cdlemg9a

	Ref	Expression
Hypotheses	cdlemg8.l	$⊢ \underset{˙}{\leq} = \leq_{K}$
	cdlemg8.j	$⊢ \underset{˙}{\lor} = join (K)$
	cdlemg8.m	$⊢ \underset{˙}{\land} = meet (K)$
	cdlemg8.a	$⊢ A = Atoms (K)$
	cdlemg8.h	$⊢ H = LHyp (K)$
	cdlemg8.t	$⊢ T = LTrn (K) (W)$
	cdlemg9.u	$⊢ U = (P \underset{˙}{\lor} Q) \underset{˙}{\land} W$
Assertion	cdlemg9a	$⊢ ((K \in HL \land W \in H) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land F \in T) \land (G \in T \land P \neq Q \land F (G (P)) \underset{˙}{\lor} F (G (Q)) \neq P \underset{˙}{\lor} Q)) \to ((P \underset{˙}{\lor} U) \underset{˙}{\land} (F (G (P)) \underset{˙}{\lor} U)) \underset{˙}{\leq} (G (P) \underset{˙}{\lor} U)$

Proof

Step	Hyp	Ref	Expression
1		cdlemg8.l	$⊢ \underset{˙}{\leq} = \leq_{K}$
2		cdlemg8.j	$⊢ \underset{˙}{\lor} = join (K)$
3		cdlemg8.m	$⊢ \underset{˙}{\land} = meet (K)$
4		cdlemg8.a	$⊢ A = Atoms (K)$
5		cdlemg8.h	$⊢ H = LHyp (K)$
6		cdlemg8.t	$⊢ T = LTrn (K) (W)$
7		cdlemg9.u	$⊢ U = (P \underset{˙}{\lor} Q) \underset{˙}{\land} W$
8		simp1l	$⊢ ((K \in HL \land W \in H) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land F \in T) \land (G \in T \land P \neq Q \land F (G (P)) \underset{˙}{\lor} F (G (Q)) \neq P \underset{˙}{\lor} Q)) \to K \in HL$
9		simp21l	$⊢ ((K \in HL \land W \in H) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land F \in T) \land (G \in T \land P \neq Q \land F (G (P)) \underset{˙}{\lor} F (G (Q)) \neq P \underset{˙}{\lor} Q)) \to P \in A$
10		simp1	$⊢ ((K \in HL \land W \in H) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land F \in T) \land (G \in T \land P \neq Q \land F (G (P)) \underset{˙}{\lor} F (G (Q)) \neq P \underset{˙}{\lor} Q)) \to (K \in HL \land W \in H)$
11		simp23	$⊢ ((K \in HL \land W \in H) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land F \in T) \land (G \in T \land P \neq Q \land F (G (P)) \underset{˙}{\lor} F (G (Q)) \neq P \underset{˙}{\lor} Q)) \to F \in T$
12		simp31	$⊢ ((K \in HL \land W \in H) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land F \in T) \land (G \in T \land P \neq Q \land F (G (P)) \underset{˙}{\lor} F (G (Q)) \neq P \underset{˙}{\lor} Q)) \to G \in T$
13	1 4 5 6	ltrncoat	$⊢ ((K \in HL \land W \in H) \land (F \in T \land G \in T) \land P \in A) \to F (G (P)) \in A$
14	10 11 12 9 13	syl121anc	$⊢ ((K \in HL \land W \in H) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land F \in T) \land (G \in T \land P \neq Q \land F (G (P)) \underset{˙}{\lor} F (G (Q)) \neq P \underset{˙}{\lor} Q)) \to F (G (P)) \in A$
15		simp1r	$⊢ ((K \in HL \land W \in H) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land F \in T) \land (G \in T \land P \neq Q \land F (G (P)) \underset{˙}{\lor} F (G (Q)) \neq P \underset{˙}{\lor} Q)) \to W \in H$
16		simp21	$⊢ ((K \in HL \land W \in H) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land F \in T) \land (G \in T \land P \neq Q \land F (G (P)) \underset{˙}{\lor} F (G (Q)) \neq P \underset{˙}{\lor} Q)) \to (P \in A \land \neg P \underset{˙}{\leq} W)$
17		simp22l	$⊢ ((K \in HL \land W \in H) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land F \in T) \land (G \in T \land P \neq Q \land F (G (P)) \underset{˙}{\lor} F (G (Q)) \neq P \underset{˙}{\lor} Q)) \to Q \in A$
18		simp32	$⊢ ((K \in HL \land W \in H) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land F \in T) \land (G \in T \land P \neq Q \land F (G (P)) \underset{˙}{\lor} F (G (Q)) \neq P \underset{˙}{\lor} Q)) \to P \neq Q$
19	1 2 3 4 5 7	cdleme0a	$⊢ ((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land P \neq Q)) \to U \in A$
20	8 15 16 17 18 19	syl212anc	$⊢ ((K \in HL \land W \in H) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land F \in T) \land (G \in T \land P \neq Q \land F (G (P)) \underset{˙}{\lor} F (G (Q)) \neq P \underset{˙}{\lor} Q)) \to U \in A$
