4atex2

Description: More general version of 4atex for a line S .\/ T not necessarily connected to P .\/ Q . (Contributed by NM, 27-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	4atex2	$⊢ ((K \in HL \land W \in H) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land (S \in A \land \neg S \underset{˙}{\leq} W)) \land (P \neq Q \land T \in A \land \exists r \in A (\neg r \underset{˙}{\leq} W \land P \underset{˙}{\lor} r = Q \underset{˙}{\lor} r))) \to \exists z \in A (\neg z \underset{˙}{\leq} W \land S \underset{˙}{\lor} z = T \underset{˙}{\lor} z)$

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		oveq1	$⊢ S = P \to S \underset{˙}{\lor} z = P \underset{˙}{\lor} z$
6	5	eqeq1d	$⊢ S = P \to (S \underset{˙}{\lor} z = T \underset{˙}{\lor} z \leftrightarrow P \underset{˙}{\lor} z = T \underset{˙}{\lor} z)$
7	6	anbi2d	$⊢ S = P \to ((\neg z \underset{˙}{\leq} W \land S \underset{˙}{\lor} z = T \underset{˙}{\lor} z) \leftrightarrow (\neg z \underset{˙}{\leq} W \land P \underset{˙}{\lor} z = T \underset{˙}{\lor} z))$
8	7	rexbidv	$⊢ S = P \to (\exists z \in A (\neg z \underset{˙}{\leq} W \land S \underset{˙}{\lor} z = T \underset{˙}{\lor} z) \leftrightarrow \exists z \in A (\neg z \underset{˙}{\leq} W \land P \underset{˙}{\lor} z = T \underset{˙}{\lor} z))$
9		simpl1	$⊢ (((K \in HL \land W \in H) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land (S \in A \land \neg S \underset{˙}{\leq} W)) \land (P \neq Q \land T \in A \land \exists r \in A (\neg r \underset{˙}{\leq} W \land P \underset{˙}{\lor} r = Q \underset{˙}{\lor} r))) \land S \neq P) \to (K \in HL \land W \in H)$
10		simpl23	$⊢ (((K \in HL \land W \in H) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land (S \in A \land \neg S \underset{˙}{\leq} W)) \land (P \neq Q \land T \in A \land \exists r \in A (\neg r \underset{˙}{\leq} W \land P \underset{˙}{\lor} r = Q \underset{˙}{\lor} r))) \land S \neq P) \to (S \in A \land \neg S \underset{˙}{\leq} W)$
11		simpl21	$⊢ (((K \in HL \land W \in H) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land (S \in A \land \neg S \underset{˙}{\leq} W)) \land (P \neq Q \land T \in A \land \exists r \in A (\neg r \underset{˙}{\leq} W \land P \underset{˙}{\lor} r = Q \underset{˙}{\lor} r))) \land S \neq P) \to (P \in A \land \neg P \underset{˙}{\leq} W)$
12		simpl32	$⊢ (((K \in HL \land W \in H) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land (S \in A \land \neg S \underset{˙}{\leq} W)) \land (P \neq Q \land T \in A \land \exists r \in A (\neg r \underset{˙}{\leq} W \land P \underset{˙}{\lor} r = Q \underset{˙}{\lor} r))) \land S \neq P) \to T \in A$
13		simpr	$⊢ (((K \in HL \land W \in H) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land (S \in A \land \neg S \underset{˙}{\leq} W)) \land (P \neq Q \land T \in A \land \exists r \in A (\neg r \underset{˙}{\leq} W \land P \underset{˙}{\lor} r = Q \underset{˙}{\lor} r))) \land S \neq P) \to S \neq P$
14		simpl22	$⊢ (((K \in HL \land W \in H) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land (S \in A \land \neg S \underset{˙}{\leq} W)) \land (P \neq Q \land T \in A \land \exists r \in A (\neg r \underset{˙}{\leq} W \land P \underset{˙}{\lor} r = Q \underset{˙}{\lor} r))) \land S \neq P) \to (Q \in A \land \neg Q \underset{˙}{\leq} W)$
15		simp23l	$⊢ ((K \in HL \land W \in H) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land (S \in A \land \neg S \underset{˙}{\leq} W)) \land (P \neq Q \land T \in A \land \exists r \in A (\neg r \underset{˙}{\leq} W \land P \underset{˙}{\lor} r = Q \underset{˙}{\lor} r))) \to S \in A$
