cdlemg17f

	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	cdlemg17f	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (F \in T \land G \in T \land P \neq Q) \land (G (P) \neq P \land R (G) \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg \exists r \in A (\neg r \underset{˙}{\leq} W \land P \underset{˙}{\lor} r = Q \underset{˙}{\lor} r))) \to F (P) \underset{˙}{\lor} F (Q) = F (P) \underset{˙}{\lor} G (F (P))$

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	1 2 3 4 5 6 7	cdlemg17e	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (F \in T \land G \in T \land P \neq Q) \land (G (P) \neq P \land R (G) \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg \exists r \in A (\neg r \underset{˙}{\leq} W \land P \underset{˙}{\lor} r = Q \underset{˙}{\lor} r))) \to F (P) \underset{˙}{\lor} F (Q) = F (P) \underset{˙}{\lor} R (G)$
9		simp11	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (F \in T \land G \in T \land P \neq Q) \land (G (P) \neq P \land R (G) \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg \exists r \in A (\neg r \underset{˙}{\leq} W \land P \underset{˙}{\lor} r = Q \underset{˙}{\lor} r))) \to (K \in HL \land W \in H)$
10		simp22	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (F \in T \land G \in T \land P \neq Q) \land (G (P) \neq P \land R (G) \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg \exists r \in A (\neg r \underset{˙}{\leq} W \land P \underset{˙}{\lor} r = Q \underset{˙}{\lor} r))) \to G \in T$
11		simp21	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (F \in T \land G \in T \land P \neq Q) \land (G (P) \neq P \land R (G) \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg \exists r \in A (\neg r \underset{˙}{\leq} W \land P \underset{˙}{\lor} r = Q \underset{˙}{\lor} r))) \to F \in T$
12		simp12	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (F \in T \land G \in T \land P \neq Q) \land (G (P) \neq P \land R (G) \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg \exists r \in A (\neg r \underset{˙}{\leq} W \land P \underset{˙}{\lor} r = Q \underset{˙}{\lor} r))) \to (P \in A \land \neg P \underset{˙}{\leq} W)$
13	1 4 5 6	ltrnel	$⊢ ((K \in HL \land W \in H) \land F \in T \land (P \in A \land \neg P \underset{˙}{\leq} W)) \to (F (P) \in A \land \neg F (P) \underset{˙}{\leq} W)$
14	9 11 12 13	syl3anc	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (F \in T \land G \in T \land P \neq Q) \land (G (P) \neq P \land R (G) \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg \exists r \in A (\neg r \underset{˙}{\leq} W \land P \underset{˙}{\lor} r = Q \underset{˙}{\lor} r))) \to (F (P) \in A \land \neg F (P) \underset{˙}{\leq} W)$
15	1 2 3 4 5 6 7	trlval2	$⊢ ((K \in HL \land W \in H) \land G \in T \land (F (P) \in A \land \neg F (P) \underset{˙}{\leq} W)) \to R (G) = (F (P) \underset{˙}{\lor} G (F (P))) \underset{˙}{\land} W$
16	9 10 14 15	syl3anc	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (F \in T \land G \in T \land P \neq Q) \land (G (P) \neq P \land R (G) \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg \exists r \in A (\neg r \underset{˙}{\leq} W \land P \underset{˙}{\lor} r = Q \underset{˙}{\lor} r))) \to R (G) = (F (P) \underset{˙}{\lor} G (F (P))) \underset{˙}{\land} W$
17	16	oveq2d	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (F \in T \land G \in T \land P \neq Q) \land (G (P) \neq P \land R (G) \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg \exists r \in A (\neg r \underset{˙}{\leq} W \land P \underset{˙}{\lor} r = Q \underset{˙}{\lor} r))) \to F (P) \underset{˙}{\lor} R (G) = F (P) \underset{˙}{\lor} ((F (P) \underset{˙}{\lor} G (F (P))) \underset{˙}{\land} W)$
18		simp12l	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (F \in T \land G \in T \land P \neq Q) \land (G (P) \neq P \land R (G) \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg \exists r \in A (\neg r \underset{˙}{\leq} W \land P \underset{˙}{\lor} r = Q \underset{˙}{\lor} r))) \to P \in A$
19	1 4 5 6	ltrncoat	$⊢ ((K \in HL \land W \in H) \land (G \in T \land F \in T) \land P \in A) \to G (F (P)) \in A$
20	9 10 11 18 19	syl121anc	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (F \in T \land G \in T \land P \neq Q) \land (G (P) \neq P \land R (G) \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg \exists r \in A (\neg r \underset{˙}{\leq} W \land P \underset{˙}{\lor} r = Q \underset{˙}{\lor} r))) \to G (F (P)) \in A$
21		eqid	$⊢ (F (P) \underset{˙}{\lor} G (F (P))) \underset{˙}{\land} W = (F (P) \underset{˙}{\lor} G (F (P))) \underset{˙}{\land} W$
22	1 2 3 4 5 21	cdleme0cp	$⊢ ((K \in HL \land W \in H) \land ((F (P) \in A \land \neg F (P) \underset{˙}{\leq} W) \land G (F (P)) \in A)) \to F (P) \underset{˙}{\lor} ((F (P) \underset{˙}{\lor} G (F (P))) \underset{˙}{\land} W) = F (P) \underset{˙}{\lor} G (F (P))$
23	9 14 20 22	syl12anc	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (F \in T \land G \in T \land P \neq Q) \land (G (P) \neq P \land R (G) \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg \exists r \in A (\neg r \underset{˙}{\leq} W \land P \underset{˙}{\lor} r = Q \underset{˙}{\lor} r))) \to F (P) \underset{˙}{\lor} ((F (P) \underset{˙}{\lor} G (F (P))) \underset{˙}{\land} W) = F (P) \underset{˙}{\lor} G (F (P))$
24	8 17 23	3eqtrd	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (F \in T \land G \in T \land P \neq Q) \land (G (P) \neq P \land R (G) \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg \exists r \in A (\neg r \underset{˙}{\leq} W \land P \underset{˙}{\lor} r = Q \underset{˙}{\lor} r))) \to F (P) \underset{˙}{\lor} F (Q) = F (P) \underset{˙}{\lor} G (F (P))$

Metamath Proof Explorer

Theorem cdlemg17f

Proof