Step |
Hyp |
Ref |
Expression |
1 |
|
remulcl |
⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ) → ( 𝐴 · 𝐵 ) ∈ ℝ ) |
2 |
1
|
3ad2antr1 |
⊢ ( ( 𝐴 ∈ ℝ ∧ ( 𝐵 ∈ ℝ ∧ 0 ≤ 𝐵 ∧ 𝐵 ≤ 1 ) ) → ( 𝐴 · 𝐵 ) ∈ ℝ ) |
3 |
2
|
3ad2antl1 |
⊢ ( ( ( 𝐴 ∈ ℝ ∧ 0 ≤ 𝐴 ∧ 𝐴 ≤ 1 ) ∧ ( 𝐵 ∈ ℝ ∧ 0 ≤ 𝐵 ∧ 𝐵 ≤ 1 ) ) → ( 𝐴 · 𝐵 ) ∈ ℝ ) |
4 |
|
mulge0 |
⊢ ( ( ( 𝐴 ∈ ℝ ∧ 0 ≤ 𝐴 ) ∧ ( 𝐵 ∈ ℝ ∧ 0 ≤ 𝐵 ) ) → 0 ≤ ( 𝐴 · 𝐵 ) ) |
5 |
4
|
3adantr3 |
⊢ ( ( ( 𝐴 ∈ ℝ ∧ 0 ≤ 𝐴 ) ∧ ( 𝐵 ∈ ℝ ∧ 0 ≤ 𝐵 ∧ 𝐵 ≤ 1 ) ) → 0 ≤ ( 𝐴 · 𝐵 ) ) |
6 |
5
|
3adantl3 |
⊢ ( ( ( 𝐴 ∈ ℝ ∧ 0 ≤ 𝐴 ∧ 𝐴 ≤ 1 ) ∧ ( 𝐵 ∈ ℝ ∧ 0 ≤ 𝐵 ∧ 𝐵 ≤ 1 ) ) → 0 ≤ ( 𝐴 · 𝐵 ) ) |
7 |
|
an6 |
⊢ ( ( ( 𝐴 ∈ ℝ ∧ 0 ≤ 𝐴 ∧ 𝐴 ≤ 1 ) ∧ ( 𝐵 ∈ ℝ ∧ 0 ≤ 𝐵 ∧ 𝐵 ≤ 1 ) ) ↔ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ) ∧ ( 0 ≤ 𝐴 ∧ 0 ≤ 𝐵 ) ∧ ( 𝐴 ≤ 1 ∧ 𝐵 ≤ 1 ) ) ) |
8 |
|
1re |
⊢ 1 ∈ ℝ |
9 |
|
lemul12a |
⊢ ( ( ( ( 𝐴 ∈ ℝ ∧ 0 ≤ 𝐴 ) ∧ 1 ∈ ℝ ) ∧ ( ( 𝐵 ∈ ℝ ∧ 0 ≤ 𝐵 ) ∧ 1 ∈ ℝ ) ) → ( ( 𝐴 ≤ 1 ∧ 𝐵 ≤ 1 ) → ( 𝐴 · 𝐵 ) ≤ ( 1 · 1 ) ) ) |
10 |
8 9
|
mpanr2 |
⊢ ( ( ( ( 𝐴 ∈ ℝ ∧ 0 ≤ 𝐴 ) ∧ 1 ∈ ℝ ) ∧ ( 𝐵 ∈ ℝ ∧ 0 ≤ 𝐵 ) ) → ( ( 𝐴 ≤ 1 ∧ 𝐵 ≤ 1 ) → ( 𝐴 · 𝐵 ) ≤ ( 1 · 1 ) ) ) |
11 |
8 10
|
mpanl2 |
⊢ ( ( ( 𝐴 ∈ ℝ ∧ 0 ≤ 𝐴 ) ∧ ( 𝐵 ∈ ℝ ∧ 0 ≤ 𝐵 ) ) → ( ( 𝐴 ≤ 1 ∧ 𝐵 ≤ 1 ) → ( 𝐴 · 𝐵 ) ≤ ( 1 · 1 ) ) ) |
