cdlemg19

Description: Show two lines intersect at an atom. TODO: fix comment. (Contributed by NM, 15-May-2013)

	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	cdlemg19	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((F \in T \land G \in T) \land P \neq Q \land G (P) \neq P) \land (R (G) \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land F (G (P)) \underset{˙}{\lor} F (G (Q)) \neq P \underset{˙}{\lor} Q \land \neg \exists r \in A (\neg r \underset{˙}{\leq} W \land P \underset{˙}{\lor} r = Q \underset{˙}{\lor} r))) \to (P \underset{˙}{\lor} F (G (P))) \underset{˙}{\land} W = (Q \underset{˙}{\lor} F (G (Q))) \underset{˙}{\land} W$

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		simp11l	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((F \in T \land G \in T) \land P \neq Q \land G (P) \neq P) \land (R (G) \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land F (G (P)) \underset{˙}{\lor} F (G (Q)) \neq P \underset{˙}{\lor} Q \land \neg \exists r \in A (\neg r \underset{˙}{\leq} W \land P \underset{˙}{\lor} r = Q \underset{˙}{\lor} r))) \to K \in HL$
9	8	hllatd	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((F \in T \land G \in T) \land P \neq Q \land G (P) \neq P) \land (R (G) \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land F (G (P)) \underset{˙}{\lor} F (G (Q)) \neq P \underset{˙}{\lor} Q \land \neg \exists r \in A (\neg r \underset{˙}{\leq} W \land P \underset{˙}{\lor} r = Q \underset{˙}{\lor} r))) \to K \in Lat$
10		simp12l	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((F \in T \land G \in T) \land P \neq Q \land G (P) \neq P) \land (R (G) \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land F (G (P)) \underset{˙}{\lor} F (G (Q)) \neq P \underset{˙}{\lor} Q \land \neg \exists r \in A (\neg r \underset{˙}{\leq} W \land P \underset{˙}{\lor} r = Q \underset{˙}{\lor} r))) \to P \in A$
11		simp11	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((F \in T \land G \in T) \land P \neq Q \land G (P) \neq P) \land (R (G) \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land F (G (P)) \underset{˙}{\lor} F (G (Q)) \neq P \underset{˙}{\lor} Q \land \neg \exists r \in A (\neg r \underset{˙}{\leq} W \land P \underset{˙}{\lor} r = Q \underset{˙}{\lor} r))) \to (K \in HL \land W \in H)$
12		simp21	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((F \in T \land G \in T) \land P \neq Q \land G (P) \neq P) \land (R (G) \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land F (G (P)) \underset{˙}{\lor} F (G (Q)) \neq P \underset{˙}{\lor} Q \land \neg \exists r \in A (\neg r \underset{˙}{\leq} W \land P \underset{˙}{\lor} r = Q \underset{˙}{\lor} r))) \to (F \in T \land G \in T)$
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	11 12 10 13	syl3anc	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((F \in T \land G \in T) \land P \neq Q \land G (P) \neq P) \land (R (G) \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land F (G (P)) \underset{˙}{\lor} F (G (Q)) \neq P \underset{˙}{\lor} Q \land \neg \exists r \in A (\neg r \underset{˙}{\leq} W \land P \underset{˙}{\lor} r = Q \underset{˙}{\lor} r))) \to F (G (P)) \in A$
15		eqid	$⊢ {Base}_{K} = {Base}_{K}$
16	15 2 4	hlatjcl	$⊢ (K \in HL \land P \in A \land F (G (P)) \in A) \to P \underset{˙}{\lor} F (G (P)) \in {Base}_{K}$
17	8 10 14 16	syl3anc	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((F \in T \land G \in T) \land P \neq Q \land G (P) \neq P) \land (R (G) \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land F (G (P)) \underset{˙}{\lor} F (G (Q)) \neq P \underset{˙}{\lor} Q \land \neg \exists r \in A (\neg r \underset{˙}{\leq} W \land P \underset{˙}{\lor} r = Q \underset{˙}{\lor} r))) \to P \underset{˙}{\lor} F (G (P)) \in {Base}_{K}$
18		simp13l	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((F \in T \land G \in T) \land P \neq Q \land G (P) \neq P) \land (R (G) \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land F (G (P)) \underset{˙}{\lor} F (G (Q)) \neq P \underset{˙}{\lor} Q \land \neg \exists r \in A (\neg r \underset{˙}{\leq} W \land P \underset{˙}{\lor} r = Q \underset{˙}{\lor} r))) \to Q \in A$
