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Metamath Proof Explorer

Theorem rpnnen2lem8

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 rpnnen2lem8 ( ( 𝐴 ⊆ ℕ ∧ 𝑀 ∈ ℕ ) → Σ 𝑘 ∈ ℕ ( ( 𝐹 ‘ 𝐴 ) ‘ 𝑘 ) = ( Σ 𝑘 ∈ ( 1 ... ( 𝑀 − 1 ) ) ( ( 𝐹 ‘ 𝐴 ) ‘ 𝑘 ) + Σ 𝑘 ∈ ( ℤ≥ ‘ 𝑀 ) ( ( 𝐹 ‘ 𝐴 ) ‘ 𝑘 ) ) )

Proof

Step Hyp Ref Expression
1 rpnnen2.1 ⊢ 𝐹 = ( 𝑥 ∈ 𝒫 ℕ ↦ ( 𝑛 ∈ ℕ ↦ if ( 𝑛 ∈ 𝑥 , ( ( 1 / 3 ) ↑ 𝑛 ) , 0 ) ) )
2 nnuz ⊢ ℕ = ( ℤ≥ ‘ 1 )
3 eqid ⊢ ( ℤ≥ ‘ 𝑀 ) = ( ℤ≥ ‘ 𝑀 )
4 simpr ⊢ ( ( 𝐴 ⊆ ℕ ∧ 𝑀 ∈ ℕ ) → 𝑀 ∈ ℕ )
5 eqidd ⊢ ( ( ( 𝐴 ⊆ ℕ ∧ 𝑀 ∈ ℕ ) ∧ 𝑘 ∈ ℕ ) → ( ( 𝐹 ‘ 𝐴 ) ‘ 𝑘 ) = ( ( 𝐹 ‘ 𝐴 ) ‘ 𝑘 ) )
6 1 rpnnen2lem2 ⊢ ( 𝐴 ⊆ ℕ → ( 𝐹 ‘ 𝐴 ) : ℕ ⟶ ℝ )
7 6 adantr ⊢ ( ( 𝐴 ⊆ ℕ ∧ 𝑀 ∈ ℕ ) → ( 𝐹 ‘ 𝐴 ) : ℕ ⟶ ℝ )
8 7 ffvelcdmda ⊢ ( ( ( 𝐴 ⊆ ℕ ∧ 𝑀 ∈ ℕ ) ∧ 𝑘 ∈ ℕ ) → ( ( 𝐹 ‘ 𝐴 ) ‘ 𝑘 ) ∈ ℝ )
9 8 recnd ⊢ ( ( ( 𝐴 ⊆ ℕ ∧ 𝑀 ∈ ℕ ) ∧ 𝑘 ∈ ℕ ) → ( ( 𝐹 ‘ 𝐴 ) ‘ 𝑘 ) ∈ ℂ )
10 1nn ⊢ 1 ∈ ℕ
11 1 rpnnen2lem5 ⊢ ( ( 𝐴 ⊆ ℕ ∧ 1 ∈ ℕ ) → seq 1 ( + , ( 𝐹 ‘ 𝐴 ) ) ∈ dom ⇝ )
12 10 11 mpan2 ⊢ ( 𝐴 ⊆ ℕ → seq 1 ( + , ( 𝐹 ‘ 𝐴 ) ) ∈ dom ⇝ )
13 12 adantr ⊢ ( ( 𝐴 ⊆ ℕ ∧ 𝑀 ∈ ℕ ) → seq 1 ( + , ( 𝐹 ‘ 𝐴 ) ) ∈ dom ⇝ )
14 2 3 4 5 9 13 isumsplit ⊢ ( ( 𝐴 ⊆ ℕ ∧ 𝑀 ∈ ℕ ) → Σ 𝑘 ∈ ℕ ( ( 𝐹 ‘ 𝐴 ) ‘ 𝑘 ) = ( Σ 𝑘 ∈ ( 1 ... ( 𝑀 − 1 ) ) ( ( 𝐹 ‘ 𝐴 ) ‘ 𝑘 ) + Σ 𝑘 ∈ ( ℤ≥ ‘ 𝑀 ) ( ( 𝐹 ‘ 𝐴 ) ‘ 𝑘 ) ) )