cdlemg4e

	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	cdlemg4e	$⊢ ((K \in HL \land W \in H) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land F \in T) \land (G \in T \land \neg Q \underset{˙}{\leq} (P \underset{˙}{\lor} V) \land F (G (P)) = P)) \to F (G (Q)) = (G (Q) \underset{˙}{\lor} R (F)) \underset{˙}{\land} (F (G (P)) \underset{˙}{\lor} ((G (P) \underset{˙}{\lor} G (Q)) \underset{˙}{\land} W))$

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		simp1	$⊢ ((K \in HL \land W \in H) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land F \in T) \land (G \in T \land \neg Q \underset{˙}{\leq} (P \underset{˙}{\lor} V) \land F (G (P)) = P)) \to (K \in HL \land W \in H)$
10		simp23	$⊢ ((K \in HL \land W \in H) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land F \in T) \land (G \in T \land \neg Q \underset{˙}{\leq} (P \underset{˙}{\lor} V) \land F (G (P)) = P)) \to F \in T$
11		simp31	$⊢ ((K \in HL \land W \in H) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land F \in T) \land (G \in T \land \neg Q \underset{˙}{\leq} (P \underset{˙}{\lor} V) \land F (G (P)) = P)) \to G \in T$
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	1 2 3 4	ltrnel	$⊢ ((K \in HL \land W \in H) \land G \in T \land (P \in A \land \neg P \underset{˙}{\leq} W)) \to (G (P) \in A \land \neg G (P) \underset{˙}{\leq} W)$
14	9 11 12 13	syl3anc	$⊢ ((K \in HL \land W \in H) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land F \in T) \land (G \in T \land \neg Q \underset{˙}{\leq} (P \underset{˙}{\lor} V) \land F (G (P)) = P)) \to (G (P) \in A \land \neg G (P) \underset{˙}{\leq} W)$
15		simp22	$⊢ ((K \in HL \land W \in H) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land F \in T) \land (G \in T \land \neg Q \underset{˙}{\leq} (P \underset{˙}{\lor} V) \land F (G (P)) = P)) \to (Q \in A \land \neg Q \underset{˙}{\leq} W)$
16	1 2 3 4	ltrnel	$⊢ ((K \in HL \land W \in H) \land G \in T \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \to (G (Q) \in A \land \neg G (Q) \underset{˙}{\leq} W)$
17	9 11 15 16	syl3anc	$⊢ ((K \in HL \land W \in H) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land F \in T) \land (G \in T \land \neg Q \underset{˙}{\leq} (P \underset{˙}{\lor} V) \land F (G (P)) = P)) \to (G (Q) \in A \land \neg G (Q) \underset{˙}{\leq} W)$
18	1 2 3 4 5 6 7	cdlemg4d	$⊢ ((K \in HL \land W \in H) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land F \in T) \land (G \in T \land \neg Q \underset{˙}{\leq} (P \underset{˙}{\lor} V) \land F (G (P)) = P)) \to \neg G (Q) \underset{˙}{\leq} (G (P) \underset{˙}{\lor} F (G (P)))$
19	1 6 8 2 3 4 5	cdlemc	$⊢ ((K \in HL \land W \in H) \land (F \in T \land (G (P) \in A \land \neg G (P) \underset{˙}{\leq} W) \land (G (Q) \in A \land \neg G (Q) \underset{˙}{\leq} W)) \land \neg G (Q) \underset{˙}{\leq} (G (P) \underset{˙}{\lor} F (G (P)))) \to F (G (Q)) = (G (Q) \underset{˙}{\lor} R (F)) \underset{˙}{\land} (F (G (P)) \underset{˙}{\lor} ((G (P) \underset{˙}{\lor} G (Q)) \underset{˙}{\land} W))$
20	9 10 14 17 18 19	syl131anc	$⊢ ((K \in HL \land W \in H) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land F \in T) \land (G \in T \land \neg Q \underset{˙}{\leq} (P \underset{˙}{\lor} V) \land F (G (P)) = P)) \to F (G (Q)) = (G (Q) \underset{˙}{\lor} R (F)) \underset{˙}{\land} (F (G (P)) \underset{˙}{\lor} ((G (P) \underset{˙}{\lor} G (Q)) \underset{˙}{\land} W))$

Metamath Proof Explorer

Theorem cdlemg4e

Proof