Metamath Proof Explorer

Theorem rlimres

Description: The restriction of a function converges if the original converges. (Contributed by Mario Carneiro, 16-Sep-2014)

Ref Expression
Assertion rlimres ( 𝐹 ⇝𝑟 𝐴 → ( 𝐹 ↾ 𝐵 ) ⇝𝑟 𝐴 )

Proof

Step Hyp Ref Expression
1 inss1 ⊢ ( dom 𝐹 ∩ 𝐵 ) ⊆ dom 𝐹
2 ssralv ⊢ ( ( dom 𝐹 ∩ 𝐵 ) ⊆ dom 𝐹 → ( ∀ 𝑧 ∈ dom 𝐹 ( 𝑦 ≤ 𝑧 → ( abs ‘ ( ( 𝐹 ‘ 𝑧 ) − 𝐴 ) ) < 𝑥 ) → ∀ 𝑧 ∈ ( dom 𝐹 ∩ 𝐵 ) ( 𝑦 ≤ 𝑧 → ( abs ‘ ( ( 𝐹 ‘ 𝑧 ) − 𝐴 ) ) < 𝑥 ) ) )
3 1 2 ax-mp ⊢ ( ∀ 𝑧 ∈ dom 𝐹 ( 𝑦 ≤ 𝑧 → ( abs ‘ ( ( 𝐹 ‘ 𝑧 ) − 𝐴 ) ) < 𝑥 ) → ∀ 𝑧 ∈ ( dom 𝐹 ∩ 𝐵 ) ( 𝑦 ≤ 𝑧 → ( abs ‘ ( ( 𝐹 ‘ 𝑧 ) − 𝐴 ) ) < 𝑥 ) )
4 3 reximi ⊢ ( ∃ 𝑦 ∈ ℝ ∀ 𝑧 ∈ dom 𝐹 ( 𝑦 ≤ 𝑧 → ( abs ‘ ( ( 𝐹 ‘ 𝑧 ) − 𝐴 ) ) < 𝑥 ) → ∃ 𝑦 ∈ ℝ ∀ 𝑧 ∈ ( dom 𝐹 ∩ 𝐵 ) ( 𝑦 ≤ 𝑧 → ( abs ‘ ( ( 𝐹 ‘ 𝑧 ) − 𝐴 ) ) < 𝑥 ) )
5 4 ralimi ⊢ ( ∀ 𝑥 ∈ ℝ+ ∃ 𝑦 ∈ ℝ ∀ 𝑧 ∈ dom 𝐹 ( 𝑦 ≤ 𝑧 → ( abs ‘ ( ( 𝐹 ‘ 𝑧 ) − 𝐴 ) ) < 𝑥 ) → ∀ 𝑥 ∈ ℝ+ ∃ 𝑦 ∈ ℝ ∀ 𝑧 ∈ ( dom 𝐹 ∩ 𝐵 ) ( 𝑦 ≤ 𝑧 → ( abs ‘ ( ( 𝐹 ‘ 𝑧 ) − 𝐴 ) ) < 𝑥 ) )
6 5 anim2i ⊢ ( ( 𝐴 ∈ ℂ ∧ ∀ 𝑥 ∈ ℝ+ ∃ 𝑦 ∈ ℝ ∀ 𝑧 ∈ dom 𝐹 ( 𝑦 ≤ 𝑧 → ( abs ‘ ( ( 𝐹 ‘ 𝑧 ) − 𝐴 ) ) < 𝑥 ) ) → ( 𝐴 ∈ ℂ ∧ ∀ 𝑥 ∈ ℝ+ ∃ 𝑦 ∈ ℝ ∀ 𝑧 ∈ ( dom 𝐹 ∩ 𝐵 ) ( 𝑦 ≤ 𝑧 → ( abs ‘ ( ( 𝐹 ‘ 𝑧 ) − 𝐴 ) ) < 𝑥 ) ) )
7 6 a1i ⊢ ( 𝐹 ⇝𝑟 𝐴 → ( ( 𝐴 ∈ ℂ ∧ ∀ 𝑥 ∈ ℝ+ ∃ 𝑦 ∈ ℝ ∀ 𝑧 ∈ dom 𝐹 ( 𝑦 ≤ 𝑧 → ( abs ‘ ( ( 𝐹 ‘ 𝑧 ) − 𝐴 ) ) < 𝑥 ) ) → ( 𝐴 ∈ ℂ ∧ ∀ 𝑥 ∈ ℝ+ ∃ 𝑦 ∈ ℝ ∀ 𝑧 ∈ ( dom 𝐹 ∩ 𝐵 ) ( 𝑦 ≤ 𝑧 → ( abs ‘ ( ( 𝐹 ‘ 𝑧 ) − 𝐴 ) ) < 𝑥 ) ) ) )
8 rlimf ⊢ ( 𝐹 ⇝𝑟 𝐴 → 𝐹 : dom 𝐹 ⟶ ℂ )
