cdlemg17dN

Description: TODO: fix comment. (Contributed by NM, 9-May-2013) (New usage is discouraged.)

	Ref	Expression
Hypotheses	cdlemg12.l	$⊢ \underset{˙}{\leq} = \leq_{K}$
	cdlemg12.j	$⊢ \underset{˙}{\lor} = join (K)$
	cdlemg12.m	$⊢ \underset{˙}{\land} = meet (K)$
	cdlemg12.a	$⊢ A = Atoms (K)$
	cdlemg12.h	$⊢ H = LHyp (K)$
	cdlemg12.t	$⊢ T = LTrn (K) (W)$
	cdlemg12b.r	$⊢ R = trL (K) (W)$
Assertion	cdlemg17dN	$⊢ ((K \in HL \land W \in H \land G \in T) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land P \neq Q) \land (R (G) \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg \exists r \in A (\neg r \underset{˙}{\leq} W \land P \underset{˙}{\lor} r = Q \underset{˙}{\lor} r) \land G (P) \neq P)) \to R (G) = (P \underset{˙}{\lor} Q) \underset{˙}{\land} W$

Proof

Step	Hyp	Ref	Expression
1		cdlemg12.l	$⊢ \underset{˙}{\leq} = \leq_{K}$
2		cdlemg12.j	$⊢ \underset{˙}{\lor} = join (K)$
3		cdlemg12.m	$⊢ \underset{˙}{\land} = meet (K)$
4		cdlemg12.a	$⊢ A = Atoms (K)$
5		cdlemg12.h	$⊢ H = LHyp (K)$
6		cdlemg12.t	$⊢ T = LTrn (K) (W)$
7		cdlemg12b.r	$⊢ R = trL (K) (W)$
8		simp1	$⊢ ((K \in HL \land W \in H \land G \in T) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land P \neq Q) \land (R (G) \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg \exists r \in A (\neg r \underset{˙}{\leq} W \land P \underset{˙}{\lor} r = Q \underset{˙}{\lor} r) \land G (P) \neq P)) \to (K \in HL \land W \in H \land G \in T)$
9		simp21	$⊢ ((K \in HL \land W \in H \land G \in T) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land P \neq Q) \land (R (G) \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg \exists r \in A (\neg r \underset{˙}{\leq} W \land P \underset{˙}{\lor} r = Q \underset{˙}{\lor} r) \land G (P) \neq P)) \to (P \in A \land \neg P \underset{˙}{\leq} W)$
10		simpl1	$⊢ ((K \in HL \land W \in H \land G \in T) \land (P \in A \land \neg P \underset{˙}{\leq} W)) \to K \in HL$
11		simpl2	$⊢ ((K \in HL \land W \in H \land G \in T) \land (P \in A \land \neg P \underset{˙}{\leq} W)) \to W \in H$
12		simpl3	$⊢ ((K \in HL \land W \in H \land G \in T) \land (P \in A \land \neg P \underset{˙}{\leq} W)) \to G \in T$
13		simpr	$⊢ ((K \in HL \land W \in H \land G \in T) \land (P \in A \land \neg P \underset{˙}{\leq} W)) \to (P \in A \land \neg P \underset{˙}{\leq} W)$
14	1 2 3 4 5 6 7	trlval2	$⊢ ((K \in HL \land W \in H) \land G \in T \land (P \in A \land \neg P \underset{˙}{\leq} W)) \to R (G) = (P \underset{˙}{\lor} G (P)) \underset{˙}{\land} W$
15	10 11 12 13 14	syl211anc	$⊢ ((K \in HL \land W \in H \land G \in T) \land (P \in A \land \neg P \underset{˙}{\leq} W)) \to R (G) = (P \underset{˙}{\lor} G (P)) \underset{˙}{\land} W$
16	8 9 15	syl2anc	$⊢ ((K \in HL \land W \in H \land G \in T) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land P \neq Q) \land (R (G) \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg \exists r \in A (\neg r \underset{˙}{\leq} W \land P \underset{˙}{\lor} r = Q \underset{˙}{\lor} r) \land G (P) \neq P)) \to R (G) = (P \underset{˙}{\lor} G (P)) \underset{˙}{\land} W$
17		simp11	$⊢ ((K \in HL \land W \in H \land G \in T) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land P \neq Q) \land (R (G) \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg \exists r \in A (\neg r \underset{˙}{\leq} W \land P \underset{˙}{\lor} r = Q \underset{˙}{\lor} r) \land G (P) \neq P)) \to K \in HL$
18		simp12	$⊢ ((K \in HL \land W \in H \land G \in T) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land P \neq Q) \land (R (G) \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg \exists r \in A (\neg r \underset{˙}{\leq} W \land P \underset{˙}{\lor} r = Q \underset{˙}{\lor} r) \land G (P) \neq P)) \to W \in H$
19	17 18	jca	$⊢ ((K \in HL \land W \in H \land G \in T) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land P \neq Q) \land (R (G) \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg \exists r \in A (\neg r \underset{˙}{\leq} W \land P \underset{˙}{\lor} r = Q \underset{˙}{\lor} r) \land G (P) \neq P)) \to (K \in HL \land W \in H)$
