4atex3

Description: More general version of 4atex for a line S .\/ T not necessarily connected to P .\/ Q . (Contributed by NM, 29-May-2013)

	Ref	Expression
Hypotheses	4that.l	$⊢ \underset{˙}{\leq} = \leq_{K}$
	4that.j	$⊢ \underset{˙}{\lor} = join (K)$
	4that.a	$⊢ A = Atoms (K)$
	4that.h	$⊢ H = LHyp (K)$
Assertion	4atex3	$⊢ ((K \in HL \land W \in H) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land (S \in A \land \neg S \underset{˙}{\leq} W)) \land (P \neq Q \land (T \in A \land S \neq T) \land \exists r \in A (\neg r \underset{˙}{\leq} W \land P \underset{˙}{\lor} r = Q \underset{˙}{\lor} r))) \to \exists z \in A (\neg z \underset{˙}{\leq} W \land (z \neq S \land z \neq T \land z \underset{˙}{\leq} (S \underset{˙}{\lor} T)))$

Proof

Step	Hyp	Ref	Expression
1		4that.l	$⊢ \underset{˙}{\leq} = \leq_{K}$
2		4that.j	$⊢ \underset{˙}{\lor} = join (K)$
3		4that.a	$⊢ A = Atoms (K)$
4		4that.h	$⊢ H = LHyp (K)$
5		simp1	$⊢ ((K \in HL \land W \in H) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land (S \in A \land \neg S \underset{˙}{\leq} W)) \land (P \neq Q \land (T \in A \land S \neq T) \land \exists r \in A (\neg r \underset{˙}{\leq} W \land P \underset{˙}{\lor} r = Q \underset{˙}{\lor} r))) \to (K \in HL \land W \in H)$
6		simp2	$⊢ ((K \in HL \land W \in H) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land (S \in A \land \neg S \underset{˙}{\leq} W)) \land (P \neq Q \land (T \in A \land S \neq T) \land \exists r \in A (\neg r \underset{˙}{\leq} W \land P \underset{˙}{\lor} r = Q \underset{˙}{\lor} r))) \to ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land (S \in A \land \neg S \underset{˙}{\leq} W))$
7		simp31	$⊢ ((K \in HL \land W \in H) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land (S \in A \land \neg S \underset{˙}{\leq} W)) \land (P \neq Q \land (T \in A \land S \neq T) \land \exists r \in A (\neg r \underset{˙}{\leq} W \land P \underset{˙}{\lor} r = Q \underset{˙}{\lor} r))) \to P \neq Q$
8		simp32l	$⊢ ((K \in HL \land W \in H) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land (S \in A \land \neg S \underset{˙}{\leq} W)) \land (P \neq Q \land (T \in A \land S \neq T) \land \exists r \in A (\neg r \underset{˙}{\leq} W \land P \underset{˙}{\lor} r = Q \underset{˙}{\lor} r))) \to T \in A$
9		simp33	$⊢ ((K \in HL \land W \in H) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land (S \in A \land \neg S \underset{˙}{\leq} W)) \land (P \neq Q \land (T \in A \land S \neq T) \land \exists r \in A (\neg r \underset{˙}{\leq} W \land P \underset{˙}{\lor} r = Q \underset{˙}{\lor} r))) \to \exists r \in A (\neg r \underset{˙}{\leq} W \land P \underset{˙}{\lor} r = Q \underset{˙}{\lor} r)$
10	1 2 3 4	4atex2	$⊢ ((K \in HL \land W \in H) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land (S \in A \land \neg S \underset{˙}{\leq} W)) \land (P \neq Q \land T \in A \land \exists r \in A (\neg r \underset{˙}{\leq} W \land P \underset{˙}{\lor} r = Q \underset{˙}{\lor} r))) \to \exists z \in A (\neg z \underset{˙}{\leq} W \land S \underset{˙}{\lor} z = T \underset{˙}{\lor} z)$
11	5 6 7 8 9 10	syl113anc	$⊢ ((K \in HL \land W \in H) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land (S \in A \land \neg S \underset{˙}{\leq} W)) \land (P \neq Q \land (T \in A \land S \neq T) \land \exists r \in A (\neg r \underset{˙}{\leq} W \land P \underset{˙}{\lor} r = Q \underset{˙}{\lor} r))) \to \exists z \in A (\neg z \underset{˙}{\leq} W \land S \underset{˙}{\lor} z = T \underset{˙}{\lor} z)$
12		simp1l	$⊢ ((K \in HL \land W \in H) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land (S \in A \land \neg S \underset{˙}{\leq} W)) \land (P \neq Q \land (T \in A \land S \neq T) \land \exists r \in A (\neg r \underset{˙}{\leq} W \land P \underset{˙}{\lor} r = Q \underset{˙}{\lor} r))) \to K \in HL$
13		hlcvl	$⊢ K \in HL \to K \in CvLat$
14	12 13	syl	$⊢ ((K \in HL \land W \in H) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land (S \in A \land \neg S \underset{˙}{\leq} W)) \land (P \neq Q \land (T \in A \land S \neq T) \land \exists r \in A (\neg r \underset{˙}{\leq} W \land P \underset{˙}{\lor} r = Q \underset{˙}{\lor} r))) \to K \in CvLat$
15	14	adantr	$⊢ (((K \in HL \land W \in H) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land (S \in A \land \neg S \underset{˙}{\leq} W)) \land (P \neq Q \land (T \in A \land S \neq T) \land \exists r \in A (\neg r \underset{˙}{\leq} W \land P \underset{˙}{\lor} r = Q \underset{˙}{\lor} r))) \land z \in A) \to K \in CvLat$
