cdlemg33d

	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	$⊢ R = trL (K) (W)$
	cdlemg31.n	$⊢ N = (P \underset{˙}{\lor} v) \underset{˙}{\land} (Q \underset{˙}{\lor} R (F))$
	cdlemg33.o	$⊢ O = (P \underset{˙}{\lor} v) \underset{˙}{\land} (Q \underset{˙}{\lor} R (G))$
Assertion	cdlemg33d	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((v \in A \land v \underset{˙}{\leq} W) \land (N = 0. (K) \land O \in A) \land (F \in T \land G \in T)) \land (P \neq Q \land v \neq R (G) \land \exists r \in A (\neg r \underset{˙}{\leq} W \land P \underset{˙}{\lor} r = Q \underset{˙}{\lor} r))) \to \exists z \in A (\neg z \underset{˙}{\leq} W \land (z \neq N \land z \neq O \land z \underset{˙}{\leq} (P \underset{˙}{\lor} v)))$

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		cdlemg31.n	$⊢ N = (P \underset{˙}{\lor} v) \underset{˙}{\land} (Q \underset{˙}{\lor} R (F))$
9		cdlemg33.o	$⊢ O = (P \underset{˙}{\lor} v) \underset{˙}{\land} (Q \underset{˙}{\lor} R (G))$
10		simp1	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((v \in A \land v \underset{˙}{\leq} W) \land (N = 0. (K) \land O \in A) \land (F \in T \land G \in T)) \land (P \neq Q \land v \neq R (G) \land \exists r \in A (\neg r \underset{˙}{\leq} W \land P \underset{˙}{\lor} r = Q \underset{˙}{\lor} r))) \to ((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W))$
11		simp21	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((v \in A \land v \underset{˙}{\leq} W) \land (N = 0. (K) \land O \in A) \land (F \in T \land G \in T)) \land (P \neq Q \land v \neq R (G) \land \exists r \in A (\neg r \underset{˙}{\leq} W \land P \underset{˙}{\lor} r = Q \underset{˙}{\lor} r))) \to (v \in A \land v \underset{˙}{\leq} W)$
12		simp22r	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((v \in A \land v \underset{˙}{\leq} W) \land (N = 0. (K) \land O \in A) \land (F \in T \land G \in T)) \land (P \neq Q \land v \neq R (G) \land \exists r \in A (\neg r \underset{˙}{\leq} W \land P \underset{˙}{\lor} r = Q \underset{˙}{\lor} r))) \to O \in A$
13		simp22l	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((v \in A \land v \underset{˙}{\leq} W) \land (N = 0. (K) \land O \in A) \land (F \in T \land G \in T)) \land (P \neq Q \land v \neq R (G) \land \exists r \in A (\neg r \underset{˙}{\leq} W \land P \underset{˙}{\lor} r = Q \underset{˙}{\lor} r))) \to N = 0. (K)$
14	12 13	jca	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((v \in A \land v \underset{˙}{\leq} W) \land (N = 0. (K) \land O \in A) \land (F \in T \land G \in T)) \land (P \neq Q \land v \neq R (G) \land \exists r \in A (\neg r \underset{˙}{\leq} W \land P \underset{˙}{\lor} r = Q \underset{˙}{\lor} r))) \to (O \in A \land N = 0. (K))$
15		simp23r	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((v \in A \land v \underset{˙}{\leq} W) \land (N = 0. (K) \land O \in A) \land (F \in T \land G \in T)) \land (P \neq Q \land v \neq R (G) \land \exists r \in A (\neg r \underset{˙}{\leq} W \land P \underset{˙}{\lor} r = Q \underset{˙}{\lor} r))) \to G \in T$
