cdlemg18d

Description: Show two lines intersect at an atom. TODO: fix comment. (Contributed by NM, 15-May-2013)

	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	cdlemg18d	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((F \in T \land G \in T) \land P \neq Q \land G (P) \neq P) \land (R (G) \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land F (G (P)) \underset{˙}{\lor} F (G (Q)) \neq P \underset{˙}{\lor} Q \land \neg \exists r \in A (\neg r \underset{˙}{\leq} W \land P \underset{˙}{\lor} r = Q \underset{˙}{\lor} r))) \to (P \underset{˙}{\lor} F (G (P))) \underset{˙}{\land} (Q \underset{˙}{\lor} F (G (Q))) \in A$

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	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((F \in T \land G \in T) \land P \neq Q \land G (P) \neq P) \land (R (G) \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land F (G (P)) \underset{˙}{\lor} F (G (Q)) \neq P \underset{˙}{\lor} Q \land \neg \exists r \in A (\neg r \underset{˙}{\leq} W \land P \underset{˙}{\lor} r = Q \underset{˙}{\lor} r))) \to ((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W))$
9		simp21r	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((F \in T \land G \in T) \land P \neq Q \land G (P) \neq P) \land (R (G) \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land F (G (P)) \underset{˙}{\lor} F (G (Q)) \neq P \underset{˙}{\lor} Q \land \neg \exists r \in A (\neg r \underset{˙}{\leq} W \land P \underset{˙}{\lor} r = Q \underset{˙}{\lor} r))) \to G \in T$
10		simp22	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((F \in T \land G \in T) \land P \neq Q \land G (P) \neq P) \land (R (G) \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land F (G (P)) \underset{˙}{\lor} F (G (Q)) \neq P \underset{˙}{\lor} Q \land \neg \exists r \in A (\neg r \underset{˙}{\leq} W \land P \underset{˙}{\lor} r = Q \underset{˙}{\lor} r))) \to P \neq Q$
11		simp23	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((F \in T \land G \in T) \land P \neq Q \land G (P) \neq P) \land (R (G) \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land F (G (P)) \underset{˙}{\lor} F (G (Q)) \neq P \underset{˙}{\lor} Q \land \neg \exists r \in A (\neg r \underset{˙}{\leq} W \land P \underset{˙}{\lor} r = Q \underset{˙}{\lor} r))) \to G (P) \neq P$
12		simp31	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((F \in T \land G \in T) \land P \neq Q \land G (P) \neq P) \land (R (G) \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land F (G (P)) \underset{˙}{\lor} F (G (Q)) \neq P \underset{˙}{\lor} Q \land \neg \exists r \in A (\neg r \underset{˙}{\leq} W \land P \underset{˙}{\lor} r = Q \underset{˙}{\lor} r))) \to R (G) \underset{˙}{\leq} (P \underset{˙}{\lor} Q)$
13		simp33	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((F \in T \land G \in T) \land P \neq Q \land G (P) \neq P) \land (R (G) \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land F (G (P)) \underset{˙}{\lor} F (G (Q)) \neq P \underset{˙}{\lor} Q \land \neg \exists r \in A (\neg r \underset{˙}{\leq} W \land P \underset{˙}{\lor} r = Q \underset{˙}{\lor} r))) \to \neg \exists r \in A (\neg r \underset{˙}{\leq} W \land P \underset{˙}{\lor} r = Q \underset{˙}{\lor} r)$
14	1 2 3 4 5 6 7	cdlemg17b	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (G \in T \land P \neq Q) \land (G (P) \neq P \land R (G) \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg \exists r \in A (\neg r \underset{˙}{\leq} W \land P \underset{˙}{\lor} r = Q \underset{˙}{\lor} r))) \to G (P) = Q$
15	8 9 10 11 12 13 14	syl123anc	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((F \in T \land G \in T) \land P \neq Q \land G (P) \neq P) \land (R (G) \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land F (G (P)) \underset{˙}{\lor} F (G (Q)) \neq P \underset{˙}{\lor} Q \land \neg \exists r \in A (\neg r \underset{˙}{\leq} W \land P \underset{˙}{\lor} r = Q \underset{˙}{\lor} r))) \to G (P) = Q$
16	15	fveq2d	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((F \in T \land G \in T) \land P \neq Q \land G (P) \neq P) \land (R (G) \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land F (G (P)) \underset{˙}{\lor} F (G (Q)) \neq P \underset{˙}{\lor} Q \land \neg \exists r \in A (\neg r \underset{˙}{\leq} W \land P \underset{˙}{\lor} r = Q \underset{˙}{\lor} r))) \to F (G (P)) = F (Q)$
