Assertion	eulerpartlemsv1	⊢ ( 𝐴 ∈ ( ( ℕ0 ↑m ℕ ) ∩ 𝑅 ) → ( 𝑆 ‘ 𝐴 ) = Σ 𝑘 ∈ ℕ ( ( 𝐴 ‘ 𝑘 ) · 𝑘 ) )

Proof

Step	Hyp	Ref	Expression
1		eulerpartlems.r	⊢ 𝑅 = { 𝑓 ∣ ( ◡ 𝑓 “ ℕ ) ∈ Fin }
2		eulerpartlems.s	⊢ 𝑆 = ( 𝑓 ∈ ( ( ℕ0 ↑m ℕ ) ∩ 𝑅 ) ↦ Σ 𝑘 ∈ ℕ ( ( 𝑓 ‘ 𝑘 ) · 𝑘 ) )
3	2	a1i	⊢ ( 𝐴 ∈ ( ( ℕ0 ↑m ℕ ) ∩ 𝑅 ) → 𝑆 = ( 𝑓 ∈ ( ( ℕ0 ↑m ℕ ) ∩ 𝑅 ) ↦ Σ 𝑘 ∈ ℕ ( ( 𝑓 ‘ 𝑘 ) · 𝑘 ) ) )
4		simplr	⊢ ( ( ( 𝐴 ∈ ( ( ℕ0 ↑m ℕ ) ∩ 𝑅 ) ∧ 𝑓 = 𝐴 ) ∧ 𝑘 ∈ ℕ ) → 𝑓 = 𝐴 )
5	4	fveq1d	⊢ ( ( ( 𝐴 ∈ ( ( ℕ0 ↑m ℕ ) ∩ 𝑅 ) ∧ 𝑓 = 𝐴 ) ∧ 𝑘 ∈ ℕ ) → ( 𝑓 ‘ 𝑘 ) = ( 𝐴 ‘ 𝑘 ) )
6	5	oveq1d	⊢ ( ( ( 𝐴 ∈ ( ( ℕ0 ↑m ℕ ) ∩ 𝑅 ) ∧ 𝑓 = 𝐴 ) ∧ 𝑘 ∈ ℕ ) → ( ( 𝑓 ‘ 𝑘 ) · 𝑘 ) = ( ( 𝐴 ‘ 𝑘 ) · 𝑘 ) )
7	6	sumeq2dv	⊢ ( ( 𝐴 ∈ ( ( ℕ0 ↑m ℕ ) ∩ 𝑅 ) ∧ 𝑓 = 𝐴 ) → Σ 𝑘 ∈ ℕ ( ( 𝑓 ‘ 𝑘 ) · 𝑘 ) = Σ 𝑘 ∈ ℕ ( ( 𝐴 ‘ 𝑘 ) · 𝑘 ) )
8		id	⊢ ( 𝐴 ∈ ( ( ℕ0 ↑m ℕ ) ∩ 𝑅 ) → 𝐴 ∈ ( ( ℕ0 ↑m ℕ ) ∩ 𝑅 ) )
9		sumex	⊢ Σ 𝑘 ∈ ℕ ( ( 𝐴 ‘ 𝑘 ) · 𝑘 ) ∈ V
10	9	a1i	⊢ ( 𝐴 ∈ ( ( ℕ0 ↑m ℕ ) ∩ 𝑅 ) → Σ 𝑘 ∈ ℕ ( ( 𝐴 ‘ 𝑘 ) · 𝑘 ) ∈ V )
11	3 7 8 10	fvmptd	⊢ ( 𝐴 ∈ ( ( ℕ0 ↑m ℕ ) ∩ 𝑅 ) → ( 𝑆 ‘ 𝐴 ) = Σ 𝑘 ∈ ℕ ( ( 𝐴 ‘ 𝑘 ) · 𝑘 ) )