Theorem rlimneg

Description: Limit of the negative of a sequence. (Contributed by Mario Carneiro, 18-May-2016)

	Ref	Expression
Hypotheses	rlimneg.1	⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝐴 ) → 𝐵 ∈ 𝑉 )
	rlimneg.2	⊢ ( 𝜑 → ( 𝑘 ∈ 𝐴 ↦ 𝐵 ) ⇝𝑟 𝐶 )
Assertion	rlimneg	⊢ ( 𝜑 → ( 𝑘 ∈ 𝐴 ↦ - 𝐵 ) ⇝𝑟 - 𝐶 )

Proof

Step	Hyp	Ref	Expression
1		rlimneg.1	⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝐴 ) → 𝐵 ∈ 𝑉 )
2		rlimneg.2	⊢ ( 𝜑 → ( 𝑘 ∈ 𝐴 ↦ 𝐵 ) ⇝𝑟 𝐶 )
3		0cnd	⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝐴 ) → 0 ∈ ℂ )
4	1 2	rlimmptrcl	⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝐴 ) → 𝐵 ∈ ℂ )
5	1	ralrimiva	⊢ ( 𝜑 → ∀ 𝑘 ∈ 𝐴 𝐵 ∈ 𝑉 )
6		dmmptg	⊢ ( ∀ 𝑘 ∈ 𝐴 𝐵 ∈ 𝑉 → dom ( 𝑘 ∈ 𝐴 ↦ 𝐵 ) = 𝐴 )
7	5 6	syl	⊢ ( 𝜑 → dom ( 𝑘 ∈ 𝐴 ↦ 𝐵 ) = 𝐴 )
8		rlimss	⊢ ( ( 𝑘 ∈ 𝐴 ↦ 𝐵 ) ⇝𝑟 𝐶 → dom ( 𝑘 ∈ 𝐴 ↦ 𝐵 ) ⊆ ℝ )
9	2 8	syl	⊢ ( 𝜑 → dom ( 𝑘 ∈ 𝐴 ↦ 𝐵 ) ⊆ ℝ )
10	7 9	eqsstrrd	⊢ ( 𝜑 → 𝐴 ⊆ ℝ )
11		0cn	⊢ 0 ∈ ℂ
12		rlimconst	⊢ ( ( 𝐴 ⊆ ℝ ∧ 0 ∈ ℂ ) → ( 𝑘 ∈ 𝐴 ↦ 0 ) ⇝𝑟 0 )
13	10 11 12	sylancl	⊢ ( 𝜑 → ( 𝑘 ∈ 𝐴 ↦ 0 ) ⇝𝑟 0 )
14	3 4 13 2	rlimsub	⊢ ( 𝜑 → ( 𝑘 ∈ 𝐴 ↦ ( 0 − 𝐵 ) ) ⇝𝑟 ( 0 − 𝐶 ) )
15		df-neg	⊢ - 𝐵 = ( 0 − 𝐵 )
16	15	mpteq2i	⊢ ( 𝑘 ∈ 𝐴 ↦ - 𝐵 ) = ( 𝑘 ∈ 𝐴 ↦ ( 0 − 𝐵 ) )
17		df-neg	⊢ - 𝐶 = ( 0 − 𝐶 )
18	14 16 17	3brtr4g	⊢ ( 𝜑 → ( 𝑘 ∈ 𝐴 ↦ - 𝐵 ) ⇝𝑟 - 𝐶 )