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Metamath Proof Explorer

Theorem cdleme17d

Description: Part of proof of Lemma E in Crawley p. 114, first part of 4th paragraph. We show, in their notation, f_s(p)=q. TODO FIX COMMENT. (Contributed by NM, 11-Apr-2013)

		Ref	Expression
	Hypotheses	cdlemef46.b	$⊢ B = {Base}_{K}$
		cdlemef46.l	$⊢ \underset{˙}{\leq} = \leq_{K}$
		cdlemef46.j	$⊢ \underset{˙}{\lor} = join (K)$
		cdlemef46.m	$⊢ \underset{˙}{\land} = meet (K)$
		cdlemef46.a	$⊢ A = Atoms (K)$
		cdlemef46.h	$⊢ H = LHyp (K)$
		cdlemef46.u	$⊢ U = (P \underset{˙}{\lor} Q) \underset{˙}{\land} W$
		cdlemef46.d	$⊢ D = (t \underset{˙}{\lor} U) \underset{˙}{\land} (Q \underset{˙}{\lor} ((P \underset{˙}{\lor} t) \underset{˙}{\land} W))$
		cdlemefs46.e	$⊢ E = (P \underset{˙}{\lor} Q) \underset{˙}{\land} (D \underset{˙}{\lor} ((s \underset{˙}{\lor} t) \underset{˙}{\land} W))$
		cdlemef46.f	$⊢ F = (x \in B ⟼ if ((P \neq Q \land \neg x \underset{˙}{\leq} W), (ι z \in B \| \forall s \in A ((\neg s \underset{˙}{\leq} W \land s \underset{˙}{\lor} (x \underset{˙}{\land} W) = x) \to z = if (s \underset{˙}{\leq} (P \underset{˙}{\lor} Q), (ι y \in B \| \forall t \in A ((\neg t \underset{˙}{\leq} W \land \neg t \underset{˙}{\leq} (P \underset{˙}{\lor} Q)) \to y = E)), ⦋ s / t ⦌ D) \underset{˙}{\lor} (x \underset{˙}{\land} W))), x))$
	Assertion	cdleme17d	$⊢ ((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \to F (P) = Q$

Proof

Step	Hyp	Ref	Expression
1		cdlemef46.b	$⊢ B = {Base}_{K}$
2		cdlemef46.l	$⊢ \underset{˙}{\leq} = \leq_{K}$
3		cdlemef46.j	$⊢ \underset{˙}{\lor} = join (K)$
4		cdlemef46.m	$⊢ \underset{˙}{\land} = meet (K)$
5		cdlemef46.a	$⊢ A = Atoms (K)$
6		cdlemef46.h	$⊢ H = LHyp (K)$
7		cdlemef46.u	$⊢ U = (P \underset{˙}{\lor} Q) \underset{˙}{\land} W$
8		cdlemef46.d	$⊢ D = (t \underset{˙}{\lor} U) \underset{˙}{\land} (Q \underset{˙}{\lor} ((P \underset{˙}{\lor} t) \underset{˙}{\land} W))$
9		cdlemefs46.e	$⊢ E = (P \underset{˙}{\lor} Q) \underset{˙}{\land} (D \underset{˙}{\lor} ((s \underset{˙}{\lor} t) \underset{˙}{\land} W))$
10		cdlemef46.f	$⊢ F = (x \in B ⟼ if ((P \neq Q \land \neg x \underset{˙}{\leq} W), (ι z \in B \| \forall s \in A ((\neg s \underset{˙}{\leq} W \land s \underset{˙}{\lor} (x \underset{˙}{\land} W) = x) \to z = if (s \underset{˙}{\leq} (P \underset{˙}{\lor} Q), (ι y \in B \| \forall t \in A ((\neg t \underset{˙}{\leq} W \land \neg t \underset{˙}{\leq} (P \underset{˙}{\lor} Q)) \to y = E)), ⦋ s / t ⦌ D) \underset{˙}{\lor} (x \underset{˙}{\land} W))), x))$
11	1 2 3 4 5 6 7 8 9 10	cdleme17d4	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land P = Q) \to F (P) = Q$
12	1 2 3 4 5 6 7 8 9 10	cdleme17d3	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land P \neq Q) \to F (P) = Q$
13	11 12	pm2.61dane	$⊢ ((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \to F (P) = Q$