cdleme0ex1N

Description: Part of proof of Lemma E in Crawley p. 113. (Contributed by NM, 9-Nov-2012) (New usage is discouraged.)

	Ref	Expression
Hypotheses	cdleme0.l	$⊢ \underset{˙}{\leq} = \leq_{K}$
	cdleme0.j	$⊢ \underset{˙}{\lor} = join (K)$
	cdleme0.m	$⊢ \underset{˙}{\land} = meet (K)$
	cdleme0.a	$⊢ A = Atoms (K)$
	cdleme0.h	$⊢ H = LHyp (K)$
	cdleme0.u	$⊢ U = (P \underset{˙}{\lor} Q) \underset{˙}{\land} W$
Assertion	cdleme0ex1N	$⊢ ((K \in HL \land W \in H) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land Q \in A) \land P \neq Q) \to \exists u \in A (u \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land u \underset{˙}{\leq} W)$

Proof

Step	Hyp	Ref	Expression
1		cdleme0.l	$⊢ \underset{˙}{\leq} = \leq_{K}$
2		cdleme0.j	$⊢ \underset{˙}{\lor} = join (K)$
3		cdleme0.m	$⊢ \underset{˙}{\land} = meet (K)$
4		cdleme0.a	$⊢ A = Atoms (K)$
5		cdleme0.h	$⊢ H = LHyp (K)$
6		cdleme0.u	$⊢ U = (P \underset{˙}{\lor} Q) \underset{˙}{\land} W$
7		simp1	$⊢ ((K \in HL \land W \in H) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land Q \in A) \land P \neq Q) \to (K \in HL \land W \in H)$
8		simp2l	$⊢ ((K \in HL \land W \in H) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land Q \in A) \land P \neq Q) \to (P \in A \land \neg P \underset{˙}{\leq} W)$
9		simp2r	$⊢ ((K \in HL \land W \in H) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land Q \in A) \land P \neq Q) \to Q \in A$
10		simp3	$⊢ ((K \in HL \land W \in H) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land Q \in A) \land P \neq Q) \to P \neq Q$
11	1 2 3 4 5 6	lhpat2	$⊢ ((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land P \neq Q)) \to U \in A$
12	7 8 9 10 11	syl112anc	$⊢ ((K \in HL \land W \in H) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land Q \in A) \land P \neq Q) \to U \in A$
13		simp2ll	$⊢ ((K \in HL \land W \in H) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land Q \in A) \land P \neq Q) \to P \in A$
14	1 2 3 4 5 6	cdlemeulpq	$⊢ ((K \in HL \land W \in H) \land (P \in A \land Q \in A)) \to U \underset{˙}{\leq} (P \underset{˙}{\lor} Q)$
15	7 13 9 14	syl12anc	$⊢ ((K \in HL \land W \in H) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land Q \in A) \land P \neq Q) \to U \underset{˙}{\leq} (P \underset{˙}{\lor} Q)$
16		simp1l	$⊢ ((K \in HL \land W \in H) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land Q \in A) \land P \neq Q) \to K \in HL$
17	16	hllatd	$⊢ ((K \in HL \land W \in H) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land Q \in A) \land P \neq Q) \to K \in Lat$
18		eqid	$⊢ {Base}_{K} = {Base}_{K}$
19	18 2 4	hlatjcl	$⊢ (K \in HL \land P \in A \land Q \in A) \to P \underset{˙}{\lor} Q \in {Base}_{K}$
20	16 13 9 19	syl3anc	$⊢ ((K \in HL \land W \in H) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land Q \in A) \land P \neq Q) \to P \underset{˙}{\lor} Q \in {Base}_{K}$
21		simp1r	$⊢ ((K \in HL \land W \in H) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land Q \in A) \land P \neq Q) \to W \in H$
22	18 5	lhpbase	$⊢ W \in H \to W \in {Base}_{K}$
23	21 22	syl	$⊢ ((K \in HL \land W \in H) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land Q \in A) \land P \neq Q) \to W \in {Base}_{K}$
24	18 1 3	latmle2	$⊢ (K \in Lat \land P \underset{˙}{\lor} Q \in {Base}_{K} \land W \in {Base}_{K}) \to ((P \underset{˙}{\lor} Q) \underset{˙}{\land} W) \underset{˙}{\leq} W$
25	17 20 23 24	syl3anc	$⊢ ((K \in HL \land W \in H) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land Q \in A) \land P \neq Q) \to ((P \underset{˙}{\lor} Q) \underset{˙}{\land} W) \underset{˙}{\leq} W$
26	6 25	eqbrtrid	$⊢ ((K \in HL \land W \in H) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land Q \in A) \land P \neq Q) \to U \underset{˙}{\leq} W$
27		breq1	$⊢ u = U \to (u \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \leftrightarrow U \underset{˙}{\leq} (P \underset{˙}{\lor} Q))$
28		breq1	$⊢ u = U \to (u \underset{˙}{\leq} W \leftrightarrow U \underset{˙}{\leq} W)$
29	27 28	anbi12d	$⊢ u = U \to ((u \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land u \underset{˙}{\leq} W) \leftrightarrow (U \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land U \underset{˙}{\leq} W))$
30	29	rspcev	$⊢ (U \in A \land (U \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land U \underset{˙}{\leq} W)) \to \exists u \in A (u \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land u \underset{˙}{\leq} W)$
31	12 15 26 30	syl12anc	$⊢ ((K \in HL \land W \in H) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land Q \in A) \land P \neq Q) \to \exists u \in A (u \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land u \underset{˙}{\leq} W)$

Metamath Proof Explorer

Theorem cdleme0ex1N

Proof