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Metamath Proof Explorer

Theorem iimulcl

Description: The unit interval is closed under multiplication. (Contributed by Jeff Madsen, 2-Sep-2009)

Ref Expression
Assertion iimulcl ( ( 𝐴 ∈ ( 0 [,] 1 ) ∧ 𝐵 ∈ ( 0 [,] 1 ) ) → ( 𝐴 · 𝐵 ) ∈ ( 0 [,] 1 ) )

Proof

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 ) )