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Metamath Proof Explorer

Theorem cdlemg1bOLDN

Description: This theorem can be used to shorten F = hypothesis that have the form of the conclusion. TODO: fix comment. (Contributed by NM, 16-Apr-2013) (New usage is discouraged.)

		Ref	Expression
	Hypotheses	cdlemg1.b	$⊢ B = {Base}_{K}$
		cdlemg1.l	$⊢ \underset{˙}{\leq} = \leq_{K}$
		cdlemg1.j	$⊢ \underset{˙}{\lor} = join (K)$
		cdlemg1.m	$⊢ \underset{˙}{\land} = meet (K)$
		cdlemg1.a	$⊢ A = Atoms (K)$
		cdlemg1.h	$⊢ H = LHyp (K)$
		cdlemg1b.u	$⊢ U = (P \underset{˙}{\lor} Q) \underset{˙}{\land} W$
		cdlemg1b.d	$⊢ D = (t \underset{˙}{\lor} U) \underset{˙}{\land} (Q \underset{˙}{\lor} ((P \underset{˙}{\lor} t) \underset{˙}{\land} W))$
		cdlemg1b.e	$⊢ E = (P \underset{˙}{\lor} Q) \underset{˙}{\land} (D \underset{˙}{\lor} ((s \underset{˙}{\lor} t) \underset{˙}{\land} W))$
		cdlemg1b.f	$⊢ F = (ι f \in T \| f (P) = Q)$
		cdlemg1b.t	$⊢ T = LTrn (K) (W)$
	Assertion	cdlemg1bOLDN	$⊢ ((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \to F = (x \in B ⟼ if ((P \neq Q \land \neg x \underset{˙}{\leq} W), (ι z \in B \| \forall s \in A ((\neg s \underset{˙}{\leq} W \land s \underset{˙}{\lor} (x \underset{˙}{\land} W) = x) \to z = if (s \underset{˙}{\leq} (P \underset{˙}{\lor} Q), (ι y \in B \| \forall t \in A ((\neg t \underset{˙}{\leq} W \land \neg t \underset{˙}{\leq} (P \underset{˙}{\lor} Q)) \to y = E)), ⦋ s / t ⦌ D) \underset{˙}{\lor} (x \underset{˙}{\land} W))), x))$

Proof

Step	Hyp	Ref	Expression
1		cdlemg1.b	$⊢ B = {Base}_{K}$
2		cdlemg1.l	$⊢ \underset{˙}{\leq} = \leq_{K}$
3		cdlemg1.j	$⊢ \underset{˙}{\lor} = join (K)$
4		cdlemg1.m	$⊢ \underset{˙}{\land} = meet (K)$
5		cdlemg1.a	$⊢ A = Atoms (K)$
6		cdlemg1.h	$⊢ H = LHyp (K)$
7		cdlemg1b.u	$⊢ U = (P \underset{˙}{\lor} Q) \underset{˙}{\land} W$
8		cdlemg1b.d	$⊢ D = (t \underset{˙}{\lor} U) \underset{˙}{\land} (Q \underset{˙}{\lor} ((P \underset{˙}{\lor} t) \underset{˙}{\land} W))$
9		cdlemg1b.e	$⊢ E = (P \underset{˙}{\lor} Q) \underset{˙}{\land} (D \underset{˙}{\lor} ((s \underset{˙}{\lor} t) \underset{˙}{\land} W))$
10		cdlemg1b.f	$⊢ F = (ι f \in T \| f (P) = Q)$
11		cdlemg1b.t	$⊢ T = LTrn (K) (W)$
12		eqid	$⊢ (x \in B ⟼ if ((P \neq Q \land \neg x \underset{˙}{\leq} W), (ι z \in B \| \forall s \in A ((\neg s \underset{˙}{\leq} W \land s \underset{˙}{\lor} (x \underset{˙}{\land} W) = x) \to z = if (s \underset{˙}{\leq} (P \underset{˙}{\lor} Q), (ι y \in B \| \forall t \in A ((\neg t \underset{˙}{\leq} W \land \neg t \underset{˙}{\leq} (P \underset{˙}{\lor} Q)) \to y = E)), ⦋ s / t ⦌ D) \underset{˙}{\lor} (x \underset{˙}{\land} W))), x)) = (x \in B ⟼ if ((P \neq Q \land \neg x \underset{˙}{\leq} W), (ι z \in B \| \forall s \in A ((\neg s \underset{˙}{\leq} W \land s \underset{˙}{\lor} (x \underset{˙}{\land} W) = x) \to z = if (s \underset{˙}{\leq} (P \underset{˙}{\lor} Q), (ι y \in B \| \forall t \in A ((\neg t \underset{˙}{\leq} W \land \neg t \underset{˙}{\leq} (P \underset{˙}{\lor} Q)) \to y = E)), ⦋ s / t ⦌ D) \underset{˙}{\lor} (x \underset{˙}{\land} W))), x))$
13	1 2 3 4 5 6 7 8 9 12 11 10	cdlemg1b2	$⊢ ((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \to F = (x \in B ⟼ if ((P \neq Q \land \neg x \underset{˙}{\leq} W), (ι z \in B \| \forall s \in A ((\neg s \underset{˙}{\leq} W \land s \underset{˙}{\lor} (x \underset{˙}{\land} W) = x) \to z = if (s \underset{˙}{\leq} (P \underset{˙}{\lor} Q), (ι y \in B \| \forall t \in A ((\neg t \underset{˙}{\leq} W \land \neg t \underset{˙}{\leq} (P \underset{˙}{\lor} Q)) \to y = E)), ⦋ s / t ⦌ D) \underset{˙}{\lor} (x \underset{˙}{\land} W))), x))$