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Metamath Proof Explorer

Theorem ltdivmul2

Description: 'Less than' relationship between division and multiplication. (Contributed by NM, 24-Feb-2005)

Ref Expression
Assertion ltdivmul2 ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ ( 𝐶 ∈ ℝ ∧ 0 < 𝐶 ) ) → ( ( 𝐴 / 𝐶 ) < 𝐵 ↔ 𝐴 < ( 𝐵 · 𝐶 ) ) )

Proof

Step Hyp Ref Expression
1 ltdivmul ⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ ( 𝐶 ∈ ℝ ∧ 0 < 𝐶 ) ) → ( ( 𝐴 / 𝐶 ) < 𝐵 ↔ 𝐴 < ( 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 ) < 𝐵 ↔ 𝐴 < ( 𝐵 · 𝐶 ) ) )