cdlemg17iqN

Description: cdlemg17i with P and Q swapped. (Contributed by NM, 13-May-2013) (New usage is discouraged.)

	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	cdlemg17iqN	$⊢ ((K \in HL \land W \in H \land (F \in T \land G \in T)) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land P \neq Q) \land (R (G) \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg \exists r \in A (\neg r \underset{˙}{\leq} W \land P \underset{˙}{\lor} r = Q \underset{˙}{\lor} r) \land G (P) \neq P)) \to G (F (Q)) = F (P)$

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		simp11	$⊢ ((K \in HL \land W \in H \land (F \in T \land G \in T)) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land P \neq Q) \land (R (G) \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg \exists r \in A (\neg r \underset{˙}{\leq} W \land P \underset{˙}{\lor} r = Q \underset{˙}{\lor} r) \land G (P) \neq P)) \to K \in HL$
9		simp12	$⊢ ((K \in HL \land W \in H \land (F \in T \land G \in T)) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land P \neq Q) \land (R (G) \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg \exists r \in A (\neg r \underset{˙}{\leq} W \land P \underset{˙}{\lor} r = Q \underset{˙}{\lor} r) \land G (P) \neq P)) \to W \in H$
10	8 9	jca	$⊢ ((K \in HL \land W \in H \land (F \in T \land G \in T)) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land P \neq Q) \land (R (G) \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg \exists r \in A (\neg r \underset{˙}{\leq} W \land P \underset{˙}{\lor} r = Q \underset{˙}{\lor} r) \land G (P) \neq P)) \to (K \in HL \land W \in H)$
11		simp21	$⊢ ((K \in HL \land W \in H \land (F \in T \land G \in T)) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land P \neq Q) \land (R (G) \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg \exists r \in A (\neg r \underset{˙}{\leq} W \land P \underset{˙}{\lor} r = Q \underset{˙}{\lor} r) \land G (P) \neq P)) \to (P \in A \land \neg P \underset{˙}{\leq} W)$
12		simp22	$⊢ ((K \in HL \land W \in H \land (F \in T \land G \in T)) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land P \neq Q) \land (R (G) \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg \exists r \in A (\neg r \underset{˙}{\leq} W \land P \underset{˙}{\lor} r = Q \underset{˙}{\lor} r) \land G (P) \neq P)) \to (Q \in A \land \neg Q \underset{˙}{\leq} W)$
13		simp13l	$⊢ ((K \in HL \land W \in H \land (F \in T \land G \in T)) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land P \neq Q) \land (R (G) \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg \exists r \in A (\neg r \underset{˙}{\leq} W \land P \underset{˙}{\lor} r = Q \underset{˙}{\lor} r) \land G (P) \neq P)) \to F \in T$
14		simp13r	$⊢ ((K \in HL \land W \in H \land (F \in T \land G \in T)) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land P \neq Q) \land (R (G) \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg \exists r \in A (\neg r \underset{˙}{\leq} W \land P \underset{˙}{\lor} r = Q \underset{˙}{\lor} r) \land G (P) \neq P)) \to G \in T$
15		simp23	$⊢ ((K \in HL \land W \in H \land (F \in T \land G \in T)) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land P \neq Q) \land (R (G) \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg \exists r \in A (\neg r \underset{˙}{\leq} W \land P \underset{˙}{\lor} r = Q \underset{˙}{\lor} r) \land G (P) \neq P)) \to P \neq Q$
16		simp33	$⊢ ((K \in HL \land W \in H \land (F \in T \land G \in T)) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land P \neq Q) \land (R (G) \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg \exists r \in A (\neg r \underset{˙}{\leq} W \land P \underset{˙}{\lor} r = Q \underset{˙}{\lor} r) \land G (P) \neq P)) \to G (P) \neq P$
17		simp31	$⊢ ((K \in HL \land W \in H \land (F \in T \land G \in T)) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land P \neq Q) \land (R (G) \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg \exists r \in A (\neg r \underset{˙}{\leq} W \land P \underset{˙}{\lor} r = Q \underset{˙}{\lor} r) \land G (P) \neq P)) \to R (G) \underset{˙}{\leq} (P \underset{˙}{\lor} Q)$
18		simp32	$⊢ ((K \in HL \land W \in H \land (F \in T \land G \in T)) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land P \neq Q) \land (R (G) \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg \exists r \in A (\neg r \underset{˙}{\leq} W \land P \underset{˙}{\lor} r = Q \underset{˙}{\lor} r) \land G (P) \neq P)) \to \neg \exists r \in A (\neg r \underset{˙}{\leq} W \land P \underset{˙}{\lor} r = Q \underset{˙}{\lor} r)$
19	1 2 3 4 5 6 7	cdlemg17pq	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (F \in T \land G \in T \land P \neq Q) \land (G (P) \neq P \land R (G) \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg \exists r \in A (\neg r \underset{˙}{\leq} W \land P \underset{˙}{\lor} r = Q \underset{˙}{\lor} r))) \to (((K \in HL \land W \in H) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land (P \in A \land \neg P \underset{˙}{\leq} W)) \land (F \in T \land G \in T \land Q \neq P) \land (G (Q) \neq Q \land R (G) \underset{˙}{\leq} (Q \underset{˙}{\lor} P) \land \neg \exists r \in A (\neg r \underset{˙}{\leq} W \land Q \underset{˙}{\lor} r = P \underset{˙}{\lor} r)))$
20	10 11 12 13 14 15 16 17 18 19	syl333anc	$⊢ ((K \in HL \land W \in H \land (F \in T \land G \in T)) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land P \neq Q) \land (R (G) \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg \exists r \in A (\neg r \underset{˙}{\leq} W \land P \underset{˙}{\lor} r = Q \underset{˙}{\lor} r) \land G (P) \neq P)) \to (((K \in HL \land W \in H) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land (P \in A \land \neg P \underset{˙}{\leq} W)) \land (F \in T \land G \in T \land Q \neq P) \land (G (Q) \neq Q \land R (G) \underset{˙}{\leq} (Q \underset{˙}{\lor} P) \land \neg \exists r \in A (\neg r \underset{˙}{\leq} W \land Q \underset{˙}{\lor} r = P \underset{˙}{\lor} r)))$
21	1 2 3 4 5 6 7	cdlemg17i	$⊢ (((K \in HL \land W \in H) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land (P \in A \land \neg P \underset{˙}{\leq} W)) \land (F \in T \land G \in T \land Q \neq P) \land (G (Q) \neq Q \land R (G) \underset{˙}{\leq} (Q \underset{˙}{\lor} P) \land \neg \exists r \in A (\neg r \underset{˙}{\leq} W \land Q \underset{˙}{\lor} r = P \underset{˙}{\lor} r))) \to G (F (Q)) = F (P)$
22	20 21	syl	$⊢ ((K \in HL \land W \in H \land (F \in T \land G \in T)) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land P \neq Q) \land (R (G) \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg \exists r \in A (\neg r \underset{˙}{\leq} W \land P \underset{˙}{\lor} r = Q \underset{˙}{\lor} r) \land G (P) \neq P)) \to G (F (Q)) = F (P)$

Metamath Proof Explorer

Theorem cdlemg17iqN

Proof