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Metamath Proof Explorer

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 ) ) )