cdlemg10a

	Ref	Expression
Hypotheses	cdlemg8.l	$⊢ \underset{˙}{\leq} = \leq_{K}$
	cdlemg8.j	$⊢ \underset{˙}{\lor} = join (K)$
	cdlemg8.m	$⊢ \underset{˙}{\land} = meet (K)$
	cdlemg8.a	$⊢ A = Atoms (K)$
	cdlemg8.h	$⊢ H = LHyp (K)$
	cdlemg8.t	$⊢ T = LTrn (K) (W)$
	cdlemg10.r	$⊢ R = trL (K) (W)$
Assertion	cdlemg10a	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (F \in T \land G \in T \land P \neq Q) \land (F (G (P)) \underset{˙}{\lor} F (G (Q)) \neq P \underset{˙}{\lor} Q \land \neg R (F) \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg R (G) \underset{˙}{\leq} (P \underset{˙}{\lor} Q))) \to ((P \underset{˙}{\lor} F (G (P))) \underset{˙}{\land} (Q \underset{˙}{\lor} F (G (Q)))) \underset{˙}{\leq} (R (F) \underset{˙}{\lor} R (G))$

Proof

Step	Hyp	Ref	Expression
1		cdlemg8.l	$⊢ \underset{˙}{\leq} = \leq_{K}$
2		cdlemg8.j	$⊢ \underset{˙}{\lor} = join (K)$
3		cdlemg8.m	$⊢ \underset{˙}{\land} = meet (K)$
4		cdlemg8.a	$⊢ A = Atoms (K)$
5		cdlemg8.h	$⊢ H = LHyp (K)$
6		cdlemg8.t	$⊢ T = LTrn (K) (W)$
7		cdlemg10.r	$⊢ R = trL (K) (W)$
8		simp11	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (F \in T \land G \in T \land P \neq Q) \land (F (G (P)) \underset{˙}{\lor} F (G (Q)) \neq P \underset{˙}{\lor} Q \land \neg R (F) \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg R (G) \underset{˙}{\leq} (P \underset{˙}{\lor} Q))) \to (K \in HL \land W \in H)$
9		simp12	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (F \in T \land G \in T \land P \neq Q) \land (F (G (P)) \underset{˙}{\lor} F (G (Q)) \neq P \underset{˙}{\lor} Q \land \neg R (F) \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg R (G) \underset{˙}{\leq} (P \underset{˙}{\lor} Q))) \to (P \in A \land \neg P \underset{˙}{\leq} W)$
10		simp13	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (F \in T \land G \in T \land P \neq Q) \land (F (G (P)) \underset{˙}{\lor} F (G (Q)) \neq P \underset{˙}{\lor} Q \land \neg R (F) \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg R (G) \underset{˙}{\leq} (P \underset{˙}{\lor} Q))) \to (Q \in A \land \neg Q \underset{˙}{\leq} W)$
11		simp21	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (F \in T \land G \in T \land P \neq Q) \land (F (G (P)) \underset{˙}{\lor} F (G (Q)) \neq P \underset{˙}{\lor} Q \land \neg R (F) \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg R (G) \underset{˙}{\leq} (P \underset{˙}{\lor} Q))) \to F \in T$
12		simp22	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (F \in T \land G \in T \land P \neq Q) \land (F (G (P)) \underset{˙}{\lor} F (G (Q)) \neq P \underset{˙}{\lor} Q \land \neg R (F) \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg R (G) \underset{˙}{\leq} (P \underset{˙}{\lor} Q))) \to G \in T$
13		simp23	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (F \in T \land G \in T \land P \neq Q) \land (F (G (P)) \underset{˙}{\lor} F (G (Q)) \neq P \underset{˙}{\lor} Q \land \neg R (F) \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg R (G) \underset{˙}{\leq} (P \underset{˙}{\lor} Q))) \to P \neq Q$
14		simp31	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (F \in T \land G \in T \land P \neq Q) \land (F (G (P)) \underset{˙}{\lor} F (G (Q)) \neq P \underset{˙}{\lor} Q \land \neg R (F) \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg R (G) \underset{˙}{\leq} (P \underset{˙}{\lor} Q))) \to F (G (P)) \underset{˙}{\lor} F (G (Q)) \neq P \underset{˙}{\lor} Q$
