Theorem clim0cf

Description: Express the predicate F converges to 0 . Similar to clim , but without the disjoint var constraint F k . (Contributed by Glauco Siliprandi, 11-Dec-2019)

	Ref	Expression
Hypotheses	clim0cf.nf	⊢ Ⅎ 𝑘 𝐹
	clim0cf.z	⊢ 𝑍 = ( ℤ≥ ‘ 𝑀 )
	clim0cf.m	⊢ ( 𝜑 → 𝑀 ∈ ℤ )
	clim0cf.f	⊢ ( 𝜑 → 𝐹 ∈ 𝑉 )
	clim0cf.fv	⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝑍 ) → ( 𝐹 ‘ 𝑘 ) = 𝐵 )
	clim0cf.b	⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝑍 ) → 𝐵 ∈ ℂ )
Assertion	clim0cf	⊢ ( 𝜑 → ( 𝐹 ⇝ 0 ↔ ∀ 𝑥 ∈ ℝ+ ∃ 𝑗 ∈ 𝑍 ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) ( abs ‘ 𝐵 ) < 𝑥 ) )

Proof

Step	Hyp	Ref	Expression
1		clim0cf.nf	⊢ Ⅎ 𝑘 𝐹
2		clim0cf.z	⊢ 𝑍 = ( ℤ≥ ‘ 𝑀 )
3		clim0cf.m	⊢ ( 𝜑 → 𝑀 ∈ ℤ )
4		clim0cf.f	⊢ ( 𝜑 → 𝐹 ∈ 𝑉 )
5		clim0cf.fv	⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝑍 ) → ( 𝐹 ‘ 𝑘 ) = 𝐵 )
6		clim0cf.b	⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝑍 ) → 𝐵 ∈ ℂ )
7		0cnd	⊢ ( 𝜑 → 0 ∈ ℂ )
8	1 2 3 4 5 7 6	clim2cf	⊢ ( 𝜑 → ( 𝐹 ⇝ 0 ↔ ∀ 𝑥 ∈ ℝ+ ∃ 𝑗 ∈ 𝑍 ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) ( abs ‘ ( 𝐵 − 0 ) ) < 𝑥 ) )
9	2	uztrn2	⊢ ( ( 𝑗 ∈ 𝑍 ∧ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) ) → 𝑘 ∈ 𝑍 )
10	6	subid1d	⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝑍 ) → ( 𝐵 − 0 ) = 𝐵 )
11	10	fveq2d	⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝑍 ) → ( abs ‘ ( 𝐵 − 0 ) ) = ( abs ‘ 𝐵 ) )
12	11	breq1d	⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝑍 ) → ( ( abs ‘ ( 𝐵 − 0 ) ) < 𝑥 ↔ ( abs ‘ 𝐵 ) < 𝑥 ) )
13	9 12	sylan2	⊢ ( ( 𝜑 ∧ ( 𝑗 ∈ 𝑍 ∧ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) ) ) → ( ( abs ‘ ( 𝐵 − 0 ) ) < 𝑥 ↔ ( abs ‘ 𝐵 ) < 𝑥 ) )
14	13	anassrs	⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ 𝑍 ) ∧ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) ) → ( ( abs ‘ ( 𝐵 − 0 ) ) < 𝑥 ↔ ( abs ‘ 𝐵 ) < 𝑥 ) )
15	14	ralbidva	⊢ ( ( 𝜑 ∧ 𝑗 ∈ 𝑍 ) → ( ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) ( abs ‘ ( 𝐵 − 0 ) ) < 𝑥 ↔ ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) ( abs ‘ 𝐵 ) < 𝑥 ) )
16	15	rexbidva	⊢ ( 𝜑 → ( ∃ 𝑗 ∈ 𝑍 ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) ( abs ‘ ( 𝐵 − 0 ) ) < 𝑥 ↔ ∃ 𝑗 ∈ 𝑍 ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) ( abs ‘ 𝐵 ) < 𝑥 ) )
17	16	ralbidv	⊢ ( 𝜑 → ( ∀ 𝑥 ∈ ℝ+ ∃ 𝑗 ∈ 𝑍 ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) ( abs ‘ ( 𝐵 − 0 ) ) < 𝑥 ↔ ∀ 𝑥 ∈ ℝ+ ∃ 𝑗 ∈ 𝑍 ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) ( abs ‘ 𝐵 ) < 𝑥 ) )
18	8 17	bitrd	⊢ ( 𝜑 → ( 𝐹 ⇝ 0 ↔ ∀ 𝑥 ∈ ℝ+ ∃ 𝑗 ∈ 𝑍 ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) ( abs ‘ 𝐵 ) < 𝑥 ) )