Theorem chabs1i

Description: Hilbert lattice absorption law. From definition of lattice in Kalmbach p. 14. (Contributed by NM, 10-Jun-2004) (New usage is discouraged.)

	Ref	Expression
Hypotheses	chabs.1	⊢ 𝐴 ∈ Cℋ
	chabs.2	⊢ 𝐵 ∈ Cℋ
Assertion	chabs1i	⊢ ( 𝐴 ∨ℋ ( 𝐴 ∩ 𝐵 ) ) = 𝐴

Proof

Step	Hyp	Ref	Expression
1		chabs.1	⊢ 𝐴 ∈ Cℋ
2		chabs.2	⊢ 𝐵 ∈ Cℋ
3		chabs1	⊢ ( ( 𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ) → ( 𝐴 ∨ℋ ( 𝐴 ∩ 𝐵 ) ) = 𝐴 )
4	1 2 3	mp2an	⊢ ( 𝐴 ∨ℋ ( 𝐴 ∩ 𝐵 ) ) = 𝐴