Metamath Proof Explorer

Theorem 4atex2-0bOLDN

Description: Same as 4atex2 except that T is zero. (Contributed by NM, 27-May-2013) (New usage is discouraged.)

		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	4atex2-0bOLDN	$⊢ ((K \in HL \land W \in H) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land (S \in A \land \neg S \underset{˙}{\leq} W)) \land (P \neq Q \land T = 0. (K) \land \exists r \in A (\neg r \underset{˙}{\leq} W \land P \underset{˙}{\lor} r = Q \underset{˙}{\lor} r))) \to \exists z \in A (\neg z \underset{˙}{\leq} W \land S \underset{˙}{\lor} z = T \underset{˙}{\lor} z)$

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		simp1	$⊢ ((K \in HL \land W \in H) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land (S \in A \land \neg S \underset{˙}{\leq} W)) \land (P \neq Q \land T = 0. (K) \land \exists r \in A (\neg r \underset{˙}{\leq} W \land P \underset{˙}{\lor} r = Q \underset{˙}{\lor} r))) \to (K \in HL \land W \in H)$
6		simp21	$⊢ ((K \in HL \land W \in H) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land (S \in A \land \neg S \underset{˙}{\leq} W)) \land (P \neq Q \land T = 0. (K) \land \exists r \in A (\neg r \underset{˙}{\leq} W \land P \underset{˙}{\lor} r = Q \underset{˙}{\lor} r))) \to (P \in A \land \neg P \underset{˙}{\leq} W)$
7		simp22	$⊢ ((K \in HL \land W \in H) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land (S \in A \land \neg S \underset{˙}{\leq} W)) \land (P \neq Q \land T = 0. (K) \land \exists r \in A (\neg r \underset{˙}{\leq} W \land P \underset{˙}{\lor} r = Q \underset{˙}{\lor} r))) \to (Q \in A \land \neg Q \underset{˙}{\leq} W)$
8		simp32	$⊢ ((K \in HL \land W \in H) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land (S \in A \land \neg S \underset{˙}{\leq} W)) \land (P \neq Q \land T = 0. (K) \land \exists r \in A (\neg r \underset{˙}{\leq} W \land P \underset{˙}{\lor} r = Q \underset{˙}{\lor} r))) \to T = 0. (K)$
9		simp31	$⊢ ((K \in HL \land W \in H) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land (S \in A \land \neg S \underset{˙}{\leq} W)) \land (P \neq Q \land T = 0. (K) \land \exists r \in A (\neg r \underset{˙}{\leq} W \land P \underset{˙}{\lor} r = Q \underset{˙}{\lor} r))) \to P \neq Q$
10		simp23	$⊢ ((K \in HL \land W \in H) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land (S \in A \land \neg S \underset{˙}{\leq} W)) \land (P \neq Q \land T = 0. (K) \land \exists r \in A (\neg r \underset{˙}{\leq} W \land P \underset{˙}{\lor} r = Q \underset{˙}{\lor} r))) \to (S \in A \land \neg S \underset{˙}{\leq} W)$
11		simp33	$⊢ ((K \in HL \land W \in H) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land (S \in A \land \neg S \underset{˙}{\leq} W)) \land (P \neq Q \land T = 0. (K) \land \exists r \in A (\neg r \underset{˙}{\leq} W \land P \underset{˙}{\lor} r = Q \underset{˙}{\lor} r))) \to \exists r \in A (\neg r \underset{˙}{\leq} W \land P \underset{˙}{\lor} r = Q \underset{˙}{\lor} r)$
12	1 2 3 4	4atex2-0aOLDN	$⊢ ((K \in HL \land W \in H) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land T = 0. (K)) \land (P \neq Q \land (S \in A \land \neg S \underset{˙}{\leq} W) \land \exists r \in A (\neg r \underset{˙}{\leq} W \land P \underset{˙}{\lor} r = Q \underset{˙}{\lor} r))) \to \exists z \in A (\neg z \underset{˙}{\leq} W \land T \underset{˙}{\lor} z = S \underset{˙}{\lor} z)$
13	5 6 7 8 9 10 11 12	syl133anc	$⊢ ((K \in HL \land W \in H) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land (S \in A \land \neg S \underset{˙}{\leq} W)) \land (P \neq Q \land T = 0. (K) \land \exists r \in A (\neg r \underset{˙}{\leq} W \land P \underset{˙}{\lor} r = Q \underset{˙}{\lor} r))) \to \exists z \in A (\neg z \underset{˙}{\leq} W \land T \underset{˙}{\lor} z = S \underset{˙}{\lor} z)$
14		eqcom	$⊢ S \underset{˙}{\lor} z = T \underset{˙}{\lor} z \leftrightarrow T \underset{˙}{\lor} z = S \underset{˙}{\lor} z$
15	14	anbi2i	$⊢ (\neg z \underset{˙}{\leq} W \land S \underset{˙}{\lor} z = T \underset{˙}{\lor} z) \leftrightarrow (\neg z \underset{˙}{\leq} W \land T \underset{˙}{\lor} z = S \underset{˙}{\lor} z)$
16	15	rexbii	$⊢ \exists z \in A (\neg z \underset{˙}{\leq} W \land S \underset{˙}{\lor} z = T \underset{˙}{\lor} z) \leftrightarrow \exists z \in A (\neg z \underset{˙}{\leq} W \land T \underset{˙}{\lor} z = S \underset{˙}{\lor} z)$
17	13 16	sylibr	$⊢ ((K \in HL \land W \in H) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land (S \in A \land \neg S \underset{˙}{\leq} W)) \land (P \neq Q \land T = 0. (K) \land \exists r \in A (\neg r \underset{˙}{\leq} W \land P \underset{˙}{\lor} r = Q \underset{˙}{\lor} r))) \to \exists z \in A (\neg z \underset{˙}{\leq} W \land S \underset{˙}{\lor} z = T \underset{˙}{\lor} z)$