19	1 4 5 6	ltrncoat	$⊢ ((K \in HL \land W \in H) \land (F \in T \land G \in T) \land Q \in A) \to F (G (Q)) \in A$
20	11 12 18 19	syl3anc	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((F \in T \land G \in T) \land P \neq Q \land G (P) \neq P) \land (R (G) \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land F (G (P)) \underset{˙}{\lor} F (G (Q)) \neq P \underset{˙}{\lor} Q \land \neg \exists r \in A (\neg r \underset{˙}{\leq} W \land P \underset{˙}{\lor} r = Q \underset{˙}{\lor} r))) \to F (G (Q)) \in A$
21	15 2 4	hlatjcl	$⊢ (K \in HL \land Q \in A \land F (G (Q)) \in A) \to Q \underset{˙}{\lor} F (G (Q)) \in {Base}_{K}$
22	8 18 20 21	syl3anc	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((F \in T \land G \in T) \land P \neq Q \land G (P) \neq P) \land (R (G) \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land F (G (P)) \underset{˙}{\lor} F (G (Q)) \neq P \underset{˙}{\lor} Q \land \neg \exists r \in A (\neg r \underset{˙}{\leq} W \land P \underset{˙}{\lor} r = Q \underset{˙}{\lor} r))) \to Q \underset{˙}{\lor} F (G (Q)) \in {Base}_{K}$
23	15 3	latmcom	$⊢ (K \in Lat \land P \underset{˙}{\lor} F (G (P)) \in {Base}_{K} \land Q \underset{˙}{\lor} F (G (Q)) \in {Base}_{K}) \to (P \underset{˙}{\lor} F (G (P))) \underset{˙}{\land} (Q \underset{˙}{\lor} F (G (Q))) = (Q \underset{˙}{\lor} F (G (Q))) \underset{˙}{\land} (P \underset{˙}{\lor} F (G (P)))$
24	9 17 22 23	syl3anc	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((F \in T \land G \in T) \land P \neq Q \land G (P) \neq P) \land (R (G) \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land F (G (P)) \underset{˙}{\lor} F (G (Q)) \neq P \underset{˙}{\lor} Q \land \neg \exists r \in A (\neg r \underset{˙}{\leq} W \land P \underset{˙}{\lor} r = Q \underset{˙}{\lor} r))) \to (P \underset{˙}{\lor} F (G (P))) \underset{˙}{\land} (Q \underset{˙}{\lor} F (G (Q))) = (Q \underset{˙}{\lor} F (G (Q))) \underset{˙}{\land} (P \underset{˙}{\lor} F (G (P)))$
25	1 2 3 4 5 6 7	cdlemg19a	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((F \in T \land G \in T) \land P \neq Q \land G (P) \neq P) \land (R (G) \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land F (G (P)) \underset{˙}{\lor} F (G (Q)) \neq P \underset{˙}{\lor} Q \land \neg \exists r \in A (\neg r \underset{˙}{\leq} W \land P \underset{˙}{\lor} r = Q \underset{˙}{\lor} r))) \to (P \underset{˙}{\lor} F (G (P))) \underset{˙}{\land} (Q \underset{˙}{\lor} F (G (Q))) = (P \underset{˙}{\lor} F (G (P))) \underset{˙}{\land} W$
26		simp13	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((F \in T \land G \in T) \land P \neq Q \land G (P) \neq P) \land (R (G) \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land F (G (P)) \underset{˙}{\lor} F (G (Q)) \neq P \underset{˙}{\lor} Q \land \neg \exists r \in A (\neg r \underset{˙}{\leq} W \land P \underset{˙}{\lor} r = Q \underset{˙}{\lor} r))) \to (Q \in A \land \neg Q \underset{˙}{\leq} W)$
27		simp12	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((F \in T \land G \in T) \land P \neq Q \land G (P) \neq P) \land (R (G) \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land F (G (P)) \underset{˙}{\lor} F (G (Q)) \neq P \underset{˙}{\lor} Q \land \neg \exists r \in A (\neg r \underset{˙}{\leq} W \land P \underset{˙}{\lor} r = Q \underset{˙}{\lor} r))) \to (P \in A \land \neg P \underset{˙}{\leq} W)$
28		simp22	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((F \in T \land G \in T) \land P \neq Q \land G (P) \neq P) \land (R (G) \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land F (G (P)) \underset{˙}{\lor} F (G (Q)) \neq P \underset{˙}{\lor} Q \land \neg \exists r \in A (\neg r \underset{˙}{\leq} W \land P \underset{˙}{\lor} r = Q \underset{˙}{\lor} r))) \to P \neq Q$
