cdlemg31b0N

Description: TODO: Fix comment. (Contributed by NM, 30-May-2013) (New usage is discouraged.)

	Ref	Expression
Hypotheses	cdlemg12.l	$⊢ \underset{˙}{\leq} = \leq_{K}$
	cdlemg12.j	$⊢ \underset{˙}{\lor} = join (K)$
	cdlemg12.m	$⊢ \underset{˙}{\land} = meet (K)$
	cdlemg12.a	$⊢ A = Atoms (K)$
	cdlemg12.h	$⊢ H = LHyp (K)$
	cdlemg12.t	$⊢ T = LTrn (K) (W)$
	cdlemg12b.r	$⊢ R = trL (K) (W)$
	cdlemg31.n	$⊢ N = (P \underset{˙}{\lor} v) \underset{˙}{\land} (Q \underset{˙}{\lor} R (F))$
Assertion	cdlemg31b0N	$⊢ ((K \in HL \land W \in H \land F \in T) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((v \in A \land v \underset{˙}{\leq} W) \land v \neq R (F) \land F (P) \neq P)) \to (N \in A \lor N = 0. (K))$

Proof

Step	Hyp	Ref	Expression
1		cdlemg12.l	$⊢ \underset{˙}{\leq} = \leq_{K}$
2		cdlemg12.j	$⊢ \underset{˙}{\lor} = join (K)$
3		cdlemg12.m	$⊢ \underset{˙}{\land} = meet (K)$
4		cdlemg12.a	$⊢ A = Atoms (K)$
5		cdlemg12.h	$⊢ H = LHyp (K)$
6		cdlemg12.t	$⊢ T = LTrn (K) (W)$
7		cdlemg12b.r	$⊢ R = trL (K) (W)$
8		cdlemg31.n	$⊢ N = (P \underset{˙}{\lor} v) \underset{˙}{\land} (Q \underset{˙}{\lor} R (F))$
9		simp11	$⊢ ((K \in HL \land W \in H \land F \in T) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((v \in A \land v \underset{˙}{\leq} W) \land v \neq R (F) \land F (P) \neq P)) \to K \in HL$
10		simp2ll	$⊢ ((K \in HL \land W \in H \land F \in T) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((v \in A \land v \underset{˙}{\leq} W) \land v \neq R (F) \land F (P) \neq P)) \to P \in A$
11		simp31l	$⊢ ((K \in HL \land W \in H \land F \in T) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((v \in A \land v \underset{˙}{\leq} W) \land v \neq R (F) \land F (P) \neq P)) \to v \in A$
12		simp2rl	$⊢ ((K \in HL \land W \in H \land F \in T) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((v \in A \land v \underset{˙}{\leq} W) \land v \neq R (F) \land F (P) \neq P)) \to Q \in A$
13		simp12	$⊢ ((K \in HL \land W \in H \land F \in T) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((v \in A \land v \underset{˙}{\leq} W) \land v \neq R (F) \land F (P) \neq P)) \to W \in H$
14	9 13	jca	$⊢ ((K \in HL \land W \in H \land F \in T) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((v \in A \land v \underset{˙}{\leq} W) \land v \neq R (F) \land F (P) \neq P)) \to (K \in HL \land W \in H)$
15		simp2l	$⊢ ((K \in HL \land W \in H \land F \in T) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((v \in A \land v \underset{˙}{\leq} W) \land v \neq R (F) \land F (P) \neq P)) \to (P \in A \land \neg P \underset{˙}{\leq} W)$
16		simp13	$⊢ ((K \in HL \land W \in H \land F \in T) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((v \in A \land v \underset{˙}{\leq} W) \land v \neq R (F) \land F (P) \neq P)) \to F \in T$
17		simp33	$⊢ ((K \in HL \land W \in H \land F \in T) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((v \in A \land v \underset{˙}{\leq} W) \land v \neq R (F) \land F (P) \neq P)) \to F (P) \neq P$
18	1 4 5 6 7	trlat	$⊢ ((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (F \in T \land F (P) \neq P)) \to R (F) \in A$
19	14 15 16 17 18	syl112anc	$⊢ ((K \in HL \land W \in H \land F \in T) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((v \in A \land v \underset{˙}{\leq} W) \land v \neq R (F) \land F (P) \neq P)) \to R (F) \in A$
20		simp2r	$⊢ ((K \in HL \land W \in H \land F \in T) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((v \in A \land v \underset{˙}{\leq} W) \land v \neq R (F) \land F (P) \neq P)) \to (Q \in A \land \neg Q \underset{˙}{\leq} W)$
