cdlemg12b

Description: The triples <. P , ( FP ) , ( F( GP ) ) >. and <. Q , ( FQ ) , ( F( GQ ) ) >. are centrally perspective. TODO: FIX COMMENT. (Contributed by NM, 5-May-2013)

	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	$⊢ R = trL (K) (W)$
Assertion	cdlemg12b	$⊢ ((K \in HL \land W \in H) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land F \in T) \land (G \in T \land P \neq Q \land \neg R (G) \underset{˙}{\leq} (P \underset{˙}{\lor} Q))) \to ((P \underset{˙}{\lor} Q) \underset{˙}{\land} (G (P) \underset{˙}{\lor} G (Q))) \underset{˙}{\leq} (F (G (P)) \underset{˙}{\lor} F (G (Q)))$

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		simp1	$⊢ ((K \in HL \land W \in H) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land F \in T) \land (G \in T \land P \neq Q \land \neg R (G) \underset{˙}{\leq} (P \underset{˙}{\lor} Q))) \to (K \in HL \land W \in H)$
9		simp2	$⊢ ((K \in HL \land W \in H) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land F \in T) \land (G \in T \land P \neq Q \land \neg R (G) \underset{˙}{\leq} (P \underset{˙}{\lor} Q))) \to ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land F \in T)$
10		simp31	$⊢ ((K \in HL \land W \in H) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land F \in T) \land (G \in T \land P \neq Q \land \neg R (G) \underset{˙}{\leq} (P \underset{˙}{\lor} Q))) \to G \in T$
11		simp32	$⊢ ((K \in HL \land W \in H) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land F \in T) \land (G \in T \land P \neq Q \land \neg R (G) \underset{˙}{\leq} (P \underset{˙}{\lor} Q))) \to P \neq Q$
12		simp21	$⊢ ((K \in HL \land W \in H) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land F \in T) \land (G \in T \land P \neq Q \land \neg R (G) \underset{˙}{\leq} (P \underset{˙}{\lor} Q))) \to (P \in A \land \neg P \underset{˙}{\leq} W)$
13		simp22l	$⊢ ((K \in HL \land W \in H) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land F \in T) \land (G \in T \land P \neq Q \land \neg R (G) \underset{˙}{\leq} (P \underset{˙}{\lor} Q))) \to Q \in A$
14		simp33	$⊢ ((K \in HL \land W \in H) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land F \in T) \land (G \in T \land P \neq Q \land \neg R (G) \underset{˙}{\leq} (P \underset{˙}{\lor} Q))) \to \neg R (G) \underset{˙}{\leq} (P \underset{˙}{\lor} Q)$
15	1 2 3 4 5 6 7	cdlemg11b	$⊢ ((K \in HL \land W \in H) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land Q \in A) \land (G \in T \land P \neq Q \land \neg R (G) \underset{˙}{\leq} (P \underset{˙}{\lor} Q))) \to P \underset{˙}{\lor} Q \neq G (P) \underset{˙}{\lor} G (Q)$
16	8 12 13 10 11 14 15	syl123anc	$⊢ ((K \in HL \land W \in H) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land F \in T) \land (G \in T \land P \neq Q \land \neg R (G) \underset{˙}{\leq} (P \underset{˙}{\lor} Q))) \to P \underset{˙}{\lor} Q \neq G (P) \underset{˙}{\lor} G (Q)$
17		simp1l	$⊢ ((K \in HL \land W \in H) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land F \in T) \land (G \in T \land P \neq Q \land \neg R (G) \underset{˙}{\leq} (P \underset{˙}{\lor} Q))) \to K \in HL$
18		simp1r	$⊢ ((K \in HL \land W \in H) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land F \in T) \land (G \in T \land P \neq Q \land \neg R (G) \underset{˙}{\leq} (P \underset{˙}{\lor} Q))) \to W \in H$
19		eqid	$⊢ (P \underset{˙}{\lor} Q) \underset{˙}{\land} W = (P \underset{˙}{\lor} Q) \underset{˙}{\land} W$
20	1 2 3 4 5 19	cdlemg3a	$⊢ ((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land Q \in A) \to P \underset{˙}{\lor} Q = P \underset{˙}{\lor} ((P \underset{˙}{\lor} Q) \underset{˙}{\land} W)$
21	17 18 12 13 20	syl211anc	$⊢ ((K \in HL \land W \in H) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land F \in T) \land (G \in T \land P \neq Q \land \neg R (G) \underset{˙}{\leq} (P \underset{˙}{\lor} Q))) \to P \underset{˙}{\lor} Q = P \underset{˙}{\lor} ((P \underset{˙}{\lor} Q) \underset{˙}{\land} W)$