29	28	necomd	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((F \in T \land G \in T) \land P \neq Q \land G (P) \neq P) \land (R (G) \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land F (G (P)) \underset{˙}{\lor} F (G (Q)) \neq P \underset{˙}{\lor} Q \land \neg \exists r \in A (\neg r \underset{˙}{\leq} W \land P \underset{˙}{\lor} r = Q \underset{˙}{\lor} r))) \to Q \neq P$
30		simp21r	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((F \in T \land G \in T) \land P \neq Q \land G (P) \neq P) \land (R (G) \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land F (G (P)) \underset{˙}{\lor} F (G (Q)) \neq P \underset{˙}{\lor} Q \land \neg \exists r \in A (\neg r \underset{˙}{\leq} W \land P \underset{˙}{\lor} r = Q \underset{˙}{\lor} r))) \to G \in T$
31		simp23	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((F \in T \land G \in T) \land P \neq Q \land G (P) \neq P) \land (R (G) \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land F (G (P)) \underset{˙}{\lor} F (G (Q)) \neq P \underset{˙}{\lor} Q \land \neg \exists r \in A (\neg r \underset{˙}{\leq} W \land P \underset{˙}{\lor} r = Q \underset{˙}{\lor} r))) \to G (P) \neq P$
32	1 4 5 6	ltrnatneq	$⊢ ((K \in HL \land W \in H) \land (G \in T \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land G (P) \neq P) \to G (Q) \neq Q$
33	11 30 27 26 31 32	syl131anc	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((F \in T \land G \in T) \land P \neq Q \land G (P) \neq P) \land (R (G) \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land F (G (P)) \underset{˙}{\lor} F (G (Q)) \neq P \underset{˙}{\lor} Q \land \neg \exists r \in A (\neg r \underset{˙}{\leq} W \land P \underset{˙}{\lor} r = Q \underset{˙}{\lor} r))) \to G (Q) \neq Q$
34		simp31	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((F \in T \land G \in T) \land P \neq Q \land G (P) \neq P) \land (R (G) \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land F (G (P)) \underset{˙}{\lor} F (G (Q)) \neq P \underset{˙}{\lor} Q \land \neg \exists r \in A (\neg r \underset{˙}{\leq} W \land P \underset{˙}{\lor} r = Q \underset{˙}{\lor} r))) \to R (G) \underset{˙}{\leq} (P \underset{˙}{\lor} Q)$
35	2 4	hlatjcom	$⊢ (K \in HL \land P \in A \land Q \in A) \to P \underset{˙}{\lor} Q = Q \underset{˙}{\lor} P$
36	8 10 18 35	syl3anc	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((F \in T \land G \in T) \land P \neq Q \land G (P) \neq P) \land (R (G) \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land F (G (P)) \underset{˙}{\lor} F (G (Q)) \neq P \underset{˙}{\lor} Q \land \neg \exists r \in A (\neg r \underset{˙}{\leq} W \land P \underset{˙}{\lor} r = Q \underset{˙}{\lor} r))) \to P \underset{˙}{\lor} Q = Q \underset{˙}{\lor} P$
37	34 36	breqtrd	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((F \in T \land G \in T) \land P \neq Q \land G (P) \neq P) \land (R (G) \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land F (G (P)) \underset{˙}{\lor} F (G (Q)) \neq P \underset{˙}{\lor} Q \land \neg \exists r \in A (\neg r \underset{˙}{\leq} W \land P \underset{˙}{\lor} r = Q \underset{˙}{\lor} r))) \to R (G) \underset{˙}{\leq} (Q \underset{˙}{\lor} P)$
38		simp32	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((F \in T \land G \in T) \land P \neq Q \land G (P) \neq P) \land (R (G) \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land F (G (P)) \underset{˙}{\lor} F (G (Q)) \neq P \underset{˙}{\lor} Q \land \neg \exists r \in A (\neg r \underset{˙}{\leq} W \land P \underset{˙}{\lor} r = Q \underset{˙}{\lor} r))) \to F (G (P)) \underset{˙}{\lor} F (G (Q)) \neq P \underset{˙}{\lor} Q$
39	2 4	hlatjcom	$⊢ (K \in HL \land F (G (P)) \in A \land F (G (Q)) \in A) \to F (G (P)) \underset{˙}{\lor} F (G (Q)) = F (G (Q)) \underset{˙}{\lor} F (G (P))$
