cdlemg12g

Description: TODO: FIX COMMENT. TODO: Combine with cdlemg12f . (Contributed by NM, 6-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	cdlemg12g	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (F \in T \land G \in T \land P \neq Q) \land ((\neg R (F) \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg R (G) \underset{˙}{\leq} (P \underset{˙}{\lor} Q)) \land R (F) \neq R (G) \land F (G (P)) \underset{˙}{\lor} F (G (Q)) \neq P \underset{˙}{\lor} Q)) \to (P \underset{˙}{\lor} F (G (P))) \underset{˙}{\land} (Q \underset{˙}{\lor} F (G (Q))) = (P \underset{˙}{\lor} F (G (P))) \underset{˙}{\land} W$

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	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (F \in T \land G \in T \land P \neq Q) \land ((\neg R (F) \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg R (G) \underset{˙}{\leq} (P \underset{˙}{\lor} Q)) \land R (F) \neq R (G) \land F (G (P)) \underset{˙}{\lor} F (G (Q)) \neq P \underset{˙}{\lor} Q)) \to K \in HL$
9		hlop	$⊢ K \in HL \to K \in OP$
10	8 9	syl	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (F \in T \land G \in T \land P \neq Q) \land ((\neg R (F) \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg R (G) \underset{˙}{\leq} (P \underset{˙}{\lor} Q)) \land R (F) \neq R (G) \land F (G (P)) \underset{˙}{\lor} F (G (Q)) \neq P \underset{˙}{\lor} Q)) \to K \in OP$
11	8	hllatd	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (F \in T \land G \in T \land P \neq Q) \land ((\neg R (F) \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg R (G) \underset{˙}{\leq} (P \underset{˙}{\lor} Q)) \land R (F) \neq R (G) \land F (G (P)) \underset{˙}{\lor} F (G (Q)) \neq P \underset{˙}{\lor} Q)) \to K \in Lat$
12		simp12l	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (F \in T \land G \in T \land P \neq Q) \land ((\neg R (F) \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg R (G) \underset{˙}{\leq} (P \underset{˙}{\lor} Q)) \land R (F) \neq R (G) \land F (G (P)) \underset{˙}{\lor} F (G (Q)) \neq P \underset{˙}{\lor} Q)) \to P \in A$
13		simp11	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (F \in T \land G \in T \land P \neq Q) \land ((\neg R (F) \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg R (G) \underset{˙}{\leq} (P \underset{˙}{\lor} Q)) \land R (F) \neq R (G) \land F (G (P)) \underset{˙}{\lor} F (G (Q)) \neq P \underset{˙}{\lor} Q)) \to (K \in HL \land W \in H)$
14		simp21	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (F \in T \land G \in T \land P \neq Q) \land ((\neg R (F) \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg R (G) \underset{˙}{\leq} (P \underset{˙}{\lor} Q)) \land R (F) \neq R (G) \land F (G (P)) \underset{˙}{\lor} F (G (Q)) \neq P \underset{˙}{\lor} Q)) \to F \in T$
15		simp22	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (F \in T \land G \in T \land P \neq Q) \land ((\neg R (F) \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg R (G) \underset{˙}{\leq} (P \underset{˙}{\lor} Q)) \land R (F) \neq R (G) \land F (G (P)) \underset{˙}{\lor} F (G (Q)) \neq P \underset{˙}{\lor} Q)) \to G \in T$
16	1 4 5 6	ltrncoat	$⊢ ((K \in HL \land W \in H) \land (F \in T \land G \in T) \land P \in A) \to F (G (P)) \in A$
17	13 14 15 12 16	syl121anc	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (F \in T \land G \in T \land P \neq Q) \land ((\neg R (F) \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg R (G) \underset{˙}{\leq} (P \underset{˙}{\lor} Q)) \land R (F) \neq R (G) \land F (G (P)) \underset{˙}{\lor} F (G (Q)) \neq P \underset{˙}{\lor} Q)) \to F (G (P)) \in A$
18		eqid	$⊢ {Base}_{K} = {Base}_{K}$
19	18 2 4	hlatjcl	$⊢ (K \in HL \land P \in A \land F (G (P)) \in A) \to P \underset{˙}{\lor} F (G (P)) \in {Base}_{K}$
