Theorem climi

Description: Convergence of a sequence of complex numbers. (Contributed by NM, 11-Jan-2007) (Revised by Mario Carneiro, 31-Jan-2014)

	Ref	Expression
Hypotheses	climi.1	⊢ 𝑍 = ( ℤ≥ ‘ 𝑀 )
	climi.2	⊢ ( 𝜑 → 𝑀 ∈ ℤ )
	climi.3	⊢ ( 𝜑 → 𝐶 ∈ ℝ+ )
	climi.4	⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝑍 ) → ( 𝐹 ‘ 𝑘 ) = 𝐵 )
	climi.5	⊢ ( 𝜑 → 𝐹 ⇝ 𝐴 )
Assertion	climi	⊢ ( 𝜑 → ∃ 𝑗 ∈ 𝑍 ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) ( 𝐵 ∈ ℂ ∧ ( abs ‘ ( 𝐵 − 𝐴 ) ) < 𝐶 ) )

Proof

Step	Hyp	Ref	Expression
1		climi.1	⊢ 𝑍 = ( ℤ≥ ‘ 𝑀 )
2		climi.2	⊢ ( 𝜑 → 𝑀 ∈ ℤ )
3		climi.3	⊢ ( 𝜑 → 𝐶 ∈ ℝ+ )
4		climi.4	⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝑍 ) → ( 𝐹 ‘ 𝑘 ) = 𝐵 )
5		climi.5	⊢ ( 𝜑 → 𝐹 ⇝ 𝐴 )
6		breq2	⊢ ( 𝑥 = 𝐶 → ( ( abs ‘ ( 𝐵 − 𝐴 ) ) < 𝑥 ↔ ( abs ‘ ( 𝐵 − 𝐴 ) ) < 𝐶 ) )
7	6	anbi2d	⊢ ( 𝑥 = 𝐶 → ( ( 𝐵 ∈ ℂ ∧ ( abs ‘ ( 𝐵 − 𝐴 ) ) < 𝑥 ) ↔ ( 𝐵 ∈ ℂ ∧ ( abs ‘ ( 𝐵 − 𝐴 ) ) < 𝐶 ) ) )
8	7	rexralbidv	⊢ ( 𝑥 = 𝐶 → ( ∃ 𝑗 ∈ 𝑍 ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) ( 𝐵 ∈ ℂ ∧ ( abs ‘ ( 𝐵 − 𝐴 ) ) < 𝑥 ) ↔ ∃ 𝑗 ∈ 𝑍 ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) ( 𝐵 ∈ ℂ ∧ ( abs ‘ ( 𝐵 − 𝐴 ) ) < 𝐶 ) ) )
9		climrel	⊢ Rel ⇝
10	9	brrelex1i	⊢ ( 𝐹 ⇝ 𝐴 → 𝐹 ∈ V )
11	5 10	syl	⊢ ( 𝜑 → 𝐹 ∈ V )
12	1 2 11 4	clim2	⊢ ( 𝜑 → ( 𝐹 ⇝ 𝐴 ↔ ( 𝐴 ∈ ℂ ∧ ∀ 𝑥 ∈ ℝ+ ∃ 𝑗 ∈ 𝑍 ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) ( 𝐵 ∈ ℂ ∧ ( abs ‘ ( 𝐵 − 𝐴 ) ) < 𝑥 ) ) ) )
13	5 12	mpbid	⊢ ( 𝜑 → ( 𝐴 ∈ ℂ ∧ ∀ 𝑥 ∈ ℝ+ ∃ 𝑗 ∈ 𝑍 ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) ( 𝐵 ∈ ℂ ∧ ( abs ‘ ( 𝐵 − 𝐴 ) ) < 𝑥 ) ) )
14	13	simprd	⊢ ( 𝜑 → ∀ 𝑥 ∈ ℝ+ ∃ 𝑗 ∈ 𝑍 ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) ( 𝐵 ∈ ℂ ∧ ( abs ‘ ( 𝐵 − 𝐴 ) ) < 𝑥 ) )
15	8 14 3	rspcdva	⊢ ( 𝜑 → ∃ 𝑗 ∈ 𝑍 ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) ( 𝐵 ∈ ℂ ∧ ( abs ‘ ( 𝐵 − 𝐴 ) ) < 𝐶 ) )