16		simp23l	$⊢ ((K \in HL \land W \in H) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land (S \in A \land \neg S \underset{˙}{\leq} W)) \land (P \neq Q \land (T \in A \land S \neq T) \land \exists r \in A (\neg r \underset{˙}{\leq} W \land P \underset{˙}{\lor} r = Q \underset{˙}{\lor} r))) \to S \in A$
17	16	adantr	$⊢ (((K \in HL \land W \in H) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land (S \in A \land \neg S \underset{˙}{\leq} W)) \land (P \neq Q \land (T \in A \land S \neq T) \land \exists r \in A (\neg r \underset{˙}{\leq} W \land P \underset{˙}{\lor} r = Q \underset{˙}{\lor} r))) \land z \in A) \to S \in A$
18	8	adantr	$⊢ (((K \in HL \land W \in H) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land (S \in A \land \neg S \underset{˙}{\leq} W)) \land (P \neq Q \land (T \in A \land S \neq T) \land \exists r \in A (\neg r \underset{˙}{\leq} W \land P \underset{˙}{\lor} r = Q \underset{˙}{\lor} r))) \land z \in A) \to T \in A$
19		simpr	$⊢ (((K \in HL \land W \in H) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land (S \in A \land \neg S \underset{˙}{\leq} W)) \land (P \neq Q \land (T \in A \land S \neq T) \land \exists r \in A (\neg r \underset{˙}{\leq} W \land P \underset{˙}{\lor} r = Q \underset{˙}{\lor} r))) \land z \in A) \to z \in A$
20		simp32r	$⊢ ((K \in HL \land W \in H) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land (S \in A \land \neg S \underset{˙}{\leq} W)) \land (P \neq Q \land (T \in A \land S \neq T) \land \exists r \in A (\neg r \underset{˙}{\leq} W \land P \underset{˙}{\lor} r = Q \underset{˙}{\lor} r))) \to S \neq T$
21	20	adantr	$⊢ (((K \in HL \land W \in H) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land (S \in A \land \neg S \underset{˙}{\leq} W)) \land (P \neq Q \land (T \in A \land S \neq T) \land \exists r \in A (\neg r \underset{˙}{\leq} W \land P \underset{˙}{\lor} r = Q \underset{˙}{\lor} r))) \land z \in A) \to S \neq T$
22	3 1 2	cvlsupr2	$⊢ (K \in CvLat \land (S \in A \land T \in A \land z \in A) \land S \neq T) \to (S \underset{˙}{\lor} z = T \underset{˙}{\lor} z \leftrightarrow (z \neq S \land z \neq T \land z \underset{˙}{\leq} (S \underset{˙}{\lor} T)))$
23	15 17 18 19 21 22	syl131anc	$⊢ (((K \in HL \land W \in H) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land (S \in A \land \neg S \underset{˙}{\leq} W)) \land (P \neq Q \land (T \in A \land S \neq T) \land \exists r \in A (\neg r \underset{˙}{\leq} W \land P \underset{˙}{\lor} r = Q \underset{˙}{\lor} r))) \land z \in A) \to (S \underset{˙}{\lor} z = T \underset{˙}{\lor} z \leftrightarrow (z \neq S \land z \neq T \land z \underset{˙}{\leq} (S \underset{˙}{\lor} T)))$
24	23	anbi2d	$⊢ (((K \in HL \land W \in H) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land (S \in A \land \neg S \underset{˙}{\leq} W)) \land (P \neq Q \land (T \in A \land S \neq T) \land \exists r \in A (\neg r \underset{˙}{\leq} W \land P \underset{˙}{\lor} r = Q \underset{˙}{\lor} r))) \land z \in A) \to ((\neg z \underset{˙}{\leq} W \land S \underset{˙}{\lor} z = T \underset{˙}{\lor} z) \leftrightarrow (\neg z \underset{˙}{\leq} W \land (z \neq S \land z \neq T \land z \underset{˙}{\leq} (S \underset{˙}{\lor} T))))$
25	24	rexbidva	$⊢ ((K \in HL \land W \in H) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land (S \in A \land \neg S \underset{˙}{\leq} W)) \land (P \neq Q \land (T \in A \land S \neq T) \land \exists r \in A (\neg r \underset{˙}{\leq} W \land P \underset{˙}{\lor} r = Q \underset{˙}{\lor} r))) \to (\exists z \in A (\neg z \underset{˙}{\leq} W \land S \underset{˙}{\lor} z = T \underset{˙}{\lor} z) \leftrightarrow \exists z \in A (\neg z \underset{˙}{\leq} W \land (z \neq S \land z \neq T \land z \underset{˙}{\leq} (S \underset{˙}{\lor} T))))$
26	11 25	mpbid	$⊢ ((K \in HL \land W \in H) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land (S \in A \land \neg S \underset{˙}{\leq} W)) \land (P \neq Q \land (T \in A \land S \neq T) \land \exists r \in A (\neg r \underset{˙}{\leq} W \land P \underset{˙}{\lor} r = Q \underset{˙}{\lor} r))) \to \exists z \in A (\neg z \underset{˙}{\leq} W \land (z \neq S \land z \neq T \land z \underset{˙}{\leq} (S \underset{˙}{\lor} T)))$

Metamath Proof Explorer

Theorem 4atex3

Proof