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Metamath Proof Explorer

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 )