20		simp22	$⊢ ((K \in HL \land W \in H \land G \in T) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land P \neq Q) \land (R (G) \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg \exists r \in A (\neg r \underset{˙}{\leq} W \land P \underset{˙}{\lor} r = Q \underset{˙}{\lor} r) \land G (P) \neq P)) \to (Q \in A \land \neg Q \underset{˙}{\leq} W)$
21		simp13	$⊢ ((K \in HL \land W \in H \land G \in T) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land P \neq Q) \land (R (G) \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg \exists r \in A (\neg r \underset{˙}{\leq} W \land P \underset{˙}{\lor} r = Q \underset{˙}{\lor} r) \land G (P) \neq P)) \to G \in T$
22		simp23	$⊢ ((K \in HL \land W \in H \land G \in T) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land P \neq Q) \land (R (G) \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg \exists r \in A (\neg r \underset{˙}{\leq} W \land P \underset{˙}{\lor} r = Q \underset{˙}{\lor} r) \land G (P) \neq P)) \to P \neq Q$
23		simp33	$⊢ ((K \in HL \land W \in H \land G \in T) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land P \neq Q) \land (R (G) \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg \exists r \in A (\neg r \underset{˙}{\leq} W \land P \underset{˙}{\lor} r = Q \underset{˙}{\lor} r) \land G (P) \neq P)) \to G (P) \neq P$
24		simp31	$⊢ ((K \in HL \land W \in H \land G \in T) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land P \neq Q) \land (R (G) \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg \exists r \in A (\neg r \underset{˙}{\leq} W \land P \underset{˙}{\lor} r = Q \underset{˙}{\lor} r) \land G (P) \neq P)) \to R (G) \underset{˙}{\leq} (P \underset{˙}{\lor} Q)$
25		simp32	$⊢ ((K \in HL \land W \in H \land G \in T) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land P \neq Q) \land (R (G) \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg \exists r \in A (\neg r \underset{˙}{\leq} W \land P \underset{˙}{\lor} r = Q \underset{˙}{\lor} r) \land G (P) \neq P)) \to \neg \exists r \in A (\neg r \underset{˙}{\leq} W \land P \underset{˙}{\lor} r = Q \underset{˙}{\lor} r)$
26	1 2 3 4 5 6 7	cdlemg17b	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (G \in T \land P \neq Q) \land (G (P) \neq P \land R (G) \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg \exists r \in A (\neg r \underset{˙}{\leq} W \land P \underset{˙}{\lor} r = Q \underset{˙}{\lor} r))) \to G (P) = Q$
27	19 9 20 21 22 23 24 25 26	syl323anc	$⊢ ((K \in HL \land W \in H \land G \in T) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land P \neq Q) \land (R (G) \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg \exists r \in A (\neg r \underset{˙}{\leq} W \land P \underset{˙}{\lor} r = Q \underset{˙}{\lor} r) \land G (P) \neq P)) \to G (P) = Q$
28	27	oveq2d	$⊢ ((K \in HL \land W \in H \land G \in T) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land P \neq Q) \land (R (G) \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg \exists r \in A (\neg r \underset{˙}{\leq} W \land P \underset{˙}{\lor} r = Q \underset{˙}{\lor} r) \land G (P) \neq P)) \to P \underset{˙}{\lor} G (P) = P \underset{˙}{\lor} Q$
29	28	oveq1d	$⊢ ((K \in HL \land W \in H \land G \in T) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land P \neq Q) \land (R (G) \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg \exists r \in A (\neg r \underset{˙}{\leq} W \land P \underset{˙}{\lor} r = Q \underset{˙}{\lor} r) \land G (P) \neq P)) \to (P \underset{˙}{\lor} G (P)) \underset{˙}{\land} W = (P \underset{˙}{\lor} Q) \underset{˙}{\land} W$
30	16 29	eqtrd	$⊢ ((K \in HL \land W \in H \land G \in T) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land P \neq Q) \land (R (G) \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg \exists r \in A (\neg r \underset{˙}{\leq} W \land P \underset{˙}{\lor} r = Q \underset{˙}{\lor} r) \land G (P) \neq P)) \to R (G) = (P \underset{˙}{\lor} Q) \underset{˙}{\land} W$

Metamath Proof Explorer

Theorem cdlemg17dN

Proof