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Metamath Proof Explorer

Theorem climcn1lem

Description: The limit of a continuous function, theorem form. (Contributed by Mario Carneiro, 9-Feb-2014)

Ref Expression
Hypotheses climcn1lem.1 ⊢ 𝑍 = ( ℤ≥ ‘ 𝑀 )
climcn1lem.2 ⊢ ( 𝜑 → 𝐹 ⇝ 𝐴 )
climcn1lem.4 ⊢ ( 𝜑 → 𝐺 ∈ 𝑊 )
climcn1lem.5 ⊢ ( 𝜑 → 𝑀 ∈ ℤ )
climcn1lem.6 ⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝑍 ) → ( 𝐹 ‘ 𝑘 ) ∈ ℂ )
climcn1lem.7 ⊢ 𝐻 : ℂ ⟶ ℂ
climcn1lem.8 ⊢ ( ( 𝐴 ∈ ℂ ∧ 𝑥 ∈ ℝ+ ) → ∃ 𝑦 ∈ ℝ+ ∀ 𝑧 ∈ ℂ ( ( abs ‘ ( 𝑧 − 𝐴 ) ) < 𝑦 → ( abs ‘ ( ( 𝐻 ‘ 𝑧 ) − ( 𝐻 ‘ 𝐴 ) ) ) < 𝑥 ) )
climcn1lem.9 ⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝑍 ) → ( 𝐺 ‘ 𝑘 ) = ( 𝐻 ‘ ( 𝐹 ‘ 𝑘 ) ) )
Assertion climcn1lem ( 𝜑 → 𝐺 ⇝ ( 𝐻 ‘ 𝐴 ) )

Proof

Step Hyp Ref Expression
1 climcn1lem.1 ⊢ 𝑍 = ( ℤ≥ ‘ 𝑀 )
2 climcn1lem.2 ⊢ ( 𝜑 → 𝐹 ⇝ 𝐴 )
3 climcn1lem.4 ⊢ ( 𝜑 → 𝐺 ∈ 𝑊 )
4 climcn1lem.5 ⊢ ( 𝜑 → 𝑀 ∈ ℤ )
5 climcn1lem.6 ⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝑍 ) → ( 𝐹 ‘ 𝑘 ) ∈ ℂ )
6 climcn1lem.7 ⊢ 𝐻 : ℂ ⟶ ℂ
7 climcn1lem.8 ⊢ ( ( 𝐴 ∈ ℂ ∧ 𝑥 ∈ ℝ+ ) → ∃ 𝑦 ∈ ℝ+ ∀ 𝑧 ∈ ℂ ( ( abs ‘ ( 𝑧 − 𝐴 ) ) < 𝑦 → ( abs ‘ ( ( 𝐻 ‘ 𝑧 ) − ( 𝐻 ‘ 𝐴 ) ) ) < 𝑥 ) )
8 climcn1lem.9 ⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝑍 ) → ( 𝐺 ‘ 𝑘 ) = ( 𝐻 ‘ ( 𝐹 ‘ 𝑘 ) ) )
9 climcl ⊢ ( 𝐹 ⇝ 𝐴 → 𝐴 ∈ ℂ )
10 2 9 syl ⊢ ( 𝜑 → 𝐴 ∈ ℂ )
11 6 ffvelcdmi ⊢ ( 𝑧 ∈ ℂ → ( 𝐻 ‘ 𝑧 ) ∈ ℂ )
12 11 adantl ⊢ ( ( 𝜑 ∧ 𝑧 ∈ ℂ ) → ( 𝐻 ‘ 𝑧 ) ∈ ℂ )
13 10 7 sylan ⊢ ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) → ∃ 𝑦 ∈ ℝ+ ∀ 𝑧 ∈ ℂ ( ( abs ‘ ( 𝑧 − 𝐴 ) ) < 𝑦 → ( abs ‘ ( ( 𝐻 ‘ 𝑧 ) − ( 𝐻 ‘ 𝐴 ) ) ) < 𝑥 ) )
14 1 4 10 12 2 3 13 5 8 climcn1 ⊢ ( 𝜑 → 𝐺 ⇝ ( 𝐻 ‘ 𝐴 ) )