Metamath Proof Explorer

Theorem rpnnen2lem7

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 rpnnen2lem7 ( ( 𝐴 ⊆ 𝐵 ∧ 𝐵 ⊆ ℕ ∧ 𝑀 ∈ ℕ ) → Σ 𝑘 ∈ ( ℤ≥ ‘ 𝑀 ) ( ( 𝐹 ‘ 𝐴 ) ‘ 𝑘 ) ≤ Σ 𝑘 ∈ ( ℤ≥ ‘ 𝑀 ) ( ( 𝐹 ‘ 𝐵 ) ‘ 𝑘 ) )

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 simp3 ⊢ ( ( 𝐴 ⊆ 𝐵 ∧ 𝐵 ⊆ ℕ ∧ 𝑀 ∈ ℕ ) → 𝑀 ∈ ℕ )
4 3 nnzd ⊢ ( ( 𝐴 ⊆ 𝐵 ∧ 𝐵 ⊆ ℕ ∧ 𝑀 ∈ ℕ ) → 𝑀 ∈ ℤ )
5 eqidd ⊢ ( ( ( 𝐴 ⊆ 𝐵 ∧ 𝐵 ⊆ ℕ ∧ 𝑀 ∈ ℕ ) ∧ 𝑘 ∈ ( ℤ≥ ‘ 𝑀 ) ) → ( ( 𝐹 ‘ 𝐴 ) ‘ 𝑘 ) = ( ( 𝐹 ‘ 𝐴 ) ‘ 𝑘 ) )
6 eluznn ⊢ ( ( 𝑀 ∈ ℕ ∧ 𝑘 ∈ ( ℤ≥ ‘ 𝑀 ) ) → 𝑘 ∈ ℕ )
7 3 6 sylan ⊢ ( ( ( 𝐴 ⊆ 𝐵 ∧ 𝐵 ⊆ ℕ ∧ 𝑀 ∈ ℕ ) ∧ 𝑘 ∈ ( ℤ≥ ‘ 𝑀 ) ) → 𝑘 ∈ ℕ )
8 sstr ⊢ ( ( 𝐴 ⊆ 𝐵 ∧ 𝐵 ⊆ ℕ ) → 𝐴 ⊆ ℕ )
9 8 3adant3 ⊢ ( ( 𝐴 ⊆ 𝐵 ∧ 𝐵 ⊆ ℕ ∧ 𝑀 ∈ ℕ ) → 𝐴 ⊆ ℕ )
10 1 rpnnen2lem2 ⊢ ( 𝐴 ⊆ ℕ → ( 𝐹 ‘ 𝐴 ) : ℕ ⟶ ℝ )
11 9 10 syl ⊢ ( ( 𝐴 ⊆ 𝐵 ∧ 𝐵 ⊆ ℕ ∧ 𝑀 ∈ ℕ ) → ( 𝐹 ‘ 𝐴 ) : ℕ ⟶ ℝ )
12 11 ffvelrnda ⊢ ( ( ( 𝐴 ⊆ 𝐵 ∧ 𝐵 ⊆ ℕ ∧ 𝑀 ∈ ℕ ) ∧ 𝑘 ∈ ℕ ) → ( ( 𝐹 ‘ 𝐴 ) ‘ 𝑘 ) ∈ ℝ )
13 7 12 syldan ⊢ ( ( ( 𝐴 ⊆ 𝐵 ∧ 𝐵 ⊆ ℕ ∧ 𝑀 ∈ ℕ ) ∧ 𝑘 ∈ ( ℤ≥ ‘ 𝑀 ) ) → ( ( 𝐹 ‘ 𝐴 ) ‘ 𝑘 ) ∈ ℝ )
14 eqidd ⊢ ( ( ( 𝐴 ⊆ 𝐵 ∧ 𝐵 ⊆ ℕ ∧ 𝑀 ∈ ℕ ) ∧ 𝑘 ∈ ( ℤ≥ ‘ 𝑀 ) ) → ( ( 𝐹 ‘ 𝐵 ) ‘ 𝑘 ) = ( ( 𝐹 ‘ 𝐵 ) ‘ 𝑘 ) )
15 1 rpnnen2lem2 ⊢ ( 𝐵 ⊆ ℕ → ( 𝐹 ‘ 𝐵 ) : ℕ ⟶ ℝ )
16 15 3ad2ant2 ⊢ ( ( 𝐴 ⊆ 𝐵 ∧ 𝐵 ⊆ ℕ ∧ 𝑀 ∈ ℕ ) → ( 𝐹 ‘ 𝐵 ) : ℕ ⟶ ℝ )
