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Metamath Proof Explorer

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 ) ) )