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Metamath Proof Explorer

Theorem chabs2i

Description: Hilbert lattice absorption law. From definition of lattice in Kalmbach p. 14. (Contributed by NM, 16-Jun-2004) (New usage is discouraged.)

	Ref	Expression
Hypotheses	chabs.1	⊢ 𝐴 ∈ C_ℋ
	chabs.2	⊢ 𝐵 ∈ C_ℋ
Assertion	chabs2i	⊢ ( 𝐴 ∩ ( 𝐴 ∨_ℋ 𝐵 ) ) = 𝐴

Proof

Step	Hyp	Ref	Expression
1		chabs.1	⊢ 𝐴 ∈ C_ℋ
2		chabs.2	⊢ 𝐵 ∈ C_ℋ
3		chabs2	⊢ ( ( 𝐴 ∈ C_ℋ ∧ 𝐵 ∈ C_ℋ ) → ( 𝐴 ∩ ( 𝐴 ∨_ℋ 𝐵 ) ) = 𝐴 )
4	1 2 3	mp2an	⊢ ( 𝐴 ∩ ( 𝐴 ∨_ℋ 𝐵 ) ) = 𝐴