Step |
Hyp |
Ref |
Expression |
1 |
|
rpdp2cl.a |
⊢ 𝐴 ∈ ℕ0 |
2 |
|
rpdp2cl.b |
⊢ 𝐵 ∈ ℝ+ |
3 |
|
df-dp2 |
⊢ _ 𝐴 𝐵 = ( 𝐴 + ( 𝐵 / ; 1 0 ) ) |
4 |
1
|
nn0rei |
⊢ 𝐴 ∈ ℝ |
5 |
|
rpssre |
⊢ ℝ+ ⊆ ℝ |
6 |
|
10nn |
⊢ ; 1 0 ∈ ℕ |
7 |
|
nnrp |
⊢ ( ; 1 0 ∈ ℕ → ; 1 0 ∈ ℝ+ ) |
8 |
6 7
|
ax-mp |
⊢ ; 1 0 ∈ ℝ+ |
9 |
|
rpdivcl |
⊢ ( ( 𝐵 ∈ ℝ+ ∧ ; 1 0 ∈ ℝ+ ) → ( 𝐵 / ; 1 0 ) ∈ ℝ+ ) |
10 |
2 8 9
|
mp2an |
⊢ ( 𝐵 / ; 1 0 ) ∈ ℝ+ |
11 |
5 10
|
sselii |
⊢ ( 𝐵 / ; 1 0 ) ∈ ℝ |
12 |
|
readdcl |
⊢ ( ( 𝐴 ∈ ℝ ∧ ( 𝐵 / ; 1 0 ) ∈ ℝ ) → ( 𝐴 + ( 𝐵 / ; 1 0 ) ) ∈ ℝ ) |
13 |
4 11 12
|
mp2an |
⊢ ( 𝐴 + ( 𝐵 / ; 1 0 ) ) ∈ ℝ |
14 |
4 11
|
pm3.2i |
⊢ ( 𝐴 ∈ ℝ ∧ ( 𝐵 / ; 1 0 ) ∈ ℝ ) |
15 |
1
|
nn0ge0i |
⊢ 0 ≤ 𝐴 |
16 |
|
rpgt0 |
⊢ ( ( 𝐵 / ; 1 0 ) ∈ ℝ+ → 0 < ( 𝐵 / ; 1 0 ) ) |
17 |
10 16
|
ax-mp |
⊢ 0 < ( 𝐵 / ; 1 0 ) |
18 |
15 17
|
pm3.2i |
⊢ ( 0 ≤ 𝐴 ∧ 0 < ( 𝐵 / ; 1 0 ) ) |
19 |
|
addgegt0 |
⊢ ( ( ( 𝐴 ∈ ℝ ∧ ( 𝐵 / ; 1 0 ) ∈ ℝ ) ∧ ( 0 ≤ 𝐴 ∧ 0 < ( 𝐵 / ; 1 0 ) ) ) → 0 < ( 𝐴 + ( 𝐵 / ; 1 0 ) ) ) |
20 |
14 18 19
|
mp2an |
⊢ 0 < ( 𝐴 + ( 𝐵 / ; 1 0 ) ) |
21 |
|
elrp |
⊢ ( ( 𝐴 + ( 𝐵 / ; 1 0 ) ) ∈ ℝ+ ↔ ( ( 𝐴 + ( 𝐵 / ; 1 0 ) ) ∈ ℝ ∧ 0 < ( 𝐴 + ( 𝐵 / ; 1 0 ) ) ) ) |
22 |
13 20 21
|
mpbir2an |
⊢ ( 𝐴 + ( 𝐵 / ; 1 0 ) ) ∈ ℝ+ |
23 |
3 22
|
eqeltri |
⊢ _ 𝐴 𝐵 ∈ ℝ+ |