Theorem xltmul1

Description: Extended real version of ltmul1 . (Contributed by Mario Carneiro, 20-Aug-2015)

	Ref	Expression
Assertion	xltmul1	⊢ ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ+ ) → ( 𝐴 < 𝐵 ↔ ( 𝐴 ·e 𝐶 ) < ( 𝐵 ·e 𝐶 ) ) )

Proof

Step	Hyp	Ref	Expression
1		xlemul1	⊢ ( ( 𝐵 ∈ ℝ* ∧ 𝐴 ∈ ℝ* ∧ 𝐶 ∈ ℝ+ ) → ( 𝐵 ≤ 𝐴 ↔ ( 𝐵 ·e 𝐶 ) ≤ ( 𝐴 ·e 𝐶 ) ) )
2	1	3com12	⊢ ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ+ ) → ( 𝐵 ≤ 𝐴 ↔ ( 𝐵 ·e 𝐶 ) ≤ ( 𝐴 ·e 𝐶 ) ) )
3	2	notbid	⊢ ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ+ ) → ( ¬ 𝐵 ≤ 𝐴 ↔ ¬ ( 𝐵 ·e 𝐶 ) ≤ ( 𝐴 ·e 𝐶 ) ) )
4		xrltnle	⊢ ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ) → ( 𝐴 < 𝐵 ↔ ¬ 𝐵 ≤ 𝐴 ) )
5	4	3adant3	⊢ ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ+ ) → ( 𝐴 < 𝐵 ↔ ¬ 𝐵 ≤ 𝐴 ) )
6		simp1	⊢ ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ+ ) → 𝐴 ∈ ℝ* )
7		rpxr	⊢ ( 𝐶 ∈ ℝ+ → 𝐶 ∈ ℝ* )
8	7	3ad2ant3	⊢ ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ+ ) → 𝐶 ∈ ℝ* )
9		xmulcl	⊢ ( ( 𝐴 ∈ ℝ* ∧ 𝐶 ∈ ℝ* ) → ( 𝐴 ·e 𝐶 ) ∈ ℝ* )
10	6 8 9	syl2anc	⊢ ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ+ ) → ( 𝐴 ·e 𝐶 ) ∈ ℝ* )
11		simp2	⊢ ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ+ ) → 𝐵 ∈ ℝ* )
12		xmulcl	⊢ ( ( 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ* ) → ( 𝐵 ·e 𝐶 ) ∈ ℝ* )
13	11 8 12	syl2anc	⊢ ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ+ ) → ( 𝐵 ·e 𝐶 ) ∈ ℝ* )
14		xrltnle	⊢ ( ( ( 𝐴 ·e 𝐶 ) ∈ ℝ* ∧ ( 𝐵 ·e 𝐶 ) ∈ ℝ* ) → ( ( 𝐴 ·e 𝐶 ) < ( 𝐵 ·e 𝐶 ) ↔ ¬ ( 𝐵 ·e 𝐶 ) ≤ ( 𝐴 ·e 𝐶 ) ) )
15	10 13 14	syl2anc	⊢ ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ+ ) → ( ( 𝐴 ·e 𝐶 ) < ( 𝐵 ·e 𝐶 ) ↔ ¬ ( 𝐵 ·e 𝐶 ) ≤ ( 𝐴 ·e 𝐶 ) ) )
16	3 5 15	3bitr4d	⊢ ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ+ ) → ( 𝐴 < 𝐵 ↔ ( 𝐴 ·e 𝐶 ) < ( 𝐵 ·e 𝐶 ) ) )