21		simp33	$⊢ ((K \in HL \land W \in H) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land F \in T) \land (G \in T \land P \neq Q \land F (G (P)) \underset{˙}{\lor} F (G (Q)) \neq P \underset{˙}{\lor} Q)) \to F (G (P)) \underset{˙}{\lor} F (G (Q)) \neq P \underset{˙}{\lor} Q$
22		simp22	$⊢ ((K \in HL \land W \in H) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land F \in T) \land (G \in T \land P \neq Q \land F (G (P)) \underset{˙}{\lor} F (G (Q)) \neq P \underset{˙}{\lor} Q)) \to (Q \in A \land \neg Q \underset{˙}{\leq} W)$
23	5 6 1 2 4 3 7	cdlemg2l	$⊢ ((K \in HL \land W \in H) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (F \in T \land G \in T)) \to F (G (P)) \underset{˙}{\lor} F (G (Q)) = F (G (P)) \underset{˙}{\lor} U$
24	10 16 22 11 12 23	syl122anc	$⊢ ((K \in HL \land W \in H) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land F \in T) \land (G \in T \land P \neq Q \land F (G (P)) \underset{˙}{\lor} F (G (Q)) \neq P \underset{˙}{\lor} Q)) \to F (G (P)) \underset{˙}{\lor} F (G (Q)) = F (G (P)) \underset{˙}{\lor} U$
25	1 2 3 4 5 7	cdlemg3a	$⊢ ((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land Q \in A) \to P \underset{˙}{\lor} Q = P \underset{˙}{\lor} U$
26	8 15 16 17 25	syl211anc	$⊢ ((K \in HL \land W \in H) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land F \in T) \land (G \in T \land P \neq Q \land F (G (P)) \underset{˙}{\lor} F (G (Q)) \neq P \underset{˙}{\lor} Q)) \to P \underset{˙}{\lor} Q = P \underset{˙}{\lor} U$
27	21 24 26	3netr3d	$⊢ ((K \in HL \land W \in H) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land F \in T) \land (G \in T \land P \neq Q \land F (G (P)) \underset{˙}{\lor} F (G (Q)) \neq P \underset{˙}{\lor} Q)) \to F (G (P)) \underset{˙}{\lor} U \neq P \underset{˙}{\lor} U$
28	27	necomd	$⊢ ((K \in HL \land W \in H) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land F \in T) \land (G \in T \land P \neq Q \land F (G (P)) \underset{˙}{\lor} F (G (Q)) \neq P \underset{˙}{\lor} Q)) \to P \underset{˙}{\lor} U \neq F (G (P)) \underset{˙}{\lor} U$
29	1 2 3 4	2llnma3r	$⊢ (K \in HL \land (P \in A \land F (G (P)) \in A \land U \in A) \land P \underset{˙}{\lor} U \neq F (G (P)) \underset{˙}{\lor} U) \to (P \underset{˙}{\lor} U) \underset{˙}{\land} (F (G (P)) \underset{˙}{\lor} U) = U$
30	8 9 14 20 28 29	syl131anc	$⊢ ((K \in HL \land W \in H) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land F \in T) \land (G \in T \land P \neq Q \land F (G (P)) \underset{˙}{\lor} F (G (Q)) \neq P \underset{˙}{\lor} Q)) \to (P \underset{˙}{\lor} U) \underset{˙}{\land} (F (G (P)) \underset{˙}{\lor} U) = U$
31	1 4 5 6	ltrnat	$⊢ ((K \in HL \land W \in H) \land G \in T \land P \in A) \to G (P) \in A$
32	10 12 9 31	syl3anc	$⊢ ((K \in HL \land W \in H) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land F \in T) \land (G \in T \land P \neq Q \land F (G (P)) \underset{˙}{\lor} F (G (Q)) \neq P \underset{˙}{\lor} Q)) \to G (P) \in A$
33	1 2 4	hlatlej2	$⊢ (K \in HL \land G (P) \in A \land U \in A) \to U \underset{˙}{\leq} (G (P) \underset{˙}{\lor} U)$
34	8 32 20 33	syl3anc	$⊢ ((K \in HL \land W \in H) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land F \in T) \land (G \in T \land P \neq Q \land F (G (P)) \underset{˙}{\lor} F (G (Q)) \neq P \underset{˙}{\lor} Q)) \to U \underset{˙}{\leq} (G (P) \underset{˙}{\lor} U)$
35	30 34	eqbrtrd	$⊢ ((K \in HL \land W \in H) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land F \in T) \land (G \in T \land P \neq Q \land F (G (P)) \underset{˙}{\lor} F (G (Q)) \neq P \underset{˙}{\lor} Q)) \to ((P \underset{˙}{\lor} U) \underset{˙}{\land} (F (G (P)) \underset{˙}{\lor} U)) \underset{˙}{\leq} (G (P) \underset{˙}{\lor} U)$

Metamath Proof Explorer

Theorem cdlemg9a

Proof