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Metamath Proof Explorer

Theorem cdleme21at

Description: Part of proof of Lemma E in Crawley p. 115. (Contributed by NM, 29-Nov-2012)

		Ref	Expression
	Hypotheses	cdleme21.l	$⊢ \underset{˙}{\leq} = \leq_{K}$
		cdleme21.j	$⊢ \underset{˙}{\lor} = join (K)$
		cdleme21.m	$⊢ \underset{˙}{\land} = meet (K)$
		cdleme21.a	$⊢ A = Atoms (K)$
		cdleme21.h	$⊢ H = LHyp (K)$
		cdleme21.u	$⊢ U = (P \underset{˙}{\lor} Q) \underset{˙}{\land} W$
	Assertion	cdleme21at	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land Q \in A) \land ((S \in A \land P \neq Q \land \neg S \underset{˙}{\leq} (P \underset{˙}{\lor} Q)) \land U \underset{˙}{\leq} (S \underset{˙}{\lor} T)) \land (z \in A \land P \underset{˙}{\lor} z = S \underset{˙}{\lor} z)) \to T \neq z$

Proof

Step	Hyp	Ref	Expression
1		cdleme21.l	$⊢ \underset{˙}{\leq} = \leq_{K}$
2		cdleme21.j	$⊢ \underset{˙}{\lor} = join (K)$
3		cdleme21.m	$⊢ \underset{˙}{\land} = meet (K)$
4		cdleme21.a	$⊢ A = Atoms (K)$
5		cdleme21.h	$⊢ H = LHyp (K)$
6		cdleme21.u	$⊢ U = (P \underset{˙}{\lor} Q) \underset{˙}{\land} W$
7	1 2 3 4 5 6	cdleme21c	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land Q \in A) \land (S \in A \land P \neq Q \land \neg S \underset{˙}{\leq} (P \underset{˙}{\lor} Q)) \land (z \in A \land P \underset{˙}{\lor} z = S \underset{˙}{\lor} z)) \to \neg U \underset{˙}{\leq} (S \underset{˙}{\lor} z)$
8	7	3adant2r	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land Q \in A) \land ((S \in A \land P \neq Q \land \neg S \underset{˙}{\leq} (P \underset{˙}{\lor} Q)) \land U \underset{˙}{\leq} (S \underset{˙}{\lor} T)) \land (z \in A \land P \underset{˙}{\lor} z = S \underset{˙}{\lor} z)) \to \neg U \underset{˙}{\leq} (S \underset{˙}{\lor} z)$
9		simp2r	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land Q \in A) \land ((S \in A \land P \neq Q \land \neg S \underset{˙}{\leq} (P \underset{˙}{\lor} Q)) \land U \underset{˙}{\leq} (S \underset{˙}{\lor} T)) \land (z \in A \land P \underset{˙}{\lor} z = S \underset{˙}{\lor} z)) \to U \underset{˙}{\leq} (S \underset{˙}{\lor} T)$
10		oveq2	$⊢ T = z \to S \underset{˙}{\lor} T = S \underset{˙}{\lor} z$
11	10	breq2d	$⊢ T = z \to (U \underset{˙}{\leq} (S \underset{˙}{\lor} T) \leftrightarrow U \underset{˙}{\leq} (S \underset{˙}{\lor} z))$
12	9 11	syl5ibcom	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land Q \in A) \land ((S \in A \land P \neq Q \land \neg S \underset{˙}{\leq} (P \underset{˙}{\lor} Q)) \land U \underset{˙}{\leq} (S \underset{˙}{\lor} T)) \land (z \in A \land P \underset{˙}{\lor} z = S \underset{˙}{\lor} z)) \to (T = z \to U \underset{˙}{\leq} (S \underset{˙}{\lor} z))$
13	12	necon3bd	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land Q \in A) \land ((S \in A \land P \neq Q \land \neg S \underset{˙}{\leq} (P \underset{˙}{\lor} Q)) \land U \underset{˙}{\leq} (S \underset{˙}{\lor} T)) \land (z \in A \land P \underset{˙}{\lor} z = S \underset{˙}{\lor} z)) \to (\neg U \underset{˙}{\leq} (S \underset{˙}{\lor} z) \to T \neq z)$
14	8 13	mpd	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land Q \in A) \land ((S \in A \land P \neq Q \land \neg S \underset{˙}{\leq} (P \underset{˙}{\lor} Q)) \land U \underset{˙}{\leq} (S \underset{˙}{\lor} T)) \land (z \in A \land P \underset{˙}{\lor} z = S \underset{˙}{\lor} z)) \to T \neq z$