cdlemg12

	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	cdlemg12	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (F \in T \land G \in T \land P \neq Q) \land ((\neg R (F) \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg R (G) \underset{˙}{\leq} (P \underset{˙}{\lor} Q)) \land R (F) \neq R (G) \land F (G (P)) \underset{˙}{\lor} F (G (Q)) \neq P \underset{˙}{\lor} Q)) \to (P \underset{˙}{\lor} F (G (P))) \underset{˙}{\land} W = (Q \underset{˙}{\lor} F (G (Q))) \underset{˙}{\land} W$

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	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (F \in T \land G \in T \land P \neq Q) \land ((\neg R (F) \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg R (G) \underset{˙}{\leq} (P \underset{˙}{\lor} Q)) \land R (F) \neq R (G) \land F (G (P)) \underset{˙}{\lor} F (G (Q)) \neq P \underset{˙}{\lor} Q)) \to K \in HL$
9	8	hllatd	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (F \in T \land G \in T \land P \neq Q) \land ((\neg R (F) \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg R (G) \underset{˙}{\leq} (P \underset{˙}{\lor} Q)) \land R (F) \neq R (G) \land F (G (P)) \underset{˙}{\lor} F (G (Q)) \neq P \underset{˙}{\lor} Q)) \to K \in Lat$
10		simp12l	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (F \in T \land G \in T \land P \neq Q) \land ((\neg R (F) \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg R (G) \underset{˙}{\leq} (P \underset{˙}{\lor} Q)) \land R (F) \neq R (G) \land F (G (P)) \underset{˙}{\lor} F (G (Q)) \neq P \underset{˙}{\lor} Q)) \to P \in A$
11		simp11	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (F \in T \land G \in T \land P \neq Q) \land ((\neg R (F) \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg R (G) \underset{˙}{\leq} (P \underset{˙}{\lor} Q)) \land R (F) \neq R (G) \land F (G (P)) \underset{˙}{\lor} F (G (Q)) \neq P \underset{˙}{\lor} Q)) \to (K \in HL \land W \in H)$
12		simp21	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (F \in T \land G \in T \land P \neq Q) \land ((\neg R (F) \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg R (G) \underset{˙}{\leq} (P \underset{˙}{\lor} Q)) \land R (F) \neq R (G) \land F (G (P)) \underset{˙}{\lor} F (G (Q)) \neq P \underset{˙}{\lor} Q)) \to F \in T$
13		simp22	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (F \in T \land G \in T \land P \neq Q) \land ((\neg R (F) \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg R (G) \underset{˙}{\leq} (P \underset{˙}{\lor} Q)) \land R (F) \neq R (G) \land F (G (P)) \underset{˙}{\lor} F (G (Q)) \neq P \underset{˙}{\lor} Q)) \to G \in T$
14	1 4 5 6	ltrncoat	$⊢ ((K \in HL \land W \in H) \land (F \in T \land G \in T) \land P \in A) \to F (G (P)) \in A$
15	11 12 13 10 14	syl121anc	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (F \in T \land G \in T \land P \neq Q) \land ((\neg R (F) \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg R (G) \underset{˙}{\leq} (P \underset{˙}{\lor} Q)) \land R (F) \neq R (G) \land F (G (P)) \underset{˙}{\lor} F (G (Q)) \neq P \underset{˙}{\lor} Q)) \to F (G (P)) \in A$
16		eqid	$⊢ {Base}_{K} = {Base}_{K}$
17	16 2 4	hlatjcl	$⊢ (K \in HL \land P \in A \land F (G (P)) \in A) \to P \underset{˙}{\lor} F (G (P)) \in {Base}_{K}$
18	8 10 15 17	syl3anc	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (F \in T \land G \in T \land P \neq Q) \land ((\neg R (F) \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg R (G) \underset{˙}{\leq} (P \underset{˙}{\lor} Q)) \land R (F) \neq R (G) \land F (G (P)) \underset{˙}{\lor} F (G (Q)) \neq P \underset{˙}{\lor} Q)) \to P \underset{˙}{\lor} F (G (P)) \in {Base}_{K}$
