Metamath Proof Explorer

Theorem cdlemg4g

Description: TODO: FIX COMMENT. (Contributed by NM, 25-Apr-2013)

		Ref	Expression
	Hypotheses	cdlemg4.l	$⊢ \underset{˙}{\leq} = \leq_{K}$
		cdlemg4.a	$⊢ A = Atoms (K)$
		cdlemg4.h	$⊢ H = LHyp (K)$
		cdlemg4.t	$⊢ T = LTrn (K) (W)$
		cdlemg4.r	$⊢ R = trL (K) (W)$
		cdlemg4.j	$⊢ \underset{˙}{\lor} = join (K)$
		cdlemg4b.v	$⊢ V = R (G)$
		cdlemg4.m	$⊢ \underset{˙}{\land} = meet (K)$
	Assertion	cdlemg4g	$⊢ ((K \in HL \land W \in H) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land F \in T) \land (G \in T \land \neg Q \underset{˙}{\leq} (P \underset{˙}{\lor} V) \land F (G (P)) = P)) \to F (G (Q)) = (Q \underset{˙}{\lor} V) \underset{˙}{\land} (P \underset{˙}{\lor} Q)$

Proof

Step	Hyp	Ref	Expression
1		cdlemg4.l	$⊢ \underset{˙}{\leq} = \leq_{K}$
2		cdlemg4.a	$⊢ A = Atoms (K)$
3		cdlemg4.h	$⊢ H = LHyp (K)$
4		cdlemg4.t	$⊢ T = LTrn (K) (W)$
5		cdlemg4.r	$⊢ R = trL (K) (W)$
6		cdlemg4.j	$⊢ \underset{˙}{\lor} = join (K)$
7		cdlemg4b.v	$⊢ V = R (G)$
8		cdlemg4.m	$⊢ \underset{˙}{\land} = meet (K)$
9	1 2 3 4 5 6 7 8	cdlemg4f	$⊢ ((K \in HL \land W \in H) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land F \in T) \land (G \in T \land \neg Q \underset{˙}{\leq} (P \underset{˙}{\lor} V) \land F (G (P)) = P)) \to F (G (Q)) = (Q \underset{˙}{\lor} V) \underset{˙}{\land} (P \underset{˙}{\lor} ((P \underset{˙}{\lor} Q) \underset{˙}{\land} W))$
10		simp1l	$⊢ ((K \in HL \land W \in H) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land F \in T) \land (G \in T \land \neg Q \underset{˙}{\leq} (P \underset{˙}{\lor} V) \land F (G (P)) = P)) \to K \in HL$
11		simp1r	$⊢ ((K \in HL \land W \in H) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land F \in T) \land (G \in T \land \neg Q \underset{˙}{\leq} (P \underset{˙}{\lor} V) \land F (G (P)) = P)) \to W \in H$
12		simp21	$⊢ ((K \in HL \land W \in H) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land F \in T) \land (G \in T \land \neg Q \underset{˙}{\leq} (P \underset{˙}{\lor} V) \land F (G (P)) = P)) \to (P \in A \land \neg P \underset{˙}{\leq} W)$
13		simp22l	$⊢ ((K \in HL \land W \in H) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land F \in T) \land (G \in T \land \neg Q \underset{˙}{\leq} (P \underset{˙}{\lor} V) \land F (G (P)) = P)) \to Q \in A$
14		eqid	$⊢ (P \underset{˙}{\lor} Q) \underset{˙}{\land} W = (P \underset{˙}{\lor} Q) \underset{˙}{\land} W$
15	1 6 8 2 3 14	cdleme0cp	$⊢ ((K \in HL \land W \in H) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land Q \in A)) \to P \underset{˙}{\lor} ((P \underset{˙}{\lor} Q) \underset{˙}{\land} W) = P \underset{˙}{\lor} Q$
16	10 11 12 13 15	syl22anc	$⊢ ((K \in HL \land W \in H) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land F \in T) \land (G \in T \land \neg Q \underset{˙}{\leq} (P \underset{˙}{\lor} V) \land F (G (P)) = P)) \to P \underset{˙}{\lor} ((P \underset{˙}{\lor} Q) \underset{˙}{\land} W) = P \underset{˙}{\lor} Q$
17	16	oveq2d	$⊢ ((K \in HL \land W \in H) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land F \in T) \land (G \in T \land \neg Q \underset{˙}{\leq} (P \underset{˙}{\lor} V) \land F (G (P)) = P)) \to (Q \underset{˙}{\lor} V) \underset{˙}{\land} (P \underset{˙}{\lor} ((P \underset{˙}{\lor} Q) \underset{˙}{\land} W)) = (Q \underset{˙}{\lor} V) \underset{˙}{\land} (P \underset{˙}{\lor} Q)$
18	9 17	eqtrd	$⊢ ((K \in HL \land W \in H) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land F \in T) \land (G \in T \land \neg Q \underset{˙}{\leq} (P \underset{˙}{\lor} V) \land F (G (P)) = P)) \to F (G (Q)) = (Q \underset{˙}{\lor} V) \underset{˙}{\land} (P \underset{˙}{\lor} Q)$