lhpmcvr5N

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	lhpmcvr5N	$⊢ ((K \in HL \land W \in H) \land (X \in B \land \neg X \underset{˙}{\leq} W) \land (Y \in B \land (X \underset{˙}{\land} Y) \underset{˙}{\leq} W)) \to \exists p \in A (\neg p \underset{˙}{\leq} W \land \neg p \underset{˙}{\leq} Y \land p \underset{˙}{\lor} (X \underset{˙}{\land} W) = X)$

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	lhpmcvr2	$⊢ ((K \in HL \land W \in H) \land (X \in B \land \neg X \underset{˙}{\leq} W)) \to \exists p \in A (\neg p \underset{˙}{\leq} W \land p \underset{˙}{\lor} (X \underset{˙}{\land} W) = X)$
8	7	3adant3	$⊢ ((K \in HL \land W \in H) \land (X \in B \land \neg X \underset{˙}{\leq} W) \land (Y \in B \land (X \underset{˙}{\land} Y) \underset{˙}{\leq} W)) \to \exists p \in A (\neg p \underset{˙}{\leq} W \land p \underset{˙}{\lor} (X \underset{˙}{\land} W) = X)$
9		simp3l	$⊢ (((K \in HL \land W \in H) \land (X \in B \land \neg X \underset{˙}{\leq} W) \land (Y \in B \land (X \underset{˙}{\land} Y) \underset{˙}{\leq} W)) \land p \in A \land (\neg p \underset{˙}{\leq} W \land p \underset{˙}{\lor} (X \underset{˙}{\land} W) = X)) \to \neg p \underset{˙}{\leq} W$
10		simp11	$⊢ (((K \in HL \land W \in H) \land (X \in B \land \neg X \underset{˙}{\leq} W) \land (Y \in B \land (X \underset{˙}{\land} Y) \underset{˙}{\leq} W)) \land p \in A \land (\neg p \underset{˙}{\leq} W \land p \underset{˙}{\lor} (X \underset{˙}{\land} W) = X)) \to (K \in HL \land W \in H)$
11		simp12	$⊢ (((K \in HL \land W \in H) \land (X \in B \land \neg X \underset{˙}{\leq} W) \land (Y \in B \land (X \underset{˙}{\land} Y) \underset{˙}{\leq} W)) \land p \in A \land (\neg p \underset{˙}{\leq} W \land p \underset{˙}{\lor} (X \underset{˙}{\land} W) = X)) \to (X \in B \land \neg X \underset{˙}{\leq} W)$
12		simp2	$⊢ (((K \in HL \land W \in H) \land (X \in B \land \neg X \underset{˙}{\leq} W) \land (Y \in B \land (X \underset{˙}{\land} Y) \underset{˙}{\leq} W)) \land p \in A \land (\neg p \underset{˙}{\leq} W \land p \underset{˙}{\lor} (X \underset{˙}{\land} W) = X)) \to p \in A$
13	12 9	jca	$⊢ (((K \in HL \land W \in H) \land (X \in B \land \neg X \underset{˙}{\leq} W) \land (Y \in B \land (X \underset{˙}{\land} Y) \underset{˙}{\leq} W)) \land p \in A \land (\neg p \underset{˙}{\leq} W \land p \underset{˙}{\lor} (X \underset{˙}{\land} W) = X)) \to (p \in A \land \neg p \underset{˙}{\leq} W)$
14		simp13l	$⊢ (((K \in HL \land W \in H) \land (X \in B \land \neg X \underset{˙}{\leq} W) \land (Y \in B \land (X \underset{˙}{\land} Y) \underset{˙}{\leq} W)) \land p \in A \land (\neg p \underset{˙}{\leq} W \land p \underset{˙}{\lor} (X \underset{˙}{\land} W) = X)) \to Y \in B$
