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Metamath Proof Explorer

Theorem cdlemeg47rv

Description: Value of g_s(r) when r is an atom under pq and s is any atom not under pq, using very compact hypotheses. TODO: FIX COMMENT. (Contributed by NM, 3-Apr-2013)

		Ref	Expression
	Hypotheses	cdlemef47.b	$⊢ B = {Base}_{K}$
		cdlemef47.l	$⊢ \underset{˙}{\leq} = \leq_{K}$
		cdlemef47.j	$⊢ \underset{˙}{\lor} = join (K)$
		cdlemef47.m	$⊢ \underset{˙}{\land} = meet (K)$
		cdlemef47.a	$⊢ A = Atoms (K)$
		cdlemef47.h	$⊢ H = LHyp (K)$
		cdlemef47.v	$⊢ V = (Q \underset{˙}{\lor} P) \underset{˙}{\land} W$
		cdlemef47.n	$⊢ N = (v \underset{˙}{\lor} V) \underset{˙}{\land} (P \underset{˙}{\lor} ((Q \underset{˙}{\lor} v) \underset{˙}{\land} W))$
		cdlemefs47.o	$⊢ O = (Q \underset{˙}{\lor} P) \underset{˙}{\land} (N \underset{˙}{\lor} ((u \underset{˙}{\lor} v) \underset{˙}{\land} W))$
		cdlemef47.g	$⊢ G = (a \in B ⟼ if ((Q \neq P \land \neg a \underset{˙}{\leq} W), (ι c \in B \| \forall u \in A ((\neg u \underset{˙}{\leq} W \land u \underset{˙}{\lor} (a \underset{˙}{\land} W) = a) \to c = if (u \underset{˙}{\leq} (Q \underset{˙}{\lor} P), (ι b \in B \| \forall v \in A ((\neg v \underset{˙}{\leq} W \land \neg v \underset{˙}{\leq} (Q \underset{˙}{\lor} P)) \to b = O)), ⦋ u / v ⦌ N) \underset{˙}{\lor} (a \underset{˙}{\land} W))), a))$
	Assertion	cdlemeg47rv	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (P \neq Q \land (R \in A \land \neg R \underset{˙}{\leq} W) \land (S \in A \land \neg S \underset{˙}{\leq} W)) \land (R \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg S \underset{˙}{\leq} (P \underset{˙}{\lor} Q))) \to G (R) = ⦋ R / u ⦌ ⦋ S / v ⦌ O$

Proof

Step	Hyp	Ref	Expression
1		cdlemef47.b	$⊢ B = {Base}_{K}$
2		cdlemef47.l	$⊢ \underset{˙}{\leq} = \leq_{K}$
3		cdlemef47.j	$⊢ \underset{˙}{\lor} = join (K)$
4		cdlemef47.m	$⊢ \underset{˙}{\land} = meet (K)$
5		cdlemef47.a	$⊢ A = Atoms (K)$
6		cdlemef47.h	$⊢ H = LHyp (K)$
7		cdlemef47.v	$⊢ V = (Q \underset{˙}{\lor} P) \underset{˙}{\land} W$
8		cdlemef47.n	$⊢ N = (v \underset{˙}{\lor} V) \underset{˙}{\land} (P \underset{˙}{\lor} ((Q \underset{˙}{\lor} v) \underset{˙}{\land} W))$
9		cdlemefs47.o	$⊢ O = (Q \underset{˙}{\lor} P) \underset{˙}{\land} (N \underset{˙}{\lor} ((u \underset{˙}{\lor} v) \underset{˙}{\land} W))$
10		cdlemef47.g	$⊢ G = (a \in B ⟼ if ((Q \neq P \land \neg a \underset{˙}{\leq} W), (ι c \in B \| \forall u \in A ((\neg u \underset{˙}{\leq} W \land u \underset{˙}{\lor} (a \underset{˙}{\land} W) = a) \to c = if (u \underset{˙}{\leq} (Q \underset{˙}{\lor} P), (ι b \in B \| \forall v \in A ((\neg v \underset{˙}{\leq} W \land \neg v \underset{˙}{\leq} (Q \underset{˙}{\lor} P)) \to b = O)), ⦋ u / v ⦌ N) \underset{˙}{\lor} (a \underset{˙}{\land} W))), a))$
11	3 5	cdleme46f2g1	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (P \neq Q \land (R \in A \land \neg R \underset{˙}{\leq} W) \land (S \in A \land \neg S \underset{˙}{\leq} W)) \land (R \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg S \underset{˙}{\leq} (P \underset{˙}{\lor} Q))) \to (((K \in HL \land W \in H) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land (P \in A \land \neg P \underset{˙}{\leq} W)) \land (Q \neq P \land (R \in A \land \neg R \underset{˙}{\leq} W) \land (S \in A \land \neg S \underset{˙}{\leq} W)) \land (R \underset{˙}{\leq} (Q \underset{˙}{\lor} P) \land \neg S \underset{˙}{\leq} (Q \underset{˙}{\lor} P)))$
12	1 2 3 4 5 6 7 8 10 9	cdlemefs45	$⊢ (((K \in HL \land W \in H) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land (P \in A \land \neg P \underset{˙}{\leq} W)) \land (Q \neq P \land (R \in A \land \neg R \underset{˙}{\leq} W) \land (S \in A \land \neg S \underset{˙}{\leq} W)) \land (R \underset{˙}{\leq} (Q \underset{˙}{\lor} P) \land \neg S \underset{˙}{\leq} (Q \underset{˙}{\lor} P))) \to G (R) = ⦋ R / u ⦌ ⦋ S / v ⦌ O$
13	11 12	syl	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (P \neq Q \land (R \in A \land \neg R \underset{˙}{\leq} W) \land (S \in A \land \neg S \underset{˙}{\leq} W)) \land (R \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg S \underset{˙}{\leq} (P \underset{˙}{\lor} Q))) \to G (R) = ⦋ R / u ⦌ ⦋ S / v ⦌ O$