cdlemg4f

	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	cdlemg4f	$⊢ ((K \in HL \land W \in H) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land F \in T) \land (G \in T \land \neg Q \underset{˙}{\leq} (P \underset{˙}{\lor} V) \land F (G (P)) = P)) \to F (G (Q)) = (Q \underset{˙}{\lor} V) \underset{˙}{\land} (P \underset{˙}{\lor} ((P \underset{˙}{\lor} Q) \underset{˙}{\land} W))$

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	cdlemg4e	$⊢ ((K \in HL \land W \in H) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land F \in T) \land (G \in T \land \neg Q \underset{˙}{\leq} (P \underset{˙}{\lor} V) \land F (G (P)) = P)) \to F (G (Q)) = (G (Q) \underset{˙}{\lor} R (F)) \underset{˙}{\land} (F (G (P)) \underset{˙}{\lor} ((G (P) \underset{˙}{\lor} G (Q)) \underset{˙}{\land} W))$
10		simp1	$⊢ ((K \in HL \land W \in H) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land F \in T) \land (G \in T \land \neg Q \underset{˙}{\leq} (P \underset{˙}{\lor} V) \land F (G (P)) = P)) \to (K \in HL \land W \in H)$
11		simp21	$⊢ ((K \in HL \land W \in H) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land F \in T) \land (G \in T \land \neg Q \underset{˙}{\leq} (P \underset{˙}{\lor} V) \land F (G (P)) = P)) \to (P \in A \land \neg P \underset{˙}{\leq} W)$
12		simp23	$⊢ ((K \in HL \land W \in H) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land F \in T) \land (G \in T \land \neg Q \underset{˙}{\leq} (P \underset{˙}{\lor} V) \land F (G (P)) = P)) \to F \in T$
13		simp31	$⊢ ((K \in HL \land W \in H) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land F \in T) \land (G \in T \land \neg Q \underset{˙}{\leq} (P \underset{˙}{\lor} V) \land F (G (P)) = P)) \to G \in T$
14		simp33	$⊢ ((K \in HL \land W \in H) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land F \in T) \land (G \in T \land \neg Q \underset{˙}{\leq} (P \underset{˙}{\lor} V) \land F (G (P)) = P)) \to F (G (P)) = P$
15	1 2 3 4 5	cdlemg4a	$⊢ ((K \in HL \land W \in H) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land F \in T \land G \in T) \land F (G (P)) = P) \to R (F) = R (G)$
16	10 11 12 13 14 15	syl131anc	$⊢ ((K \in HL \land W \in H) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land F \in T) \land (G \in T \land \neg Q \underset{˙}{\leq} (P \underset{˙}{\lor} V) \land F (G (P)) = P)) \to R (F) = R (G)$
17	7 16	eqtr4id	$⊢ ((K \in HL \land W \in H) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land F \in T) \land (G \in T \land \neg Q \underset{˙}{\leq} (P \underset{˙}{\lor} V) \land F (G (P)) = P)) \to V = R (F)$
18	17	oveq2d	$⊢ ((K \in HL \land W \in H) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land F \in T) \land (G \in T \land \neg Q \underset{˙}{\leq} (P \underset{˙}{\lor} V) \land F (G (P)) = P)) \to G (Q) \underset{˙}{\lor} V = G (Q) \underset{˙}{\lor} R (F)$
19		simp22	$⊢ ((K \in HL \land W \in H) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land F \in T) \land (G \in T \land \neg Q \underset{˙}{\leq} (P \underset{˙}{\lor} V) \land F (G (P)) = P)) \to (Q \in A \land \neg Q \underset{˙}{\leq} W)$
20	1 2 3 4 5 6 7	cdlemg4b12	$⊢ ((K \in HL \land W \in H) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land G \in T) \to G (Q) \underset{˙}{\lor} V = Q \underset{˙}{\lor} V$
21	10 19 13 20	syl3anc	$⊢ ((K \in HL \land W \in H) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land F \in T) \land (G \in T \land \neg Q \underset{˙}{\leq} (P \underset{˙}{\lor} V) \land F (G (P)) = P)) \to G (Q) \underset{˙}{\lor} V = Q \underset{˙}{\lor} V$
22	18 21	eqtr3d	$⊢ ((K \in HL \land W \in H) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land F \in T) \land (G \in T \land \neg Q \underset{˙}{\leq} (P \underset{˙}{\lor} V) \land F (G (P)) = P)) \to G (Q) \underset{˙}{\lor} R (F) = Q \underset{˙}{\lor} V$
23		eqid	$⊢ (P \underset{˙}{\lor} Q) \underset{˙}{\land} W = (P \underset{˙}{\lor} Q) \underset{˙}{\land} W$
24	3 4 1 6 2 8 23	cdlemg2m	$⊢ ((K \in HL \land W \in H) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land G \in T) \to (G (P) \underset{˙}{\lor} G (Q)) \underset{˙}{\land} W = (P \underset{˙}{\lor} Q) \underset{˙}{\land} W$
25	10 11 19 13 24	syl121anc	$⊢ ((K \in HL \land W \in H) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land F \in T) \land (G \in T \land \neg Q \underset{˙}{\leq} (P \underset{˙}{\lor} V) \land F (G (P)) = P)) \to (G (P) \underset{˙}{\lor} G (Q)) \underset{˙}{\land} W = (P \underset{˙}{\lor} Q) \underset{˙}{\land} W$
26	14 25	oveq12d	$⊢ ((K \in HL \land W \in H) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land F \in T) \land (G \in T \land \neg Q \underset{˙}{\leq} (P \underset{˙}{\lor} V) \land F (G (P)) = P)) \to F (G (P)) \underset{˙}{\lor} ((G (P) \underset{˙}{\lor} G (Q)) \underset{˙}{\land} W) = P \underset{˙}{\lor} ((P \underset{˙}{\lor} Q) \underset{˙}{\land} W)$
27	22 26	oveq12d	$⊢ ((K \in HL \land W \in H) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land F \in T) \land (G \in T \land \neg Q \underset{˙}{\leq} (P \underset{˙}{\lor} V) \land F (G (P)) = P)) \to (G (Q) \underset{˙}{\lor} R (F)) \underset{˙}{\land} (F (G (P)) \underset{˙}{\lor} ((G (P) \underset{˙}{\lor} G (Q)) \underset{˙}{\land} W)) = (Q \underset{˙}{\lor} V) \underset{˙}{\land} (P \underset{˙}{\lor} ((P \underset{˙}{\lor} Q) \underset{˙}{\land} W))$
28	9 27	eqtrd	$⊢ ((K \in HL \land W \in H) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land F \in T) \land (G \in T \land \neg Q \underset{˙}{\leq} (P \underset{˙}{\lor} V) \land F (G (P)) = P)) \to F (G (Q)) = (Q \underset{˙}{\lor} V) \underset{˙}{\land} (P \underset{˙}{\lor} ((P \underset{˙}{\lor} Q) \underset{˙}{\land} W))$

Metamath Proof Explorer

Theorem cdlemg4f

Proof