19		simp13l	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (F \in T \land G \in T \land P \neq Q) \land ((\neg R (F) \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg R (G) \underset{˙}{\leq} (P \underset{˙}{\lor} Q)) \land R (F) \neq R (G) \land F (G (P)) \underset{˙}{\lor} F (G (Q)) \neq P \underset{˙}{\lor} Q)) \to Q \in A$
20	1 4 5 6	ltrncoat	$⊢ ((K \in HL \land W \in H) \land (F \in T \land G \in T) \land Q \in A) \to F (G (Q)) \in A$
21	11 12 13 19 20	syl121anc	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (F \in T \land G \in T \land P \neq Q) \land ((\neg R (F) \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg R (G) \underset{˙}{\leq} (P \underset{˙}{\lor} Q)) \land R (F) \neq R (G) \land F (G (P)) \underset{˙}{\lor} F (G (Q)) \neq P \underset{˙}{\lor} Q)) \to F (G (Q)) \in A$
22	16 2 4	hlatjcl	$⊢ (K \in HL \land Q \in A \land F (G (Q)) \in A) \to Q \underset{˙}{\lor} F (G (Q)) \in {Base}_{K}$
23	8 19 21 22	syl3anc	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (F \in T \land G \in T \land P \neq Q) \land ((\neg R (F) \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg R (G) \underset{˙}{\leq} (P \underset{˙}{\lor} Q)) \land R (F) \neq R (G) \land F (G (P)) \underset{˙}{\lor} F (G (Q)) \neq P \underset{˙}{\lor} Q)) \to Q \underset{˙}{\lor} F (G (Q)) \in {Base}_{K}$
24	16 3	latmcom	$⊢ (K \in Lat \land P \underset{˙}{\lor} F (G (P)) \in {Base}_{K} \land Q \underset{˙}{\lor} F (G (Q)) \in {Base}_{K}) \to (P \underset{˙}{\lor} F (G (P))) \underset{˙}{\land} (Q \underset{˙}{\lor} F (G (Q))) = (Q \underset{˙}{\lor} F (G (Q))) \underset{˙}{\land} (P \underset{˙}{\lor} F (G (P)))$
25	9 18 23 24	syl3anc	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (F \in T \land G \in T \land P \neq Q) \land ((\neg R (F) \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg R (G) \underset{˙}{\leq} (P \underset{˙}{\lor} Q)) \land R (F) \neq R (G) \land F (G (P)) \underset{˙}{\lor} F (G (Q)) \neq P \underset{˙}{\lor} Q)) \to (P \underset{˙}{\lor} F (G (P))) \underset{˙}{\land} (Q \underset{˙}{\lor} F (G (Q))) = (Q \underset{˙}{\lor} F (G (Q))) \underset{˙}{\land} (P \underset{˙}{\lor} F (G (P)))$
26	1 2 3 4 5 6 7	cdlemg12g	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (F \in T \land G \in T \land P \neq Q) \land ((\neg R (F) \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg R (G) \underset{˙}{\leq} (P \underset{˙}{\lor} Q)) \land R (F) \neq R (G) \land F (G (P)) \underset{˙}{\lor} F (G (Q)) \neq P \underset{˙}{\lor} Q)) \to (P \underset{˙}{\lor} F (G (P))) \underset{˙}{\land} (Q \underset{˙}{\lor} F (G (Q))) = (P \underset{˙}{\lor} F (G (P))) \underset{˙}{\land} W$
27		simp13	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (F \in T \land G \in T \land P \neq Q) \land ((\neg R (F) \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg R (G) \underset{˙}{\leq} (P \underset{˙}{\lor} Q)) \land R (F) \neq R (G) \land F (G (P)) \underset{˙}{\lor} F (G (Q)) \neq P \underset{˙}{\lor} Q)) \to (Q \in A \land \neg Q \underset{˙}{\leq} W)$
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		simp23	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (F \in T \land G \in T \land P \neq Q) \land ((\neg R (F) \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg R (G) \underset{˙}{\leq} (P \underset{˙}{\lor} Q)) \land R (F) \neq R (G) \land F (G (P)) \underset{˙}{\lor} F (G (Q)) \neq P \underset{˙}{\lor} Q)) \to P \neq Q$
