Theorem radcnvlem3

Description: Lemma for radcnvlt1 , radcnvle . If X is a point closer to zero than Y and the power series converges at Y , then it converges at X . (Contributed by Mario Carneiro, 31-Mar-2015)

	Ref	Expression
Hypotheses	pser.g	⊢ 𝐺 = ( 𝑥 ∈ ℂ ↦ ( 𝑛 ∈ ℕ0 ↦ ( ( 𝐴 ‘ 𝑛 ) · ( 𝑥 ↑ 𝑛 ) ) ) )
	radcnv.a	⊢ ( 𝜑 → 𝐴 : ℕ0 ⟶ ℂ )
	psergf.x	⊢ ( 𝜑 → 𝑋 ∈ ℂ )
	radcnvlem2.y	⊢ ( 𝜑 → 𝑌 ∈ ℂ )
	radcnvlem2.a	⊢ ( 𝜑 → ( abs ‘ 𝑋 ) < ( abs ‘ 𝑌 ) )
	radcnvlem2.c	⊢ ( 𝜑 → seq 0 ( + , ( 𝐺 ‘ 𝑌 ) ) ∈ dom ⇝ )
Assertion	radcnvlem3	⊢ ( 𝜑 → seq 0 ( + , ( 𝐺 ‘ 𝑋 ) ) ∈ dom ⇝ )

Proof

Step	Hyp	Ref	Expression
1		pser.g	⊢ 𝐺 = ( 𝑥 ∈ ℂ ↦ ( 𝑛 ∈ ℕ0 ↦ ( ( 𝐴 ‘ 𝑛 ) · ( 𝑥 ↑ 𝑛 ) ) ) )
2		radcnv.a	⊢ ( 𝜑 → 𝐴 : ℕ0 ⟶ ℂ )
3		psergf.x	⊢ ( 𝜑 → 𝑋 ∈ ℂ )
4		radcnvlem2.y	⊢ ( 𝜑 → 𝑌 ∈ ℂ )
5		radcnvlem2.a	⊢ ( 𝜑 → ( abs ‘ 𝑋 ) < ( abs ‘ 𝑌 ) )
6		radcnvlem2.c	⊢ ( 𝜑 → seq 0 ( + , ( 𝐺 ‘ 𝑌 ) ) ∈ dom ⇝ )
7		nn0uz	⊢ ℕ0 = ( ℤ≥ ‘ 0 )
8		0zd	⊢ ( 𝜑 → 0 ∈ ℤ )
9	1 2 3	psergf	⊢ ( 𝜑 → ( 𝐺 ‘ 𝑋 ) : ℕ0 ⟶ ℂ )
10		fvco3	⊢ ( ( ( 𝐺 ‘ 𝑋 ) : ℕ0 ⟶ ℂ ∧ 𝑘 ∈ ℕ0 ) → ( ( abs ∘ ( 𝐺 ‘ 𝑋 ) ) ‘ 𝑘 ) = ( abs ‘ ( ( 𝐺 ‘ 𝑋 ) ‘ 𝑘 ) ) )
11	9 10	sylan	⊢ ( ( 𝜑 ∧ 𝑘 ∈ ℕ0 ) → ( ( abs ∘ ( 𝐺 ‘ 𝑋 ) ) ‘ 𝑘 ) = ( abs ‘ ( ( 𝐺 ‘ 𝑋 ) ‘ 𝑘 ) ) )
12	9	ffvelrnda	⊢ ( ( 𝜑 ∧ 𝑘 ∈ ℕ0 ) → ( ( 𝐺 ‘ 𝑋 ) ‘ 𝑘 ) ∈ ℂ )
13	1 2 3 4 5 6	radcnvlem2	⊢ ( 𝜑 → seq 0 ( + , ( abs ∘ ( 𝐺 ‘ 𝑋 ) ) ) ∈ dom ⇝ )
14	7 8 11 12 13	abscvgcvg	⊢ ( 𝜑 → seq 0 ( + , ( 𝐺 ‘ 𝑋 ) ) ∈ dom ⇝ )