16	15	adantr	$⊢ (((K \in HL \land W \in H) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land (S \in A \land \neg S \underset{˙}{\leq} W)) \land (P \neq Q \land T \in A \land \exists r \in A (\neg r \underset{˙}{\leq} W \land P \underset{˙}{\lor} r = Q \underset{˙}{\lor} r))) \land S \neq P) \to S \in A$
17		simpl31	$⊢ (((K \in HL \land W \in H) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land (S \in A \land \neg S \underset{˙}{\leq} W)) \land (P \neq Q \land T \in A \land \exists r \in A (\neg r \underset{˙}{\leq} W \land P \underset{˙}{\lor} r = Q \underset{˙}{\lor} r))) \land S \neq P) \to P \neq Q$
18		simpl33	$⊢ (((K \in HL \land W \in H) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land (S \in A \land \neg S \underset{˙}{\leq} W)) \land (P \neq Q \land T \in A \land \exists r \in A (\neg r \underset{˙}{\leq} W \land P \underset{˙}{\lor} r = Q \underset{˙}{\lor} r))) \land S \neq P) \to \exists r \in A (\neg r \underset{˙}{\leq} W \land P \underset{˙}{\lor} r = Q \underset{˙}{\lor} r)$
19	1 2 3 4	4atex	$⊢ ((K \in HL \land W \in H) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land S \in A) \land (P \neq Q \land \exists r \in A (\neg r \underset{˙}{\leq} W \land P \underset{˙}{\lor} r = Q \underset{˙}{\lor} r))) \to \exists y \in A (\neg y \underset{˙}{\leq} W \land P \underset{˙}{\lor} y = S \underset{˙}{\lor} y)$
20	9 11 14 16 17 18 19	syl132anc	$⊢ (((K \in HL \land W \in H) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land (S \in A \land \neg S \underset{˙}{\leq} W)) \land (P \neq Q \land T \in A \land \exists r \in A (\neg r \underset{˙}{\leq} W \land P \underset{˙}{\lor} r = Q \underset{˙}{\lor} r))) \land S \neq P) \to \exists y \in A (\neg y \underset{˙}{\leq} W \land P \underset{˙}{\lor} y = S \underset{˙}{\lor} y)$
21		eqcom	$⊢ P \underset{˙}{\lor} y = S \underset{˙}{\lor} y \leftrightarrow S \underset{˙}{\lor} y = P \underset{˙}{\lor} y$
22	21	anbi2i	$⊢ (\neg y \underset{˙}{\leq} W \land P \underset{˙}{\lor} y = S \underset{˙}{\lor} y) \leftrightarrow (\neg y \underset{˙}{\leq} W \land S \underset{˙}{\lor} y = P \underset{˙}{\lor} y)$
23	22	rexbii	$⊢ \exists y \in A (\neg y \underset{˙}{\leq} W \land P \underset{˙}{\lor} y = S \underset{˙}{\lor} y) \leftrightarrow \exists y \in A (\neg y \underset{˙}{\leq} W \land S \underset{˙}{\lor} y = P \underset{˙}{\lor} y)$
24	20 23	sylib	$⊢ (((K \in HL \land W \in H) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land (S \in A \land \neg S \underset{˙}{\leq} W)) \land (P \neq Q \land T \in A \land \exists r \in A (\neg r \underset{˙}{\leq} W \land P \underset{˙}{\lor} r = Q \underset{˙}{\lor} r))) \land S \neq P) \to \exists y \in A (\neg y \underset{˙}{\leq} W \land S \underset{˙}{\lor} y = P \underset{˙}{\lor} y)$
25	1 2 3 4	4atex	$⊢ ((K \in HL \land W \in H) \land ((S \in A \land \neg S \underset{˙}{\leq} W) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land T \in A) \land (S \neq P \land \exists y \in A (\neg y \underset{˙}{\leq} W \land S \underset{˙}{\lor} y = P \underset{˙}{\lor} y))) \to \exists z \in A (\neg z \underset{˙}{\leq} W \land S \underset{˙}{\lor} z = T \underset{˙}{\lor} z)$
