Theorem rpnnen2lem6

Description: Lemma for rpnnen2 . (Contributed by Mario Carneiro, 13-May-2013) (Revised by Mario Carneiro, 30-Apr-2014)

	Ref	Expression
Hypothesis	rpnnen2.1	⊢ 𝐹 = ( 𝑥 ∈ 𝒫 ℕ ↦ ( 𝑛 ∈ ℕ ↦ if ( 𝑛 ∈ 𝑥 , ( ( 1 / 3 ) ↑ 𝑛 ) , 0 ) ) )
Assertion	rpnnen2lem6	⊢ ( ( 𝐴 ⊆ ℕ ∧ 𝑀 ∈ ℕ ) → Σ 𝑘 ∈ ( ℤ≥ ‘ 𝑀 ) ( ( 𝐹 ‘ 𝐴 ) ‘ 𝑘 ) ∈ ℝ )

Proof

Step	Hyp	Ref	Expression
1		rpnnen2.1	⊢ 𝐹 = ( 𝑥 ∈ 𝒫 ℕ ↦ ( 𝑛 ∈ ℕ ↦ if ( 𝑛 ∈ 𝑥 , ( ( 1 / 3 ) ↑ 𝑛 ) , 0 ) ) )
2		eqid	⊢ ( ℤ≥ ‘ 𝑀 ) = ( ℤ≥ ‘ 𝑀 )
3		nnz	⊢ ( 𝑀 ∈ ℕ → 𝑀 ∈ ℤ )
4	3	adantl	⊢ ( ( 𝐴 ⊆ ℕ ∧ 𝑀 ∈ ℕ ) → 𝑀 ∈ ℤ )
5		eqidd	⊢ ( ( ( 𝐴 ⊆ ℕ ∧ 𝑀 ∈ ℕ ) ∧ 𝑘 ∈ ( ℤ≥ ‘ 𝑀 ) ) → ( ( 𝐹 ‘ 𝐴 ) ‘ 𝑘 ) = ( ( 𝐹 ‘ 𝐴 ) ‘ 𝑘 ) )
6	1	rpnnen2lem2	⊢ ( 𝐴 ⊆ ℕ → ( 𝐹 ‘ 𝐴 ) : ℕ ⟶ ℝ )
7	6	ad2antrr	⊢ ( ( ( 𝐴 ⊆ ℕ ∧ 𝑀 ∈ ℕ ) ∧ 𝑘 ∈ ( ℤ≥ ‘ 𝑀 ) ) → ( 𝐹 ‘ 𝐴 ) : ℕ ⟶ ℝ )
8		eluznn	⊢ ( ( 𝑀 ∈ ℕ ∧ 𝑘 ∈ ( ℤ≥ ‘ 𝑀 ) ) → 𝑘 ∈ ℕ )
9	8	adantll	⊢ ( ( ( 𝐴 ⊆ ℕ ∧ 𝑀 ∈ ℕ ) ∧ 𝑘 ∈ ( ℤ≥ ‘ 𝑀 ) ) → 𝑘 ∈ ℕ )
10	7 9	ffvelcdmd	⊢ ( ( ( 𝐴 ⊆ ℕ ∧ 𝑀 ∈ ℕ ) ∧ 𝑘 ∈ ( ℤ≥ ‘ 𝑀 ) ) → ( ( 𝐹 ‘ 𝐴 ) ‘ 𝑘 ) ∈ ℝ )
11	1	rpnnen2lem5	⊢ ( ( 𝐴 ⊆ ℕ ∧ 𝑀 ∈ ℕ ) → seq 𝑀 ( + , ( 𝐹 ‘ 𝐴 ) ) ∈ dom ⇝ )
12	2 4 5 10 11	isumrecl	⊢ ( ( 𝐴 ⊆ ℕ ∧ 𝑀 ∈ ℕ ) → Σ 𝑘 ∈ ( ℤ≥ ‘ 𝑀 ) ( ( 𝐹 ‘ 𝐴 ) ‘ 𝑘 ) ∈ ℝ )