Theorem xltmul2

Description: Extended real version of ltmul2 . (Contributed by Mario Carneiro, 8-Sep-2015)

	Ref	Expression
Assertion	xltmul2	⊢ ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ+ ) → ( 𝐴 < 𝐵 ↔ ( 𝐶 ·e 𝐴 ) < ( 𝐶 ·e 𝐵 ) ) )

Proof

Step	Hyp	Ref	Expression
1		xltmul1	⊢ ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ+ ) → ( 𝐴 < 𝐵 ↔ ( 𝐴 ·e 𝐶 ) < ( 𝐵 ·e 𝐶 ) ) )
2		rpxr	⊢ ( 𝐶 ∈ ℝ+ → 𝐶 ∈ ℝ* )
3		xmulcom	⊢ ( ( 𝐴 ∈ ℝ* ∧ 𝐶 ∈ ℝ* ) → ( 𝐴 ·e 𝐶 ) = ( 𝐶 ·e 𝐴 ) )
4	3	3adant2	⊢ ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ* ) → ( 𝐴 ·e 𝐶 ) = ( 𝐶 ·e 𝐴 ) )
5		xmulcom	⊢ ( ( 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ* ) → ( 𝐵 ·e 𝐶 ) = ( 𝐶 ·e 𝐵 ) )
6	5	3adant1	⊢ ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ* ) → ( 𝐵 ·e 𝐶 ) = ( 𝐶 ·e 𝐵 ) )
7	4 6	breq12d	⊢ ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ* ) → ( ( 𝐴 ·e 𝐶 ) < ( 𝐵 ·e 𝐶 ) ↔ ( 𝐶 ·e 𝐴 ) < ( 𝐶 ·e 𝐵 ) ) )
8	2 7	syl3an3	⊢ ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ+ ) → ( ( 𝐴 ·e 𝐶 ) < ( 𝐵 ·e 𝐶 ) ↔ ( 𝐶 ·e 𝐴 ) < ( 𝐶 ·e 𝐵 ) ) )
9	1 8	bitrd	⊢ ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ+ ) → ( 𝐴 < 𝐵 ↔ ( 𝐶 ·e 𝐴 ) < ( 𝐶 ·e 𝐵 ) ) )