cdlemeg46nlpq

Description: Show that ( GS ) is not under P .\/ Q when S isn't. (Contributed by NM, 3-Apr-2013)

	Ref	Expression
Hypotheses	cdlemef46g.b	$⊢ B = {Base}_{K}$
	cdlemef46g.l	$⊢ \underset{˙}{\leq} = \leq_{K}$
	cdlemef46g.j	$⊢ \underset{˙}{\lor} = join (K)$
	cdlemef46g.m	$⊢ \underset{˙}{\land} = meet (K)$
	cdlemef46g.a	$⊢ A = Atoms (K)$
	cdlemef46g.h	$⊢ H = LHyp (K)$
	cdlemef46g.u	$⊢ U = (P \underset{˙}{\lor} Q) \underset{˙}{\land} W$
	cdlemef46g.d	$⊢ D = (t \underset{˙}{\lor} U) \underset{˙}{\land} (Q \underset{˙}{\lor} ((P \underset{˙}{\lor} t) \underset{˙}{\land} W))$
	cdlemefs46g.e	$⊢ E = (P \underset{˙}{\lor} Q) \underset{˙}{\land} (D \underset{˙}{\lor} ((s \underset{˙}{\lor} t) \underset{˙}{\land} W))$
	cdlemef46g.f	$⊢ F = (x \in B ⟼ if ((P \neq Q \land \neg x \underset{˙}{\leq} W), (ι z \in B \| \forall s \in A ((\neg s \underset{˙}{\leq} W \land s \underset{˙}{\lor} (x \underset{˙}{\land} W) = x) \to z = if (s \underset{˙}{\leq} (P \underset{˙}{\lor} Q), (ι y \in B \| \forall t \in A ((\neg t \underset{˙}{\leq} W \land \neg t \underset{˙}{\leq} (P \underset{˙}{\lor} Q)) \to y = E)), ⦋ s / t ⦌ D) \underset{˙}{\lor} (x \underset{˙}{\land} W))), x))$
	cdlemef46.v	$⊢ V = (Q \underset{˙}{\lor} P) \underset{˙}{\land} W$
	cdlemef46.n	$⊢ N = (v \underset{˙}{\lor} V) \underset{˙}{\land} (P \underset{˙}{\lor} ((Q \underset{˙}{\lor} v) \underset{˙}{\land} W))$
	cdlemefs46.o	$⊢ O = (Q \underset{˙}{\lor} P) \underset{˙}{\land} (N \underset{˙}{\lor} ((u \underset{˙}{\lor} v) \underset{˙}{\land} W))$
	cdlemef46.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	cdlemeg46nlpq	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (P \neq Q \land (S \in A \land \neg S \underset{˙}{\leq} W)) \land \neg S \underset{˙}{\leq} (P \underset{˙}{\lor} Q)) \to \neg G (S) \underset{˙}{\leq} (P \underset{˙}{\lor} Q)$

