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Metamath Proof Explorer

Theorem lemul2

Description: Multiplication of both sides of 'less than or equal to' by a positive number. (Contributed by NM, 16-Mar-2005)

Ref Expression
Assertion lemul2 ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ ( 𝐶 ∈ ℝ ∧ 0 < 𝐶 ) ) → ( 𝐴 ≤ 𝐵 ↔ ( 𝐶 · 𝐴 ) ≤ ( 𝐶 · 𝐵 ) ) )

Proof

Step Hyp Ref Expression
1 lemul1 ⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ ( 𝐶 ∈ ℝ ∧ 0 < 𝐶 ) ) → ( 𝐴 ≤ 𝐵 ↔ ( 𝐴 · 𝐶 ) ≤ ( 𝐵 · 𝐶 ) ) )
2 recn ⊢ ( 𝐴 ∈ ℝ → 𝐴 ∈ ℂ )
3 recn ⊢ ( 𝐶 ∈ ℝ → 𝐶 ∈ ℂ )
4 mulcom ⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐶 ∈ ℂ ) → ( 𝐴 · 𝐶 ) = ( 𝐶 · 𝐴 ) )
5 2 3 4 syl2an ⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐶 ∈ ℝ ) → ( 𝐴 · 𝐶 ) = ( 𝐶 · 𝐴 ) )
6 5 3adant2 ⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ ) → ( 𝐴 · 𝐶 ) = ( 𝐶 · 𝐴 ) )
7 recn ⊢ ( 𝐵 ∈ ℝ → 𝐵 ∈ ℂ )
8 mulcom ⊢ ( ( 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ ) → ( 𝐵 · 𝐶 ) = ( 𝐶 · 𝐵 ) )
9 7 3 8 syl2an ⊢ ( ( 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ ) → ( 𝐵 · 𝐶 ) = ( 𝐶 · 𝐵 ) )
10 9 3adant1 ⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ ) → ( 𝐵 · 𝐶 ) = ( 𝐶 · 𝐵 ) )
11 6 10 breq12d ⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ ) → ( ( 𝐴 · 𝐶 ) ≤ ( 𝐵 · 𝐶 ) ↔ ( 𝐶 · 𝐴 ) ≤ ( 𝐶 · 𝐵 ) ) )
12 11 3adant3r ⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ ( 𝐶 ∈ ℝ ∧ 0 < 𝐶 ) ) → ( ( 𝐴 · 𝐶 ) ≤ ( 𝐵 · 𝐶 ) ↔ ( 𝐶 · 𝐴 ) ≤ ( 𝐶 · 𝐵 ) ) )
13 1 12 bitrd ⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ ( 𝐶 ∈ ℝ ∧ 0 < 𝐶 ) ) → ( 𝐴 ≤ 𝐵 ↔ ( 𝐶 · 𝐴 ) ≤ ( 𝐶 · 𝐵 ) ) )