Description: Closure of half-sum and half-difference. (Contributed by Paul Chapman, 12-Oct-2007)
Ref | Expression | ||
---|---|---|---|
Assertion | halfaddsubcl | ⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ) → ( ( ( 𝐴 + 𝐵 ) / 2 ) ∈ ℂ ∧ ( ( 𝐴 − 𝐵 ) / 2 ) ∈ ℂ ) ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | addcl | ⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ) → ( 𝐴 + 𝐵 ) ∈ ℂ ) | |
2 | halfcl | ⊢ ( ( 𝐴 + 𝐵 ) ∈ ℂ → ( ( 𝐴 + 𝐵 ) / 2 ) ∈ ℂ ) | |
3 | 1 2 | syl | ⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ) → ( ( 𝐴 + 𝐵 ) / 2 ) ∈ ℂ ) |
4 | subcl | ⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ) → ( 𝐴 − 𝐵 ) ∈ ℂ ) | |
5 | halfcl | ⊢ ( ( 𝐴 − 𝐵 ) ∈ ℂ → ( ( 𝐴 − 𝐵 ) / 2 ) ∈ ℂ ) | |
6 | 4 5 | syl | ⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ) → ( ( 𝐴 − 𝐵 ) / 2 ) ∈ ℂ ) |
7 | 3 6 | jca | ⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ) → ( ( ( 𝐴 + 𝐵 ) / 2 ) ∈ ℂ ∧ ( ( 𝐴 − 𝐵 ) / 2 ) ∈ ℂ ) ) |