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Metamath Proof Explorer

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	$⊢ (A \in C_{ℋ} \land B \in C_{ℋ}) \to A \lor_{ℋ} (A \cap B) = A$

Proof

Step	Hyp	Ref	Expression
1		ssid	$⊢ A \subseteq A$
2		inss1	$⊢ A \cap B \subseteq A$
3	1 2	pm3.2i	$⊢ (A \subseteq A \land A \cap B \subseteq A)$
4		simpl	$⊢ (A \in C_{ℋ} \land B \in C_{ℋ}) \to A \in C_{ℋ}$
5		chincl	$⊢ (A \in C_{ℋ} \land B \in C_{ℋ}) \to A \cap B \in C_{ℋ}$
6		chlub	$⊢ (A \in C_{ℋ} \land A \cap B \in C_{ℋ} \land A \in C_{ℋ}) \to ((A \subseteq A \land A \cap B \subseteq A) \leftrightarrow A \lor_{ℋ} (A \cap B) \subseteq A)$
7	4 5 4 6	syl3anc	$⊢ (A \in C_{ℋ} \land B \in C_{ℋ}) \to ((A \subseteq A \land A \cap B \subseteq A) \leftrightarrow A \lor_{ℋ} (A \cap B) \subseteq A)$
8	3 7	mpbii	$⊢ (A \in C_{ℋ} \land B \in C_{ℋ}) \to A \lor_{ℋ} (A \cap B) \subseteq A$
9		chub1	$⊢ (A \in C_{ℋ} \land A \cap B \in C_{ℋ}) \to A \subseteq A \lor_{ℋ} (A \cap B)$
10	5 9	syldan	$⊢ (A \in C_{ℋ} \land B \in C_{ℋ}) \to A \subseteq A \lor_{ℋ} (A \cap B)$
11	8 10	eqssd	$⊢ (A \in C_{ℋ} \land B \in C_{ℋ}) \to A \lor_{ℋ} (A \cap B) = A$