cdlemg33e

	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	cdlemg33e	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((v \in A \land v \underset{˙}{\leq} W) \land (N = 0. (K) \land O = 0. (K)) \land (F \in T \land G \in T)) \land (P \neq Q \land v \neq R (F) \land \exists r \in A (\neg r \underset{˙}{\leq} W \land P \underset{˙}{\lor} r = Q \underset{˙}{\lor} r))) \to \exists z \in A (\neg z \underset{˙}{\leq} W \land (z \neq N \land z \neq O \land z \underset{˙}{\leq} (P \underset{˙}{\lor} v)))$

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	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((v \in A \land v \underset{˙}{\leq} W) \land (N = 0. (K) \land O = 0. (K)) \land (F \in T \land G \in T)) \land (P \neq Q \land v \neq R (F) \land \exists r \in A (\neg r \underset{˙}{\leq} W \land P \underset{˙}{\lor} r = Q \underset{˙}{\lor} r))) \to ((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W))$
11		simp21	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((v \in A \land v \underset{˙}{\leq} W) \land (N = 0. (K) \land O = 0. (K)) \land (F \in T \land G \in T)) \land (P \neq Q \land v \neq R (F) \land \exists r \in A (\neg r \underset{˙}{\leq} W \land P \underset{˙}{\lor} r = Q \underset{˙}{\lor} r))) \to (v \in A \land v \underset{˙}{\leq} W)$
12		simp23l	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((v \in A \land v \underset{˙}{\leq} W) \land (N = 0. (K) \land O = 0. (K)) \land (F \in T \land G \in T)) \land (P \neq Q \land v \neq R (F) \land \exists r \in A (\neg r \underset{˙}{\leq} W \land P \underset{˙}{\lor} r = Q \underset{˙}{\lor} r))) \to F \in T$
13		simp3	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((v \in A \land v \underset{˙}{\leq} W) \land (N = 0. (K) \land O = 0. (K)) \land (F \in T \land G \in T)) \land (P \neq Q \land v \neq R (F) \land \exists r \in A (\neg r \underset{˙}{\leq} W \land P \underset{˙}{\lor} r = Q \underset{˙}{\lor} r))) \to (P \neq Q \land v \neq R (F) \land \exists r \in A (\neg r \underset{˙}{\leq} W \land P \underset{˙}{\lor} r = Q \underset{˙}{\lor} r))$
14	1 2 3 4 5 6 7 8	cdlemg33c0	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((v \in A \land v \underset{˙}{\leq} W) \land F \in T) \land (P \neq Q \land v \neq R (F) \land \exists r \in A (\neg r \underset{˙}{\leq} W \land P \underset{˙}{\lor} r = Q \underset{˙}{\lor} r))) \to \exists z \in A (\neg z \underset{˙}{\leq} W \land z \underset{˙}{\leq} (P \underset{˙}{\lor} v))$
15	10 11 12 13 14	syl121anc	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((v \in A \land v \underset{˙}{\leq} W) \land (N = 0. (K) \land O = 0. (K)) \land (F \in T \land G \in T)) \land (P \neq Q \land v \neq R (F) \land \exists r \in A (\neg r \underset{˙}{\leq} W \land P \underset{˙}{\lor} r = Q \underset{˙}{\lor} r))) \to \exists z \in A (\neg z \underset{˙}{\leq} W \land z \underset{˙}{\leq} (P \underset{˙}{\lor} v))$
16		simp11l	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((v \in A \land v \underset{˙}{\leq} W) \land (N = 0. (K) \land O = 0. (K)) \land (F \in T \land G \in T)) \land (P \neq Q \land v \neq R (F) \land \exists r \in A (\neg r \underset{˙}{\leq} W \land P \underset{˙}{\lor} r = Q \underset{˙}{\lor} r))) \to K \in HL$
