lhpmcvr6N

Description: Specialization of lhpmcvr2 . (Contributed by NM, 6-Apr-2014) (New usage is discouraged.)

	Ref	Expression
Hypotheses	lhpmcvr2.b	$⊢ B = {Base}_{K}$
	lhpmcvr2.l	$⊢ \underset{˙}{\leq} = \leq_{K}$
	lhpmcvr2.j	$⊢ \underset{˙}{\lor} = join (K)$
	lhpmcvr2.m	$⊢ \underset{˙}{\land} = meet (K)$
	lhpmcvr2.a	$⊢ A = Atoms (K)$
	lhpmcvr2.h	$⊢ H = LHyp (K)$
Assertion	lhpmcvr6N	$⊢ ((K \in HL \land W \in H) \land (X \in B \land \neg X \underset{˙}{\leq} W) \land (Y \in B \land (X \underset{˙}{\land} Y) \underset{˙}{\leq} W)) \to \exists p \in A (\neg p \underset{˙}{\leq} W \land \neg p \underset{˙}{\leq} Y \land p \underset{˙}{\leq} X)$

Proof

Step	Hyp	Ref	Expression
1		lhpmcvr2.b	$⊢ B = {Base}_{K}$
2		lhpmcvr2.l	$⊢ \underset{˙}{\leq} = \leq_{K}$
3		lhpmcvr2.j	$⊢ \underset{˙}{\lor} = join (K)$
4		lhpmcvr2.m	$⊢ \underset{˙}{\land} = meet (K)$
5		lhpmcvr2.a	$⊢ A = Atoms (K)$
6		lhpmcvr2.h	$⊢ H = LHyp (K)$
7	1 2 3 4 5 6	lhpmcvr5N	$⊢ ((K \in HL \land W \in H) \land (X \in B \land \neg X \underset{˙}{\leq} W) \land (Y \in B \land (X \underset{˙}{\land} Y) \underset{˙}{\leq} W)) \to \exists p \in A (\neg p \underset{˙}{\leq} W \land \neg p \underset{˙}{\leq} Y \land p \underset{˙}{\lor} (X \underset{˙}{\land} W) = X)$
8		simp31	$⊢ (((K \in HL \land W \in H) \land (X \in B \land \neg X \underset{˙}{\leq} W) \land (Y \in B \land (X \underset{˙}{\land} Y) \underset{˙}{\leq} W)) \land p \in A \land (\neg p \underset{˙}{\leq} W \land \neg p \underset{˙}{\leq} Y \land p \underset{˙}{\lor} (X \underset{˙}{\land} W) = X)) \to \neg p \underset{˙}{\leq} W$
9		simp32	$⊢ (((K \in HL \land W \in H) \land (X \in B \land \neg X \underset{˙}{\leq} W) \land (Y \in B \land (X \underset{˙}{\land} Y) \underset{˙}{\leq} W)) \land p \in A \land (\neg p \underset{˙}{\leq} W \land \neg p \underset{˙}{\leq} Y \land p \underset{˙}{\lor} (X \underset{˙}{\land} W) = X)) \to \neg p \underset{˙}{\leq} Y$
10		simp11l	$⊢ (((K \in HL \land W \in H) \land (X \in B \land \neg X \underset{˙}{\leq} W) \land (Y \in B \land (X \underset{˙}{\land} Y) \underset{˙}{\leq} W)) \land p \in A \land (\neg p \underset{˙}{\leq} W \land \neg p \underset{˙}{\leq} Y \land p \underset{˙}{\lor} (X \underset{˙}{\land} W) = X)) \to K \in HL$
11	10	hllatd	$⊢ (((K \in HL \land W \in H) \land (X \in B \land \neg X \underset{˙}{\leq} W) \land (Y \in B \land (X \underset{˙}{\land} Y) \underset{˙}{\leq} W)) \land p \in A \land (\neg p \underset{˙}{\leq} W \land \neg p \underset{˙}{\leq} Y \land p \underset{˙}{\lor} (X \underset{˙}{\land} W) = X)) \to K \in Lat$
12	1 5	atbase	$⊢ p \in A \to p \in B$
13	12	3ad2ant2	$⊢ (((K \in HL \land W \in H) \land (X \in B \land \neg X \underset{˙}{\leq} W) \land (Y \in B \land (X \underset{˙}{\land} Y) \underset{˙}{\leq} W)) \land p \in A \land (\neg p \underset{˙}{\leq} W \land \neg p \underset{˙}{\leq} Y \land p \underset{˙}{\lor} (X \underset{˙}{\land} W) = X)) \to p \in B$
14		simp12l	$⊢ (((K \in HL \land W \in H) \land (X \in B \land \neg X \underset{˙}{\leq} W) \land (Y \in B \land (X \underset{˙}{\land} Y) \underset{˙}{\leq} W)) \land p \in A \land (\neg p \underset{˙}{\leq} W \land \neg p \underset{˙}{\leq} Y \land p \underset{˙}{\lor} (X \underset{˙}{\land} W) = X)) \to X \in B$
15		simp11r	$⊢ (((K \in HL \land W \in H) \land (X \in B \land \neg X \underset{˙}{\leq} W) \land (Y \in B \land (X \underset{˙}{\land} Y) \underset{˙}{\leq} W)) \land p \in A \land (\neg p \underset{˙}{\leq} W \land \neg p \underset{˙}{\leq} Y \land p \underset{˙}{\lor} (X \underset{˙}{\land} W) = X)) \to W \in H$
16	1 6	lhpbase	$⊢ W \in H \to W \in B$