20	8 12 17 19	syl3anc	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (F \in T \land G \in T \land P \neq Q) \land ((\neg R (F) \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg R (G) \underset{˙}{\leq} (P \underset{˙}{\lor} Q)) \land R (F) \neq R (G) \land F (G (P)) \underset{˙}{\lor} F (G (Q)) \neq P \underset{˙}{\lor} Q)) \to P \underset{˙}{\lor} F (G (P)) \in {Base}_{K}$
21		simp13l	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (F \in T \land G \in T \land P \neq Q) \land ((\neg R (F) \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg R (G) \underset{˙}{\leq} (P \underset{˙}{\lor} Q)) \land R (F) \neq R (G) \land F (G (P)) \underset{˙}{\lor} F (G (Q)) \neq P \underset{˙}{\lor} Q)) \to Q \in A$
22	1 4 5 6	ltrncoat	$⊢ ((K \in HL \land W \in H) \land (F \in T \land G \in T) \land Q \in A) \to F (G (Q)) \in A$
23	13 14 15 21 22	syl121anc	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (F \in T \land G \in T \land P \neq Q) \land ((\neg R (F) \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg R (G) \underset{˙}{\leq} (P \underset{˙}{\lor} Q)) \land R (F) \neq R (G) \land F (G (P)) \underset{˙}{\lor} F (G (Q)) \neq P \underset{˙}{\lor} Q)) \to F (G (Q)) \in A$
24	18 2 4	hlatjcl	$⊢ (K \in HL \land Q \in A \land F (G (Q)) \in A) \to Q \underset{˙}{\lor} F (G (Q)) \in {Base}_{K}$
25	8 21 23 24	syl3anc	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (F \in T \land G \in T \land P \neq Q) \land ((\neg R (F) \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg R (G) \underset{˙}{\leq} (P \underset{˙}{\lor} Q)) \land R (F) \neq R (G) \land F (G (P)) \underset{˙}{\lor} F (G (Q)) \neq P \underset{˙}{\lor} Q)) \to Q \underset{˙}{\lor} F (G (Q)) \in {Base}_{K}$
26	18 3	latmcl	$⊢ (K \in Lat \land P \underset{˙}{\lor} F (G (P)) \in {Base}_{K} \land Q \underset{˙}{\lor} F (G (Q)) \in {Base}_{K}) \to (P \underset{˙}{\lor} F (G (P))) \underset{˙}{\land} (Q \underset{˙}{\lor} F (G (Q))) \in {Base}_{K}$
27	11 20 25 26	syl3anc	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (F \in T \land G \in T \land P \neq Q) \land ((\neg R (F) \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg R (G) \underset{˙}{\leq} (P \underset{˙}{\lor} Q)) \land R (F) \neq R (G) \land F (G (P)) \underset{˙}{\lor} F (G (Q)) \neq P \underset{˙}{\lor} Q)) \to (P \underset{˙}{\lor} F (G (P))) \underset{˙}{\land} (Q \underset{˙}{\lor} F (G (Q))) \in {Base}_{K}$
28		simp12	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (F \in T \land G \in T \land P \neq Q) \land ((\neg R (F) \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg R (G) \underset{˙}{\leq} (P \underset{˙}{\lor} Q)) \land R (F) \neq R (G) \land F (G (P)) \underset{˙}{\lor} F (G (Q)) \neq P \underset{˙}{\lor} Q)) \to (P \in A \land \neg P \underset{˙}{\leq} W)$
29		simp13	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (F \in T \land G \in T \land P \neq Q) \land ((\neg R (F) \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg R (G) \underset{˙}{\leq} (P \underset{˙}{\lor} Q)) \land R (F) \neq R (G) \land F (G (P)) \underset{˙}{\lor} F (G (Q)) \neq P \underset{˙}{\lor} Q)) \to (Q \in A \land \neg Q \underset{˙}{\leq} W)$
30		simp33	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (F \in T \land G \in T \land P \neq Q) \land ((\neg R (F) \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg R (G) \underset{˙}{\leq} (P \underset{˙}{\lor} Q)) \land R (F) \neq R (G) \land F (G (P)) \underset{˙}{\lor} F (G (Q)) \neq P \underset{˙}{\lor} Q)) \to F (G (P)) \underset{˙}{\lor} F (G (Q)) \neq P \underset{˙}{\lor} Q$
31	1 2 3 4 5 6	cdlemg11a	$⊢ ((K \in HL \land W \in H) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (F \in T \land G \in T \land F (G (P)) \underset{˙}{\lor} F (G (Q)) \neq P \underset{˙}{\lor} Q)) \to F (G (P)) \neq P$
