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Metamath Proof Explorer

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 )