cdlemg8c

	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	$⊢ T = LTrn (K) (W)$
Assertion	cdlemg8c	$⊢ ((K \in HL \land W \in H) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land F \in T) \land (G \in T \land F (G (P)) \underset{˙}{\lor} F (G (Q)) = P \underset{˙}{\lor} Q \land F (G (P)) \neq P)) \to Q \underset{˙}{\lor} F (G (Q)) = P \underset{˙}{\lor} Q$

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		simp1	$⊢ ((K \in HL \land W \in H) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land F \in T) \land (G \in T \land F (G (P)) \underset{˙}{\lor} F (G (Q)) = P \underset{˙}{\lor} Q \land F (G (P)) \neq P)) \to (K \in HL \land W \in H)$
8		simp22	$⊢ ((K \in HL \land W \in H) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land F \in T) \land (G \in T \land F (G (P)) \underset{˙}{\lor} F (G (Q)) = P \underset{˙}{\lor} Q \land F (G (P)) \neq P)) \to (Q \in A \land \neg Q \underset{˙}{\leq} W)$
9		simp21	$⊢ ((K \in HL \land W \in H) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land F \in T) \land (G \in T \land F (G (P)) \underset{˙}{\lor} F (G (Q)) = P \underset{˙}{\lor} Q \land F (G (P)) \neq P)) \to (P \in A \land \neg P \underset{˙}{\leq} W)$
10		simp23	$⊢ ((K \in HL \land W \in H) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land F \in T) \land (G \in T \land F (G (P)) \underset{˙}{\lor} F (G (Q)) = P \underset{˙}{\lor} Q \land F (G (P)) \neq P)) \to F \in T$
11		simp31	$⊢ ((K \in HL \land W \in H) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land F \in T) \land (G \in T \land F (G (P)) \underset{˙}{\lor} F (G (Q)) = P \underset{˙}{\lor} Q \land F (G (P)) \neq P)) \to G \in T$
12		simp32	$⊢ ((K \in HL \land W \in H) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land F \in T) \land (G \in T \land F (G (P)) \underset{˙}{\lor} F (G (Q)) = P \underset{˙}{\lor} Q \land F (G (P)) \neq P)) \to F (G (P)) \underset{˙}{\lor} F (G (Q)) = P \underset{˙}{\lor} Q$
13		simp1l	$⊢ ((K \in HL \land W \in H) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land F \in T) \land (G \in T \land F (G (P)) \underset{˙}{\lor} F (G (Q)) = P \underset{˙}{\lor} Q \land F (G (P)) \neq P)) \to K \in HL$
14	1 4 5 6	ltrnel	$⊢ ((K \in HL \land W \in H) \land G \in T \land (P \in A \land \neg P \underset{˙}{\leq} W)) \to (G (P) \in A \land \neg G (P) \underset{˙}{\leq} W)$
15	7 11 9 14	syl3anc	$⊢ ((K \in HL \land W \in H) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land F \in T) \land (G \in T \land F (G (P)) \underset{˙}{\lor} F (G (Q)) = P \underset{˙}{\lor} Q \land F (G (P)) \neq P)) \to (G (P) \in A \land \neg G (P) \underset{˙}{\leq} W)$
16	1 4 5 6	ltrnel	$⊢ ((K \in HL \land W \in H) \land F \in T \land (G (P) \in A \land \neg G (P) \underset{˙}{\leq} W)) \to (F (G (P)) \in A \land \neg F (G (P)) \underset{˙}{\leq} W)$
17	16	simpld	$⊢ ((K \in HL \land W \in H) \land F \in T \land (G (P) \in A \land \neg G (P) \underset{˙}{\leq} W)) \to F (G (P)) \in A$
18	7 10 15 17	syl3anc	$⊢ ((K \in HL \land W \in H) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land F \in T) \land (G \in T \land F (G (P)) \underset{˙}{\lor} F (G (Q)) = P \underset{˙}{\lor} Q \land F (G (P)) \neq P)) \to F (G (P)) \in A$
