cdlemg43

Description: Part of proof of Lemma G of Crawley p. 116, third line of third paragraph on p. 117. (Contributed by NM, 3-Jun-2013)

	Ref	Expression
Hypotheses	cdlemg42.l	$⊢ \underset{˙}{\leq} = \leq_{K}$
	cdlemg42.j	$⊢ \underset{˙}{\lor} = join (K)$
	cdlemg42.a	$⊢ A = Atoms (K)$
	cdlemg42.h	$⊢ H = LHyp (K)$
	cdlemg42.t	$⊢ T = LTrn (K) (W)$
	cdlemg42.r	$⊢ R = trL (K) (W)$
	cdlemg42.m	$⊢ \underset{˙}{\land} = meet (K)$
Assertion	cdlemg43	$⊢ ((K \in HL \land W \in H) \land (F \in T \land G \in T) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land G (P) \neq P \land R (F) \neq R (G))) \to F (G (P)) = (G (P) \underset{˙}{\lor} R (F)) \underset{˙}{\land} (F (P) \underset{˙}{\lor} R (G))$

Proof

Step	Hyp	Ref	Expression
1		cdlemg42.l	$⊢ \underset{˙}{\leq} = \leq_{K}$
2		cdlemg42.j	$⊢ \underset{˙}{\lor} = join (K)$
3		cdlemg42.a	$⊢ A = Atoms (K)$
4		cdlemg42.h	$⊢ H = LHyp (K)$
5		cdlemg42.t	$⊢ T = LTrn (K) (W)$
6		cdlemg42.r	$⊢ R = trL (K) (W)$
7		cdlemg42.m	$⊢ \underset{˙}{\land} = meet (K)$
8		simp1	$⊢ ((K \in HL \land W \in H) \land (F \in T \land G \in T) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land G (P) \neq P \land R (F) \neq R (G))) \to (K \in HL \land W \in H)$
9		simp2l	$⊢ ((K \in HL \land W \in H) \land (F \in T \land G \in T) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land G (P) \neq P \land R (F) \neq R (G))) \to F \in T$
10		simp31	$⊢ ((K \in HL \land W \in H) \land (F \in T \land G \in T) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land G (P) \neq P \land R (F) \neq R (G))) \to (P \in A \land \neg P \underset{˙}{\leq} W)$
11		simp2r	$⊢ ((K \in HL \land W \in H) \land (F \in T \land G \in T) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land G (P) \neq P \land R (F) \neq R (G))) \to G \in T$
12	1 3 4 5	ltrnel	$⊢ ((K \in HL \land W \in H) \land G \in T \land (P \in A \land \neg P \underset{˙}{\leq} W)) \to (G (P) \in A \land \neg G (P) \underset{˙}{\leq} W)$
13	8 11 10 12	syl3anc	$⊢ ((K \in HL \land W \in H) \land (F \in T \land G \in T) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land G (P) \neq P \land R (F) \neq R (G))) \to (G (P) \in A \land \neg G (P) \underset{˙}{\leq} W)$
14	1 2 3 4 5 6	cdlemg42	$⊢ ((K \in HL \land W \in H) \land (F \in T \land G \in T) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land G (P) \neq P \land R (F) \neq R (G))) \to \neg G (P) \underset{˙}{\leq} (P \underset{˙}{\lor} F (P))$
15	1 2 7 3 4 5 6	cdlemc	$⊢ ((K \in HL \land W \in H) \land (F \in T \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (G (P) \in A \land \neg G (P) \underset{˙}{\leq} W)) \land \neg G (P) \underset{˙}{\leq} (P \underset{˙}{\lor} F (P))) \to F (G (P)) = (G (P) \underset{˙}{\lor} R (F)) \underset{˙}{\land} (F (P) \underset{˙}{\lor} ((P \underset{˙}{\lor} G (P)) \underset{˙}{\land} W))$
16	8 9 10 13 14 15	syl131anc	$⊢ ((K \in HL \land W \in H) \land (F \in T \land G \in T) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land G (P) \neq P \land R (F) \neq R (G))) \to F (G (P)) = (G (P) \underset{˙}{\lor} R (F)) \underset{˙}{\land} (F (P) \underset{˙}{\lor} ((P \underset{˙}{\lor} G (P)) \underset{˙}{\land} W))$
17	1 2 7 3 4 5 6	trlval2	$⊢ ((K \in HL \land W \in H) \land G \in T \land (P \in A \land \neg P \underset{˙}{\leq} W)) \to R (G) = (P \underset{˙}{\lor} G (P)) \underset{˙}{\land} W$
18	8 11 10 17	syl3anc	$⊢ ((K \in HL \land W \in H) \land (F \in T \land G \in T) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land G (P) \neq P \land R (F) \neq R (G))) \to R (G) = (P \underset{˙}{\lor} G (P)) \underset{˙}{\land} W$
19	18	oveq2d	$⊢ ((K \in HL \land W \in H) \land (F \in T \land G \in T) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land G (P) \neq P \land R (F) \neq R (G))) \to F (P) \underset{˙}{\lor} R (G) = F (P) \underset{˙}{\lor} ((P \underset{˙}{\lor} G (P)) \underset{˙}{\land} W)$
20	19	oveq2d	$⊢ ((K \in HL \land W \in H) \land (F \in T \land G \in T) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land G (P) \neq P \land R (F) \neq R (G))) \to (G (P) \underset{˙}{\lor} R (F)) \underset{˙}{\land} (F (P) \underset{˙}{\lor} R (G)) = (G (P) \underset{˙}{\lor} R (F)) \underset{˙}{\land} (F (P) \underset{˙}{\lor} ((P \underset{˙}{\lor} G (P)) \underset{˙}{\land} W))$
21	16 20	eqtr4d	$⊢ ((K \in HL \land W \in H) \land (F \in T \land G \in T) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land G (P) \neq P \land R (F) \neq R (G))) \to F (G (P)) = (G (P) \underset{˙}{\lor} R (F)) \underset{˙}{\land} (F (P) \underset{˙}{\lor} R (G))$

Metamath Proof Explorer

Theorem cdlemg43

Proof