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Metamath Proof Explorer

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 ) )