12 |
11
|
an4s |
⊢ ( ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ) ∧ ( 0 ≤ 𝐴 ∧ 0 ≤ 𝐵 ) ) → ( ( 𝐴 ≤ 1 ∧ 𝐵 ≤ 1 ) → ( 𝐴 · 𝐵 ) ≤ ( 1 · 1 ) ) ) |
13 |
12
|
3impia |
⊢ ( ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ) ∧ ( 0 ≤ 𝐴 ∧ 0 ≤ 𝐵 ) ∧ ( 𝐴 ≤ 1 ∧ 𝐵 ≤ 1 ) ) → ( 𝐴 · 𝐵 ) ≤ ( 1 · 1 ) ) |
14 |
7 13
|
sylbi |
⊢ ( ( ( 𝐴 ∈ ℝ ∧ 0 ≤ 𝐴 ∧ 𝐴 ≤ 1 ) ∧ ( 𝐵 ∈ ℝ ∧ 0 ≤ 𝐵 ∧ 𝐵 ≤ 1 ) ) → ( 𝐴 · 𝐵 ) ≤ ( 1 · 1 ) ) |
15 |
|
1t1e1 |
⊢ ( 1 · 1 ) = 1 |
16 |
14 15
|
breqtrdi |
⊢ ( ( ( 𝐴 ∈ ℝ ∧ 0 ≤ 𝐴 ∧ 𝐴 ≤ 1 ) ∧ ( 𝐵 ∈ ℝ ∧ 0 ≤ 𝐵 ∧ 𝐵 ≤ 1 ) ) → ( 𝐴 · 𝐵 ) ≤ 1 ) |
17 |
3 6 16
|
3jca |
⊢ ( ( ( 𝐴 ∈ ℝ ∧ 0 ≤ 𝐴 ∧ 𝐴 ≤ 1 ) ∧ ( 𝐵 ∈ ℝ ∧ 0 ≤ 𝐵 ∧ 𝐵 ≤ 1 ) ) → ( ( 𝐴 · 𝐵 ) ∈ ℝ ∧ 0 ≤ ( 𝐴 · 𝐵 ) ∧ ( 𝐴 · 𝐵 ) ≤ 1 ) ) |
18 |
|
elicc01 |
⊢ ( 𝐴 ∈ ( 0 [,] 1 ) ↔ ( 𝐴 ∈ ℝ ∧ 0 ≤ 𝐴 ∧ 𝐴 ≤ 1 ) ) |
19 |
|
elicc01 |
⊢ ( 𝐵 ∈ ( 0 [,] 1 ) ↔ ( 𝐵 ∈ ℝ ∧ 0 ≤ 𝐵 ∧ 𝐵 ≤ 1 ) ) |
20 |
18 19
|
anbi12i |
⊢ ( ( 𝐴 ∈ ( 0 [,] 1 ) ∧ 𝐵 ∈ ( 0 [,] 1 ) ) ↔ ( ( 𝐴 ∈ ℝ ∧ 0 ≤ 𝐴 ∧ 𝐴 ≤ 1 ) ∧ ( 𝐵 ∈ ℝ ∧ 0 ≤ 𝐵 ∧ 𝐵 ≤ 1 ) ) ) |
21 |
|
elicc01 |
⊢ ( ( 𝐴 · 𝐵 ) ∈ ( 0 [,] 1 ) ↔ ( ( 𝐴 · 𝐵 ) ∈ ℝ ∧ 0 ≤ ( 𝐴 · 𝐵 ) ∧ ( 𝐴 · 𝐵 ) ≤ 1 ) ) |
22 |
17 20 21
|
3imtr4i |
⊢ ( ( 𝐴 ∈ ( 0 [,] 1 ) ∧ 𝐵 ∈ ( 0 [,] 1 ) ) → ( 𝐴 · 𝐵 ) ∈ ( 0 [,] 1 ) ) |