Theorem xlemul2a

Description: Extended real version of lemul2a . (Contributed by Mario Carneiro, 8-Sep-2015)

	Ref	Expression
Assertion	xlemul2a	⊢ ( ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ ( 𝐶 ∈ ℝ* ∧ 0 ≤ 𝐶 ) ) ∧ 𝐴 ≤ 𝐵 ) → ( 𝐶 ·e 𝐴 ) ≤ ( 𝐶 ·e 𝐵 ) )

Proof

Step	Hyp	Ref	Expression
1		xlemul1a	⊢ ( ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ ( 𝐶 ∈ ℝ* ∧ 0 ≤ 𝐶 ) ) ∧ 𝐴 ≤ 𝐵 ) → ( 𝐴 ·e 𝐶 ) ≤ ( 𝐵 ·e 𝐶 ) )
2		simpl1	⊢ ( ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ ( 𝐶 ∈ ℝ* ∧ 0 ≤ 𝐶 ) ) ∧ 𝐴 ≤ 𝐵 ) → 𝐴 ∈ ℝ* )
3		simpl3l	⊢ ( ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ ( 𝐶 ∈ ℝ* ∧ 0 ≤ 𝐶 ) ) ∧ 𝐴 ≤ 𝐵 ) → 𝐶 ∈ ℝ* )
4		xmulcom	⊢ ( ( 𝐴 ∈ ℝ* ∧ 𝐶 ∈ ℝ* ) → ( 𝐴 ·e 𝐶 ) = ( 𝐶 ·e 𝐴 ) )
5	2 3 4	syl2anc	⊢ ( ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ ( 𝐶 ∈ ℝ* ∧ 0 ≤ 𝐶 ) ) ∧ 𝐴 ≤ 𝐵 ) → ( 𝐴 ·e 𝐶 ) = ( 𝐶 ·e 𝐴 ) )
6		simpl2	⊢ ( ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ ( 𝐶 ∈ ℝ* ∧ 0 ≤ 𝐶 ) ) ∧ 𝐴 ≤ 𝐵 ) → 𝐵 ∈ ℝ* )
7		xmulcom	⊢ ( ( 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ* ) → ( 𝐵 ·e 𝐶 ) = ( 𝐶 ·e 𝐵 ) )
8	6 3 7	syl2anc	⊢ ( ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ ( 𝐶 ∈ ℝ* ∧ 0 ≤ 𝐶 ) ) ∧ 𝐴 ≤ 𝐵 ) → ( 𝐵 ·e 𝐶 ) = ( 𝐶 ·e 𝐵 ) )
9	1 5 8	3brtr3d	⊢ ( ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ ( 𝐶 ∈ ℝ* ∧ 0 ≤ 𝐶 ) ) ∧ 𝐴 ≤ 𝐵 ) → ( 𝐶 ·e 𝐴 ) ≤ ( 𝐶 ·e 𝐵 ) )