30	29	necomd	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (F \in T \land G \in T \land P \neq Q) \land ((\neg R (F) \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg R (G) \underset{˙}{\leq} (P \underset{˙}{\lor} Q)) \land R (F) \neq R (G) \land F (G (P)) \underset{˙}{\lor} F (G (Q)) \neq P \underset{˙}{\lor} Q)) \to Q \neq P$
31		simp31l	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (F \in T \land G \in T \land P \neq Q) \land ((\neg R (F) \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg R (G) \underset{˙}{\leq} (P \underset{˙}{\lor} Q)) \land R (F) \neq R (G) \land F (G (P)) \underset{˙}{\lor} F (G (Q)) \neq P \underset{˙}{\lor} Q)) \to \neg R (F) \underset{˙}{\leq} (P \underset{˙}{\lor} Q)$
32	2 4	hlatjcom	$⊢ (K \in HL \land P \in A \land Q \in A) \to P \underset{˙}{\lor} Q = Q \underset{˙}{\lor} P$
33	8 10 19 32	syl3anc	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (F \in T \land G \in T \land P \neq Q) \land ((\neg R (F) \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg R (G) \underset{˙}{\leq} (P \underset{˙}{\lor} Q)) \land R (F) \neq R (G) \land F (G (P)) \underset{˙}{\lor} F (G (Q)) \neq P \underset{˙}{\lor} Q)) \to P \underset{˙}{\lor} Q = Q \underset{˙}{\lor} P$
34	33	breq2d	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (F \in T \land G \in T \land P \neq Q) \land ((\neg R (F) \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg R (G) \underset{˙}{\leq} (P \underset{˙}{\lor} Q)) \land R (F) \neq R (G) \land F (G (P)) \underset{˙}{\lor} F (G (Q)) \neq P \underset{˙}{\lor} Q)) \to (R (F) \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \leftrightarrow R (F) \underset{˙}{\leq} (Q \underset{˙}{\lor} P))$
35	31 34	mtbid	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (F \in T \land G \in T \land P \neq Q) \land ((\neg R (F) \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg R (G) \underset{˙}{\leq} (P \underset{˙}{\lor} Q)) \land R (F) \neq R (G) \land F (G (P)) \underset{˙}{\lor} F (G (Q)) \neq P \underset{˙}{\lor} Q)) \to \neg R (F) \underset{˙}{\leq} (Q \underset{˙}{\lor} P)$
36		simp31r	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (F \in T \land G \in T \land P \neq Q) \land ((\neg R (F) \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg R (G) \underset{˙}{\leq} (P \underset{˙}{\lor} Q)) \land R (F) \neq R (G) \land F (G (P)) \underset{˙}{\lor} F (G (Q)) \neq P \underset{˙}{\lor} Q)) \to \neg R (G) \underset{˙}{\leq} (P \underset{˙}{\lor} Q)$
37	33	breq2d	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (F \in T \land G \in T \land P \neq Q) \land ((\neg R (F) \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg R (G) \underset{˙}{\leq} (P \underset{˙}{\lor} Q)) \land R (F) \neq R (G) \land F (G (P)) \underset{˙}{\lor} F (G (Q)) \neq P \underset{˙}{\lor} Q)) \to (R (G) \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \leftrightarrow R (G) \underset{˙}{\leq} (Q \underset{˙}{\lor} P))$
38	36 37	mtbid	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (F \in T \land G \in T \land P \neq Q) \land ((\neg R (F) \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg R (G) \underset{˙}{\leq} (P \underset{˙}{\lor} Q)) \land R (F) \neq R (G) \land F (G (P)) \underset{˙}{\lor} F (G (Q)) \neq P \underset{˙}{\lor} Q)) \to \neg R (G) \underset{˙}{\leq} (Q \underset{˙}{\lor} P)$
39	35 38	jca	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (F \in T \land G \in T \land P \neq Q) \land ((\neg R (F) \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg R (G) \underset{˙}{\leq} (P \underset{˙}{\lor} Q)) \land R (F) \neq R (G) \land F (G (P)) \underset{˙}{\lor} F (G (Q)) \neq P \underset{˙}{\lor} Q)) \to (\neg R (F) \underset{˙}{\leq} (Q \underset{˙}{\lor} P) \land \neg R (G) \underset{˙}{\leq} (Q \underset{˙}{\lor} P))$