17	16	oveq2d	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((F \in T \land G \in T) \land P \neq Q \land G (P) \neq P) \land (R (G) \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land F (G (P)) \underset{˙}{\lor} F (G (Q)) \neq P \underset{˙}{\lor} Q \land \neg \exists r \in A (\neg r \underset{˙}{\leq} W \land P \underset{˙}{\lor} r = Q \underset{˙}{\lor} r))) \to P \underset{˙}{\lor} F (G (P)) = P \underset{˙}{\lor} F (Q)$
18		simp21l	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((F \in T \land G \in T) \land P \neq Q \land G (P) \neq P) \land (R (G) \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land F (G (P)) \underset{˙}{\lor} F (G (Q)) \neq P \underset{˙}{\lor} Q \land \neg \exists r \in A (\neg r \underset{˙}{\leq} W \land P \underset{˙}{\lor} r = Q \underset{˙}{\lor} r))) \to F \in T$
19	1 2 3 4 5 6 7	cdlemg17bq	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (F \in T \land G \in T \land P \neq Q) \land (G (P) \neq P \land R (G) \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg \exists r \in A (\neg r \underset{˙}{\leq} W \land P \underset{˙}{\lor} r = Q \underset{˙}{\lor} r))) \to G (Q) = P$
20	8 18 9 10 11 12 13 19	syl133anc	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((F \in T \land G \in T) \land P \neq Q \land G (P) \neq P) \land (R (G) \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land F (G (P)) \underset{˙}{\lor} F (G (Q)) \neq P \underset{˙}{\lor} Q \land \neg \exists r \in A (\neg r \underset{˙}{\leq} W \land P \underset{˙}{\lor} r = Q \underset{˙}{\lor} r))) \to G (Q) = P$
21	20	fveq2d	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((F \in T \land G \in T) \land P \neq Q \land G (P) \neq P) \land (R (G) \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land F (G (P)) \underset{˙}{\lor} F (G (Q)) \neq P \underset{˙}{\lor} Q \land \neg \exists r \in A (\neg r \underset{˙}{\leq} W \land P \underset{˙}{\lor} r = Q \underset{˙}{\lor} r))) \to F (G (Q)) = F (P)$
22	21	oveq2d	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((F \in T \land G \in T) \land P \neq Q \land G (P) \neq P) \land (R (G) \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land F (G (P)) \underset{˙}{\lor} F (G (Q)) \neq P \underset{˙}{\lor} Q \land \neg \exists r \in A (\neg r \underset{˙}{\leq} W \land P \underset{˙}{\lor} r = Q \underset{˙}{\lor} r))) \to Q \underset{˙}{\lor} F (G (Q)) = Q \underset{˙}{\lor} F (P)$
23	17 22	oveq12d	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((F \in T \land G \in T) \land P \neq Q \land G (P) \neq P) \land (R (G) \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land F (G (P)) \underset{˙}{\lor} F (G (Q)) \neq P \underset{˙}{\lor} Q \land \neg \exists r \in A (\neg r \underset{˙}{\leq} W \land P \underset{˙}{\lor} r = Q \underset{˙}{\lor} r))) \to (P \underset{˙}{\lor} F (G (P))) \underset{˙}{\land} (Q \underset{˙}{\lor} F (G (Q))) = (P \underset{˙}{\lor} F (Q)) \underset{˙}{\land} (Q \underset{˙}{\lor} F (P))$
24		simp11	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((F \in T \land G \in T) \land P \neq Q \land G (P) \neq P) \land (R (G) \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land F (G (P)) \underset{˙}{\lor} F (G (Q)) \neq P \underset{˙}{\lor} Q \land \neg \exists r \in A (\neg r \underset{˙}{\leq} W \land P \underset{˙}{\lor} r = Q \underset{˙}{\lor} r))) \to (K \in HL \land W \in H)$
25		simp12	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((F \in T \land G \in T) \land P \neq Q \land G (P) \neq P) \land (R (G) \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land F (G (P)) \underset{˙}{\lor} F (G (Q)) \neq P \underset{˙}{\lor} Q \land \neg \exists r \in A (\neg r \underset{˙}{\leq} W \land P \underset{˙}{\lor} r = Q \underset{˙}{\lor} r))) \to (P \in A \land \neg P \underset{˙}{\leq} W)$
26		simp13	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((F \in T \land G \in T) \land P \neq Q \land G (P) \neq P) \land (R (G) \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land F (G (P)) \underset{˙}{\lor} F (G (Q)) \neq P \underset{˙}{\lor} Q \land \neg \exists r \in A (\neg r \underset{˙}{\leq} W \land P \underset{˙}{\lor} r = Q \underset{˙}{\lor} r))) \to (Q \in A \land \neg Q \underset{˙}{\leq} W)$
