Theorem chjval

Description: Value of join in CH . (Contributed by NM, 9-Aug-2000) (New usage is discouraged.)

	Ref	Expression
Assertion	chjval	⊢ ( ( 𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ) → ( 𝐴 ∨ℋ 𝐵 ) = ( ⊥ ‘ ( ⊥ ‘ ( 𝐴 ∪ 𝐵 ) ) ) )

Proof

Step	Hyp	Ref	Expression
1		chsh	⊢ ( 𝐴 ∈ Cℋ → 𝐴 ∈ Sℋ )
2		chsh	⊢ ( 𝐵 ∈ Cℋ → 𝐵 ∈ Sℋ )
3		shjval	⊢ ( ( 𝐴 ∈ Sℋ ∧ 𝐵 ∈ Sℋ ) → ( 𝐴 ∨ℋ 𝐵 ) = ( ⊥ ‘ ( ⊥ ‘ ( 𝐴 ∪ 𝐵 ) ) ) )
4	1 2 3	syl2an	⊢ ( ( 𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ) → ( 𝐴 ∨ℋ 𝐵 ) = ( ⊥ ‘ ( ⊥ ‘ ( 𝐴 ∪ 𝐵 ) ) ) )