Step |
Hyp |
Ref |
Expression |
1 |
|
rpcn |
⊢ ( 𝐴 ∈ ℝ+ → 𝐴 ∈ ℂ ) |
2 |
1
|
mulid2d |
⊢ ( 𝐴 ∈ ℝ+ → ( 1 · 𝐴 ) = 𝐴 ) |
3 |
2
|
eqcomd |
⊢ ( 𝐴 ∈ ℝ+ → 𝐴 = ( 1 · 𝐴 ) ) |
4 |
3
|
adantr |
⊢ ( ( 𝐴 ∈ ℝ+ ∧ 𝐵 ∈ ℝ ) → 𝐴 = ( 1 · 𝐴 ) ) |
5 |
4
|
breq1d |
⊢ ( ( 𝐴 ∈ ℝ+ ∧ 𝐵 ∈ ℝ ) → ( 𝐴 ≤ 𝐵 ↔ ( 1 · 𝐴 ) ≤ 𝐵 ) ) |
6 |
|
1red |
⊢ ( ( 𝐴 ∈ ℝ+ ∧ 𝐵 ∈ ℝ ) → 1 ∈ ℝ ) |
7 |
|
simpr |
⊢ ( ( 𝐴 ∈ ℝ+ ∧ 𝐵 ∈ ℝ ) → 𝐵 ∈ ℝ ) |
8 |
|
rpregt0 |
⊢ ( 𝐴 ∈ ℝ+ → ( 𝐴 ∈ ℝ ∧ 0 < 𝐴 ) ) |
9 |
8
|
adantr |
⊢ ( ( 𝐴 ∈ ℝ+ ∧ 𝐵 ∈ ℝ ) → ( 𝐴 ∈ ℝ ∧ 0 < 𝐴 ) ) |
10 |
|
lemuldiv |
⊢ ( ( 1 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ ( 𝐴 ∈ ℝ ∧ 0 < 𝐴 ) ) → ( ( 1 · 𝐴 ) ≤ 𝐵 ↔ 1 ≤ ( 𝐵 / 𝐴 ) ) ) |
11 |
6 7 9 10
|
syl3anc |
⊢ ( ( 𝐴 ∈ ℝ+ ∧ 𝐵 ∈ ℝ ) → ( ( 1 · 𝐴 ) ≤ 𝐵 ↔ 1 ≤ ( 𝐵 / 𝐴 ) ) ) |
12 |
5 11
|
bitrd |
⊢ ( ( 𝐴 ∈ ℝ+ ∧ 𝐵 ∈ ℝ ) → ( 𝐴 ≤ 𝐵 ↔ 1 ≤ ( 𝐵 / 𝐴 ) ) ) |