Description: Prove that the closure of the decimal point is RR as we have defined it. See df-dp . (Contributed by David A. Wheeler, 15-May-2015)
Ref | Expression | ||
---|---|---|---|
Assertion | dpcl | ⊢ ( ( 𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℝ ) → ( 𝐴 . 𝐵 ) ∈ ℝ ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dpval | ⊢ ( ( 𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℝ ) → ( 𝐴 . 𝐵 ) = _ 𝐴 𝐵 ) | |
2 | nn0re | ⊢ ( 𝐴 ∈ ℕ0 → 𝐴 ∈ ℝ ) | |
3 | dp2cl | ⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ) → _ 𝐴 𝐵 ∈ ℝ ) | |
4 | 2 3 | sylan | ⊢ ( ( 𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℝ ) → _ 𝐴 𝐵 ∈ ℝ ) |
5 | 1 4 | eqeltrd | ⊢ ( ( 𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℝ ) → ( 𝐴 . 𝐵 ) ∈ ℝ ) |