Step |
Hyp |
Ref |
Expression |
1 |
|
0re |
⊢ 0 ∈ ℝ |
2 |
|
leadd2 |
⊢ ( ( 0 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 ∈ ℝ ) → ( 0 ≤ 𝐵 ↔ ( 𝐴 + 0 ) ≤ ( 𝐴 + 𝐵 ) ) ) |
3 |
1 2
|
mp3an1 |
⊢ ( ( 𝐵 ∈ ℝ ∧ 𝐴 ∈ ℝ ) → ( 0 ≤ 𝐵 ↔ ( 𝐴 + 0 ) ≤ ( 𝐴 + 𝐵 ) ) ) |
4 |
3
|
ancoms |
⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ) → ( 0 ≤ 𝐵 ↔ ( 𝐴 + 0 ) ≤ ( 𝐴 + 𝐵 ) ) ) |
5 |
|
recn |
⊢ ( 𝐴 ∈ ℝ → 𝐴 ∈ ℂ ) |
6 |
5
|
addid1d |
⊢ ( 𝐴 ∈ ℝ → ( 𝐴 + 0 ) = 𝐴 ) |
7 |
6
|
adantr |
⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ) → ( 𝐴 + 0 ) = 𝐴 ) |
8 |
7
|
breq1d |
⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ) → ( ( 𝐴 + 0 ) ≤ ( 𝐴 + 𝐵 ) ↔ 𝐴 ≤ ( 𝐴 + 𝐵 ) ) ) |
9 |
4 8
|
bitrd |
⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ) → ( 0 ≤ 𝐵 ↔ 𝐴 ≤ ( 𝐴 + 𝐵 ) ) ) |