15		simp13r	$⊢ (((K \in HL \land W \in H) \land (X \in B \land \neg X \underset{˙}{\leq} W) \land (Y \in B \land (X \underset{˙}{\land} Y) \underset{˙}{\leq} W)) \land p \in A \land (\neg p \underset{˙}{\leq} W \land p \underset{˙}{\lor} (X \underset{˙}{\land} W) = X)) \to (X \underset{˙}{\land} Y) \underset{˙}{\leq} W$
16		simp11l	$⊢ (((K \in HL \land W \in H) \land (X \in B \land \neg X \underset{˙}{\leq} W) \land (Y \in B \land (X \underset{˙}{\land} Y) \underset{˙}{\leq} W)) \land p \in A \land (\neg p \underset{˙}{\leq} W \land p \underset{˙}{\lor} (X \underset{˙}{\land} W) = X)) \to K \in HL$
17	16	hllatd	$⊢ (((K \in HL \land W \in H) \land (X \in B \land \neg X \underset{˙}{\leq} W) \land (Y \in B \land (X \underset{˙}{\land} Y) \underset{˙}{\leq} W)) \land p \in A \land (\neg p \underset{˙}{\leq} W \land p \underset{˙}{\lor} (X \underset{˙}{\land} W) = X)) \to K \in Lat$
18	1 5	atbase	$⊢ p \in A \to p \in B$
19	18	3ad2ant2	$⊢ (((K \in HL \land W \in H) \land (X \in B \land \neg X \underset{˙}{\leq} W) \land (Y \in B \land (X \underset{˙}{\land} Y) \underset{˙}{\leq} W)) \land p \in A \land (\neg p \underset{˙}{\leq} W \land p \underset{˙}{\lor} (X \underset{˙}{\land} W) = X)) \to p \in B$
20		simp12l	$⊢ (((K \in HL \land W \in H) \land (X \in B \land \neg X \underset{˙}{\leq} W) \land (Y \in B \land (X \underset{˙}{\land} Y) \underset{˙}{\leq} W)) \land p \in A \land (\neg p \underset{˙}{\leq} W \land p \underset{˙}{\lor} (X \underset{˙}{\land} W) = X)) \to X \in B$
21		simp11r	$⊢ (((K \in HL \land W \in H) \land (X \in B \land \neg X \underset{˙}{\leq} W) \land (Y \in B \land (X \underset{˙}{\land} Y) \underset{˙}{\leq} W)) \land p \in A \land (\neg p \underset{˙}{\leq} W \land p \underset{˙}{\lor} (X \underset{˙}{\land} W) = X)) \to W \in H$
22	1 6	lhpbase	$⊢ W \in H \to W \in B$
23	21 22	syl	$⊢ (((K \in HL \land W \in H) \land (X \in B \land \neg X \underset{˙}{\leq} W) \land (Y \in B \land (X \underset{˙}{\land} Y) \underset{˙}{\leq} W)) \land p \in A \land (\neg p \underset{˙}{\leq} W \land p \underset{˙}{\lor} (X \underset{˙}{\land} W) = X)) \to W \in B$
24	1 4	latmcl	$⊢ (K \in Lat \land X \in B \land W \in B) \to X \underset{˙}{\land} W \in B$
25	17 20 23 24	syl3anc	$⊢ (((K \in HL \land W \in H) \land (X \in B \land \neg X \underset{˙}{\leq} W) \land (Y \in B \land (X \underset{˙}{\land} Y) \underset{˙}{\leq} W)) \land p \in A \land (\neg p \underset{˙}{\leq} W \land p \underset{˙}{\lor} (X \underset{˙}{\land} W) = X)) \to X \underset{˙}{\land} W \in B$
26	1 2 3	latlej1	$⊢ (K \in Lat \land p \in B \land X \underset{˙}{\land} W \in B) \to p \underset{˙}{\leq} (p \underset{˙}{\lor} (X \underset{˙}{\land} W))$
27	17 19 25 26	syl3anc	$⊢ (((K \in HL \land W \in H) \land (X \in B \land \neg X \underset{˙}{\leq} W) \land (Y \in B \land (X \underset{˙}{\land} Y) \underset{˙}{\leq} W)) \land p \in A \land (\neg p \underset{˙}{\leq} W \land p \underset{˙}{\lor} (X \underset{˙}{\land} W) = X)) \to p \underset{˙}{\leq} (p \underset{˙}{\lor} (X \underset{˙}{\land} W))$