40	8 14 20 39	syl3anc	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((F \in T \land G \in T) \land P \neq Q \land G (P) \neq P) \land (R (G) \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land F (G (P)) \underset{˙}{\lor} F (G (Q)) \neq P \underset{˙}{\lor} Q \land \neg \exists r \in A (\neg r \underset{˙}{\leq} W \land P \underset{˙}{\lor} r = Q \underset{˙}{\lor} r))) \to F (G (P)) \underset{˙}{\lor} F (G (Q)) = F (G (Q)) \underset{˙}{\lor} F (G (P))$
41	38 40 36	3netr3d	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((F \in T \land G \in T) \land P \neq Q \land G (P) \neq P) \land (R (G) \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land F (G (P)) \underset{˙}{\lor} F (G (Q)) \neq P \underset{˙}{\lor} Q \land \neg \exists r \in A (\neg r \underset{˙}{\leq} W \land P \underset{˙}{\lor} r = Q \underset{˙}{\lor} r))) \to F (G (Q)) \underset{˙}{\lor} F (G (P)) \neq Q \underset{˙}{\lor} P$
42		simp33	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((F \in T \land G \in T) \land P \neq Q \land G (P) \neq P) \land (R (G) \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land F (G (P)) \underset{˙}{\lor} F (G (Q)) \neq P \underset{˙}{\lor} Q \land \neg \exists r \in A (\neg r \underset{˙}{\leq} W \land P \underset{˙}{\lor} r = Q \underset{˙}{\lor} r))) \to \neg \exists r \in A (\neg r \underset{˙}{\leq} W \land P \underset{˙}{\lor} r = Q \underset{˙}{\lor} r)$
43		eqcom	$⊢ P \underset{˙}{\lor} r = Q \underset{˙}{\lor} r \leftrightarrow Q \underset{˙}{\lor} r = P \underset{˙}{\lor} r$
44	43	anbi2i	$⊢ (\neg r \underset{˙}{\leq} W \land P \underset{˙}{\lor} r = Q \underset{˙}{\lor} r) \leftrightarrow (\neg r \underset{˙}{\leq} W \land Q \underset{˙}{\lor} r = P \underset{˙}{\lor} r)$
45	44	rexbii	$⊢ \exists r \in A (\neg r \underset{˙}{\leq} W \land P \underset{˙}{\lor} r = Q \underset{˙}{\lor} r) \leftrightarrow \exists r \in A (\neg r \underset{˙}{\leq} W \land Q \underset{˙}{\lor} r = P \underset{˙}{\lor} r)$
46	42 45	sylnib	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((F \in T \land G \in T) \land P \neq Q \land G (P) \neq P) \land (R (G) \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land F (G (P)) \underset{˙}{\lor} F (G (Q)) \neq P \underset{˙}{\lor} Q \land \neg \exists r \in A (\neg r \underset{˙}{\leq} W \land P \underset{˙}{\lor} r = Q \underset{˙}{\lor} r))) \to \neg \exists r \in A (\neg r \underset{˙}{\leq} W \land Q \underset{˙}{\lor} r = P \underset{˙}{\lor} r)$
47	1 2 3 4 5 6 7	cdlemg19a	$⊢ (((K \in HL \land W \in H) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land (P \in A \land \neg P \underset{˙}{\leq} W)) \land ((F \in T \land G \in T) \land Q \neq P \land G (Q) \neq Q) \land (R (G) \underset{˙}{\leq} (Q \underset{˙}{\lor} P) \land F (G (Q)) \underset{˙}{\lor} F (G (P)) \neq Q \underset{˙}{\lor} P \land \neg \exists r \in A (\neg r \underset{˙}{\leq} W \land Q \underset{˙}{\lor} r = P \underset{˙}{\lor} r))) \to (Q \underset{˙}{\lor} F (G (Q))) \underset{˙}{\land} (P \underset{˙}{\lor} F (G (P))) = (Q \underset{˙}{\lor} F (G (Q))) \underset{˙}{\land} W$
48	11 26 27 12 29 33 37 41 46 47	syl333anc	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((F \in T \land G \in T) \land P \neq Q \land G (P) \neq P) \land (R (G) \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land F (G (P)) \underset{˙}{\lor} F (G (Q)) \neq P \underset{˙}{\lor} Q \land \neg \exists r \in A (\neg r \underset{˙}{\leq} W \land P \underset{˙}{\lor} r = Q \underset{˙}{\lor} r))) \to (Q \underset{˙}{\lor} F (G (Q))) \underset{˙}{\land} (P \underset{˙}{\lor} F (G (P))) = (Q \underset{˙}{\lor} F (G (Q))) \underset{˙}{\land} W$
49	24 25 48	3eqtr3d	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((F \in T \land G \in T) \land P \neq Q \land G (P) \neq P) \land (R (G) \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land F (G (P)) \underset{˙}{\lor} F (G (Q)) \neq P \underset{˙}{\lor} Q \land \neg \exists r \in A (\neg r \underset{˙}{\leq} W \land P \underset{˙}{\lor} r = Q \underset{˙}{\lor} r))) \to (P \underset{˙}{\lor} F (G (P))) \underset{˙}{\land} W = (Q \underset{˙}{\lor} F (G (Q))) \underset{˙}{\land} W$

Metamath Proof Explorer

Theorem cdlemg19

Proof