32	31	necomd	$⊢ ((K \in HL \land W \in H) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (F \in T \land G \in T \land F (G (P)) \underset{˙}{\lor} F (G (Q)) \neq P \underset{˙}{\lor} Q)) \to P \neq F (G (P))$
33	13 28 29 14 15 30 32	syl123anc	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (F \in T \land G \in T \land P \neq Q) \land ((\neg R (F) \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg R (G) \underset{˙}{\leq} (P \underset{˙}{\lor} Q)) \land R (F) \neq R (G) \land F (G (P)) \underset{˙}{\lor} F (G (Q)) \neq P \underset{˙}{\lor} Q)) \to P \neq F (G (P))$
34	1 2 3 4 5	lhpat	$⊢ ((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (F (G (P)) \in A \land P \neq F (G (P)))) \to (P \underset{˙}{\lor} F (G (P))) \underset{˙}{\land} W \in A$
35	13 28 17 33 34	syl112anc	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (F \in T \land G \in T \land P \neq Q) \land ((\neg R (F) \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg R (G) \underset{˙}{\leq} (P \underset{˙}{\lor} Q)) \land R (F) \neq R (G) \land F (G (P)) \underset{˙}{\lor} F (G (Q)) \neq P \underset{˙}{\lor} Q)) \to (P \underset{˙}{\lor} F (G (P))) \underset{˙}{\land} W \in A$
36	2 4	hlatjcom	$⊢ (K \in HL \land P \in A \land F (G (P)) \in A) \to P \underset{˙}{\lor} F (G (P)) = F (G (P)) \underset{˙}{\lor} P$
37	8 12 17 36	syl3anc	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (F \in T \land G \in T \land P \neq Q) \land ((\neg R (F) \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg R (G) \underset{˙}{\leq} (P \underset{˙}{\lor} Q)) \land R (F) \neq R (G) \land F (G (P)) \underset{˙}{\lor} F (G (Q)) \neq P \underset{˙}{\lor} Q)) \to P \underset{˙}{\lor} F (G (P)) = F (G (P)) \underset{˙}{\lor} P$
38	2 4	hlatjcom	$⊢ (K \in HL \land Q \in A \land F (G (Q)) \in A) \to Q \underset{˙}{\lor} F (G (Q)) = F (G (Q)) \underset{˙}{\lor} Q$
39	8 21 23 38	syl3anc	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (F \in T \land G \in T \land P \neq Q) \land ((\neg R (F) \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg R (G) \underset{˙}{\leq} (P \underset{˙}{\lor} Q)) \land R (F) \neq R (G) \land F (G (P)) \underset{˙}{\lor} F (G (Q)) \neq P \underset{˙}{\lor} Q)) \to Q \underset{˙}{\lor} F (G (Q)) = F (G (Q)) \underset{˙}{\lor} Q$
40	37 39	oveq12d	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (F \in T \land G \in T \land P \neq Q) \land ((\neg R (F) \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg R (G) \underset{˙}{\leq} (P \underset{˙}{\lor} Q)) \land R (F) \neq R (G) \land F (G (P)) \underset{˙}{\lor} F (G (Q)) \neq P \underset{˙}{\lor} Q)) \to (P \underset{˙}{\lor} F (G (P))) \underset{˙}{\land} (Q \underset{˙}{\lor} F (G (Q))) = (F (G (P)) \underset{˙}{\lor} P) \underset{˙}{\land} (F (G (Q)) \underset{˙}{\lor} Q)$
41		simp1	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (F \in T \land G \in T \land P \neq Q) \land ((\neg R (F) \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg R (G) \underset{˙}{\leq} (P \underset{˙}{\lor} Q)) \land R (F) \neq R (G) \land F (G (P)) \underset{˙}{\lor} F (G (Q)) \neq P \underset{˙}{\lor} Q)) \to ((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W))$
42		simp2	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (F \in T \land G \in T \land P \neq Q) \land ((\neg R (F) \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg R (G) \underset{˙}{\leq} (P \underset{˙}{\lor} Q)) \land R (F) \neq R (G) \land F (G (P)) \underset{˙}{\lor} F (G (Q)) \neq P \underset{˙}{\lor} Q)) \to (F \in T \land G \in T \land P \neq Q)$
43		simp31l	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (F \in T \land G \in T \land P \neq Q) \land ((\neg R (F) \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg R (G) \underset{˙}{\leq} (P \underset{˙}{\lor} Q)) \land R (F) \neq R (G) \land F (G (P)) \underset{˙}{\lor} F (G (Q)) \neq P \underset{˙}{\lor} Q)) \to \neg R (F) \underset{˙}{\leq} (P \underset{˙}{\lor} Q)$
