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Metamath Proof Explorer

Theorem dalem61

Description: Lemma for dath . Show that atoms D , E , and F lie on the same line (axis of perspectivity). Eliminate hypotheses containing dummy atoms c and d . (Contributed by NM, 11-Aug-2012)

		Ref	Expression
	Hypotheses	dalem.ph	$⊢ φ \leftrightarrow (((K \in HL \land C \in {Base}_{K}) \land (P \in A \land Q \in A \land R \in A) \land (S \in A \land T \in A \land U \in A)) \land (Y \in O \land Z \in O) \land ((\neg C \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg C \underset{˙}{\leq} (Q \underset{˙}{\lor} R) \land \neg C \underset{˙}{\leq} (R \underset{˙}{\lor} P)) \land (\neg C \underset{˙}{\leq} (S \underset{˙}{\lor} T) \land \neg C \underset{˙}{\leq} (T \underset{˙}{\lor} U) \land \neg C \underset{˙}{\leq} (U \underset{˙}{\lor} S)) \land (C \underset{˙}{\leq} (P \underset{˙}{\lor} S) \land C \underset{˙}{\leq} (Q \underset{˙}{\lor} T) \land C \underset{˙}{\leq} (R \underset{˙}{\lor} U))))$
		dalem.l	$⊢ \underset{˙}{\leq} = \leq_{K}$
		dalem.j	$⊢ \underset{˙}{\lor} = join (K)$
		dalem.a	$⊢ A = Atoms (K)$
		dalem.ps	$⊢ ψ \leftrightarrow ((c \in A \land d \in A) \land \neg c \underset{˙}{\leq} Y \land (d \neq c \land \neg d \underset{˙}{\leq} Y \land C \underset{˙}{\leq} (c \underset{˙}{\lor} d)))$
		dalem61.m	$⊢ \underset{˙}{\land} = meet (K)$
		dalem61.o	$⊢ O = LPlanes (K)$
		dalem61.y	$⊢ Y = (P \underset{˙}{\lor} Q) \underset{˙}{\lor} R$
		dalem61.z	$⊢ Z = (S \underset{˙}{\lor} T) \underset{˙}{\lor} U$
		dalem61.d	$⊢ D = (P \underset{˙}{\lor} Q) \underset{˙}{\land} (S \underset{˙}{\lor} T)$
		dalem61.e	$⊢ E = (Q \underset{˙}{\lor} R) \underset{˙}{\land} (T \underset{˙}{\lor} U)$
		dalem61.f	$⊢ F = (R \underset{˙}{\lor} P) \underset{˙}{\land} (U \underset{˙}{\lor} S)$
	Assertion	dalem61	$⊢ (φ \land Y = Z \land ψ) \to F \underset{˙}{\leq} (D \underset{˙}{\lor} E)$

Proof

Step	Hyp	Ref	Expression
1		dalem.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		dalem.l	$⊢ \underset{˙}{\leq} = \leq_{K}$
3		dalem.j	$⊢ \underset{˙}{\lor} = join (K)$
4		dalem.a	$⊢ A = Atoms (K)$
5		dalem.ps	$⊢ ψ \leftrightarrow ((c \in A \land d \in A) \land \neg c \underset{˙}{\leq} Y \land (d \neq c \land \neg d \underset{˙}{\leq} Y \land C \underset{˙}{\leq} (c \underset{˙}{\lor} d)))$
6		dalem61.m	$⊢ \underset{˙}{\land} = meet (K)$
7		dalem61.o	$⊢ O = LPlanes (K)$
8		dalem61.y	$⊢ Y = (P \underset{˙}{\lor} Q) \underset{˙}{\lor} R$
9		dalem61.z	$⊢ Z = (S \underset{˙}{\lor} T) \underset{˙}{\lor} U$
10		dalem61.d	$⊢ D = (P \underset{˙}{\lor} Q) \underset{˙}{\land} (S \underset{˙}{\lor} T)$
11		dalem61.e	$⊢ E = (Q \underset{˙}{\lor} R) \underset{˙}{\land} (T \underset{˙}{\lor} U)$
12		dalem61.f	$⊢ F = (R \underset{˙}{\lor} P) \underset{˙}{\land} (U \underset{˙}{\lor} S)$
13		eqid	$⊢ (c \underset{˙}{\lor} P) \underset{˙}{\land} (d \underset{˙}{\lor} S) = (c \underset{˙}{\lor} P) \underset{˙}{\land} (d \underset{˙}{\lor} S)$
14		eqid	$⊢ (c \underset{˙}{\lor} Q) \underset{˙}{\land} (d \underset{˙}{\lor} T) = (c \underset{˙}{\lor} Q) \underset{˙}{\land} (d \underset{˙}{\lor} T)$
15		eqid	$⊢ (c \underset{˙}{\lor} R) \underset{˙}{\land} (d \underset{˙}{\lor} U) = (c \underset{˙}{\lor} R) \underset{˙}{\land} (d \underset{˙}{\lor} U)$
16		eqid	$⊢ ((((c \underset{˙}{\lor} P) \underset{˙}{\land} (d \underset{˙}{\lor} S)) \underset{˙}{\lor} ((c \underset{˙}{\lor} Q) \underset{˙}{\land} (d \underset{˙}{\lor} T))) \underset{˙}{\lor} ((c \underset{˙}{\lor} R) \underset{˙}{\land} (d \underset{˙}{\lor} U))) \underset{˙}{\land} Y = ((((c \underset{˙}{\lor} P) \underset{˙}{\land} (d \underset{˙}{\lor} S)) \underset{˙}{\lor} ((c \underset{˙}{\lor} Q) \underset{˙}{\land} (d \underset{˙}{\lor} T))) \underset{˙}{\lor} ((c \underset{˙}{\lor} R) \underset{˙}{\land} (d \underset{˙}{\lor} U))) \underset{˙}{\land} Y$
17	1 2 3 4 5 6 7 8 9 12 13 14 15 16	dalem59	$⊢ (φ \land Y = Z \land ψ) \to F \underset{˙}{\leq} (((((c \underset{˙}{\lor} P) \underset{˙}{\land} (d \underset{˙}{\lor} S)) \underset{˙}{\lor} ((c \underset{˙}{\lor} Q) \underset{˙}{\land} (d \underset{˙}{\lor} T))) \underset{˙}{\lor} ((c \underset{˙}{\lor} R) \underset{˙}{\land} (d \underset{˙}{\lor} U))) \underset{˙}{\land} Y)$
18	1 2 3 4 5 6 7 8 9 10 11 13 14 15 16	dalem60	$⊢ (φ \land Y = Z \land ψ) \to D \underset{˙}{\lor} E = ((((c \underset{˙}{\lor} P) \underset{˙}{\land} (d \underset{˙}{\lor} S)) \underset{˙}{\lor} ((c \underset{˙}{\lor} Q) \underset{˙}{\land} (d \underset{˙}{\lor} T))) \underset{˙}{\lor} ((c \underset{˙}{\lor} R) \underset{˙}{\land} (d \underset{˙}{\lor} U))) \underset{˙}{\land} Y$
19	17 18	breqtrrd	$⊢ (φ \land Y = Z \land ψ) \to F \underset{˙}{\leq} (D \underset{˙}{\lor} E)$