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	7 11 8 19	syl3anc	$⊢ ((K \in HL \land W \in H) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land F \in T) \land (G \in T \land F (G (P)) \underset{˙}{\lor} F (G (Q)) = P \underset{˙}{\lor} Q \land F (G (P)) \neq P)) \to (G (Q) \in A \land \neg G (Q) \underset{˙}{\leq} W)$
21	1 4 5 6	ltrnel	$⊢ ((K \in HL \land W \in H) \land F \in T \land (G (Q) \in A \land \neg G (Q) \underset{˙}{\leq} W)) \to (F (G (Q)) \in A \land \neg F (G (Q)) \underset{˙}{\leq} W)$
22	21	simpld	$⊢ ((K \in HL \land W \in H) \land F \in T \land (G (Q) \in A \land \neg G (Q) \underset{˙}{\leq} W)) \to F (G (Q)) \in A$
23	7 10 20 22	syl3anc	$⊢ ((K \in HL \land W \in H) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land F \in T) \land (G \in T \land F (G (P)) \underset{˙}{\lor} F (G (Q)) = P \underset{˙}{\lor} Q \land F (G (P)) \neq P)) \to F (G (Q)) \in A$
24	2 4	hlatjcom	$⊢ (K \in HL \land F (G (P)) \in A \land F (G (Q)) \in A) \to F (G (P)) \underset{˙}{\lor} F (G (Q)) = F (G (Q)) \underset{˙}{\lor} F (G (P))$
25	13 18 23 24	syl3anc	$⊢ ((K \in HL \land W \in H) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land F \in T) \land (G \in T \land F (G (P)) \underset{˙}{\lor} F (G (Q)) = P \underset{˙}{\lor} Q \land F (G (P)) \neq P)) \to F (G (P)) \underset{˙}{\lor} F (G (Q)) = F (G (Q)) \underset{˙}{\lor} F (G (P))$
26		simp21l	$⊢ ((K \in HL \land W \in H) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land F \in T) \land (G \in T \land F (G (P)) \underset{˙}{\lor} F (G (Q)) = P \underset{˙}{\lor} Q \land F (G (P)) \neq P)) \to P \in A$
27		simp22l	$⊢ ((K \in HL \land W \in H) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land F \in T) \land (G \in T \land F (G (P)) \underset{˙}{\lor} F (G (Q)) = P \underset{˙}{\lor} Q \land F (G (P)) \neq P)) \to Q \in A$
28	2 4	hlatjcom	$⊢ (K \in HL \land P \in A \land Q \in A) \to P \underset{˙}{\lor} Q = Q \underset{˙}{\lor} P$
29	13 26 27 28	syl3anc	$⊢ ((K \in HL \land W \in H) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land F \in T) \land (G \in T \land F (G (P)) \underset{˙}{\lor} F (G (Q)) = P \underset{˙}{\lor} Q \land F (G (P)) \neq P)) \to P \underset{˙}{\lor} Q = Q \underset{˙}{\lor} P$
30	12 25 29	3eqtr3d	$⊢ ((K \in HL \land W \in H) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land F \in T) \land (G \in T \land F (G (P)) \underset{˙}{\lor} F (G (Q)) = P \underset{˙}{\lor} Q \land F (G (P)) \neq P)) \to F (G (Q)) \underset{˙}{\lor} F (G (P)) = Q \underset{˙}{\lor} P$
31		simp33	$⊢ ((K \in HL \land W \in H) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land F \in T) \land (G \in T \land F (G (P)) \underset{˙}{\lor} F (G (Q)) = P \underset{˙}{\lor} Q \land F (G (P)) \neq P)) \to F (G (P)) \neq P$
32		simpl1	$⊢ (((K \in HL \land W \in H) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land F \in T) \land (G \in T \land F (G (P)) \underset{˙}{\lor} F (G (Q)) = P \underset{˙}{\lor} Q \land F (G (P)) \neq P)) \land F (G (Q)) = Q) \to (K \in HL \land W \in H)$
33		simpl22	$⊢ (((K \in HL \land W \in H) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land F \in T) \land (G \in T \land F (G (P)) \underset{˙}{\lor} F (G (Q)) = P \underset{˙}{\lor} Q \land F (G (P)) \neq P)) \land F (G (Q)) = Q) \to (Q \in A \land \neg Q \underset{˙}{\leq} W)$