21	1 5 6 7	trlle	$⊢ ((K \in HL \land W \in H) \land F \in T) \to R (F) \underset{˙}{\leq} W$
22	14 16 21	syl2anc	$⊢ ((K \in HL \land W \in H \land F \in T) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((v \in A \land v \underset{˙}{\leq} W) \land v \neq R (F) \land F (P) \neq P)) \to R (F) \underset{˙}{\leq} W$
23	19 22	jca	$⊢ ((K \in HL \land W \in H \land F \in T) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((v \in A \land v \underset{˙}{\leq} W) \land v \neq R (F) \land F (P) \neq P)) \to (R (F) \in A \land R (F) \underset{˙}{\leq} W)$
24		simp31	$⊢ ((K \in HL \land W \in H \land F \in T) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((v \in A \land v \underset{˙}{\leq} W) \land v \neq R (F) \land F (P) \neq P)) \to (v \in A \land v \underset{˙}{\leq} W)$
25		simp32	$⊢ ((K \in HL \land W \in H \land F \in T) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((v \in A \land v \underset{˙}{\leq} W) \land v \neq R (F) \land F (P) \neq P)) \to v \neq R (F)$
26	25	necomd	$⊢ ((K \in HL \land W \in H \land F \in T) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((v \in A \land v \underset{˙}{\leq} W) \land v \neq R (F) \land F (P) \neq P)) \to R (F) \neq v$
27	1 2 4 5	lhp2atne	$⊢ (((K \in HL \land W \in H) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land P \in A) \land ((R (F) \in A \land R (F) \underset{˙}{\leq} W) \land (v \in A \land v \underset{˙}{\leq} W)) \land R (F) \neq v) \to Q \underset{˙}{\lor} R (F) \neq P \underset{˙}{\lor} v$
28	14 20 10 23 24 26 27	syl321anc	$⊢ ((K \in HL \land W \in H \land F \in T) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((v \in A \land v \underset{˙}{\leq} W) \land v \neq R (F) \land F (P) \neq P)) \to Q \underset{˙}{\lor} R (F) \neq P \underset{˙}{\lor} v$
29	28	necomd	$⊢ ((K \in HL \land W \in H \land F \in T) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((v \in A \land v \underset{˙}{\leq} W) \land v \neq R (F) \land F (P) \neq P)) \to P \underset{˙}{\lor} v \neq Q \underset{˙}{\lor} R (F)$
30		eqid	$⊢ 0. (K) = 0. (K)$
31	2 3 30 4	2atmat0	$⊢ ((K \in HL \land P \in A \land v \in A) \land (Q \in A \land R (F) \in A \land P \underset{˙}{\lor} v \neq Q \underset{˙}{\lor} R (F))) \to ((P \underset{˙}{\lor} v) \underset{˙}{\land} (Q \underset{˙}{\lor} R (F)) \in A \lor (P \underset{˙}{\lor} v) \underset{˙}{\land} (Q \underset{˙}{\lor} R (F)) = 0. (K))$
32	9 10 11 12 19 29 31	syl33anc	$⊢ ((K \in HL \land W \in H \land F \in T) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((v \in A \land v \underset{˙}{\leq} W) \land v \neq R (F) \land F (P) \neq P)) \to ((P \underset{˙}{\lor} v) \underset{˙}{\land} (Q \underset{˙}{\lor} R (F)) \in A \lor (P \underset{˙}{\lor} v) \underset{˙}{\land} (Q \underset{˙}{\lor} R (F)) = 0. (K))$
33	8	eleq1i	$⊢ N \in A \leftrightarrow (P \underset{˙}{\lor} v) \underset{˙}{\land} (Q \underset{˙}{\lor} R (F)) \in A$
34	8	eqeq1i	$⊢ N = 0. (K) \leftrightarrow (P \underset{˙}{\lor} v) \underset{˙}{\land} (Q \underset{˙}{\lor} R (F)) = 0. (K)$
35	33 34	orbi12i	$⊢ (N \in A \lor N = 0. (K)) \leftrightarrow ((P \underset{˙}{\lor} v) \underset{˙}{\land} (Q \underset{˙}{\lor} R (F)) \in A \lor (P \underset{˙}{\lor} v) \underset{˙}{\land} (Q \underset{˙}{\lor} R (F)) = 0. (K))$
36	32 35	sylibr	$⊢ ((K \in HL \land W \in H \land F \in T) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((v \in A \land v \underset{˙}{\leq} W) \land v \neq R (F) \land F (P) \neq P)) \to (N \in A \lor N = 0. (K))$

Metamath Proof Explorer

Theorem cdlemg31b0N

Proof