Theorem clim2c

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

	Ref	Expression
Hypotheses	clim2.1	⊢ 𝑍 = ( ℤ≥ ‘ 𝑀 )
	clim2.2	⊢ ( 𝜑 → 𝑀 ∈ ℤ )
	clim2.3	⊢ ( 𝜑 → 𝐹 ∈ 𝑉 )
	clim2.4	⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝑍 ) → ( 𝐹 ‘ 𝑘 ) = 𝐵 )
	clim2c.5	⊢ ( 𝜑 → 𝐴 ∈ ℂ )
	clim2c.6	⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝑍 ) → 𝐵 ∈ ℂ )
Assertion	clim2c	⊢ ( 𝜑 → ( 𝐹 ⇝ 𝐴 ↔ ∀ 𝑥 ∈ ℝ+ ∃ 𝑗 ∈ 𝑍 ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) ( abs ‘ ( 𝐵 − 𝐴 ) ) < 𝑥 ) )

Proof

Step	Hyp	Ref	Expression
1		clim2.1	⊢ 𝑍 = ( ℤ≥ ‘ 𝑀 )
2		clim2.2	⊢ ( 𝜑 → 𝑀 ∈ ℤ )
3		clim2.3	⊢ ( 𝜑 → 𝐹 ∈ 𝑉 )
4		clim2.4	⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝑍 ) → ( 𝐹 ‘ 𝑘 ) = 𝐵 )
5		clim2c.5	⊢ ( 𝜑 → 𝐴 ∈ ℂ )
6		clim2c.6	⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝑍 ) → 𝐵 ∈ ℂ )
7	5	biantrurd	⊢ ( 𝜑 → ( ∀ 𝑥 ∈ ℝ+ ∃ 𝑗 ∈ 𝑍 ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) ( 𝐵 ∈ ℂ ∧ ( abs ‘ ( 𝐵 − 𝐴 ) ) < 𝑥 ) ↔ ( 𝐴 ∈ ℂ ∧ ∀ 𝑥 ∈ ℝ+ ∃ 𝑗 ∈ 𝑍 ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) ( 𝐵 ∈ ℂ ∧ ( abs ‘ ( 𝐵 − 𝐴 ) ) < 𝑥 ) ) ) )
8	1	uztrn2	⊢ ( ( 𝑗 ∈ 𝑍 ∧ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) ) → 𝑘 ∈ 𝑍 )
9	6	biantrurd	⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝑍 ) → ( ( abs ‘ ( 𝐵 − 𝐴 ) ) < 𝑥 ↔ ( 𝐵 ∈ ℂ ∧ ( abs ‘ ( 𝐵 − 𝐴 ) ) < 𝑥 ) ) )
10	8 9	sylan2	⊢ ( ( 𝜑 ∧ ( 𝑗 ∈ 𝑍 ∧ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) ) ) → ( ( abs ‘ ( 𝐵 − 𝐴 ) ) < 𝑥 ↔ ( 𝐵 ∈ ℂ ∧ ( abs ‘ ( 𝐵 − 𝐴 ) ) < 𝑥 ) ) )
11	10	anassrs	⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ 𝑍 ) ∧ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) ) → ( ( abs ‘ ( 𝐵 − 𝐴 ) ) < 𝑥 ↔ ( 𝐵 ∈ ℂ ∧ ( abs ‘ ( 𝐵 − 𝐴 ) ) < 𝑥 ) ) )
12	11	ralbidva	⊢ ( ( 𝜑 ∧ 𝑗 ∈ 𝑍 ) → ( ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) ( abs ‘ ( 𝐵 − 𝐴 ) ) < 𝑥 ↔ ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) ( 𝐵 ∈ ℂ ∧ ( abs ‘ ( 𝐵 − 𝐴 ) ) < 𝑥 ) ) )
13	12	rexbidva	⊢ ( 𝜑 → ( ∃ 𝑗 ∈ 𝑍 ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) ( abs ‘ ( 𝐵 − 𝐴 ) ) < 𝑥 ↔ ∃ 𝑗 ∈ 𝑍 ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) ( 𝐵 ∈ ℂ ∧ ( abs ‘ ( 𝐵 − 𝐴 ) ) < 𝑥 ) ) )
14	13	ralbidv	⊢ ( 𝜑 → ( ∀ 𝑥 ∈ ℝ+ ∃ 𝑗 ∈ 𝑍 ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) ( abs ‘ ( 𝐵 − 𝐴 ) ) < 𝑥 ↔ ∀ 𝑥 ∈ ℝ+ ∃ 𝑗 ∈ 𝑍 ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) ( 𝐵 ∈ ℂ ∧ ( abs ‘ ( 𝐵 − 𝐴 ) ) < 𝑥 ) ) )
15	1 2 3 4	clim2	⊢ ( 𝜑 → ( 𝐹 ⇝ 𝐴 ↔ ( 𝐴 ∈ ℂ ∧ ∀ 𝑥 ∈ ℝ+ ∃ 𝑗 ∈ 𝑍 ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) ( 𝐵 ∈ ℂ ∧ ( abs ‘ ( 𝐵 − 𝐴 ) ) < 𝑥 ) ) ) )
16	7 14 15	3bitr4rd	⊢ ( 𝜑 → ( 𝐹 ⇝ 𝐴 ↔ ∀ 𝑥 ∈ ℝ+ ∃ 𝑗 ∈ 𝑍 ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) ( abs ‘ ( 𝐵 − 𝐴 ) ) < 𝑥 ) )