Metamath Proof Explorer


Theorem ledivmul2

Description: 'Less than or equal to' relationship between division and multiplication. (Contributed by NM, 9-Dec-2005)

Ref Expression
Assertion ledivmul2 ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ ( 𝐶 ∈ ℝ ∧ 0 < 𝐶 ) ) → ( ( 𝐴 / 𝐶 ) ≤ 𝐵𝐴 ≤ ( 𝐵 · 𝐶 ) ) )

Proof

Step Hyp Ref Expression
1 ledivmul ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ ( 𝐶 ∈ ℝ ∧ 0 < 𝐶 ) ) → ( ( 𝐴 / 𝐶 ) ≤ 𝐵𝐴 ≤ ( 𝐶 · 𝐵 ) ) )
2 recn ( 𝐵 ∈ ℝ → 𝐵 ∈ ℂ )
3 recn ( 𝐶 ∈ ℝ → 𝐶 ∈ ℂ )
4 mulcom ( ( 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ ) → ( 𝐵 · 𝐶 ) = ( 𝐶 · 𝐵 ) )
5 2 3 4 syl2an ( ( 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ ) → ( 𝐵 · 𝐶 ) = ( 𝐶 · 𝐵 ) )
6 5 adantrr ( ( 𝐵 ∈ ℝ ∧ ( 𝐶 ∈ ℝ ∧ 0 < 𝐶 ) ) → ( 𝐵 · 𝐶 ) = ( 𝐶 · 𝐵 ) )
7 6 3adant1 ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ ( 𝐶 ∈ ℝ ∧ 0 < 𝐶 ) ) → ( 𝐵 · 𝐶 ) = ( 𝐶 · 𝐵 ) )
8 7 breq2d ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ ( 𝐶 ∈ ℝ ∧ 0 < 𝐶 ) ) → ( 𝐴 ≤ ( 𝐵 · 𝐶 ) ↔ 𝐴 ≤ ( 𝐶 · 𝐵 ) ) )
9 1 8 bitr4d ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ ( 𝐶 ∈ ℝ ∧ 0 < 𝐶 ) ) → ( ( 𝐴 / 𝐶 ) ≤ 𝐵𝐴 ≤ ( 𝐵 · 𝐶 ) ) )