cdlemg2l

	Ref	Expression
Hypotheses	cdlemg2inv.h	$⊢ H = LHyp (K)$
	cdlemg2inv.t	$⊢ T = LTrn (K) (W)$
	cdlemg2j.l	$⊢ \underset{˙}{\leq} = \leq_{K}$
	cdlemg2j.j	$⊢ \underset{˙}{\lor} = join (K)$
	cdlemg2j.a	$⊢ A = Atoms (K)$
	cdlemg2j.m	$⊢ \underset{˙}{\land} = meet (K)$
	cdlemg2j.u	$⊢ U = (P \underset{˙}{\lor} Q) \underset{˙}{\land} W$
Assertion	cdlemg2l	$⊢ ((K \in HL \land W \in H) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (F \in T \land G \in T)) \to F (G (P)) \underset{˙}{\lor} F (G (Q)) = F (G (P)) \underset{˙}{\lor} U$

Proof

Step	Hyp	Ref	Expression
1		cdlemg2inv.h	$⊢ H = LHyp (K)$
2		cdlemg2inv.t	$⊢ T = LTrn (K) (W)$
3		cdlemg2j.l	$⊢ \underset{˙}{\leq} = \leq_{K}$
4		cdlemg2j.j	$⊢ \underset{˙}{\lor} = join (K)$
5		cdlemg2j.a	$⊢ A = Atoms (K)$
6		cdlemg2j.m	$⊢ \underset{˙}{\land} = meet (K)$
7		cdlemg2j.u	$⊢ U = (P \underset{˙}{\lor} Q) \underset{˙}{\land} W$
8	1 2 3 4 5 6 7	cdlemg2k	$⊢ ((K \in HL \land W \in H) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land G \in T) \to G (P) \underset{˙}{\lor} G (Q) = G (P) \underset{˙}{\lor} U$
9	8	3adant3l	$⊢ ((K \in HL \land W \in H) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (F \in T \land G \in T)) \to G (P) \underset{˙}{\lor} G (Q) = G (P) \underset{˙}{\lor} U$
10	9	fveq2d	$⊢ ((K \in HL \land W \in H) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (F \in T \land G \in T)) \to F (G (P) \underset{˙}{\lor} G (Q)) = F (G (P) \underset{˙}{\lor} U)$
11		simp1	$⊢ ((K \in HL \land W \in H) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (F \in T \land G \in T)) \to (K \in HL \land W \in H)$
12		simp3l	$⊢ ((K \in HL \land W \in H) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (F \in T \land G \in T)) \to F \in T$
13		simp3r	$⊢ ((K \in HL \land W \in H) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (F \in T \land G \in T)) \to G \in T$
14		simp2l	$⊢ ((K \in HL \land W \in H) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (F \in T \land G \in T)) \to (P \in A \land \neg P \underset{˙}{\leq} W)$
15	3 5 1 2	ltrnel	$⊢ ((K \in HL \land W \in H) \land G \in T \land (P \in A \land \neg P \underset{˙}{\leq} W)) \to (G (P) \in A \land \neg G (P) \underset{˙}{\leq} W)$
16	11 13 14 15	syl3anc	$⊢ ((K \in HL \land W \in H) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (F \in T \land G \in T)) \to (G (P) \in A \land \neg G (P) \underset{˙}{\leq} W)$
17	16	simpld	$⊢ ((K \in HL \land W \in H) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (F \in T \land G \in T)) \to G (P) \in A$
18		eqid	$⊢ {Base}_{K} = {Base}_{K}$
19	18 5	atbase	$⊢ G (P) \in A \to G (P) \in {Base}_{K}$
20	17 19	syl	$⊢ ((K \in HL \land W \in H) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (F \in T \land G \in T)) \to G (P) \in {Base}_{K}$
21		simp2r	$⊢ ((K \in HL \land W \in H) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (F \in T \land G \in T)) \to (Q \in A \land \neg Q \underset{˙}{\leq} W)$
22	3 5 1 2	ltrnel	$⊢ ((K \in HL \land W \in H) \land G \in T \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \to (G (Q) \in A \land \neg G (Q) \underset{˙}{\leq} W)$
23	11 13 21 22	syl3anc	$⊢ ((K \in HL \land W \in H) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (F \in T \land G \in T)) \to (G (Q) \in A \land \neg G (Q) \underset{˙}{\leq} W)$
24	23	simpld	$⊢ ((K \in HL \land W \in H) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (F \in T \land G \in T)) \to G (Q) \in A$
25	18 5	atbase	$⊢ G (Q) \in A \to G (Q) \in {Base}_{K}$
26	24 25	syl	$⊢ ((K \in HL \land W \in H) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (F \in T \land G \in T)) \to G (Q) \in {Base}_{K}$
27	18 4 1 2	ltrnj	$⊢ ((K \in HL \land W \in H) \land F \in T \land (G (P) \in {Base}_{K} \land G (Q) \in {Base}_{K})) \to F (G (P) \underset{˙}{\lor} G (Q)) = F (G (P)) \underset{˙}{\lor} F (G (Q))$
28	11 12 20 26 27	syl112anc	$⊢ ((K \in HL \land W \in H) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (F \in T \land G \in T)) \to F (G (P) \underset{˙}{\lor} G (Q)) = F (G (P)) \underset{˙}{\lor} F (G (Q))$
29	1 2 3 4 5 6 7	cdlemg2fv2	$⊢ ((K \in HL \land W \in H) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land (G (P) \in A \land \neg G (P) \underset{˙}{\leq} W)) \land F \in T) \to F (G (P) \underset{˙}{\lor} U) = F (G (P)) \underset{˙}{\lor} U$
30	11 14 21 16 12 29	syl131anc	$⊢ ((K \in HL \land W \in H) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (F \in T \land G \in T)) \to F (G (P) \underset{˙}{\lor} U) = F (G (P)) \underset{˙}{\lor} U$
31	10 28 30	3eqtr3d	$⊢ ((K \in HL \land W \in H) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (F \in T \land G \in T)) \to F (G (P)) \underset{˙}{\lor} F (G (Q)) = F (G (P)) \underset{˙}{\lor} U$

Metamath Proof Explorer

Theorem cdlemg2l

Proof