15	1 2 3 4 5 6	cdlemg9	$⊢ ((K \in HL \land W \in H) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land F \in T) \land (G \in T \land P \neq Q \land F (G (P)) \underset{˙}{\lor} F (G (Q)) \neq P \underset{˙}{\lor} Q)) \to ((P \underset{˙}{\lor} F (G (P))) \underset{˙}{\land} (Q \underset{˙}{\lor} F (G (Q)))) \underset{˙}{\leq} (((F (G (P)) \underset{˙}{\lor} G (P)) \underset{˙}{\land} (F (G (Q)) \underset{˙}{\lor} G (Q))) \underset{˙}{\lor} ((G (P) \underset{˙}{\lor} P) \underset{˙}{\land} (G (Q) \underset{˙}{\lor} Q)))$
16	8 9 10 11 12 13 14 15	syl133anc	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (F \in T \land G \in T \land P \neq Q) \land (F (G (P)) \underset{˙}{\lor} F (G (Q)) \neq P \underset{˙}{\lor} Q \land \neg R (F) \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg R (G) \underset{˙}{\leq} (P \underset{˙}{\lor} Q))) \to ((P \underset{˙}{\lor} F (G (P))) \underset{˙}{\land} (Q \underset{˙}{\lor} F (G (Q)))) \underset{˙}{\leq} (((F (G (P)) \underset{˙}{\lor} G (P)) \underset{˙}{\land} (F (G (Q)) \underset{˙}{\lor} G (Q))) \underset{˙}{\lor} ((G (P) \underset{˙}{\lor} P) \underset{˙}{\land} (G (Q) \underset{˙}{\lor} Q)))$
17	1 4 5 6	ltrnel	$⊢ ((K \in HL \land W \in H) \land G \in T \land (P \in A \land \neg P \underset{˙}{\leq} W)) \to (G (P) \in A \land \neg G (P) \underset{˙}{\leq} W)$
18	8 12 9 17	syl3anc	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (F \in T \land G \in T \land P \neq Q) \land (F (G (P)) \underset{˙}{\lor} F (G (Q)) \neq P \underset{˙}{\lor} Q \land \neg R (F) \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg R (G) \underset{˙}{\leq} (P \underset{˙}{\lor} Q))) \to (G (P) \in A \land \neg G (P) \underset{˙}{\leq} W)$
19	1 4 5 6	ltrnel	$⊢ ((K \in HL \land W \in H) \land G \in T \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \to (G (Q) \in A \land \neg G (Q) \underset{˙}{\leq} W)$
20	8 12 10 19	syl3anc	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (F \in T \land G \in T \land P \neq Q) \land (F (G (P)) \underset{˙}{\lor} F (G (Q)) \neq P \underset{˙}{\lor} Q \land \neg R (F) \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg R (G) \underset{˙}{\leq} (P \underset{˙}{\lor} Q))) \to (G (Q) \in A \land \neg G (Q) \underset{˙}{\leq} W)$
21		simp12l	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (F \in T \land G \in T \land P \neq Q) \land (F (G (P)) \underset{˙}{\lor} F (G (Q)) \neq P \underset{˙}{\lor} Q \land \neg R (F) \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg R (G) \underset{˙}{\leq} (P \underset{˙}{\lor} Q))) \to P \in A$
22		simp13l	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (F \in T \land G \in T \land P \neq Q) \land (F (G (P)) \underset{˙}{\lor} F (G (Q)) \neq P \underset{˙}{\lor} Q \land \neg R (F) \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg R (G) \underset{˙}{\leq} (P \underset{˙}{\lor} Q))) \to Q \in A$
23	4 5 6	ltrn11at	$⊢ ((K \in HL \land W \in H) \land G \in T \land (P \in A \land Q \in A \land P \neq Q)) \to G (P) \neq G (Q)$
24	8 12 21 22 13 23	syl113anc	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (F \in T \land G \in T \land P \neq Q) \land (F (G (P)) \underset{˙}{\lor} F (G (Q)) \neq P \underset{˙}{\lor} Q \land \neg R (F) \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg R (G) \underset{˙}{\leq} (P \underset{˙}{\lor} Q))) \to G (P) \neq G (Q)$
