Metamath Proof Explorer

Theorem rpnnen2lem5

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 rpnnen2lem5 ( ( 𝐴 ⊆ ℕ ∧ 𝑀 ∈ ℕ ) → seq 𝑀 ( + , ( 𝐹 ‘ 𝐴 ) ) ∈ dom ⇝ )

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 1nn ⊢ 1 ∈ ℕ
4 3 a1i ⊢ ( 𝐴 ⊆ ℕ → 1 ∈ ℕ )
5 ssid ⊢ ℕ ⊆ ℕ
6 1 rpnnen2lem2 ⊢ ( ℕ ⊆ ℕ → ( 𝐹 ‘ ℕ ) : ℕ ⟶ ℝ )
7 5 6 mp1i ⊢ ( 𝐴 ⊆ ℕ → ( 𝐹 ‘ ℕ ) : ℕ ⟶ ℝ )
8 7 ffvelcdmda ⊢ ( ( 𝐴 ⊆ ℕ ∧ 𝑘 ∈ ℕ ) → ( ( 𝐹 ‘ ℕ ) ‘ 𝑘 ) ∈ ℝ )
9 1 rpnnen2lem2 ⊢ ( 𝐴 ⊆ ℕ → ( 𝐹 ‘ 𝐴 ) : ℕ ⟶ ℝ )
10 9 ffvelcdmda ⊢ ( ( 𝐴 ⊆ ℕ ∧ 𝑘 ∈ ℕ ) → ( ( 𝐹 ‘ 𝐴 ) ‘ 𝑘 ) ∈ ℝ )
11 1 rpnnen2lem3 ⊢ seq 1 ( + , ( 𝐹 ‘ ℕ ) ) ⇝ ( 1 / 2 )
12 seqex ⊢ seq 1 ( + , ( 𝐹 ‘ ℕ ) ) ∈ V
13 ovex ⊢ ( 1 / 2 ) ∈ V
14 12 13 breldm ⊢ ( seq 1 ( + , ( 𝐹 ‘ ℕ ) ) ⇝ ( 1 / 2 ) → seq 1 ( + , ( 𝐹 ‘ ℕ ) ) ∈ dom ⇝ )
15 11 14 mp1i ⊢ ( 𝐴 ⊆ ℕ → seq 1 ( + , ( 𝐹 ‘ ℕ ) ) ∈ dom ⇝ )
16 elnnuz ⊢ ( 𝑘 ∈ ℕ ↔ 𝑘 ∈ ( ℤ≥ ‘ 1 ) )
17 1 rpnnen2lem4 ⊢ ( ( 𝐴 ⊆ ℕ ∧ ℕ ⊆ ℕ ∧ 𝑘 ∈ ℕ ) → ( 0 ≤ ( ( 𝐹 ‘ 𝐴 ) ‘ 𝑘 ) ∧ ( ( 𝐹 ‘ 𝐴 ) ‘ 𝑘 ) ≤ ( ( 𝐹 ‘ ℕ ) ‘ 𝑘 ) ) )
18 5 17 mp3an2 ⊢ ( ( 𝐴 ⊆ ℕ ∧ 𝑘 ∈ ℕ ) → ( 0 ≤ ( ( 𝐹 ‘ 𝐴 ) ‘ 𝑘 ) ∧ ( ( 𝐹 ‘ 𝐴 ) ‘ 𝑘 ) ≤ ( ( 𝐹 ‘ ℕ ) ‘ 𝑘 ) ) )
19 16 18 sylan2br ⊢ ( ( 𝐴 ⊆ ℕ ∧ 𝑘 ∈ ( ℤ≥ ‘ 1 ) ) → ( 0 ≤ ( ( 𝐹 ‘ 𝐴 ) ‘ 𝑘 ) ∧ ( ( 𝐹 ‘ 𝐴 ) ‘ 𝑘 ) ≤ ( ( 𝐹 ‘ ℕ ) ‘ 𝑘 ) ) )
20 19 simpld ⊢ ( ( 𝐴 ⊆ ℕ ∧ 𝑘 ∈ ( ℤ≥ ‘ 1 ) ) → 0 ≤ ( ( 𝐹 ‘ 𝐴 ) ‘ 𝑘 ) )
21 19 simprd ⊢ ( ( 𝐴 ⊆ ℕ ∧ 𝑘 ∈ ( ℤ≥ ‘ 1 ) ) → ( ( 𝐹 ‘ 𝐴 ) ‘ 𝑘 ) ≤ ( ( 𝐹 ‘ ℕ ) ‘ 𝑘 ) )
22 2 4 8 10 15 20 21 cvgcmp ⊢ ( 𝐴 ⊆ ℕ → seq 1 ( + , ( 𝐹 ‘ 𝐴 ) ) ∈ dom ⇝ )
23 22 adantr ⊢ ( ( 𝐴 ⊆ ℕ ∧ 𝑀 ∈ ℕ ) → seq 1 ( + , ( 𝐹 ‘ 𝐴 ) ) ∈ dom ⇝ )
24 simpr ⊢ ( ( 𝐴 ⊆ ℕ ∧ 𝑀 ∈ ℕ ) → 𝑀 ∈ ℕ )
25 10 adantlr ⊢ ( ( ( 𝐴 ⊆ ℕ ∧ 𝑀 ∈ ℕ ) ∧ 𝑘 ∈ ℕ ) → ( ( 𝐹 ‘ 𝐴 ) ‘ 𝑘 ) ∈ ℝ )
26 25 recnd ⊢ ( ( ( 𝐴 ⊆ ℕ ∧ 𝑀 ∈ ℕ ) ∧ 𝑘 ∈ ℕ ) → ( ( 𝐹 ‘ 𝐴 ) ‘ 𝑘 ) ∈ ℂ )
27 2 24 26 iserex ⊢ ( ( 𝐴 ⊆ ℕ ∧ 𝑀 ∈ ℕ ) → ( seq 1 ( + , ( 𝐹 ‘ 𝐴 ) ) ∈ dom ⇝ ↔ seq 𝑀 ( + , ( 𝐹 ‘ 𝐴 ) ) ∈ dom ⇝ ) )
28 23 27 mpbid ⊢ ( ( 𝐴 ⊆ ℕ ∧ 𝑀 ∈ ℕ ) → seq 𝑀 ( + , ( 𝐹 ‘ 𝐴 ) ) ∈ dom ⇝ )