26	9 10 11 12 13 24 25	syl132anc	$⊢ (((K \in HL \land W \in H) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land (S \in A \land \neg S \underset{˙}{\leq} W)) \land (P \neq Q \land T \in A \land \exists r \in A (\neg r \underset{˙}{\leq} W \land P \underset{˙}{\lor} r = Q \underset{˙}{\lor} r))) \land S \neq P) \to \exists z \in A (\neg z \underset{˙}{\leq} W \land S \underset{˙}{\lor} z = T \underset{˙}{\lor} z)$
27		simp1	$⊢ ((K \in HL \land W \in H) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land (S \in A \land \neg S \underset{˙}{\leq} W)) \land (P \neq Q \land T \in A \land \exists r \in A (\neg r \underset{˙}{\leq} W \land P \underset{˙}{\lor} r = Q \underset{˙}{\lor} r))) \to (K \in HL \land W \in H)$
28		simp21	$⊢ ((K \in HL \land W \in H) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land (S \in A \land \neg S \underset{˙}{\leq} W)) \land (P \neq Q \land T \in A \land \exists r \in A (\neg r \underset{˙}{\leq} W \land P \underset{˙}{\lor} r = Q \underset{˙}{\lor} r))) \to (P \in A \land \neg P \underset{˙}{\leq} W)$
29		simp22	$⊢ ((K \in HL \land W \in H) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land (S \in A \land \neg S \underset{˙}{\leq} W)) \land (P \neq Q \land T \in A \land \exists r \in A (\neg r \underset{˙}{\leq} W \land P \underset{˙}{\lor} r = Q \underset{˙}{\lor} r))) \to (Q \in A \land \neg Q \underset{˙}{\leq} W)$
30		simp32	$⊢ ((K \in HL \land W \in H) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land (S \in A \land \neg S \underset{˙}{\leq} W)) \land (P \neq Q \land T \in A \land \exists r \in A (\neg r \underset{˙}{\leq} W \land P \underset{˙}{\lor} r = Q \underset{˙}{\lor} r))) \to T \in A$
31		simp31	$⊢ ((K \in HL \land W \in H) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land (S \in A \land \neg S \underset{˙}{\leq} W)) \land (P \neq Q \land T \in A \land \exists r \in A (\neg r \underset{˙}{\leq} W \land P \underset{˙}{\lor} r = Q \underset{˙}{\lor} r))) \to P \neq Q$
32		simp33	$⊢ ((K \in HL \land W \in H) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land (S \in A \land \neg S \underset{˙}{\leq} W)) \land (P \neq Q \land T \in A \land \exists r \in A (\neg r \underset{˙}{\leq} W \land P \underset{˙}{\lor} r = Q \underset{˙}{\lor} r))) \to \exists r \in A (\neg r \underset{˙}{\leq} W \land P \underset{˙}{\lor} r = Q \underset{˙}{\lor} r)$
33	1 2 3 4	4atex	$⊢ ((K \in HL \land W \in H) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land T \in A) \land (P \neq Q \land \exists r \in A (\neg r \underset{˙}{\leq} W \land P \underset{˙}{\lor} r = Q \underset{˙}{\lor} r))) \to \exists z \in A (\neg z \underset{˙}{\leq} W \land P \underset{˙}{\lor} z = T \underset{˙}{\lor} z)$
34	27 28 29 30 31 32 33	syl132anc	$⊢ ((K \in HL \land W \in H) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land (S \in A \land \neg S \underset{˙}{\leq} W)) \land (P \neq Q \land T \in A \land \exists r \in A (\neg r \underset{˙}{\leq} W \land P \underset{˙}{\lor} r = Q \underset{˙}{\lor} r))) \to \exists z \in A (\neg z \underset{˙}{\leq} W \land P \underset{˙}{\lor} z = T \underset{˙}{\lor} z)$
35	8 26 34	pm2.61ne	$⊢ ((K \in HL \land W \in H) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land (S \in A \land \neg S \underset{˙}{\leq} W)) \land (P \neq Q \land T \in A \land \exists r \in A (\neg r \underset{˙}{\leq} W \land P \underset{˙}{\lor} r = Q \underset{˙}{\lor} r))) \to \exists z \in A (\neg z \underset{˙}{\leq} W \land S \underset{˙}{\lor} z = T \underset{˙}{\lor} z)$

Metamath Proof Explorer

Theorem 4atex2

Proof