25		simp32	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (F \in T \land G \in T \land P \neq Q) \land (F (G (P)) \underset{˙}{\lor} F (G (Q)) \neq P \underset{˙}{\lor} Q \land \neg R (F) \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg R (G) \underset{˙}{\leq} (P \underset{˙}{\lor} Q))) \to \neg R (F) \underset{˙}{\leq} (P \underset{˙}{\lor} Q)$
26	1 2 3 4 5 6 7	cdlemg10c	$⊢ ((K \in HL \land W \in H) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (F \in T \land G \in T)) \to (R (F) \underset{˙}{\leq} (G (P) \underset{˙}{\lor} G (Q)) \leftrightarrow R (F) \underset{˙}{\leq} (P \underset{˙}{\lor} Q))$
27	8 9 10 11 12 26	syl122anc	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (F \in T \land G \in T \land P \neq Q) \land (F (G (P)) \underset{˙}{\lor} F (G (Q)) \neq P \underset{˙}{\lor} Q \land \neg R (F) \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg R (G) \underset{˙}{\leq} (P \underset{˙}{\lor} Q))) \to (R (F) \underset{˙}{\leq} (G (P) \underset{˙}{\lor} G (Q)) \leftrightarrow R (F) \underset{˙}{\leq} (P \underset{˙}{\lor} Q))$
28	25 27	mtbird	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (F \in T \land G \in T \land P \neq Q) \land (F (G (P)) \underset{˙}{\lor} F (G (Q)) \neq P \underset{˙}{\lor} Q \land \neg R (F) \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg R (G) \underset{˙}{\leq} (P \underset{˙}{\lor} Q))) \to \neg R (F) \underset{˙}{\leq} (G (P) \underset{˙}{\lor} G (Q))$
29	1 2 3 4 5 6 7	trlval4	$⊢ ((K \in HL \land W \in H) \land (F \in T \land (G (P) \in A \land \neg G (P) \underset{˙}{\leq} W) \land (G (Q) \in A \land \neg G (Q) \underset{˙}{\leq} W)) \land (G (P) \neq G (Q) \land \neg R (F) \underset{˙}{\leq} (G (P) \underset{˙}{\lor} G (Q)))) \to R (F) = (G (P) \underset{˙}{\lor} F (G (P))) \underset{˙}{\land} (G (Q) \underset{˙}{\lor} F (G (Q)))$
30	8 11 18 20 24 28 29	syl132anc	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (F \in T \land G \in T \land P \neq Q) \land (F (G (P)) \underset{˙}{\lor} F (G (Q)) \neq P \underset{˙}{\lor} Q \land \neg R (F) \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg R (G) \underset{˙}{\leq} (P \underset{˙}{\lor} Q))) \to R (F) = (G (P) \underset{˙}{\lor} F (G (P))) \underset{˙}{\land} (G (Q) \underset{˙}{\lor} F (G (Q)))$
31		simp11l	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (F \in T \land G \in T \land P \neq Q) \land (F (G (P)) \underset{˙}{\lor} F (G (Q)) \neq P \underset{˙}{\lor} Q \land \neg R (F) \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg R (G) \underset{˙}{\leq} (P \underset{˙}{\lor} Q))) \to K \in HL$
32	1 4 5 6	ltrnat	$⊢ ((K \in HL \land W \in H) \land G \in T \land P \in A) \to G (P) \in A$
33	8 12 21 32	syl3anc	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (F \in T \land G \in T \land P \neq Q) \land (F (G (P)) \underset{˙}{\lor} F (G (Q)) \neq P \underset{˙}{\lor} Q \land \neg R (F) \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg R (G) \underset{˙}{\leq} (P \underset{˙}{\lor} Q))) \to G (P) \in A$
34	1 4 5 6	ltrnat	$⊢ ((K \in HL \land W \in H) \land F \in T \land G (P) \in A) \to F (G (P)) \in A$
35	8 11 33 34	syl3anc	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (F \in T \land G \in T \land P \neq Q) \land (F (G (P)) \underset{˙}{\lor} F (G (Q)) \neq P \underset{˙}{\lor} Q \land \neg R (F) \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg R (G) \underset{˙}{\leq} (P \underset{˙}{\lor} Q))) \to F (G (P)) \in A$
