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Metamath Proof Explorer

Theorem climi0

Description: Convergence of a sequence of complex numbers to zero. (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 )
climi0.5 ⊢ ( 𝜑 → 𝐹 ⇝ 0 )
Assertion climi0 ( 𝜑 → ∃ 𝑗 ∈ 𝑍 ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) ( abs ‘ 𝐵 ) < 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 climi0.5 ⊢ ( 𝜑 → 𝐹 ⇝ 0 )
6 1 2 3 4 5 climi ⊢ ( 𝜑 → ∃ 𝑗 ∈ 𝑍 ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) ( 𝐵 ∈ ℂ ∧ ( abs ‘ ( 𝐵 − 0 ) ) < 𝐶 ) )
7 subid1 ⊢ ( 𝐵 ∈ ℂ → ( 𝐵 − 0 ) = 𝐵 )
8 7 fveq2d ⊢ ( 𝐵 ∈ ℂ → ( abs ‘ ( 𝐵 − 0 ) ) = ( abs ‘ 𝐵 ) )
9 8 breq1d ⊢ ( 𝐵 ∈ ℂ → ( ( abs ‘ ( 𝐵 − 0 ) ) < 𝐶 ↔ ( abs ‘ 𝐵 ) < 𝐶 ) )
10 9 biimpa ⊢ ( ( 𝐵 ∈ ℂ ∧ ( abs ‘ ( 𝐵 − 0 ) ) < 𝐶 ) → ( abs ‘ 𝐵 ) < 𝐶 )
11 10 ralimi ⊢ ( ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) ( 𝐵 ∈ ℂ ∧ ( abs ‘ ( 𝐵 − 0 ) ) < 𝐶 ) → ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) ( abs ‘ 𝐵 ) < 𝐶 )
12 11 reximi ⊢ ( ∃ 𝑗 ∈ 𝑍 ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) ( 𝐵 ∈ ℂ ∧ ( abs ‘ ( 𝐵 − 0 ) ) < 𝐶 ) → ∃ 𝑗 ∈ 𝑍 ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) ( abs ‘ 𝐵 ) < 𝐶 )
13 6 12 syl ⊢ ( 𝜑 → ∃ 𝑗 ∈ 𝑍 ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) ( abs ‘ 𝐵 ) < 𝐶 )