Description: Two ways of saying a number is less than the minimum of two others. (Contributed by NM, 1-Sep-2006)
Ref | Expression | ||
---|---|---|---|
Assertion | ltmin | ⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ ) → ( 𝐴 < if ( 𝐵 ≤ 𝐶 , 𝐵 , 𝐶 ) ↔ ( 𝐴 < 𝐵 ∧ 𝐴 < 𝐶 ) ) ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rexr | ⊢ ( 𝐴 ∈ ℝ → 𝐴 ∈ ℝ* ) | |
2 | rexr | ⊢ ( 𝐵 ∈ ℝ → 𝐵 ∈ ℝ* ) | |
3 | rexr | ⊢ ( 𝐶 ∈ ℝ → 𝐶 ∈ ℝ* ) | |
4 | xrltmin | ⊢ ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ* ) → ( 𝐴 < if ( 𝐵 ≤ 𝐶 , 𝐵 , 𝐶 ) ↔ ( 𝐴 < 𝐵 ∧ 𝐴 < 𝐶 ) ) ) | |
5 | 1 2 3 4 | syl3an | ⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ ) → ( 𝐴 < if ( 𝐵 ≤ 𝐶 , 𝐵 , 𝐶 ) ↔ ( 𝐴 < 𝐵 ∧ 𝐴 < 𝐶 ) ) ) |