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Metamath Proof Explorer

Theorem ltmul2

Description: Multiplication of both sides of 'less than' by a positive number. Theorem I.19 of Apostol p. 20. (Contributed by NM, 13-Feb-2005)

Ref Expression
Assertion ltmul2 ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ ( 𝐶 ∈ ℝ ∧ 0 < 𝐶 ) ) → ( 𝐴 < 𝐵 ↔ ( 𝐶 · 𝐴 ) < ( 𝐶 · 𝐵 ) ) )

Proof

Step Hyp Ref Expression
1 ltmul1 ⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ ( 𝐶 ∈ ℝ ∧ 0 < 𝐶 ) ) → ( 𝐴 < 𝐵 ↔ ( 𝐴 · 𝐶 ) < ( 𝐵 · 𝐶 ) ) )
2 recn ⊢ ( 𝐶 ∈ ℝ → 𝐶 ∈ ℂ )
3 recn ⊢ ( 𝐴 ∈ ℝ → 𝐴 ∈ ℂ )
4 mulcom ⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐶 ∈ ℂ ) → ( 𝐴 · 𝐶 ) = ( 𝐶 · 𝐴 ) )
5 3 4 sylan ⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐶 ∈ ℂ ) → ( 𝐴 · 𝐶 ) = ( 𝐶 · 𝐴 ) )
6 5 3adant2 ⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℂ ) → ( 𝐴 · 𝐶 ) = ( 𝐶 · 𝐴 ) )
7 recn ⊢ ( 𝐵 ∈ ℝ → 𝐵 ∈ ℂ )
8 mulcom ⊢ ( ( 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ ) → ( 𝐵 · 𝐶 ) = ( 𝐶 · 𝐵 ) )
9 7 8 sylan ⊢ ( ( 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℂ ) → ( 𝐵 · 𝐶 ) = ( 𝐶 · 𝐵 ) )
10 9 3adant1 ⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℂ ) → ( 𝐵 · 𝐶 ) = ( 𝐶 · 𝐵 ) )
11 6 10 breq12d ⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℂ ) → ( ( 𝐴 · 𝐶 ) < ( 𝐵 · 𝐶 ) ↔ ( 𝐶 · 𝐴 ) < ( 𝐶 · 𝐵 ) ) )
12 2 11 syl3an3 ⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ ) → ( ( 𝐴 · 𝐶 ) < ( 𝐵 · 𝐶 ) ↔ ( 𝐶 · 𝐴 ) < ( 𝐶 · 𝐵 ) ) )
13 12 3adant3r ⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ ( 𝐶 ∈ ℝ ∧ 0 < 𝐶 ) ) → ( ( 𝐴 · 𝐶 ) < ( 𝐵 · 𝐶 ) ↔ ( 𝐶 · 𝐴 ) < ( 𝐶 · 𝐵 ) ) )
14 1 13 bitrd ⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ ( 𝐶 ∈ ℝ ∧ 0 < 𝐶 ) ) → ( 𝐴 < 𝐵 ↔ ( 𝐶 · 𝐴 ) < ( 𝐶 · 𝐵 ) ) )