Theorem xlemul2

Description: Extended real version of lemul2 . (Contributed by Mario Carneiro, 20-Aug-2015)

	Ref	Expression
Assertion	xlemul2	⊢ ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ+ ) → ( 𝐴 ≤ 𝐵 ↔ ( 𝐶 ·e 𝐴 ) ≤ ( 𝐶 ·e 𝐵 ) ) )

Proof

Step	Hyp	Ref	Expression
1		xlemul1	⊢ ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ+ ) → ( 𝐴 ≤ 𝐵 ↔ ( 𝐴 ·e 𝐶 ) ≤ ( 𝐵 ·e 𝐶 ) ) )
2		simp1	⊢ ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ+ ) → 𝐴 ∈ ℝ* )
3		rpxr	⊢ ( 𝐶 ∈ ℝ+ → 𝐶 ∈ ℝ* )
4	3	3ad2ant3	⊢ ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ+ ) → 𝐶 ∈ ℝ* )
5		xmulcom	⊢ ( ( 𝐴 ∈ ℝ* ∧ 𝐶 ∈ ℝ* ) → ( 𝐴 ·e 𝐶 ) = ( 𝐶 ·e 𝐴 ) )
6	2 4 5	syl2anc	⊢ ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ+ ) → ( 𝐴 ·e 𝐶 ) = ( 𝐶 ·e 𝐴 ) )
7		simp2	⊢ ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ+ ) → 𝐵 ∈ ℝ* )
8		xmulcom	⊢ ( ( 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ* ) → ( 𝐵 ·e 𝐶 ) = ( 𝐶 ·e 𝐵 ) )
9	7 4 8	syl2anc	⊢ ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ+ ) → ( 𝐵 ·e 𝐶 ) = ( 𝐶 ·e 𝐵 ) )
10	6 9	breq12d	⊢ ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ+ ) → ( ( 𝐴 ·e 𝐶 ) ≤ ( 𝐵 ·e 𝐶 ) ↔ ( 𝐶 ·e 𝐴 ) ≤ ( 𝐶 ·e 𝐵 ) ) )
11	1 10	bitrd	⊢ ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ+ ) → ( 𝐴 ≤ 𝐵 ↔ ( 𝐶 ·e 𝐴 ) ≤ ( 𝐶 ·e 𝐵 ) ) )