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Metamath Proof Explorer

Theorem ltdivmul

Description: 'Less than' relationship between division and multiplication. (Contributed by NM, 18-Nov-2004)

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

Proof

Step Hyp Ref Expression
1 remulcl ⊢ ( ( 𝐶 ∈ ℝ ∧ 𝐵 ∈ ℝ ) → ( 𝐶 · 𝐵 ) ∈ ℝ )
2 1 ancoms ⊢ ( ( 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ ) → ( 𝐶 · 𝐵 ) ∈ ℝ )
3 2 adantrr ⊢ ( ( 𝐵 ∈ ℝ ∧ ( 𝐶 ∈ ℝ ∧ 0 < 𝐶 ) ) → ( 𝐶 · 𝐵 ) ∈ ℝ )
4 3 3adant1 ⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ ( 𝐶 ∈ ℝ ∧ 0 < 𝐶 ) ) → ( 𝐶 · 𝐵 ) ∈ ℝ )
5 ltdiv1 ⊢ ( ( 𝐴 ∈ ℝ ∧ ( 𝐶 · 𝐵 ) ∈ ℝ ∧ ( 𝐶 ∈ ℝ ∧ 0 < 𝐶 ) ) → ( 𝐴 < ( 𝐶 · 𝐵 ) ↔ ( 𝐴 / 𝐶 ) < ( ( 𝐶 · 𝐵 ) / 𝐶 ) ) )
6 4 5 syld3an2 ⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ ( 𝐶 ∈ ℝ ∧ 0 < 𝐶 ) ) → ( 𝐴 < ( 𝐶 · 𝐵 ) ↔ ( 𝐴 / 𝐶 ) < ( ( 𝐶 · 𝐵 ) / 𝐶 ) ) )
7 recn ⊢ ( 𝐵 ∈ ℝ → 𝐵 ∈ ℂ )
8 7 adantr ⊢ ( ( 𝐵 ∈ ℝ ∧ ( 𝐶 ∈ ℝ ∧ 0 < 𝐶 ) ) → 𝐵 ∈ ℂ )
9 recn ⊢ ( 𝐶 ∈ ℝ → 𝐶 ∈ ℂ )
10 9 ad2antrl ⊢ ( ( 𝐵 ∈ ℝ ∧ ( 𝐶 ∈ ℝ ∧ 0 < 𝐶 ) ) → 𝐶 ∈ ℂ )
11 gt0ne0 ⊢ ( ( 𝐶 ∈ ℝ ∧ 0 < 𝐶 ) → 𝐶 ≠ 0 )
12 11 adantl ⊢ ( ( 𝐵 ∈ ℝ ∧ ( 𝐶 ∈ ℝ ∧ 0 < 𝐶 ) ) → 𝐶 ≠ 0 )
13 8 10 12 divcan3d ⊢ ( ( 𝐵 ∈ ℝ ∧ ( 𝐶 ∈ ℝ ∧ 0 < 𝐶 ) ) → ( ( 𝐶 · 𝐵 ) / 𝐶 ) = 𝐵 )
14 13 3adant1 ⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ ( 𝐶 ∈ ℝ ∧ 0 < 𝐶 ) ) → ( ( 𝐶 · 𝐵 ) / 𝐶 ) = 𝐵 )
15 14 breq2d ⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ ( 𝐶 ∈ ℝ ∧ 0 < 𝐶 ) ) → ( ( 𝐴 / 𝐶 ) < ( ( 𝐶 · 𝐵 ) / 𝐶 ) ↔ ( 𝐴 / 𝐶 ) < 𝐵 ) )
16 6 15 bitr2d ⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ ( 𝐶 ∈ ℝ ∧ 0 < 𝐶 ) ) → ( ( 𝐴 / 𝐶 ) < 𝐵 ↔ 𝐴 < ( 𝐶 · 𝐵 ) ) )