34		simpl21	$⊢ (((K \in HL \land W \in H) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land F \in T) \land (G \in T \land F (G (P)) \underset{˙}{\lor} F (G (Q)) = P \underset{˙}{\lor} Q \land F (G (P)) \neq P)) \land F (G (Q)) = Q) \to (P \in A \land \neg P \underset{˙}{\leq} W)$
35		simpl23	$⊢ (((K \in HL \land W \in H) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land F \in T) \land (G \in T \land F (G (P)) \underset{˙}{\lor} F (G (Q)) = P \underset{˙}{\lor} Q \land F (G (P)) \neq P)) \land F (G (Q)) = Q) \to F \in T$
36		simpl31	$⊢ (((K \in HL \land W \in H) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land F \in T) \land (G \in T \land F (G (P)) \underset{˙}{\lor} F (G (Q)) = P \underset{˙}{\lor} Q \land F (G (P)) \neq P)) \land F (G (Q)) = Q) \to G \in T$
37		simpr	$⊢ (((K \in HL \land W \in H) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land F \in T) \land (G \in T \land F (G (P)) \underset{˙}{\lor} F (G (Q)) = P \underset{˙}{\lor} Q \land F (G (P)) \neq P)) \land F (G (Q)) = Q) \to F (G (Q)) = Q$
38	1 4 5 6	cdlemg6	$⊢ ((K \in HL \land W \in H) \land ((Q \in A \land \neg Q \underset{˙}{\leq} W) \land (P \in A \land \neg P \underset{˙}{\leq} W)) \land (F \in T \land G \in T \land F (G (Q)) = Q)) \to F (G (P)) = P$
39	32 33 34 35 36 37 38	syl123anc	$⊢ (((K \in HL \land W \in H) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land F \in T) \land (G \in T \land F (G (P)) \underset{˙}{\lor} F (G (Q)) = P \underset{˙}{\lor} Q \land F (G (P)) \neq P)) \land F (G (Q)) = Q) \to F (G (P)) = P$
40	39	ex	$⊢ ((K \in HL \land W \in H) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land F \in T) \land (G \in T \land F (G (P)) \underset{˙}{\lor} F (G (Q)) = P \underset{˙}{\lor} Q \land F (G (P)) \neq P)) \to (F (G (Q)) = Q \to F (G (P)) = P)$
41	40	necon3d	$⊢ ((K \in HL \land W \in H) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land F \in T) \land (G \in T \land F (G (P)) \underset{˙}{\lor} F (G (Q)) = P \underset{˙}{\lor} Q \land F (G (P)) \neq P)) \to (F (G (P)) \neq P \to F (G (Q)) \neq Q)$
42	31 41	mpd	$⊢ ((K \in HL \land W \in H) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land F \in T) \land (G \in T \land F (G (P)) \underset{˙}{\lor} F (G (Q)) = P \underset{˙}{\lor} Q \land F (G (P)) \neq P)) \to F (G (Q)) \neq Q$
43	1 2 3 4 5 6	cdlemg8b	$⊢ ((K \in HL \land W \in H) \land ((Q \in A \land \neg Q \underset{˙}{\leq} W) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land F \in T) \land (G \in T \land F (G (Q)) \underset{˙}{\lor} F (G (P)) = Q \underset{˙}{\lor} P \land F (G (Q)) \neq Q)) \to Q \underset{˙}{\lor} F (G (Q)) = Q \underset{˙}{\lor} P$
44	7 8 9 10 11 30 42 43	syl133anc	$⊢ ((K \in HL \land W \in H) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land F \in T) \land (G \in T \land F (G (P)) \underset{˙}{\lor} F (G (Q)) = P \underset{˙}{\lor} Q \land F (G (P)) \neq P)) \to Q \underset{˙}{\lor} F (G (Q)) = Q \underset{˙}{\lor} P$
45	44 29	eqtr4d	$⊢ ((K \in HL \land W \in H) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land F \in T) \land (G \in T \land F (G (P)) \underset{˙}{\lor} F (G (Q)) = P \underset{˙}{\lor} Q \land F (G (P)) \neq P)) \to Q \underset{˙}{\lor} F (G (Q)) = P \underset{˙}{\lor} Q$

Metamath Proof Explorer

Theorem cdlemg8c

Proof