cdleme4gfv

Description: Part of proof of Lemma D in Crawley p. 113. TODO: Can this replace uses of cdleme32a ? TODO: Can this be used to help prove the R or S case where X is an atom? TODO: Would an antecedent transformer like cdleme46f2g2 help? (Contributed by NM, 8-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	cdleme4gfv	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (P \neq Q \land (X \in B \land \neg X \underset{˙}{\leq} W)) \land ((S \in A \land \neg S \underset{˙}{\leq} W) \land S \underset{˙}{\lor} (X \underset{˙}{\land} W) = X)) \to G (X) = G (S) \underset{˙}{\lor} (X \underset{˙}{\land} W)$

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		simp11	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (P \neq Q \land (X \in B \land \neg X \underset{˙}{\leq} W)) \land ((S \in A \land \neg S \underset{˙}{\leq} W) \land S \underset{˙}{\lor} (X \underset{˙}{\land} W) = X)) \to (K \in HL \land W \in H)$
12		simp13	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (P \neq Q \land (X \in B \land \neg X \underset{˙}{\leq} W)) \land ((S \in A \land \neg S \underset{˙}{\leq} W) \land S \underset{˙}{\lor} (X \underset{˙}{\land} W) = X)) \to (Q \in A \land \neg Q \underset{˙}{\leq} W)$
13		simp12	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (P \neq Q \land (X \in B \land \neg X \underset{˙}{\leq} W)) \land ((S \in A \land \neg S \underset{˙}{\leq} W) \land S \underset{˙}{\lor} (X \underset{˙}{\land} W) = X)) \to (P \in A \land \neg P \underset{˙}{\leq} W)$
14		simp2l	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (P \neq Q \land (X \in B \land \neg X \underset{˙}{\leq} W)) \land ((S \in A \land \neg S \underset{˙}{\leq} W) \land S \underset{˙}{\lor} (X \underset{˙}{\land} W) = X)) \to P \neq Q$
15	14	necomd	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (P \neq Q \land (X \in B \land \neg X \underset{˙}{\leq} W)) \land ((S \in A \land \neg S \underset{˙}{\leq} W) \land S \underset{˙}{\lor} (X \underset{˙}{\land} W) = X)) \to Q \neq P$
16		simp2r	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (P \neq Q \land (X \in B \land \neg X \underset{˙}{\leq} W)) \land ((S \in A \land \neg S \underset{˙}{\leq} W) \land S \underset{˙}{\lor} (X \underset{˙}{\land} W) = X)) \to (X \in B \land \neg X \underset{˙}{\leq} W)$
17		simp3	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (P \neq Q \land (X \in B \land \neg X \underset{˙}{\leq} W)) \land ((S \in A \land \neg S \underset{˙}{\leq} W) \land S \underset{˙}{\lor} (X \underset{˙}{\land} W) = X)) \to ((S \in A \land \neg S \underset{˙}{\leq} W) \land S \underset{˙}{\lor} (X \underset{˙}{\land} W) = X)$
18	1 2 3 4 5 6 7 8 9 10	cdleme48fv	$⊢ (((K \in HL \land W \in H) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land (P \in A \land \neg P \underset{˙}{\leq} W)) \land (Q \neq P \land (X \in B \land \neg X \underset{˙}{\leq} W)) \land ((S \in A \land \neg S \underset{˙}{\leq} W) \land S \underset{˙}{\lor} (X \underset{˙}{\land} W) = X)) \to G (X) = G (S) \underset{˙}{\lor} (X \underset{˙}{\land} W)$
19	11 12 13 15 16 17 18	syl321anc	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (P \neq Q \land (X \in B \land \neg X \underset{˙}{\leq} W)) \land ((S \in A \land \neg S \underset{˙}{\leq} W) \land S \underset{˙}{\lor} (X \underset{˙}{\land} W) = X)) \to G (X) = G (S) \underset{˙}{\lor} (X \underset{˙}{\land} W)$

Metamath Proof Explorer

Theorem cdleme4gfv

Proof