44		simp31r	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (F \in T \land G \in T \land P \neq Q) \land ((\neg R (F) \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg R (G) \underset{˙}{\leq} (P \underset{˙}{\lor} Q)) \land R (F) \neq R (G) \land F (G (P)) \underset{˙}{\lor} F (G (Q)) \neq P \underset{˙}{\lor} Q)) \to \neg R (G) \underset{˙}{\leq} (P \underset{˙}{\lor} Q)$
45		simp32	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (F \in T \land G \in T \land P \neq Q) \land ((\neg R (F) \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg R (G) \underset{˙}{\leq} (P \underset{˙}{\lor} Q)) \land R (F) \neq R (G) \land F (G (P)) \underset{˙}{\lor} F (G (Q)) \neq P \underset{˙}{\lor} Q)) \to R (F) \neq R (G)$
46		eqid	$⊢ 0. (K) = 0. (K)$
47	1 2 3 4 5 6 7 46	cdlemg12e	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (F \in T \land G \in T \land P \neq Q) \land (\neg R (F) \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg R (G) \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land R (F) \neq R (G))) \to (F (G (P)) \underset{˙}{\lor} P) \underset{˙}{\land} (F (G (Q)) \underset{˙}{\lor} Q) \neq 0. (K)$
48	41 42 43 44 45 47	syl113anc	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (F \in T \land G \in T \land P \neq Q) \land ((\neg R (F) \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg R (G) \underset{˙}{\leq} (P \underset{˙}{\lor} Q)) \land R (F) \neq R (G) \land F (G (P)) \underset{˙}{\lor} F (G (Q)) \neq P \underset{˙}{\lor} Q)) \to (F (G (P)) \underset{˙}{\lor} P) \underset{˙}{\land} (F (G (Q)) \underset{˙}{\lor} Q) \neq 0. (K)$
49	40 48	eqnetrd	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (F \in T \land G \in T \land P \neq Q) \land ((\neg R (F) \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg R (G) \underset{˙}{\leq} (P \underset{˙}{\lor} Q)) \land R (F) \neq R (G) \land F (G (P)) \underset{˙}{\lor} F (G (Q)) \neq P \underset{˙}{\lor} Q)) \to (P \underset{˙}{\lor} F (G (P))) \underset{˙}{\land} (Q \underset{˙}{\lor} F (G (Q))) \neq 0. (K)$
50	1 2 3 4 5 6 7	cdlemg12f	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (F \in T \land G \in T \land P \neq Q) \land ((\neg R (F) \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg R (G) \underset{˙}{\leq} (P \underset{˙}{\lor} Q)) \land R (F) \neq R (G) \land F (G (P)) \underset{˙}{\lor} F (G (Q)) \neq P \underset{˙}{\lor} Q)) \to ((P \underset{˙}{\lor} F (G (P))) \underset{˙}{\land} (Q \underset{˙}{\lor} F (G (Q)))) \underset{˙}{\leq} ((P \underset{˙}{\lor} F (G (P))) \underset{˙}{\land} W)$
51	18 1 46 4	leat2	$⊢ ((K \in OP \land (P \underset{˙}{\lor} F (G (P))) \underset{˙}{\land} (Q \underset{˙}{\lor} F (G (Q))) \in {Base}_{K} \land (P \underset{˙}{\lor} F (G (P))) \underset{˙}{\land} W \in A) \land ((P \underset{˙}{\lor} F (G (P))) \underset{˙}{\land} (Q \underset{˙}{\lor} F (G (Q))) \neq 0. (K) \land ((P \underset{˙}{\lor} F (G (P))) \underset{˙}{\land} (Q \underset{˙}{\lor} F (G (Q)))) \underset{˙}{\leq} ((P \underset{˙}{\lor} F (G (P))) \underset{˙}{\land} W))) \to (P \underset{˙}{\lor} F (G (P))) \underset{˙}{\land} (Q \underset{˙}{\lor} F (G (Q))) = (P \underset{˙}{\lor} F (G (P))) \underset{˙}{\land} W$
52	10 27 35 49 50 51	syl32anc	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (F \in T \land G \in T \land P \neq Q) \land ((\neg R (F) \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg R (G) \underset{˙}{\leq} (P \underset{˙}{\lor} Q)) \land R (F) \neq R (G) \land F (G (P)) \underset{˙}{\lor} F (G (Q)) \neq P \underset{˙}{\lor} Q)) \to (P \underset{˙}{\lor} F (G (P))) \underset{˙}{\land} (Q \underset{˙}{\lor} F (G (Q))) = (P \underset{˙}{\lor} F (G (P))) \underset{˙}{\land} W$

Metamath Proof Explorer

Theorem cdlemg12g

Proof