17		hlatl	$⊢ K \in HL \to K \in AtLat$
18	16 17	syl	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((v \in A \land v \underset{˙}{\leq} W) \land (N = 0. (K) \land O = 0. (K)) \land (F \in T \land G \in T)) \land (P \neq Q \land v \neq R (F) \land \exists r \in A (\neg r \underset{˙}{\leq} W \land P \underset{˙}{\lor} r = Q \underset{˙}{\lor} r))) \to K \in AtLat$
19		eqid	$⊢ 0. (K) = 0. (K)$
20	19 4	atn0	$⊢ (K \in AtLat \land z \in A) \to z \neq 0. (K)$
21	18 20	sylan	$⊢ ((((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((v \in A \land v \underset{˙}{\leq} W) \land (N = 0. (K) \land O = 0. (K)) \land (F \in T \land G \in T)) \land (P \neq Q \land v \neq R (F) \land \exists r \in A (\neg r \underset{˙}{\leq} W \land P \underset{˙}{\lor} r = Q \underset{˙}{\lor} r))) \land z \in A) \to z \neq 0. (K)$
22		simp22l	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((v \in A \land v \underset{˙}{\leq} W) \land (N = 0. (K) \land O = 0. (K)) \land (F \in T \land G \in T)) \land (P \neq Q \land v \neq R (F) \land \exists r \in A (\neg r \underset{˙}{\leq} W \land P \underset{˙}{\lor} r = Q \underset{˙}{\lor} r))) \to N = 0. (K)$
23	22	adantr	$⊢ ((((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((v \in A \land v \underset{˙}{\leq} W) \land (N = 0. (K) \land O = 0. (K)) \land (F \in T \land G \in T)) \land (P \neq Q \land v \neq R (F) \land \exists r \in A (\neg r \underset{˙}{\leq} W \land P \underset{˙}{\lor} r = Q \underset{˙}{\lor} r))) \land z \in A) \to N = 0. (K)$
24	21 23	neeqtrrd	$⊢ ((((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((v \in A \land v \underset{˙}{\leq} W) \land (N = 0. (K) \land O = 0. (K)) \land (F \in T \land G \in T)) \land (P \neq Q \land v \neq R (F) \land \exists r \in A (\neg r \underset{˙}{\leq} W \land P \underset{˙}{\lor} r = Q \underset{˙}{\lor} r))) \land z \in A) \to z \neq N$
25		simp22r	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((v \in A \land v \underset{˙}{\leq} W) \land (N = 0. (K) \land O = 0. (K)) \land (F \in T \land G \in T)) \land (P \neq Q \land v \neq R (F) \land \exists r \in A (\neg r \underset{˙}{\leq} W \land P \underset{˙}{\lor} r = Q \underset{˙}{\lor} r))) \to O = 0. (K)$
26	25	adantr	$⊢ ((((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((v \in A \land v \underset{˙}{\leq} W) \land (N = 0. (K) \land O = 0. (K)) \land (F \in T \land G \in T)) \land (P \neq Q \land v \neq R (F) \land \exists r \in A (\neg r \underset{˙}{\leq} W \land P \underset{˙}{\lor} r = Q \underset{˙}{\lor} r))) \land z \in A) \to O = 0. (K)$
27	21 26	neeqtrrd	$⊢ ((((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((v \in A \land v \underset{˙}{\leq} W) \land (N = 0. (K) \land O = 0. (K)) \land (F \in T \land G \in T)) \land (P \neq Q \land v \neq R (F) \land \exists r \in A (\neg r \underset{˙}{\leq} W \land P \underset{˙}{\lor} r = Q \underset{˙}{\lor} r))) \land z \in A) \to z \neq O$
