cdlemg12a

	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	$⊢ T = LTrn (K) (W)$
	cdlemg12.u	$⊢ U = (P \underset{˙}{\lor} Q) \underset{˙}{\land} W$
Assertion	cdlemg12a	$⊢ ((K \in HL \land W \in H) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land F \in T) \land (G \in T \land P \neq Q \land P \underset{˙}{\lor} U \neq G (P) \underset{˙}{\lor} U)) \to ((P \underset{˙}{\lor} U) \underset{˙}{\land} (G (P) \underset{˙}{\lor} U)) \underset{˙}{\leq} (F (G (P)) \underset{˙}{\lor} U)$

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		cdlemg12.u	$⊢ U = (P \underset{˙}{\lor} Q) \underset{˙}{\land} W$
8		simp1l	$⊢ ((K \in HL \land W \in H) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land F \in T) \land (G \in T \land P \neq Q \land P \underset{˙}{\lor} U \neq G (P) \underset{˙}{\lor} U)) \to K \in HL$
9		simp21l	$⊢ ((K \in HL \land W \in H) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land F \in T) \land (G \in T \land P \neq Q \land P \underset{˙}{\lor} U \neq G (P) \underset{˙}{\lor} U)) \to P \in A$
10		simp1	$⊢ ((K \in HL \land W \in H) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land F \in T) \land (G \in T \land P \neq Q \land P \underset{˙}{\lor} U \neq G (P) \underset{˙}{\lor} U)) \to (K \in HL \land W \in H)$
11		simp31	$⊢ ((K \in HL \land W \in H) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land F \in T) \land (G \in T \land P \neq Q \land P \underset{˙}{\lor} U \neq G (P) \underset{˙}{\lor} U)) \to G \in T$
12	1 4 5 6	ltrnat	$⊢ ((K \in HL \land W \in H) \land G \in T \land P \in A) \to G (P) \in A$
13	10 11 9 12	syl3anc	$⊢ ((K \in HL \land W \in H) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land F \in T) \land (G \in T \land P \neq Q \land P \underset{˙}{\lor} U \neq G (P) \underset{˙}{\lor} U)) \to G (P) \in A$
14		simp1r	$⊢ ((K \in HL \land W \in H) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land F \in T) \land (G \in T \land P \neq Q \land P \underset{˙}{\lor} U \neq G (P) \underset{˙}{\lor} U)) \to W \in H$
15		simp21	$⊢ ((K \in HL \land W \in H) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land F \in T) \land (G \in T \land P \neq Q \land P \underset{˙}{\lor} U \neq G (P) \underset{˙}{\lor} U)) \to (P \in A \land \neg P \underset{˙}{\leq} W)$
16		simp22l	$⊢ ((K \in HL \land W \in H) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land F \in T) \land (G \in T \land P \neq Q \land P \underset{˙}{\lor} U \neq G (P) \underset{˙}{\lor} U)) \to Q \in A$
17		simp32	$⊢ ((K \in HL \land W \in H) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land F \in T) \land (G \in T \land P \neq Q \land P \underset{˙}{\lor} U \neq G (P) \underset{˙}{\lor} U)) \to P \neq Q$
18	1 2 3 4 5 7	cdleme0a	$⊢ ((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land P \neq Q)) \to U \in A$
19	8 14 15 16 17 18	syl212anc	$⊢ ((K \in HL \land W \in H) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land F \in T) \land (G \in T \land P \neq Q \land P \underset{˙}{\lor} U \neq G (P) \underset{˙}{\lor} U)) \to U \in A$
20		simp33	$⊢ ((K \in HL \land W \in H) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land F \in T) \land (G \in T \land P \neq Q \land P \underset{˙}{\lor} U \neq G (P) \underset{˙}{\lor} U)) \to P \underset{˙}{\lor} U \neq G (P) \underset{˙}{\lor} U$
21	1 2 3 4	2llnma3r	$⊢ (K \in HL \land (P \in A \land G (P) \in A \land U \in A) \land P \underset{˙}{\lor} U \neq G (P) \underset{˙}{\lor} U) \to (P \underset{˙}{\lor} U) \underset{˙}{\land} (G (P) \underset{˙}{\lor} U) = U$
22	8 9 13 19 20 21	syl131anc	$⊢ ((K \in HL \land W \in H) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land F \in T) \land (G \in T \land P \neq Q \land P \underset{˙}{\lor} U \neq G (P) \underset{˙}{\lor} U)) \to (P \underset{˙}{\lor} U) \underset{˙}{\land} (G (P) \underset{˙}{\lor} U) = U$
23		simp23	$⊢ ((K \in HL \land W \in H) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land F \in T) \land (G \in T \land P \neq Q \land P \underset{˙}{\lor} U \neq G (P) \underset{˙}{\lor} U)) \to F \in T$
24	1 4 5 6	ltrncoat	$⊢ ((K \in HL \land W \in H) \land (F \in T \land G \in T) \land P \in A) \to F (G (P)) \in A$
25	10 23 11 9 24	syl121anc	$⊢ ((K \in HL \land W \in H) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land F \in T) \land (G \in T \land P \neq Q \land P \underset{˙}{\lor} U \neq G (P) \underset{˙}{\lor} U)) \to F (G (P)) \in A$
26	1 2 4	hlatlej2	$⊢ (K \in HL \land F (G (P)) \in A \land U \in A) \to U \underset{˙}{\leq} (F (G (P)) \underset{˙}{\lor} U)$
27	8 25 19 26	syl3anc	$⊢ ((K \in HL \land W \in H) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land F \in T) \land (G \in T \land P \neq Q \land P \underset{˙}{\lor} U \neq G (P) \underset{˙}{\lor} U)) \to U \underset{˙}{\leq} (F (G (P)) \underset{˙}{\lor} U)$
28	22 27	eqbrtrd	$⊢ ((K \in HL \land W \in H) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land F \in T) \land (G \in T \land P \neq Q \land P \underset{˙}{\lor} U \neq G (P) \underset{˙}{\lor} U)) \to ((P \underset{˙}{\lor} U) \underset{˙}{\land} (G (P) \underset{˙}{\lor} U)) \underset{˙}{\leq} (F (G (P)) \underset{˙}{\lor} U)$

Metamath Proof Explorer

Theorem cdlemg12a

Proof