Metamath Proof Explorer

Theorem cdlemk1

Description: Part of proof of Lemma K of Crawley p. 118. (Contributed by NM, 22-Jun-2013)

		Ref	Expression
	Hypotheses	cdlemk.b	$⊢ B = {Base}_{K}$
		cdlemk.l	$⊢ \underset{˙}{\leq} = \leq_{K}$
		cdlemk.j	$⊢ \underset{˙}{\lor} = join (K)$
		cdlemk.a	$⊢ A = Atoms (K)$
		cdlemk.h	$⊢ H = LHyp (K)$
		cdlemk.t	$⊢ T = LTrn (K) (W)$
		cdlemk.r	$⊢ R = trL (K) (W)$
	Assertion	cdlemk1	$⊢ ((K \in HL \land W \in H) \land (F \in T \land N \in T) \land (R (F) = R (N) \land (P \in A \land \neg P \underset{˙}{\leq} W))) \to P \underset{˙}{\lor} N (P) = F (P) \underset{˙}{\lor} R (F)$

Proof

Step	Hyp	Ref	Expression
1		cdlemk.b	$⊢ B = {Base}_{K}$
2		cdlemk.l	$⊢ \underset{˙}{\leq} = \leq_{K}$
3		cdlemk.j	$⊢ \underset{˙}{\lor} = join (K)$
4		cdlemk.a	$⊢ A = Atoms (K)$
5		cdlemk.h	$⊢ H = LHyp (K)$
6		cdlemk.t	$⊢ T = LTrn (K) (W)$
7		cdlemk.r	$⊢ R = trL (K) (W)$
8		simp3l	$⊢ ((K \in HL \land W \in H) \land (F \in T \land N \in T) \land (R (F) = R (N) \land (P \in A \land \neg P \underset{˙}{\leq} W))) \to R (F) = R (N)$
9	8	oveq2d	$⊢ ((K \in HL \land W \in H) \land (F \in T \land N \in T) \land (R (F) = R (N) \land (P \in A \land \neg P \underset{˙}{\leq} W))) \to P \underset{˙}{\lor} R (F) = P \underset{˙}{\lor} R (N)$
10		simp1	$⊢ ((K \in HL \land W \in H) \land (F \in T \land N \in T) \land (R (F) = R (N) \land (P \in A \land \neg P \underset{˙}{\leq} W))) \to (K \in HL \land W \in H)$
11		simp2l	$⊢ ((K \in HL \land W \in H) \land (F \in T \land N \in T) \land (R (F) = R (N) \land (P \in A \land \neg P \underset{˙}{\leq} W))) \to F \in T$
12		simp3r	$⊢ ((K \in HL \land W \in H) \land (F \in T \land N \in T) \land (R (F) = R (N) \land (P \in A \land \neg P \underset{˙}{\leq} W))) \to (P \in A \land \neg P \underset{˙}{\leq} W)$
13	2 3 4 5 6 7	trljat3	$⊢ ((K \in HL \land W \in H) \land F \in T \land (P \in A \land \neg P \underset{˙}{\leq} W)) \to P \underset{˙}{\lor} R (F) = F (P) \underset{˙}{\lor} R (F)$
14	10 11 12 13	syl3anc	$⊢ ((K \in HL \land W \in H) \land (F \in T \land N \in T) \land (R (F) = R (N) \land (P \in A \land \neg P \underset{˙}{\leq} W))) \to P \underset{˙}{\lor} R (F) = F (P) \underset{˙}{\lor} R (F)$
15		simp2r	$⊢ ((K \in HL \land W \in H) \land (F \in T \land N \in T) \land (R (F) = R (N) \land (P \in A \land \neg P \underset{˙}{\leq} W))) \to N \in T$
16	2 3 4 5 6 7	trljat1	$⊢ ((K \in HL \land W \in H) \land N \in T \land (P \in A \land \neg P \underset{˙}{\leq} W)) \to P \underset{˙}{\lor} R (N) = P \underset{˙}{\lor} N (P)$
17	10 15 12 16	syl3anc	$⊢ ((K \in HL \land W \in H) \land (F \in T \land N \in T) \land (R (F) = R (N) \land (P \in A \land \neg P \underset{˙}{\leq} W))) \to P \underset{˙}{\lor} R (N) = P \underset{˙}{\lor} N (P)$
18	9 14 17	3eqtr3rd	$⊢ ((K \in HL \land W \in H) \land (F \in T \land N \in T) \land (R (F) = R (N) \land (P \in A \land \neg P \underset{˙}{\leq} W))) \to P \underset{˙}{\lor} N (P) = F (P) \underset{˙}{\lor} R (F)$