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Metamath Proof Explorer

Theorem dp2clq

Description: Closure for a decimal fraction. (Contributed by Thierry Arnoux, 16-Dec-2021)

Ref Expression
Hypotheses dp2clq.a ⊢ 𝐴 ∈ ℕ0
dp2clq.b ⊢ 𝐵 ∈ ℚ
Assertion dp2clq 𝐴 𝐵 ∈ ℚ

Proof

Step Hyp Ref Expression
1 dp2clq.a ⊢ 𝐴 ∈ ℕ0
2 dp2clq.b ⊢ 𝐵 ∈ ℚ
3 df-dp2 ⊢ 𝐴 𝐵 = ( 𝐴 + ( 𝐵 / 1 0 ) )
4 nn0ssq ⊢ ℕ0 ⊆ ℚ
5 4 1 sselii ⊢ 𝐴 ∈ ℚ
6 10nn0 ⊢ 1 0 ∈ ℕ0
7 4 6 sselii ⊢ 1 0 ∈ ℚ
8 0re ⊢ 0 ∈ ℝ
9 10pos ⊢ 0 < 1 0
10 8 9 gtneii ⊢ 1 0 ≠ 0
11 qdivcl ⊢ ( ( 𝐵 ∈ ℚ ∧ 1 0 ∈ ℚ ∧ 1 0 ≠ 0 ) → ( 𝐵 / 1 0 ) ∈ ℚ )
12 2 7 10 11 mp3an ⊢ ( 𝐵 / 1 0 ) ∈ ℚ
13 qaddcl ⊢ ( ( 𝐴 ∈ ℚ ∧ ( 𝐵 / 1 0 ) ∈ ℚ ) → ( 𝐴 + ( 𝐵 / 1 0 ) ) ∈ ℚ )
14 5 12 13 mp2an ⊢ ( 𝐴 + ( 𝐵 / 1 0 ) ) ∈ ℚ
15 3 14 eqeltri ⊢ 𝐴 𝐵 ∈ ℚ