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	ffvelrni	⊢ ( 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 ) )