cdlemg31b0a

	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	cdlemg31b0a	$⊢ ((K \in HL \land W \in H) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land (v \in A \land v \underset{˙}{\leq} W)) \land (F \in T \land v \neq R (F))) \to (N \in A \lor N = 0. (K))$

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		simp1l	$⊢ ((K \in HL \land W \in H) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land (v \in A \land v \underset{˙}{\leq} W)) \land (F \in T \land v \neq R (F))) \to K \in HL$
10		simp21l	$⊢ ((K \in HL \land W \in H) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land (v \in A \land v \underset{˙}{\leq} W)) \land (F \in T \land v \neq R (F))) \to P \in A$
11		simp23l	$⊢ ((K \in HL \land W \in H) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land (v \in A \land v \underset{˙}{\leq} W)) \land (F \in T \land v \neq R (F))) \to v \in A$
12		simp22l	$⊢ ((K \in HL \land W \in H) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land (v \in A \land v \underset{˙}{\leq} W)) \land (F \in T \land v \neq R (F))) \to Q \in A$
13		simp1	$⊢ ((K \in HL \land W \in H) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land (v \in A \land v \underset{˙}{\leq} W)) \land (F \in T \land v \neq R (F))) \to (K \in HL \land W \in H)$
14		simp3l	$⊢ ((K \in HL \land W \in H) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land (v \in A \land v \underset{˙}{\leq} W)) \land (F \in T \land v \neq R (F))) \to F \in T$
15		eqid	$⊢ 0. (K) = 0. (K)$
16	15 4 5 6 7	trlator0	$⊢ ((K \in HL \land W \in H) \land F \in T) \to (R (F) \in A \lor R (F) = 0. (K))$
17	13 14 16	syl2anc	$⊢ ((K \in HL \land W \in H) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land (v \in A \land v \underset{˙}{\leq} W)) \land (F \in T \land v \neq R (F))) \to (R (F) \in A \lor R (F) = 0. (K))$
18		simp22	$⊢ ((K \in HL \land W \in H) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land (v \in A \land v \underset{˙}{\leq} W)) \land (F \in T \land v \neq R (F))) \to (Q \in A \land \neg Q \underset{˙}{\leq} W)$
19	1 5 6 7	trlle	$⊢ ((K \in HL \land W \in H) \land F \in T) \to R (F) \underset{˙}{\leq} W$
20	13 14 19	syl2anc	$⊢ ((K \in HL \land W \in H) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land (v \in A \land v \underset{˙}{\leq} W)) \land (F \in T \land v \neq R (F))) \to R (F) \underset{˙}{\leq} W$
21	17 20	jca	$⊢ ((K \in HL \land W \in H) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land (v \in A \land v \underset{˙}{\leq} W)) \land (F \in T \land v \neq R (F))) \to ((R (F) \in A \lor R (F) = 0. (K)) \land R (F) \underset{˙}{\leq} W)$
22		simp23	$⊢ ((K \in HL \land W \in H) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land (v \in A \land v \underset{˙}{\leq} W)) \land (F \in T \land v \neq R (F))) \to (v \in A \land v \underset{˙}{\leq} W)$
23		simp3r	$⊢ ((K \in HL \land W \in H) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land (v \in A \land v \underset{˙}{\leq} W)) \land (F \in T \land v \neq R (F))) \to v \neq R (F)$
24	23	necomd	$⊢ ((K \in HL \land W \in H) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land (v \in A \land v \underset{˙}{\leq} W)) \land (F \in T \land v \neq R (F))) \to R (F) \neq v$
25	1 2 15 4 5	lhp2at0ne	$⊢ (((K \in HL \land W \in H) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land P \in A) \land (((R (F) \in A \lor R (F) = 0. (K)) \land R (F) \underset{˙}{\leq} W) \land (v \in A \land v \underset{˙}{\leq} W)) \land R (F) \neq v) \to Q \underset{˙}{\lor} R (F) \neq P \underset{˙}{\lor} v$
26	13 18 10 21 22 24 25	syl321anc	$⊢ ((K \in HL \land W \in H) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land (v \in A \land v \underset{˙}{\leq} W)) \land (F \in T \land v \neq R (F))) \to Q \underset{˙}{\lor} R (F) \neq P \underset{˙}{\lor} v$
27	26	necomd	$⊢ ((K \in HL \land W \in H) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land (v \in A \land v \underset{˙}{\leq} W)) \land (F \in T \land v \neq R (F))) \to P \underset{˙}{\lor} v \neq Q \underset{˙}{\lor} R (F)$
28	2 3 15 4	2at0mat0	$⊢ ((K \in HL \land P \in A \land v \in A) \land (Q \in A \land (R (F) \in A \lor R (F) = 0. (K)) \land P \underset{˙}{\lor} v \neq Q \underset{˙}{\lor} R (F))) \to ((P \underset{˙}{\lor} v) \underset{˙}{\land} (Q \underset{˙}{\lor} R (F)) \in A \lor (P \underset{˙}{\lor} v) \underset{˙}{\land} (Q \underset{˙}{\lor} R (F)) = 0. (K))$
29	9 10 11 12 17 27 28	syl33anc	$⊢ ((K \in HL \land W \in H) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land (v \in A \land v \underset{˙}{\leq} W)) \land (F \in T \land v \neq R (F))) \to ((P \underset{˙}{\lor} v) \underset{˙}{\land} (Q \underset{˙}{\lor} R (F)) \in A \lor (P \underset{˙}{\lor} v) \underset{˙}{\land} (Q \underset{˙}{\lor} R (F)) = 0. (K))$
30	8	eleq1i	$⊢ N \in A \leftrightarrow (P \underset{˙}{\lor} v) \underset{˙}{\land} (Q \underset{˙}{\lor} R (F)) \in A$
31	8	eqeq1i	$⊢ N = 0. (K) \leftrightarrow (P \underset{˙}{\lor} v) \underset{˙}{\land} (Q \underset{˙}{\lor} R (F)) = 0. (K)$
32	30 31	orbi12i	$⊢ (N \in A \lor N = 0. (K)) \leftrightarrow ((P \underset{˙}{\lor} v) \underset{˙}{\land} (Q \underset{˙}{\lor} R (F)) \in A \lor (P \underset{˙}{\lor} v) \underset{˙}{\land} (Q \underset{˙}{\lor} R (F)) = 0. (K))$
33	29 32	sylibr	$⊢ ((K \in HL \land W \in H) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land (v \in A \land v \underset{˙}{\leq} W)) \land (F \in T \land v \neq R (F))) \to (N \in A \lor N = 0. (K))$

Metamath Proof Explorer

Theorem cdlemg31b0a

Proof