27		simp32	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((F \in T \land G \in T) \land P \neq Q \land G (P) \neq P) \land (R (G) \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land F (G (P)) \underset{˙}{\lor} F (G (Q)) \neq P \underset{˙}{\lor} Q \land \neg \exists r \in A (\neg r \underset{˙}{\leq} W \land P \underset{˙}{\lor} r = Q \underset{˙}{\lor} r))) \to F (G (P)) \underset{˙}{\lor} F (G (Q)) \neq P \underset{˙}{\lor} Q$
28	1 2 3 4 5 6	cdlemg11aq	$⊢ ((K \in HL \land W \in H) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (F \in T \land G \in T \land F (G (P)) \underset{˙}{\lor} F (G (Q)) \neq P \underset{˙}{\lor} Q)) \to F (G (Q)) \neq Q$
29	24 25 26 18 9 27 28	syl123anc	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((F \in T \land G \in T) \land P \neq Q \land G (P) \neq P) \land (R (G) \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land F (G (P)) \underset{˙}{\lor} F (G (Q)) \neq P \underset{˙}{\lor} Q \land \neg \exists r \in A (\neg r \underset{˙}{\leq} W \land P \underset{˙}{\lor} r = Q \underset{˙}{\lor} r))) \to F (G (Q)) \neq Q$
30	21 29	eqnetrrd	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((F \in T \land G \in T) \land P \neq Q \land G (P) \neq P) \land (R (G) \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land F (G (P)) \underset{˙}{\lor} F (G (Q)) \neq P \underset{˙}{\lor} Q \land \neg \exists r \in A (\neg r \underset{˙}{\leq} W \land P \underset{˙}{\lor} r = Q \underset{˙}{\lor} r))) \to F (P) \neq Q$
31	1 2 3 4 5 6 7	cdlemg17irq	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (F \in T \land G \in T \land P \neq Q) \land (G (P) \neq P \land R (G) \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg \exists r \in A (\neg r \underset{˙}{\leq} W \land P \underset{˙}{\lor} r = Q \underset{˙}{\lor} r))) \to F (G (Q)) = F (P)$
32	8 18 9 10 11 12 13 31	syl133anc	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((F \in T \land G \in T) \land P \neq Q \land G (P) \neq P) \land (R (G) \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land F (G (P)) \underset{˙}{\lor} F (G (Q)) \neq P \underset{˙}{\lor} Q \land \neg \exists r \in A (\neg r \underset{˙}{\leq} W \land P \underset{˙}{\lor} r = Q \underset{˙}{\lor} r))) \to F (G (Q)) = F (P)$
33	16 32	oveq12d	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((F \in T \land G \in T) \land P \neq Q \land G (P) \neq P) \land (R (G) \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land F (G (P)) \underset{˙}{\lor} F (G (Q)) \neq P \underset{˙}{\lor} Q \land \neg \exists r \in A (\neg r \underset{˙}{\leq} W \land P \underset{˙}{\lor} r = Q \underset{˙}{\lor} r))) \to F (G (P)) \underset{˙}{\lor} F (G (Q)) = F (Q) \underset{˙}{\lor} F (P)$
34	33 27	eqnetrrd	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((F \in T \land G \in T) \land P \neq Q \land G (P) \neq P) \land (R (G) \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land F (G (P)) \underset{˙}{\lor} F (G (Q)) \neq P \underset{˙}{\lor} Q \land \neg \exists r \in A (\neg r \underset{˙}{\leq} W \land P \underset{˙}{\lor} r = Q \underset{˙}{\lor} r))) \to F (Q) \underset{˙}{\lor} F (P) \neq P \underset{˙}{\lor} Q$
35		eqid	$⊢ (P \underset{˙}{\lor} Q) \underset{˙}{\land} W = (P \underset{˙}{\lor} Q) \underset{˙}{\land} W$
36	1 2 3 4 5 6 7 35	cdlemg18c	$⊢ ((K \in HL \land W \in H) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land F \in T) \land (P \neq Q \land F (P) \neq Q \land F (Q) \underset{˙}{\lor} F (P) \neq P \underset{˙}{\lor} Q)) \to (P \underset{˙}{\lor} F (Q)) \underset{˙}{\land} (Q \underset{˙}{\lor} F (P)) \in A$
37	24 25 26 18 10 30 34 36	syl133anc	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((F \in T \land G \in T) \land P \neq Q \land G (P) \neq P) \land (R (G) \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land F (G (P)) \underset{˙}{\lor} F (G (Q)) \neq P \underset{˙}{\lor} Q \land \neg \exists r \in A (\neg r \underset{˙}{\leq} W \land P \underset{˙}{\lor} r = Q \underset{˙}{\lor} r))) \to (P \underset{˙}{\lor} F (Q)) \underset{˙}{\land} (Q \underset{˙}{\lor} F (P)) \in A$
38	23 37	eqeltrd	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((F \in T \land G \in T) \land P \neq Q \land G (P) \neq P) \land (R (G) \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land F (G (P)) \underset{˙}{\lor} F (G (Q)) \neq P \underset{˙}{\lor} Q \land \neg \exists r \in A (\neg r \underset{˙}{\leq} W \land P \underset{˙}{\lor} r = Q \underset{˙}{\lor} r))) \to (P \underset{˙}{\lor} F (G (P))) \underset{˙}{\land} (Q \underset{˙}{\lor} F (G (Q))) \in A$

Metamath Proof Explorer

Theorem cdlemg18d

Proof