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Metamath Proof Explorer

Theorem dalemdnee

Description: Lemma for dath . Axis of perspectivity points D and E are different. (Contributed by NM, 10-Aug-2012)

		Ref	Expression
	Hypotheses	dalema.ph	$⊢ φ \leftrightarrow (((K \in HL \land C \in {Base}_{K}) \land (P \in A \land Q \in A \land R \in A) \land (S \in A \land T \in A \land U \in A)) \land (Y \in O \land Z \in O) \land ((\neg C \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg C \underset{˙}{\leq} (Q \underset{˙}{\lor} R) \land \neg C \underset{˙}{\leq} (R \underset{˙}{\lor} P)) \land (\neg C \underset{˙}{\leq} (S \underset{˙}{\lor} T) \land \neg C \underset{˙}{\leq} (T \underset{˙}{\lor} U) \land \neg C \underset{˙}{\leq} (U \underset{˙}{\lor} S)) \land (C \underset{˙}{\leq} (P \underset{˙}{\lor} S) \land C \underset{˙}{\leq} (Q \underset{˙}{\lor} T) \land C \underset{˙}{\leq} (R \underset{˙}{\lor} U))))$
		dalemc.l	$⊢ \underset{˙}{\leq} = \leq_{K}$
		dalemc.j	$⊢ \underset{˙}{\lor} = join (K)$
		dalemc.a	$⊢ A = Atoms (K)$
		dalem3.m	$⊢ \underset{˙}{\land} = meet (K)$
		dalem3.o	$⊢ O = LPlanes (K)$
		dalem3.y	$⊢ Y = (P \underset{˙}{\lor} Q) \underset{˙}{\lor} R$
		dalem3.z	$⊢ Z = (S \underset{˙}{\lor} T) \underset{˙}{\lor} U$
		dalem3.d	$⊢ D = (P \underset{˙}{\lor} Q) \underset{˙}{\land} (S \underset{˙}{\lor} T)$
		dalem3.e	$⊢ E = (Q \underset{˙}{\lor} R) \underset{˙}{\land} (T \underset{˙}{\lor} U)$
	Assertion	dalemdnee	$⊢ φ \to D \neq E$

Proof

Step	Hyp	Ref	Expression
1		dalema.ph	$⊢ φ \leftrightarrow (((K \in HL \land C \in {Base}_{K}) \land (P \in A \land Q \in A \land R \in A) \land (S \in A \land T \in A \land U \in A)) \land (Y \in O \land Z \in O) \land ((\neg C \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg C \underset{˙}{\leq} (Q \underset{˙}{\lor} R) \land \neg C \underset{˙}{\leq} (R \underset{˙}{\lor} P)) \land (\neg C \underset{˙}{\leq} (S \underset{˙}{\lor} T) \land \neg C \underset{˙}{\leq} (T \underset{˙}{\lor} U) \land \neg C \underset{˙}{\leq} (U \underset{˙}{\lor} S)) \land (C \underset{˙}{\leq} (P \underset{˙}{\lor} S) \land C \underset{˙}{\leq} (Q \underset{˙}{\lor} T) \land C \underset{˙}{\leq} (R \underset{˙}{\lor} U))))$
2		dalemc.l	$⊢ \underset{˙}{\leq} = \leq_{K}$
3		dalemc.j	$⊢ \underset{˙}{\lor} = join (K)$
4		dalemc.a	$⊢ A = Atoms (K)$
5		dalem3.m	$⊢ \underset{˙}{\land} = meet (K)$
6		dalem3.o	$⊢ O = LPlanes (K)$
7		dalem3.y	$⊢ Y = (P \underset{˙}{\lor} Q) \underset{˙}{\lor} R$
8		dalem3.z	$⊢ Z = (S \underset{˙}{\lor} T) \underset{˙}{\lor} U$
9		dalem3.d	$⊢ D = (P \underset{˙}{\lor} Q) \underset{˙}{\land} (S \underset{˙}{\lor} T)$
10		dalem3.e	$⊢ E = (Q \underset{˙}{\lor} R) \underset{˙}{\land} (T \underset{˙}{\lor} U)$
11		simpr	$⊢ (φ \land D = Q) \to D = Q$
12	1 2 3 4 6 7	dalemqnet	$⊢ φ \to Q \neq T$
13	12	adantr	$⊢ (φ \land D = Q) \to Q \neq T$
14	11 13	eqnetrd	$⊢ (φ \land D = Q) \to D \neq T$
15	1 2 3 4 5 6 7 8 9 10	dalem4	$⊢ (φ \land D \neq T) \to D \neq E$
16	14 15	syldan	$⊢ (φ \land D = Q) \to D \neq E$
17	1 2 3 4 5 6 7 8 9 10	dalem3	$⊢ (φ \land D \neq Q) \to D \neq E$
18	16 17	pm2.61dane	$⊢ φ \to D \neq E$