22		simp22	$⊢ ((K \in HL \land W \in H) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land F \in T) \land (G \in T \land P \neq Q \land \neg R (G) \underset{˙}{\leq} (P \underset{˙}{\lor} Q))) \to (Q \in A \land \neg Q \underset{˙}{\leq} W)$
23	5 6 1 2 4 3 19	cdlemg2k	$⊢ ((K \in HL \land W \in H) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land G \in T) \to G (P) \underset{˙}{\lor} G (Q) = G (P) \underset{˙}{\lor} ((P \underset{˙}{\lor} Q) \underset{˙}{\land} W)$
24	8 12 22 10 23	syl121anc	$⊢ ((K \in HL \land W \in H) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land F \in T) \land (G \in T \land P \neq Q \land \neg R (G) \underset{˙}{\leq} (P \underset{˙}{\lor} Q))) \to G (P) \underset{˙}{\lor} G (Q) = G (P) \underset{˙}{\lor} ((P \underset{˙}{\lor} Q) \underset{˙}{\land} W)$
25	16 21 24	3netr3d	$⊢ ((K \in HL \land W \in H) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land F \in T) \land (G \in T \land P \neq Q \land \neg R (G) \underset{˙}{\leq} (P \underset{˙}{\lor} Q))) \to P \underset{˙}{\lor} ((P \underset{˙}{\lor} Q) \underset{˙}{\land} W) \neq G (P) \underset{˙}{\lor} ((P \underset{˙}{\lor} Q) \underset{˙}{\land} W)$
26	1 2 3 4 5 6 19	cdlemg12a	$⊢ ((K \in HL \land W \in H) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land F \in T) \land (G \in T \land P \neq Q \land P \underset{˙}{\lor} ((P \underset{˙}{\lor} Q) \underset{˙}{\land} W) \neq G (P) \underset{˙}{\lor} ((P \underset{˙}{\lor} Q) \underset{˙}{\land} W))) \to ((P \underset{˙}{\lor} ((P \underset{˙}{\lor} Q) \underset{˙}{\land} W)) \underset{˙}{\land} (G (P) \underset{˙}{\lor} ((P \underset{˙}{\lor} Q) \underset{˙}{\land} W))) \underset{˙}{\leq} (F (G (P)) \underset{˙}{\lor} ((P \underset{˙}{\lor} Q) \underset{˙}{\land} W))$
27	8 9 10 11 25 26	syl113anc	$⊢ ((K \in HL \land W \in H) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land F \in T) \land (G \in T \land P \neq Q \land \neg R (G) \underset{˙}{\leq} (P \underset{˙}{\lor} Q))) \to ((P \underset{˙}{\lor} ((P \underset{˙}{\lor} Q) \underset{˙}{\land} W)) \underset{˙}{\land} (G (P) \underset{˙}{\lor} ((P \underset{˙}{\lor} Q) \underset{˙}{\land} W))) \underset{˙}{\leq} (F (G (P)) \underset{˙}{\lor} ((P \underset{˙}{\lor} Q) \underset{˙}{\land} W))$
28	21 24	oveq12d	$⊢ ((K \in HL \land W \in H) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land F \in T) \land (G \in T \land P \neq Q \land \neg R (G) \underset{˙}{\leq} (P \underset{˙}{\lor} Q))) \to (P \underset{˙}{\lor} Q) \underset{˙}{\land} (G (P) \underset{˙}{\lor} G (Q)) = (P \underset{˙}{\lor} ((P \underset{˙}{\lor} Q) \underset{˙}{\land} W)) \underset{˙}{\land} (G (P) \underset{˙}{\lor} ((P \underset{˙}{\lor} Q) \underset{˙}{\land} W))$
29		simp23	$⊢ ((K \in HL \land W \in H) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land F \in T) \land (G \in T \land P \neq Q \land \neg R (G) \underset{˙}{\leq} (P \underset{˙}{\lor} Q))) \to F \in T$
30	5 6 1 2 4 3 19	cdlemg2l	$⊢ ((K \in HL \land W \in H) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (F \in T \land G \in T)) \to F (G (P)) \underset{˙}{\lor} F (G (Q)) = F (G (P)) \underset{˙}{\lor} ((P \underset{˙}{\lor} Q) \underset{˙}{\land} W)$
31	8 12 22 29 10 30	syl122anc	$⊢ ((K \in HL \land W \in H) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land F \in T) \land (G \in T \land P \neq Q \land \neg R (G) \underset{˙}{\leq} (P \underset{˙}{\lor} Q))) \to F (G (P)) \underset{˙}{\lor} F (G (Q)) = F (G (P)) \underset{˙}{\lor} ((P \underset{˙}{\lor} Q) \underset{˙}{\land} W)$
32	27 28 31	3brtr4d	$⊢ ((K \in HL \land W \in H) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land F \in T) \land (G \in T \land P \neq Q \land \neg R (G) \underset{˙}{\leq} (P \underset{˙}{\lor} Q))) \to ((P \underset{˙}{\lor} Q) \underset{˙}{\land} (G (P) \underset{˙}{\lor} G (Q))) \underset{˙}{\leq} (F (G (P)) \underset{˙}{\lor} F (G (Q)))$

Metamath Proof Explorer

Theorem cdlemg12b

Proof