9 rlimss ⊢ ( 𝐹 ⇝𝑟 𝐴 → dom 𝐹 ⊆ ℝ )
10 eqidd ⊢ ( ( 𝐹 ⇝𝑟 𝐴 ∧ 𝑧 ∈ dom 𝐹 ) → ( 𝐹 ‘ 𝑧 ) = ( 𝐹 ‘ 𝑧 ) )
11 8 9 10 rlim ⊢ ( 𝐹 ⇝𝑟 𝐴 → ( 𝐹 ⇝𝑟 𝐴 ↔ ( 𝐴 ∈ ℂ ∧ ∀ 𝑥 ∈ ℝ+ ∃ 𝑦 ∈ ℝ ∀ 𝑧 ∈ dom 𝐹 ( 𝑦 ≤ 𝑧 → ( abs ‘ ( ( 𝐹 ‘ 𝑧 ) − 𝐴 ) ) < 𝑥 ) ) ) )
12 fssres ⊢ ( ( 𝐹 : dom 𝐹 ⟶ ℂ ∧ ( dom 𝐹 ∩ 𝐵 ) ⊆ dom 𝐹 ) → ( 𝐹 ↾ ( dom 𝐹 ∩ 𝐵 ) ) : ( dom 𝐹 ∩ 𝐵 ) ⟶ ℂ )
13 8 1 12 sylancl ⊢ ( 𝐹 ⇝𝑟 𝐴 → ( 𝐹 ↾ ( dom 𝐹 ∩ 𝐵 ) ) : ( dom 𝐹 ∩ 𝐵 ) ⟶ ℂ )
14 resres ⊢ ( ( 𝐹 ↾ dom 𝐹 ) ↾ 𝐵 ) = ( 𝐹 ↾ ( dom 𝐹 ∩ 𝐵 ) )
15 ffn ⊢ ( 𝐹 : dom 𝐹 ⟶ ℂ → 𝐹 Fn dom 𝐹 )
16 fnresdm ⊢ ( 𝐹 Fn dom 𝐹 → ( 𝐹 ↾ dom 𝐹 ) = 𝐹 )
17 8 15 16 3syl ⊢ ( 𝐹 ⇝𝑟 𝐴 → ( 𝐹 ↾ dom 𝐹 ) = 𝐹 )
18 17 reseq1d ⊢ ( 𝐹 ⇝𝑟 𝐴 → ( ( 𝐹 ↾ dom 𝐹 ) ↾ 𝐵 ) = ( 𝐹 ↾ 𝐵 ) )
19 14 18 eqtr3id ⊢ ( 𝐹 ⇝𝑟 𝐴 → ( 𝐹 ↾ ( dom 𝐹 ∩ 𝐵 ) ) = ( 𝐹 ↾ 𝐵 ) )
20 19 feq1d ⊢ ( 𝐹 ⇝𝑟 𝐴 → ( ( 𝐹 ↾ ( dom 𝐹 ∩ 𝐵 ) ) : ( dom 𝐹 ∩ 𝐵 ) ⟶ ℂ ↔ ( 𝐹 ↾ 𝐵 ) : ( dom 𝐹 ∩ 𝐵 ) ⟶ ℂ ) )
21 13 20 mpbid ⊢ ( 𝐹 ⇝𝑟 𝐴 → ( 𝐹 ↾ 𝐵 ) : ( dom 𝐹 ∩ 𝐵 ) ⟶ ℂ )
22 1 9 sstrid ⊢ ( 𝐹 ⇝𝑟 𝐴 → ( dom 𝐹 ∩ 𝐵 ) ⊆ ℝ )
23 elinel2 ⊢ ( 𝑧 ∈ ( dom 𝐹 ∩ 𝐵 ) → 𝑧 ∈ 𝐵 )
24 23 fvresd ⊢ ( 𝑧 ∈ ( dom 𝐹 ∩ 𝐵 ) → ( ( 𝐹 ↾ 𝐵 ) ‘ 𝑧 ) = ( 𝐹 ‘ 𝑧 ) )
25 24 adantl ⊢ ( ( 𝐹 ⇝𝑟 𝐴 ∧ 𝑧 ∈ ( dom 𝐹 ∩ 𝐵 ) ) → ( ( 𝐹 ↾ 𝐵 ) ‘ 𝑧 ) = ( 𝐹 ‘ 𝑧 ) )
26 21 22 25 rlim ⊢ ( 𝐹 ⇝𝑟 𝐴 → ( ( 𝐹 ↾ 𝐵 ) ⇝𝑟 𝐴 ↔ ( 𝐴 ∈ ℂ ∧ ∀ 𝑥 ∈ ℝ+ ∃ 𝑦 ∈ ℝ ∀ 𝑧 ∈ ( dom 𝐹 ∩ 𝐵 ) ( 𝑦 ≤ 𝑧 → ( abs ‘ ( ( 𝐹 ‘ 𝑧 ) − 𝐴 ) ) < 𝑥 ) ) ) )
27 7 11 26 3imtr4d ⊢ ( 𝐹 ⇝𝑟 𝐴 → ( 𝐹 ⇝𝑟 𝐴 → ( 𝐹 ↾ 𝐵 ) ⇝𝑟 𝐴 ) )
28 27 pm2.43i ⊢ ( 𝐹 ⇝𝑟 𝐴 → ( 𝐹 ↾ 𝐵 ) ⇝𝑟 𝐴 )