28	24 27	jca	$⊢ ((((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((v \in A \land v \underset{˙}{\leq} W) \land (N = 0. (K) \land O = 0. (K)) \land (F \in T \land G \in T)) \land (P \neq Q \land v \neq R (F) \land \exists r \in A (\neg r \underset{˙}{\leq} W \land P \underset{˙}{\lor} r = Q \underset{˙}{\lor} r))) \land z \in A) \to (z \neq N \land z \neq O)$
29	28	biantrurd	$⊢ ((((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((v \in A \land v \underset{˙}{\leq} W) \land (N = 0. (K) \land O = 0. (K)) \land (F \in T \land G \in T)) \land (P \neq Q \land v \neq R (F) \land \exists r \in A (\neg r \underset{˙}{\leq} W \land P \underset{˙}{\lor} r = Q \underset{˙}{\lor} r))) \land z \in A) \to (z \underset{˙}{\leq} (P \underset{˙}{\lor} v) \leftrightarrow ((z \neq N \land z \neq O) \land z \underset{˙}{\leq} (P \underset{˙}{\lor} v)))$
30		df-3an	$⊢ (z \neq N \land z \neq O \land z \underset{˙}{\leq} (P \underset{˙}{\lor} v)) \leftrightarrow ((z \neq N \land z \neq O) \land z \underset{˙}{\leq} (P \underset{˙}{\lor} v))$
31	29 30	bitr4di	$⊢ ((((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((v \in A \land v \underset{˙}{\leq} W) \land (N = 0. (K) \land O = 0. (K)) \land (F \in T \land G \in T)) \land (P \neq Q \land v \neq R (F) \land \exists r \in A (\neg r \underset{˙}{\leq} W \land P \underset{˙}{\lor} r = Q \underset{˙}{\lor} r))) \land z \in A) \to (z \underset{˙}{\leq} (P \underset{˙}{\lor} v) \leftrightarrow (z \neq N \land z \neq O \land z \underset{˙}{\leq} (P \underset{˙}{\lor} v)))$
32	31	anbi2d	$⊢ ((((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((v \in A \land v \underset{˙}{\leq} W) \land (N = 0. (K) \land O = 0. (K)) \land (F \in T \land G \in T)) \land (P \neq Q \land v \neq R (F) \land \exists r \in A (\neg r \underset{˙}{\leq} W \land P \underset{˙}{\lor} r = Q \underset{˙}{\lor} r))) \land z \in A) \to ((\neg z \underset{˙}{\leq} W \land z \underset{˙}{\leq} (P \underset{˙}{\lor} v)) \leftrightarrow (\neg z \underset{˙}{\leq} W \land (z \neq N \land z \neq O \land z \underset{˙}{\leq} (P \underset{˙}{\lor} v))))$
33	32	rexbidva	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((v \in A \land v \underset{˙}{\leq} W) \land (N = 0. (K) \land O = 0. (K)) \land (F \in T \land G \in T)) \land (P \neq Q \land v \neq R (F) \land \exists r \in A (\neg r \underset{˙}{\leq} W \land P \underset{˙}{\lor} r = Q \underset{˙}{\lor} r))) \to (\exists z \in A (\neg z \underset{˙}{\leq} W \land z \underset{˙}{\leq} (P \underset{˙}{\lor} v)) \leftrightarrow \exists z \in A (\neg z \underset{˙}{\leq} W \land (z \neq N \land z \neq O \land z \underset{˙}{\leq} (P \underset{˙}{\lor} v))))$
34	15 33	mpbid	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((v \in A \land v \underset{˙}{\leq} W) \land (N = 0. (K) \land O = 0. (K)) \land (F \in T \land G \in T)) \land (P \neq Q \land v \neq R (F) \land \exists r \in A (\neg r \underset{˙}{\leq} W \land P \underset{˙}{\lor} r = Q \underset{˙}{\lor} r))) \to \exists z \in A (\neg z \underset{˙}{\leq} W \land (z \neq N \land z \neq O \land z \underset{˙}{\leq} (P \underset{˙}{\lor} v)))$

Metamath Proof Explorer

Theorem cdlemg33e

Proof