Proof

Step	Hyp	Ref	Expression
1		cdlemef46g.b	$⊢ B = {Base}_{K}$
2		cdlemef46g.l	$⊢ \underset{˙}{\leq} = \leq_{K}$
3		cdlemef46g.j	$⊢ \underset{˙}{\lor} = join (K)$
4		cdlemef46g.m	$⊢ \underset{˙}{\land} = meet (K)$
5		cdlemef46g.a	$⊢ A = Atoms (K)$
6		cdlemef46g.h	$⊢ H = LHyp (K)$
7		cdlemef46g.u	$⊢ U = (P \underset{˙}{\lor} Q) \underset{˙}{\land} W$
8		cdlemef46g.d	$⊢ D = (t \underset{˙}{\lor} U) \underset{˙}{\land} (Q \underset{˙}{\lor} ((P \underset{˙}{\lor} t) \underset{˙}{\land} W))$
9		cdlemefs46g.e	$⊢ E = (P \underset{˙}{\lor} Q) \underset{˙}{\land} (D \underset{˙}{\lor} ((s \underset{˙}{\lor} t) \underset{˙}{\land} W))$
10		cdlemef46g.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		cdlemef46.v	$⊢ V = (Q \underset{˙}{\lor} P) \underset{˙}{\land} W$
12		cdlemef46.n	$⊢ N = (v \underset{˙}{\lor} V) \underset{˙}{\land} (P \underset{˙}{\lor} ((Q \underset{˙}{\lor} v) \underset{˙}{\land} W))$
13		cdlemefs46.o	$⊢ O = (Q \underset{˙}{\lor} P) \underset{˙}{\land} (N \underset{˙}{\lor} ((u \underset{˙}{\lor} v) \underset{˙}{\land} W))$
14		cdlemef46.g	$⊢ G = (a \in B ⟼ if ((Q \neq P \land \neg a \underset{˙}{\leq} W), (ι c \in B \| \forall u \in A ((\neg u \underset{˙}{\leq} W \land u \underset{˙}{\lor} (a \underset{˙}{\land} W) = a) \to c = if (u \underset{˙}{\leq} (Q \underset{˙}{\lor} P), (ι b \in B \| \forall v \in A ((\neg v \underset{˙}{\leq} W \land \neg v \underset{˙}{\leq} (Q \underset{˙}{\lor} P)) \to b = O)), ⦋ u / v ⦌ N) \underset{˙}{\lor} (a \underset{˙}{\land} W))), a))$
15		simp11	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (P \neq Q \land (S \in A \land \neg S \underset{˙}{\leq} W)) \land \neg S \underset{˙}{\leq} (P \underset{˙}{\lor} Q)) \to (K \in HL \land W \in H)$
16		simp13	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (P \neq Q \land (S \in A \land \neg S \underset{˙}{\leq} W)) \land \neg S \underset{˙}{\leq} (P \underset{˙}{\lor} Q)) \to (Q \in A \land \neg Q \underset{˙}{\leq} W)$
17		simp12	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (P \neq Q \land (S \in A \land \neg S \underset{˙}{\leq} W)) \land \neg S \underset{˙}{\leq} (P \underset{˙}{\lor} Q)) \to (P \in A \land \neg P \underset{˙}{\leq} W)$
18		simp2l	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (P \neq Q \land (S \in A \land \neg S \underset{˙}{\leq} W)) \land \neg S \underset{˙}{\leq} (P \underset{˙}{\lor} Q)) \to P \neq Q$
19	18	necomd	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (P \neq Q \land (S \in A \land \neg S \underset{˙}{\leq} W)) \land \neg S \underset{˙}{\leq} (P \underset{˙}{\lor} Q)) \to Q \neq P$
20		simp2r	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (P \neq Q \land (S \in A \land \neg S \underset{˙}{\leq} W)) \land \neg S \underset{˙}{\leq} (P \underset{˙}{\lor} Q)) \to (S \in A \land \neg S \underset{˙}{\leq} W)$
21		simp3	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (P \neq Q \land (S \in A \land \neg S \underset{˙}{\leq} W)) \land \neg S \underset{˙}{\leq} (P \underset{˙}{\lor} Q)) \to \neg S \underset{˙}{\leq} (P \underset{˙}{\lor} Q)$
22		simp11l	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (P \neq Q \land (S \in A \land \neg S \underset{˙}{\leq} W)) \land \neg S \underset{˙}{\leq} (P \underset{˙}{\lor} Q)) \to K \in HL$
23		simp12l	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (P \neq Q \land (S \in A \land \neg S \underset{˙}{\leq} W)) \land \neg S \underset{˙}{\leq} (P \underset{˙}{\lor} Q)) \to P \in A$
24		simp13l	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (P \neq Q \land (S \in A \land \neg S \underset{˙}{\leq} W)) \land \neg S \underset{˙}{\leq} (P \underset{˙}{\lor} Q)) \to Q \in A$
25	3 5	hlatjcom	$⊢ (K \in HL \land P \in A \land Q \in A) \to P \underset{˙}{\lor} Q = Q \underset{˙}{\lor} P$
26	22 23 24 25	syl3anc	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (P \neq Q \land (S \in A \land \neg S \underset{˙}{\leq} W)) \land \neg S \underset{˙}{\leq} (P \underset{˙}{\lor} Q)) \to P \underset{˙}{\lor} Q = Q \underset{˙}{\lor} P$
27	26	breq2d	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (P \neq Q \land (S \in A \land \neg S \underset{˙}{\leq} W)) \land \neg S \underset{˙}{\leq} (P \underset{˙}{\lor} Q)) \to (S \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \leftrightarrow S \underset{˙}{\leq} (Q \underset{˙}{\lor} P))$
28	21 27	mtbid	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (P \neq Q \land (S \in A \land \neg S \underset{˙}{\leq} W)) \land \neg S \underset{˙}{\leq} (P \underset{˙}{\lor} Q)) \to \neg S \underset{˙}{\leq} (Q \underset{˙}{\lor} P)$
29	1 2 3 4 5 6 11 12 13 14	cdleme46frvlpq	$⊢ (((K \in HL \land W \in H) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land (P \in A \land \neg P \underset{˙}{\leq} W)) \land (Q \neq P \land (S \in A \land \neg S \underset{˙}{\leq} W)) \land \neg S \underset{˙}{\leq} (Q \underset{˙}{\lor} P)) \to \neg G (S) \underset{˙}{\leq} (Q \underset{˙}{\lor} P)$
30	15 16 17 19 20 28 29	syl321anc	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (P \neq Q \land (S \in A \land \neg S \underset{˙}{\leq} W)) \land \neg S \underset{˙}{\leq} (P \underset{˙}{\lor} Q)) \to \neg G (S) \underset{˙}{\leq} (Q \underset{˙}{\lor} P)$
31	26	breq2d	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (P \neq Q \land (S \in A \land \neg S \underset{˙}{\leq} W)) \land \neg S \underset{˙}{\leq} (P \underset{˙}{\lor} Q)) \to (G (S) \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \leftrightarrow G (S) \underset{˙}{\leq} (Q \underset{˙}{\lor} P))$
32	30 31	mtbird	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (P \neq Q \land (S \in A \land \neg S \underset{˙}{\leq} W)) \land \neg S \underset{˙}{\leq} (P \underset{˙}{\lor} Q)) \to \neg G (S) \underset{˙}{\leq} (P \underset{˙}{\lor} Q)$

Metamath Proof Explorer

Theorem cdlemeg46nlpq

Proof