Description: An element of a closed interval is more than or equal to its lower bound. (Contributed by Glauco Siliprandi, 26-Jun-2021)
Ref | Expression | ||
---|---|---|---|
Hypotheses | iccgelbd.1 | ⊢ ( 𝜑 → 𝐴 ∈ ℝ* ) | |
iccgelbd.2 | ⊢ ( 𝜑 → 𝐵 ∈ ℝ* ) | ||
iccgelbd.3 | ⊢ ( 𝜑 → 𝐶 ∈ ( 𝐴 [,] 𝐵 ) ) | ||
Assertion | iccgelbd | ⊢ ( 𝜑 → 𝐴 ≤ 𝐶 ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | iccgelbd.1 | ⊢ ( 𝜑 → 𝐴 ∈ ℝ* ) | |
2 | iccgelbd.2 | ⊢ ( 𝜑 → 𝐵 ∈ ℝ* ) | |
3 | iccgelbd.3 | ⊢ ( 𝜑 → 𝐶 ∈ ( 𝐴 [,] 𝐵 ) ) | |
4 | iccgelb | ⊢ ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ( 𝐴 [,] 𝐵 ) ) → 𝐴 ≤ 𝐶 ) | |
5 | 1 2 3 4 | syl3anc | ⊢ ( 𝜑 → 𝐴 ≤ 𝐶 ) |