17	15 16	syl	$⊢ (((K \in HL \land W \in H) \land (X \in B \land \neg X \underset{˙}{\leq} W) \land (Y \in B \land (X \underset{˙}{\land} Y) \underset{˙}{\leq} W)) \land p \in A \land (\neg p \underset{˙}{\leq} W \land \neg p \underset{˙}{\leq} Y \land p \underset{˙}{\lor} (X \underset{˙}{\land} W) = X)) \to W \in B$
18	1 4	latmcl	$⊢ (K \in Lat \land X \in B \land W \in B) \to X \underset{˙}{\land} W \in B$
19	11 14 17 18	syl3anc	$⊢ (((K \in HL \land W \in H) \land (X \in B \land \neg X \underset{˙}{\leq} W) \land (Y \in B \land (X \underset{˙}{\land} Y) \underset{˙}{\leq} W)) \land p \in A \land (\neg p \underset{˙}{\leq} W \land \neg p \underset{˙}{\leq} Y \land p \underset{˙}{\lor} (X \underset{˙}{\land} W) = X)) \to X \underset{˙}{\land} W \in B$
20	1 2 3	latlej1	$⊢ (K \in Lat \land p \in B \land X \underset{˙}{\land} W \in B) \to p \underset{˙}{\leq} (p \underset{˙}{\lor} (X \underset{˙}{\land} W))$
21	11 13 19 20	syl3anc	$⊢ (((K \in HL \land W \in H) \land (X \in B \land \neg X \underset{˙}{\leq} W) \land (Y \in B \land (X \underset{˙}{\land} Y) \underset{˙}{\leq} W)) \land p \in A \land (\neg p \underset{˙}{\leq} W \land \neg p \underset{˙}{\leq} Y \land p \underset{˙}{\lor} (X \underset{˙}{\land} W) = X)) \to p \underset{˙}{\leq} (p \underset{˙}{\lor} (X \underset{˙}{\land} W))$
22		simp33	$⊢ (((K \in HL \land W \in H) \land (X \in B \land \neg X \underset{˙}{\leq} W) \land (Y \in B \land (X \underset{˙}{\land} Y) \underset{˙}{\leq} W)) \land p \in A \land (\neg p \underset{˙}{\leq} W \land \neg p \underset{˙}{\leq} Y \land p \underset{˙}{\lor} (X \underset{˙}{\land} W) = X)) \to p \underset{˙}{\lor} (X \underset{˙}{\land} W) = X$
23	21 22	breqtrd	$⊢ (((K \in HL \land W \in H) \land (X \in B \land \neg X \underset{˙}{\leq} W) \land (Y \in B \land (X \underset{˙}{\land} Y) \underset{˙}{\leq} W)) \land p \in A \land (\neg p \underset{˙}{\leq} W \land \neg p \underset{˙}{\leq} Y \land p \underset{˙}{\lor} (X \underset{˙}{\land} W) = X)) \to p \underset{˙}{\leq} X$
24	8 9 23	3jca	$⊢ (((K \in HL \land W \in H) \land (X \in B \land \neg X \underset{˙}{\leq} W) \land (Y \in B \land (X \underset{˙}{\land} Y) \underset{˙}{\leq} W)) \land p \in A \land (\neg p \underset{˙}{\leq} W \land \neg p \underset{˙}{\leq} Y \land p \underset{˙}{\lor} (X \underset{˙}{\land} W) = X)) \to (\neg p \underset{˙}{\leq} W \land \neg p \underset{˙}{\leq} Y \land p \underset{˙}{\leq} X)$
25	24	3expia	$⊢ (((K \in HL \land W \in H) \land (X \in B \land \neg X \underset{˙}{\leq} W) \land (Y \in B \land (X \underset{˙}{\land} Y) \underset{˙}{\leq} W)) \land p \in A) \to ((\neg p \underset{˙}{\leq} W \land \neg p \underset{˙}{\leq} Y \land p \underset{˙}{\lor} (X \underset{˙}{\land} W) = X) \to (\neg p \underset{˙}{\leq} W \land \neg p \underset{˙}{\leq} Y \land p \underset{˙}{\leq} X))$
26	25	reximdva	$⊢ ((K \in HL \land W \in H) \land (X \in B \land \neg X \underset{˙}{\leq} W) \land (Y \in B \land (X \underset{˙}{\land} Y) \underset{˙}{\leq} W)) \to (\exists p \in A (\neg p \underset{˙}{\leq} W \land \neg p \underset{˙}{\leq} Y \land p \underset{˙}{\lor} (X \underset{˙}{\land} W) = X) \to \exists p \in A (\neg p \underset{˙}{\leq} W \land \neg p \underset{˙}{\leq} Y \land p \underset{˙}{\leq} X))$
27	7 26	mpd	$⊢ ((K \in HL \land W \in H) \land (X \in B \land \neg X \underset{˙}{\leq} W) \land (Y \in B \land (X \underset{˙}{\land} Y) \underset{˙}{\leq} W)) \to \exists p \in A (\neg p \underset{˙}{\leq} W \land \neg p \underset{˙}{\leq} Y \land p \underset{˙}{\leq} X)$

Metamath Proof Explorer

Theorem lhpmcvr6N

Proof