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Metamath Proof Explorer

Theorem dihmeetlem8N

Description: Lemma for isomorphism H of a lattice meet. TODO: shorter proof if we change .\/ order of ( X ./\ Y ) .\/ p here and down? (Contributed by NM, 6-Apr-2014) (New usage is discouraged.)

		Ref	Expression
	Hypotheses	dihmeetlem8.b	$⊢ B = {Base}_{K}$
		dihmeetlem8.l	$⊢ \underset{˙}{\leq} = \leq_{K}$
		dihmeetlem8.h	$⊢ H = LHyp (K)$
		dihmeetlem8.j	$⊢ \underset{˙}{\lor} = join (K)$
		dihmeetlem8.m	$⊢ \underset{˙}{\land} = meet (K)$
		dihmeetlem8.a	$⊢ A = Atoms (K)$
		dihmeetlem8.u	$⊢ U = DVecH (K) (W)$
		dihmeetlem8.s	$⊢ \underset{˙}{\oplus} = LSSum (U)$
		dihmeetlem8.i	$⊢ I = DIsoH (K) (W)$
	Assertion	dihmeetlem8N	$⊢ (((K \in HL \land W \in H) \land X \in B \land Y \in B) \land (p \in A \land \neg p \underset{˙}{\leq} W) \land (p \underset{˙}{\leq} X \land (X \underset{˙}{\land} Y) \underset{˙}{\leq} W)) \to I ((X \underset{˙}{\land} Y) \underset{˙}{\lor} p) = I (p) \underset{˙}{\oplus} I (X \underset{˙}{\land} Y)$

Proof

Step	Hyp	Ref	Expression
1		dihmeetlem8.b	$⊢ B = {Base}_{K}$
2		dihmeetlem8.l	$⊢ \underset{˙}{\leq} = \leq_{K}$
3		dihmeetlem8.h	$⊢ H = LHyp (K)$
4		dihmeetlem8.j	$⊢ \underset{˙}{\lor} = join (K)$
5		dihmeetlem8.m	$⊢ \underset{˙}{\land} = meet (K)$
6		dihmeetlem8.a	$⊢ A = Atoms (K)$
7		dihmeetlem8.u	$⊢ U = DVecH (K) (W)$
8		dihmeetlem8.s	$⊢ \underset{˙}{\oplus} = LSSum (U)$
9		dihmeetlem8.i	$⊢ I = DIsoH (K) (W)$
10	1 2 3 4 5 6 7 8 9	dihjatc1	$⊢ (((K \in HL \land W \in H) \land X \in B \land Y \in B) \land (p \in A \land \neg p \underset{˙}{\leq} W) \land (p \underset{˙}{\leq} X \land (X \underset{˙}{\land} Y) \underset{˙}{\leq} W)) \to I ((X \underset{˙}{\land} Y) \underset{˙}{\lor} p) = I (p) \underset{˙}{\oplus} I (X \underset{˙}{\land} Y)$