17 16 ffvelrnda ⊢ ( ( ( 𝐴 ⊆ 𝐵 ∧ 𝐵 ⊆ ℕ ∧ 𝑀 ∈ ℕ ) ∧ 𝑘 ∈ ℕ ) → ( ( 𝐹 ‘ 𝐵 ) ‘ 𝑘 ) ∈ ℝ )
18 7 17 syldan ⊢ ( ( ( 𝐴 ⊆ 𝐵 ∧ 𝐵 ⊆ ℕ ∧ 𝑀 ∈ ℕ ) ∧ 𝑘 ∈ ( ℤ≥ ‘ 𝑀 ) ) → ( ( 𝐹 ‘ 𝐵 ) ‘ 𝑘 ) ∈ ℝ )
19 1 rpnnen2lem4 ⊢ ( ( 𝐴 ⊆ 𝐵 ∧ 𝐵 ⊆ ℕ ∧ 𝑘 ∈ ℕ ) → ( 0 ≤ ( ( 𝐹 ‘ 𝐴 ) ‘ 𝑘 ) ∧ ( ( 𝐹 ‘ 𝐴 ) ‘ 𝑘 ) ≤ ( ( 𝐹 ‘ 𝐵 ) ‘ 𝑘 ) ) )
20 19 simprd ⊢ ( ( 𝐴 ⊆ 𝐵 ∧ 𝐵 ⊆ ℕ ∧ 𝑘 ∈ ℕ ) → ( ( 𝐹 ‘ 𝐴 ) ‘ 𝑘 ) ≤ ( ( 𝐹 ‘ 𝐵 ) ‘ 𝑘 ) )
21 20 3expa ⊢ ( ( ( 𝐴 ⊆ 𝐵 ∧ 𝐵 ⊆ ℕ ) ∧ 𝑘 ∈ ℕ ) → ( ( 𝐹 ‘ 𝐴 ) ‘ 𝑘 ) ≤ ( ( 𝐹 ‘ 𝐵 ) ‘ 𝑘 ) )
22 21 3adantl3 ⊢ ( ( ( 𝐴 ⊆ 𝐵 ∧ 𝐵 ⊆ ℕ ∧ 𝑀 ∈ ℕ ) ∧ 𝑘 ∈ ℕ ) → ( ( 𝐹 ‘ 𝐴 ) ‘ 𝑘 ) ≤ ( ( 𝐹 ‘ 𝐵 ) ‘ 𝑘 ) )
23 7 22 syldan ⊢ ( ( ( 𝐴 ⊆ 𝐵 ∧ 𝐵 ⊆ ℕ ∧ 𝑀 ∈ ℕ ) ∧ 𝑘 ∈ ( ℤ≥ ‘ 𝑀 ) ) → ( ( 𝐹 ‘ 𝐴 ) ‘ 𝑘 ) ≤ ( ( 𝐹 ‘ 𝐵 ) ‘ 𝑘 ) )
24 1 rpnnen2lem5 ⊢ ( ( 𝐴 ⊆ ℕ ∧ 𝑀 ∈ ℕ ) → seq 𝑀 ( + , ( 𝐹 ‘ 𝐴 ) ) ∈ dom ⇝ )
25 8 24 stoic3 ⊢ ( ( 𝐴 ⊆ 𝐵 ∧ 𝐵 ⊆ ℕ ∧ 𝑀 ∈ ℕ ) → seq 𝑀 ( + , ( 𝐹 ‘ 𝐴 ) ) ∈ dom ⇝ )
26 1 rpnnen2lem5 ⊢ ( ( 𝐵 ⊆ ℕ ∧ 𝑀 ∈ ℕ ) → seq 𝑀 ( + , ( 𝐹 ‘ 𝐵 ) ) ∈ dom ⇝ )
27 26 3adant1 ⊢ ( ( 𝐴 ⊆ 𝐵 ∧ 𝐵 ⊆ ℕ ∧ 𝑀 ∈ ℕ ) → seq 𝑀 ( + , ( 𝐹 ‘ 𝐵 ) ) ∈ dom ⇝ )
28 2 4 5 13 14 18 23 25 27 isumle ⊢ ( ( 𝐴 ⊆ 𝐵 ∧ 𝐵 ⊆ ℕ ∧ 𝑀 ∈ ℕ ) → Σ 𝑘 ∈ ( ℤ≥ ‘ 𝑀 ) ( ( 𝐹 ‘ 𝐴 ) ‘ 𝑘 ) ≤ Σ 𝑘 ∈ ( ℤ≥ ‘ 𝑀 ) ( ( 𝐹 ‘ 𝐵 ) ‘ 𝑘 ) )