36	2 4	hlatjcom	$⊢ (K \in HL \land G (P) \in A \land F (G (P)) \in A) \to G (P) \underset{˙}{\lor} F (G (P)) = F (G (P)) \underset{˙}{\lor} G (P)$
37	31 33 35 36	syl3anc	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (F \in T \land G \in T \land P \neq Q) \land (F (G (P)) \underset{˙}{\lor} F (G (Q)) \neq P \underset{˙}{\lor} Q \land \neg R (F) \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg R (G) \underset{˙}{\leq} (P \underset{˙}{\lor} Q))) \to G (P) \underset{˙}{\lor} F (G (P)) = F (G (P)) \underset{˙}{\lor} G (P)$
38	1 4 5 6	ltrnat	$⊢ ((K \in HL \land W \in H) \land G \in T \land Q \in A) \to G (Q) \in A$
39	8 12 22 38	syl3anc	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (F \in T \land G \in T \land P \neq Q) \land (F (G (P)) \underset{˙}{\lor} F (G (Q)) \neq P \underset{˙}{\lor} Q \land \neg R (F) \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg R (G) \underset{˙}{\leq} (P \underset{˙}{\lor} Q))) \to G (Q) \in A$
40	1 4 5 6	ltrnat	$⊢ ((K \in HL \land W \in H) \land F \in T \land G (Q) \in A) \to F (G (Q)) \in A$
41	8 11 39 40	syl3anc	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (F \in T \land G \in T \land P \neq Q) \land (F (G (P)) \underset{˙}{\lor} F (G (Q)) \neq P \underset{˙}{\lor} Q \land \neg R (F) \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg R (G) \underset{˙}{\leq} (P \underset{˙}{\lor} Q))) \to F (G (Q)) \in A$
42	2 4	hlatjcom	$⊢ (K \in HL \land G (Q) \in A \land F (G (Q)) \in A) \to G (Q) \underset{˙}{\lor} F (G (Q)) = F (G (Q)) \underset{˙}{\lor} G (Q)$
43	31 39 41 42	syl3anc	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (F \in T \land G \in T \land P \neq Q) \land (F (G (P)) \underset{˙}{\lor} F (G (Q)) \neq P \underset{˙}{\lor} Q \land \neg R (F) \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg R (G) \underset{˙}{\leq} (P \underset{˙}{\lor} Q))) \to G (Q) \underset{˙}{\lor} F (G (Q)) = F (G (Q)) \underset{˙}{\lor} G (Q)$
44	37 43	oveq12d	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (F \in T \land G \in T \land P \neq Q) \land (F (G (P)) \underset{˙}{\lor} F (G (Q)) \neq P \underset{˙}{\lor} Q \land \neg R (F) \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg R (G) \underset{˙}{\leq} (P \underset{˙}{\lor} Q))) \to (G (P) \underset{˙}{\lor} F (G (P))) \underset{˙}{\land} (G (Q) \underset{˙}{\lor} F (G (Q))) = (F (G (P)) \underset{˙}{\lor} G (P)) \underset{˙}{\land} (F (G (Q)) \underset{˙}{\lor} G (Q))$
45	30 44	eqtrd	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (F \in T \land G \in T \land P \neq Q) \land (F (G (P)) \underset{˙}{\lor} F (G (Q)) \neq P \underset{˙}{\lor} Q \land \neg R (F) \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg R (G) \underset{˙}{\leq} (P \underset{˙}{\lor} Q))) \to R (F) = (F (G (P)) \underset{˙}{\lor} G (P)) \underset{˙}{\land} (F (G (Q)) \underset{˙}{\lor} G (Q))$
46		simp33	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (F \in T \land G \in T \land P \neq Q) \land (F (G (P)) \underset{˙}{\lor} F (G (Q)) \neq P \underset{˙}{\lor} Q \land \neg R (F) \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg R (G) \underset{˙}{\leq} (P \underset{˙}{\lor} Q))) \to \neg R (G) \underset{˙}{\leq} (P \underset{˙}{\lor} Q)$
47	1 2 3 4 5 6 7	trlval4	$⊢ ((K \in HL \land W \in H) \land (G \in T \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (P \neq Q \land \neg R (G) \underset{˙}{\leq} (P \underset{˙}{\lor} Q))) \to R (G) = (P \underset{˙}{\lor} G (P)) \underset{˙}{\land} (Q \underset{˙}{\lor} G (Q))$