16		simp23l	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((v \in A \land v \underset{˙}{\leq} W) \land (N = 0. (K) \land O \in A) \land (F \in T \land G \in T)) \land (P \neq Q \land v \neq R (G) \land \exists r \in A (\neg r \underset{˙}{\leq} W \land P \underset{˙}{\lor} r = Q \underset{˙}{\lor} r))) \to F \in T$
17	15 16	jca	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((v \in A \land v \underset{˙}{\leq} W) \land (N = 0. (K) \land O \in A) \land (F \in T \land G \in T)) \land (P \neq Q \land v \neq R (G) \land \exists r \in A (\neg r \underset{˙}{\leq} W \land P \underset{˙}{\lor} r = Q \underset{˙}{\lor} r))) \to (G \in T \land F \in T)$
18		simp3	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((v \in A \land v \underset{˙}{\leq} W) \land (N = 0. (K) \land O \in A) \land (F \in T \land G \in T)) \land (P \neq Q \land v \neq R (G) \land \exists r \in A (\neg r \underset{˙}{\leq} W \land P \underset{˙}{\lor} r = Q \underset{˙}{\lor} r))) \to (P \neq Q \land v \neq R (G) \land \exists r \in A (\neg r \underset{˙}{\leq} W \land P \underset{˙}{\lor} r = Q \underset{˙}{\lor} r))$
19	1 2 3 4 5 6 7 9 8	cdlemg33c	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((v \in A \land v \underset{˙}{\leq} W) \land (O \in A \land N = 0. (K)) \land (G \in T \land F \in T)) \land (P \neq Q \land v \neq R (G) \land \exists r \in A (\neg r \underset{˙}{\leq} W \land P \underset{˙}{\lor} r = Q \underset{˙}{\lor} r))) \to \exists z \in A (\neg z \underset{˙}{\leq} W \land (z \neq O \land z \neq N \land z \underset{˙}{\leq} (P \underset{˙}{\lor} v)))$
20	10 11 14 17 18 19	syl131anc	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((v \in A \land v \underset{˙}{\leq} W) \land (N = 0. (K) \land O \in A) \land (F \in T \land G \in T)) \land (P \neq Q \land v \neq R (G) \land \exists r \in A (\neg r \underset{˙}{\leq} W \land P \underset{˙}{\lor} r = Q \underset{˙}{\lor} r))) \to \exists z \in A (\neg z \underset{˙}{\leq} W \land (z \neq O \land z \neq N \land z \underset{˙}{\leq} (P \underset{˙}{\lor} v)))$
21		3ancoma	$⊢ (z \neq O \land z \neq N \land z \underset{˙}{\leq} (P \underset{˙}{\lor} v)) \leftrightarrow (z \neq N \land z \neq O \land z \underset{˙}{\leq} (P \underset{˙}{\lor} v))$
22	21	anbi2i	$⊢ (\neg z \underset{˙}{\leq} W \land (z \neq O \land z \neq N \land z \underset{˙}{\leq} (P \underset{˙}{\lor} v))) \leftrightarrow (\neg z \underset{˙}{\leq} W \land (z \neq N \land z \neq O \land z \underset{˙}{\leq} (P \underset{˙}{\lor} v)))$
23	22	rexbii	$⊢ \exists z \in A (\neg z \underset{˙}{\leq} W \land (z \neq O \land z \neq N \land z \underset{˙}{\leq} (P \underset{˙}{\lor} v))) \leftrightarrow \exists z \in A (\neg z \underset{˙}{\leq} W \land (z \neq N \land z \neq O \land z \underset{˙}{\leq} (P \underset{˙}{\lor} v)))$
24	20 23	sylib	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((v \in A \land v \underset{˙}{\leq} W) \land (N = 0. (K) \land O \in A) \land (F \in T \land G \in T)) \land (P \neq Q \land v \neq R (G) \land \exists r \in A (\neg r \underset{˙}{\leq} W \land P \underset{˙}{\lor} r = Q \underset{˙}{\lor} r))) \to \exists z \in A (\neg z \underset{˙}{\leq} W \land (z \neq N \land z \neq O \land z \underset{˙}{\leq} (P \underset{˙}{\lor} v)))$

Metamath Proof Explorer

Theorem cdlemg33d

Proof