40		simp32	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (F \in T \land G \in T \land P \neq Q) \land ((\neg R (F) \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg R (G) \underset{˙}{\leq} (P \underset{˙}{\lor} Q)) \land R (F) \neq R (G) \land F (G (P)) \underset{˙}{\lor} F (G (Q)) \neq P \underset{˙}{\lor} Q)) \to R (F) \neq R (G)$
41		simp33	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (F \in T \land G \in T \land P \neq Q) \land ((\neg R (F) \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg R (G) \underset{˙}{\leq} (P \underset{˙}{\lor} Q)) \land R (F) \neq R (G) \land F (G (P)) \underset{˙}{\lor} F (G (Q)) \neq P \underset{˙}{\lor} Q)) \to F (G (P)) \underset{˙}{\lor} F (G (Q)) \neq P \underset{˙}{\lor} Q$
42	2 4	hlatjcom	$⊢ (K \in HL \land F (G (P)) \in A \land F (G (Q)) \in A) \to F (G (P)) \underset{˙}{\lor} F (G (Q)) = F (G (Q)) \underset{˙}{\lor} F (G (P))$
43	8 15 21 42	syl3anc	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (F \in T \land G \in T \land P \neq Q) \land ((\neg R (F) \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg R (G) \underset{˙}{\leq} (P \underset{˙}{\lor} Q)) \land R (F) \neq R (G) \land F (G (P)) \underset{˙}{\lor} F (G (Q)) \neq P \underset{˙}{\lor} Q)) \to F (G (P)) \underset{˙}{\lor} F (G (Q)) = F (G (Q)) \underset{˙}{\lor} F (G (P))$
44	41 43 33	3netr3d	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (F \in T \land G \in T \land P \neq Q) \land ((\neg R (F) \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg R (G) \underset{˙}{\leq} (P \underset{˙}{\lor} Q)) \land R (F) \neq R (G) \land F (G (P)) \underset{˙}{\lor} F (G (Q)) \neq P \underset{˙}{\lor} Q)) \to F (G (Q)) \underset{˙}{\lor} F (G (P)) \neq Q \underset{˙}{\lor} P$
45	1 2 3 4 5 6 7	cdlemg12g	$⊢ (((K \in HL \land W \in H) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land (P \in A \land \neg P \underset{˙}{\leq} W)) \land (F \in T \land G \in T \land Q \neq P) \land ((\neg R (F) \underset{˙}{\leq} (Q \underset{˙}{\lor} P) \land \neg R (G) \underset{˙}{\leq} (Q \underset{˙}{\lor} P)) \land R (F) \neq R (G) \land F (G (Q)) \underset{˙}{\lor} F (G (P)) \neq Q \underset{˙}{\lor} P)) \to (Q \underset{˙}{\lor} F (G (Q))) \underset{˙}{\land} (P \underset{˙}{\lor} F (G (P))) = (Q \underset{˙}{\lor} F (G (Q))) \underset{˙}{\land} W$
46	11 27 28 12 13 30 39 40 44 45	syl333anc	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (F \in T \land G \in T \land P \neq Q) \land ((\neg R (F) \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg R (G) \underset{˙}{\leq} (P \underset{˙}{\lor} Q)) \land R (F) \neq R (G) \land F (G (P)) \underset{˙}{\lor} F (G (Q)) \neq P \underset{˙}{\lor} Q)) \to (Q \underset{˙}{\lor} F (G (Q))) \underset{˙}{\land} (P \underset{˙}{\lor} F (G (P))) = (Q \underset{˙}{\lor} F (G (Q))) \underset{˙}{\land} W$
47	25 26 46	3eqtr3d	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (F \in T \land G \in T \land P \neq Q) \land ((\neg R (F) \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg R (G) \underset{˙}{\leq} (P \underset{˙}{\lor} Q)) \land R (F) \neq R (G) \land F (G (P)) \underset{˙}{\lor} F (G (Q)) \neq P \underset{˙}{\lor} Q)) \to (P \underset{˙}{\lor} F (G (P))) \underset{˙}{\land} W = (Q \underset{˙}{\lor} F (G (Q))) \underset{˙}{\land} W$

Metamath Proof Explorer

Theorem cdlemg12

Proof