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Metamath Proof Explorer

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 ) ) )