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Metamath Proof Explorer

Theorem cdleme50rn

Description: Part of proof of Lemma D in Crawley p. 113. TODO: fix comment. (Contributed by NM, 9-Apr-2013)

		Ref	Expression
	Hypotheses	cdlemef50.b	$⊢ B = {Base}_{K}$
		cdlemef50.l	$⊢ \underset{˙}{\leq} = \leq_{K}$
		cdlemef50.j	$⊢ \underset{˙}{\lor} = join (K)$
		cdlemef50.m	$⊢ \underset{˙}{\land} = meet (K)$
		cdlemef50.a	$⊢ A = Atoms (K)$
		cdlemef50.h	$⊢ H = LHyp (K)$
		cdlemef50.u	$⊢ U = (P \underset{˙}{\lor} Q) \underset{˙}{\land} W$
		cdlemef50.d	$⊢ D = (t \underset{˙}{\lor} U) \underset{˙}{\land} (Q \underset{˙}{\lor} ((P \underset{˙}{\lor} t) \underset{˙}{\land} W))$
		cdlemefs50.e	$⊢ E = (P \underset{˙}{\lor} Q) \underset{˙}{\land} (D \underset{˙}{\lor} ((s \underset{˙}{\lor} t) \underset{˙}{\land} W))$
		cdlemef50.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	cdleme50rn	$⊢ ((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \to ran F = B$

Proof

Step	Hyp	Ref	Expression
1		cdlemef50.b	$⊢ B = {Base}_{K}$
2		cdlemef50.l	$⊢ \underset{˙}{\leq} = \leq_{K}$
3		cdlemef50.j	$⊢ \underset{˙}{\lor} = join (K)$
4		cdlemef50.m	$⊢ \underset{˙}{\land} = meet (K)$
5		cdlemef50.a	$⊢ A = Atoms (K)$
6		cdlemef50.h	$⊢ H = LHyp (K)$
7		cdlemef50.u	$⊢ U = (P \underset{˙}{\lor} Q) \underset{˙}{\land} W$
8		cdlemef50.d	$⊢ D = (t \underset{˙}{\lor} U) \underset{˙}{\land} (Q \underset{˙}{\lor} ((P \underset{˙}{\lor} t) \underset{˙}{\land} W))$
9		cdlemefs50.e	$⊢ E = (P \underset{˙}{\lor} Q) \underset{˙}{\land} (D \underset{˙}{\lor} ((s \underset{˙}{\lor} t) \underset{˙}{\land} W))$
10		cdlemef50.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		eqid	$⊢ (Q \underset{˙}{\lor} P) \underset{˙}{\land} W = (Q \underset{˙}{\lor} P) \underset{˙}{\land} W$
12		eqid	$⊢ (v \underset{˙}{\lor} ((Q \underset{˙}{\lor} P) \underset{˙}{\land} W)) \underset{˙}{\land} (P \underset{˙}{\lor} ((Q \underset{˙}{\lor} v) \underset{˙}{\land} W)) = (v \underset{˙}{\lor} ((Q \underset{˙}{\lor} P) \underset{˙}{\land} W)) \underset{˙}{\land} (P \underset{˙}{\lor} ((Q \underset{˙}{\lor} v) \underset{˙}{\land} W))$
13		eqid	$⊢ (Q \underset{˙}{\lor} P) \underset{˙}{\land} (((v \underset{˙}{\lor} ((Q \underset{˙}{\lor} P) \underset{˙}{\land} W)) \underset{˙}{\land} (P \underset{˙}{\lor} ((Q \underset{˙}{\lor} v) \underset{˙}{\land} W))) \underset{˙}{\lor} ((u \underset{˙}{\lor} v) \underset{˙}{\land} W)) = (Q \underset{˙}{\lor} P) \underset{˙}{\land} (((v \underset{˙}{\lor} ((Q \underset{˙}{\lor} P) \underset{˙}{\land} W)) \underset{˙}{\land} (P \underset{˙}{\lor} ((Q \underset{˙}{\lor} v) \underset{˙}{\land} W))) \underset{˙}{\lor} ((u \underset{˙}{\lor} v) \underset{˙}{\land} W))$
14		eqid	$⊢ (a \in B ⟼ if ((Q \neq P \land \neg a \underset{˙}{\leq} W), (ι c \in B \| \forall u \in A ((\neg u \underset{˙}{\leq} W \land u \underset{˙}{\lor} (a \underset{˙}{\land} W) = a) \to c = if (u \underset{˙}{\leq} (Q \underset{˙}{\lor} P), (ι b \in B \| \forall v \in A ((\neg v \underset{˙}{\leq} W \land \neg v \underset{˙}{\leq} (Q \underset{˙}{\lor} P)) \to b = (Q \underset{˙}{\lor} P) \underset{˙}{\land} (((v \underset{˙}{\lor} ((Q \underset{˙}{\lor} P) \underset{˙}{\land} W)) \underset{˙}{\land} (P \underset{˙}{\lor} ((Q \underset{˙}{\lor} v) \underset{˙}{\land} W))) \underset{˙}{\lor} ((u \underset{˙}{\lor} v) \underset{˙}{\land} W)))), ⦋ u / v ⦌ ((v \underset{˙}{\lor} ((Q \underset{˙}{\lor} P) \underset{˙}{\land} W)) \underset{˙}{\land} (P \underset{˙}{\lor} ((Q \underset{˙}{\lor} v) \underset{˙}{\land} W)))) \underset{˙}{\lor} (a \underset{˙}{\land} W))), a)) = (a \in B ⟼ if ((Q \neq P \land \neg a \underset{˙}{\leq} W), (ι c \in B \| \forall u \in A ((\neg u \underset{˙}{\leq} W \land u \underset{˙}{\lor} (a \underset{˙}{\land} W) = a) \to c = if (u \underset{˙}{\leq} (Q \underset{˙}{\lor} P), (ι b \in B \| \forall v \in A ((\neg v \underset{˙}{\leq} W \land \neg v \underset{˙}{\leq} (Q \underset{˙}{\lor} P)) \to b = (Q \underset{˙}{\lor} P) \underset{˙}{\land} (((v \underset{˙}{\lor} ((Q \underset{˙}{\lor} P) \underset{˙}{\land} W)) \underset{˙}{\land} (P \underset{˙}{\lor} ((Q \underset{˙}{\lor} v) \underset{˙}{\land} W))) \underset{˙}{\lor} ((u \underset{˙}{\lor} v) \underset{˙}{\land} W)))), ⦋ u / v ⦌ ((v \underset{˙}{\lor} ((Q \underset{˙}{\lor} P) \underset{˙}{\land} W)) \underset{˙}{\land} (P \underset{˙}{\lor} ((Q \underset{˙}{\lor} v) \underset{˙}{\land} W)))) \underset{˙}{\lor} (a \underset{˙}{\land} W))), a))$
15	1 2 3 4 5 6 7 8 9 10 11 12 13 14	cdleme50rnlem	$⊢ ((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \to ran F = B$