48	8 12 9 10 13 46 47	syl132anc	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (F \in T \land G \in T \land P \neq Q) \land (F (G (P)) \underset{˙}{\lor} F (G (Q)) \neq P \underset{˙}{\lor} Q \land \neg R (F) \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg R (G) \underset{˙}{\leq} (P \underset{˙}{\lor} Q))) \to R (G) = (P \underset{˙}{\lor} G (P)) \underset{˙}{\land} (Q \underset{˙}{\lor} G (Q))$
49	2 4	hlatjcom	$⊢ (K \in HL \land P \in A \land G (P) \in A) \to P \underset{˙}{\lor} G (P) = G (P) \underset{˙}{\lor} P$
50	31 21 33 49	syl3anc	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (F \in T \land G \in T \land P \neq Q) \land (F (G (P)) \underset{˙}{\lor} F (G (Q)) \neq P \underset{˙}{\lor} Q \land \neg R (F) \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg R (G) \underset{˙}{\leq} (P \underset{˙}{\lor} Q))) \to P \underset{˙}{\lor} G (P) = G (P) \underset{˙}{\lor} P$
51	2 4	hlatjcom	$⊢ (K \in HL \land Q \in A \land G (Q) \in A) \to Q \underset{˙}{\lor} G (Q) = G (Q) \underset{˙}{\lor} Q$
52	31 22 39 51	syl3anc	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (F \in T \land G \in T \land P \neq Q) \land (F (G (P)) \underset{˙}{\lor} F (G (Q)) \neq P \underset{˙}{\lor} Q \land \neg R (F) \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg R (G) \underset{˙}{\leq} (P \underset{˙}{\lor} Q))) \to Q \underset{˙}{\lor} G (Q) = G (Q) \underset{˙}{\lor} Q$
53	50 52	oveq12d	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (F \in T \land G \in T \land P \neq Q) \land (F (G (P)) \underset{˙}{\lor} F (G (Q)) \neq P \underset{˙}{\lor} Q \land \neg R (F) \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg R (G) \underset{˙}{\leq} (P \underset{˙}{\lor} Q))) \to (P \underset{˙}{\lor} G (P)) \underset{˙}{\land} (Q \underset{˙}{\lor} G (Q)) = (G (P) \underset{˙}{\lor} P) \underset{˙}{\land} (G (Q) \underset{˙}{\lor} Q)$
54	48 53	eqtrd	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (F \in T \land G \in T \land P \neq Q) \land (F (G (P)) \underset{˙}{\lor} F (G (Q)) \neq P \underset{˙}{\lor} Q \land \neg R (F) \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg R (G) \underset{˙}{\leq} (P \underset{˙}{\lor} Q))) \to R (G) = (G (P) \underset{˙}{\lor} P) \underset{˙}{\land} (G (Q) \underset{˙}{\lor} Q)$
55	45 54	oveq12d	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (F \in T \land G \in T \land P \neq Q) \land (F (G (P)) \underset{˙}{\lor} F (G (Q)) \neq P \underset{˙}{\lor} Q \land \neg R (F) \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg R (G) \underset{˙}{\leq} (P \underset{˙}{\lor} Q))) \to R (F) \underset{˙}{\lor} R (G) = ((F (G (P)) \underset{˙}{\lor} G (P)) \underset{˙}{\land} (F (G (Q)) \underset{˙}{\lor} G (Q))) \underset{˙}{\lor} ((G (P) \underset{˙}{\lor} P) \underset{˙}{\land} (G (Q) \underset{˙}{\lor} Q))$
56	16 55	breqtrrd	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (F \in T \land G \in T \land P \neq Q) \land (F (G (P)) \underset{˙}{\lor} F (G (Q)) \neq P \underset{˙}{\lor} Q \land \neg R (F) \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg R (G) \underset{˙}{\leq} (P \underset{˙}{\lor} Q))) \to ((P \underset{˙}{\lor} F (G (P))) \underset{˙}{\land} (Q \underset{˙}{\lor} F (G (Q)))) \underset{˙}{\leq} (R (F) \underset{˙}{\lor} R (G))$

Metamath Proof Explorer

Theorem cdlemg10a

Proof