Theorem clim0c

Description: Express the predicate F converges to 0 . (Contributed by NM, 24-Feb-2008) (Revised by Mario Carneiro, 31-Jan-2014)

	Ref	Expression
Hypotheses	clim0.1	⊢ 𝑍 = ( ℤ≥ ‘ 𝑀 )
	clim0.2	⊢ ( 𝜑 → 𝑀 ∈ ℤ )
	clim0.3	⊢ ( 𝜑 → 𝐹 ∈ 𝑉 )
	clim0.4	⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝑍 ) → ( 𝐹 ‘ 𝑘 ) = 𝐵 )
	clim0c.6	⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝑍 ) → 𝐵 ∈ ℂ )
Assertion	clim0c	⊢ ( 𝜑 → ( 𝐹 ⇝ 0 ↔ ∀ 𝑥 ∈ ℝ+ ∃ 𝑗 ∈ 𝑍 ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) ( abs ‘ 𝐵 ) < 𝑥 ) )

Proof

Step	Hyp	Ref	Expression
1		clim0.1	⊢ 𝑍 = ( ℤ≥ ‘ 𝑀 )
2		clim0.2	⊢ ( 𝜑 → 𝑀 ∈ ℤ )
3		clim0.3	⊢ ( 𝜑 → 𝐹 ∈ 𝑉 )
4		clim0.4	⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝑍 ) → ( 𝐹 ‘ 𝑘 ) = 𝐵 )
5		clim0c.6	⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝑍 ) → 𝐵 ∈ ℂ )
6		0cnd	⊢ ( 𝜑 → 0 ∈ ℂ )
7	1 2 3 4 6 5	clim2c	⊢ ( 𝜑 → ( 𝐹 ⇝ 0 ↔ ∀ 𝑥 ∈ ℝ+ ∃ 𝑗 ∈ 𝑍 ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) ( abs ‘ ( 𝐵 − 0 ) ) < 𝑥 ) )
8	1	uztrn2	⊢ ( ( 𝑗 ∈ 𝑍 ∧ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) ) → 𝑘 ∈ 𝑍 )
9	5	subid1d	⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝑍 ) → ( 𝐵 − 0 ) = 𝐵 )
10	9	fveq2d	⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝑍 ) → ( abs ‘ ( 𝐵 − 0 ) ) = ( abs ‘ 𝐵 ) )
11	10	breq1d	⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝑍 ) → ( ( abs ‘ ( 𝐵 − 0 ) ) < 𝑥 ↔ ( abs ‘ 𝐵 ) < 𝑥 ) )
12	8 11	sylan2	⊢ ( ( 𝜑 ∧ ( 𝑗 ∈ 𝑍 ∧ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) ) ) → ( ( abs ‘ ( 𝐵 − 0 ) ) < 𝑥 ↔ ( abs ‘ 𝐵 ) < 𝑥 ) )
13	12	anassrs	⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ 𝑍 ) ∧ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) ) → ( ( abs ‘ ( 𝐵 − 0 ) ) < 𝑥 ↔ ( abs ‘ 𝐵 ) < 𝑥 ) )
14	13	ralbidva	⊢ ( ( 𝜑 ∧ 𝑗 ∈ 𝑍 ) → ( ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) ( abs ‘ ( 𝐵 − 0 ) ) < 𝑥 ↔ ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) ( abs ‘ 𝐵 ) < 𝑥 ) )
15	14	rexbidva	⊢ ( 𝜑 → ( ∃ 𝑗 ∈ 𝑍 ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) ( abs ‘ ( 𝐵 − 0 ) ) < 𝑥 ↔ ∃ 𝑗 ∈ 𝑍 ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) ( abs ‘ 𝐵 ) < 𝑥 ) )
16	15	ralbidv	⊢ ( 𝜑 → ( ∀ 𝑥 ∈ ℝ+ ∃ 𝑗 ∈ 𝑍 ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) ( abs ‘ ( 𝐵 − 0 ) ) < 𝑥 ↔ ∀ 𝑥 ∈ ℝ+ ∃ 𝑗 ∈ 𝑍 ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) ( abs ‘ 𝐵 ) < 𝑥 ) )
17	7 16	bitrd	⊢ ( 𝜑 → ( 𝐹 ⇝ 0 ↔ ∀ 𝑥 ∈ ℝ+ ∃ 𝑗 ∈ 𝑍 ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) ( abs ‘ 𝐵 ) < 𝑥 ) )