Metamath Proof Explorer

Theorem cdlemg5

Description: TODO: Is there a simpler more direct proof, that could be placed earlier e.g. near lhpexle ? TODO: The .\/ hypothesis is unused. FIX COMMENT. (Contributed by NM, 26-Apr-2013)

		Ref	Expression
	Hypotheses	cdlemg5.l	$⊢ \underset{˙}{\leq} = \leq_{K}$
		cdlemg5.j	$⊢ \underset{˙}{\lor} = join (K)$
		cdlemg5.a	$⊢ A = Atoms (K)$
		cdlemg5.h	$⊢ H = LHyp (K)$
	Assertion	cdlemg5	$⊢ ((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W)) \to \exists q \in A (P \neq q \land \neg q \underset{˙}{\leq} W)$

Proof

Step	Hyp	Ref	Expression
1		cdlemg5.l	$⊢ \underset{˙}{\leq} = \leq_{K}$
2		cdlemg5.j	$⊢ \underset{˙}{\lor} = join (K)$
3		cdlemg5.a	$⊢ A = Atoms (K)$
4		cdlemg5.h	$⊢ H = LHyp (K)$
5	1 3 4	lhpexle	$⊢ (K \in HL \land W \in H) \to \exists r \in A r \underset{˙}{\leq} W$
6	5	adantr	$⊢ ((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W)) \to \exists r \in A r \underset{˙}{\leq} W$
7		simpll	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W)) \land (r \in A \land r \underset{˙}{\leq} W)) \to (K \in HL \land W \in H)$
8		simpr	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W)) \land (r \in A \land r \underset{˙}{\leq} W)) \to (r \in A \land r \underset{˙}{\leq} W)$
9		simplr	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W)) \land (r \in A \land r \underset{˙}{\leq} W)) \to (P \in A \land \neg P \underset{˙}{\leq} W)$
10	1 2 3 4	cdlemf1	$⊢ ((K \in HL \land W \in H) \land (r \in A \land r \underset{˙}{\leq} W) \land (P \in A \land \neg P \underset{˙}{\leq} W)) \to \exists q \in A (P \neq q \land \neg q \underset{˙}{\leq} W \land r \underset{˙}{\leq} (P \underset{˙}{\lor} q))$
11	7 8 9 10	syl3anc	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W)) \land (r \in A \land r \underset{˙}{\leq} W)) \to \exists q \in A (P \neq q \land \neg q \underset{˙}{\leq} W \land r \underset{˙}{\leq} (P \underset{˙}{\lor} q))$
12		3simpa	$⊢ (P \neq q \land \neg q \underset{˙}{\leq} W \land r \underset{˙}{\leq} (P \underset{˙}{\lor} q)) \to (P \neq q \land \neg q \underset{˙}{\leq} W)$
13	12	reximi	$⊢ \exists q \in A (P \neq q \land \neg q \underset{˙}{\leq} W \land r \underset{˙}{\leq} (P \underset{˙}{\lor} q)) \to \exists q \in A (P \neq q \land \neg q \underset{˙}{\leq} W)$
14	11 13	syl	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W)) \land (r \in A \land r \underset{˙}{\leq} W)) \to \exists q \in A (P \neq q \land \neg q \underset{˙}{\leq} W)$
15	6 14	rexlimddv	$⊢ ((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W)) \to \exists q \in A (P \neq q \land \neg q \underset{˙}{\leq} W)$