28		simp3r	$⊢ (((K \in HL \land W \in H) \land (X \in B \land \neg X \underset{˙}{\leq} W) \land (Y \in B \land (X \underset{˙}{\land} Y) \underset{˙}{\leq} W)) \land p \in A \land (\neg p \underset{˙}{\leq} W \land p \underset{˙}{\lor} (X \underset{˙}{\land} W) = X)) \to p \underset{˙}{\lor} (X \underset{˙}{\land} W) = X$
29	27 28	breqtrd	$⊢ (((K \in HL \land W \in H) \land (X \in B \land \neg X \underset{˙}{\leq} W) \land (Y \in B \land (X \underset{˙}{\land} Y) \underset{˙}{\leq} W)) \land p \in A \land (\neg p \underset{˙}{\leq} W \land p \underset{˙}{\lor} (X \underset{˙}{\land} W) = X)) \to p \underset{˙}{\leq} X$
30	1 2 3 4 5 6	lhpmcvr4N	$⊢ ((K \in HL \land W \in H) \land ((X \in B \land \neg X \underset{˙}{\leq} W) \land (p \in A \land \neg p \underset{˙}{\leq} W)) \land (Y \in B \land (X \underset{˙}{\land} Y) \underset{˙}{\leq} W \land p \underset{˙}{\leq} X)) \to \neg p \underset{˙}{\leq} Y$
31	10 11 13 14 15 29 30	syl123anc	$⊢ (((K \in HL \land W \in H) \land (X \in B \land \neg X \underset{˙}{\leq} W) \land (Y \in B \land (X \underset{˙}{\land} Y) \underset{˙}{\leq} W)) \land p \in A \land (\neg p \underset{˙}{\leq} W \land p \underset{˙}{\lor} (X \underset{˙}{\land} W) = X)) \to \neg p \underset{˙}{\leq} Y$
32	9 31 28	3jca	$⊢ (((K \in HL \land W \in H) \land (X \in B \land \neg X \underset{˙}{\leq} W) \land (Y \in B \land (X \underset{˙}{\land} Y) \underset{˙}{\leq} W)) \land p \in A \land (\neg p \underset{˙}{\leq} W \land p \underset{˙}{\lor} (X \underset{˙}{\land} W) = X)) \to (\neg p \underset{˙}{\leq} W \land \neg p \underset{˙}{\leq} Y \land p \underset{˙}{\lor} (X \underset{˙}{\land} W) = X)$
33	32	3expia	$⊢ (((K \in HL \land W \in H) \land (X \in B \land \neg X \underset{˙}{\leq} W) \land (Y \in B \land (X \underset{˙}{\land} Y) \underset{˙}{\leq} W)) \land p \in A) \to ((\neg p \underset{˙}{\leq} W \land p \underset{˙}{\lor} (X \underset{˙}{\land} W) = X) \to (\neg p \underset{˙}{\leq} W \land \neg p \underset{˙}{\leq} Y \land p \underset{˙}{\lor} (X \underset{˙}{\land} W) = X))$
34	33	reximdva	$⊢ ((K \in HL \land W \in H) \land (X \in B \land \neg X \underset{˙}{\leq} W) \land (Y \in B \land (X \underset{˙}{\land} Y) \underset{˙}{\leq} W)) \to (\exists p \in A (\neg p \underset{˙}{\leq} W \land p \underset{˙}{\lor} (X \underset{˙}{\land} W) = X) \to \exists p \in A (\neg p \underset{˙}{\leq} W \land \neg p \underset{˙}{\leq} Y \land p \underset{˙}{\lor} (X \underset{˙}{\land} W) = X))$
35	8 34	mpd	$⊢ ((K \in HL \land W \in H) \land (X \in B \land \neg X \underset{˙}{\leq} W) \land (Y \in B \land (X \underset{˙}{\land} Y) \underset{˙}{\leq} W)) \to \exists p \in A (\neg p \underset{˙}{\leq} W \land \neg p \underset{˙}{\leq} Y \land p \underset{˙}{\lor} (X \underset{˙}{\land} W) = X)$

Metamath Proof Explorer

Theorem lhpmcvr5N

Proof