Theorem chabs1

Description: Hilbert lattice absorption law. From definition of lattice in Kalmbach p. 14. (Contributed by NM, 15-Jun-2004) (New usage is discouraged.)

	Ref	Expression
Assertion	chabs1	⊢ ( ( 𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ) → ( 𝐴 ∨ℋ ( 𝐴 ∩ 𝐵 ) ) = 𝐴 )

Proof

Step	Hyp	Ref	Expression
1		ssid	⊢ 𝐴 ⊆ 𝐴
2		inss1	⊢ ( 𝐴 ∩ 𝐵 ) ⊆ 𝐴
3	1 2	pm3.2i	⊢ ( 𝐴 ⊆ 𝐴 ∧ ( 𝐴 ∩ 𝐵 ) ⊆ 𝐴 )
4		simpl	⊢ ( ( 𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ) → 𝐴 ∈ Cℋ )
5		chincl	⊢ ( ( 𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ) → ( 𝐴 ∩ 𝐵 ) ∈ Cℋ )
6		chlub	⊢ ( ( 𝐴 ∈ Cℋ ∧ ( 𝐴 ∩ 𝐵 ) ∈ Cℋ ∧ 𝐴 ∈ Cℋ ) → ( ( 𝐴 ⊆ 𝐴 ∧ ( 𝐴 ∩ 𝐵 ) ⊆ 𝐴 ) ↔ ( 𝐴 ∨ℋ ( 𝐴 ∩ 𝐵 ) ) ⊆ 𝐴 ) )
7	4 5 4 6	syl3anc	⊢ ( ( 𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ) → ( ( 𝐴 ⊆ 𝐴 ∧ ( 𝐴 ∩ 𝐵 ) ⊆ 𝐴 ) ↔ ( 𝐴 ∨ℋ ( 𝐴 ∩ 𝐵 ) ) ⊆ 𝐴 ) )
8	3 7	mpbii	⊢ ( ( 𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ) → ( 𝐴 ∨ℋ ( 𝐴 ∩ 𝐵 ) ) ⊆ 𝐴 )
9		chub1	⊢ ( ( 𝐴 ∈ Cℋ ∧ ( 𝐴 ∩ 𝐵 ) ∈ Cℋ ) → 𝐴 ⊆ ( 𝐴 ∨ℋ ( 𝐴 ∩ 𝐵 ) ) )
10	5 9	syldan	⊢ ( ( 𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ) → 𝐴 ⊆ ( 𝐴 ∨ℋ ( 𝐴 ∩ 𝐵 ) ) )
11	8 10	eqssd	⊢ ( ( 𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ) → ( 𝐴 ∨ℋ ( 𝐴 ∩ 𝐵 ) ) = 𝐴 )