cdlemg4d

	Ref	Expression
Hypotheses	cdlemg4.l	$⊢ \underset{˙}{\leq} = \leq_{K}$
	cdlemg4.a	$⊢ A = Atoms (K)$
	cdlemg4.h	$⊢ H = LHyp (K)$
	cdlemg4.t	$⊢ T = LTrn (K) (W)$
	cdlemg4.r	$⊢ R = trL (K) (W)$
	cdlemg4.j	$⊢ \underset{˙}{\lor} = join (K)$
	cdlemg4b.v	$⊢ V = R (G)$
Assertion	cdlemg4d	$⊢ ((K \in HL \land W \in H) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land F \in T) \land (G \in T \land \neg Q \underset{˙}{\leq} (P \underset{˙}{\lor} V) \land F (G (P)) = P)) \to \neg G (Q) \underset{˙}{\leq} (G (P) \underset{˙}{\lor} F (G (P)))$

Proof

Step	Hyp	Ref	Expression
1		cdlemg4.l	$⊢ \underset{˙}{\leq} = \leq_{K}$
2		cdlemg4.a	$⊢ A = Atoms (K)$
3		cdlemg4.h	$⊢ H = LHyp (K)$
4		cdlemg4.t	$⊢ T = LTrn (K) (W)$
5		cdlemg4.r	$⊢ R = trL (K) (W)$
6		cdlemg4.j	$⊢ \underset{˙}{\lor} = join (K)$
7		cdlemg4b.v	$⊢ V = R (G)$
8		simp1	$⊢ ((K \in HL \land W \in H) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land F \in T) \land (G \in T \land \neg Q \underset{˙}{\leq} (P \underset{˙}{\lor} V) \land F (G (P)) = P)) \to (K \in HL \land W \in H)$
9		simp21	$⊢ ((K \in HL \land W \in H) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land F \in T) \land (G \in T \land \neg Q \underset{˙}{\leq} (P \underset{˙}{\lor} V) \land F (G (P)) = P)) \to (P \in A \land \neg P \underset{˙}{\leq} W)$
10		simp22	$⊢ ((K \in HL \land W \in H) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land F \in T) \land (G \in T \land \neg Q \underset{˙}{\leq} (P \underset{˙}{\lor} V) \land F (G (P)) = P)) \to (Q \in A \land \neg Q \underset{˙}{\leq} W)$
11		simp31	$⊢ ((K \in HL \land W \in H) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land F \in T) \land (G \in T \land \neg Q \underset{˙}{\leq} (P \underset{˙}{\lor} V) \land F (G (P)) = P)) \to G \in T$
12		simp32	$⊢ ((K \in HL \land W \in H) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land F \in T) \land (G \in T \land \neg Q \underset{˙}{\leq} (P \underset{˙}{\lor} V) \land F (G (P)) = P)) \to \neg Q \underset{˙}{\leq} (P \underset{˙}{\lor} V)$
13	1 2 3 4 5 6 7	cdlemg4c	$⊢ ((K \in HL \land W \in H) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land G \in T) \land \neg Q \underset{˙}{\leq} (P \underset{˙}{\lor} V)) \to \neg G (Q) \underset{˙}{\leq} (P \underset{˙}{\lor} V)$
14	8 9 10 11 12 13	syl131anc	$⊢ ((K \in HL \land W \in H) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land F \in T) \land (G \in T \land \neg Q \underset{˙}{\leq} (P \underset{˙}{\lor} V) \land F (G (P)) = P)) \to \neg G (Q) \underset{˙}{\leq} (P \underset{˙}{\lor} V)$
15		simp1l	$⊢ ((K \in HL \land W \in H) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land F \in T) \land (G \in T \land \neg Q \underset{˙}{\leq} (P \underset{˙}{\lor} V) \land F (G (P)) = P)) \to K \in HL$
16		simp21l	$⊢ ((K \in HL \land W \in H) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land F \in T) \land (G \in T \land \neg Q \underset{˙}{\leq} (P \underset{˙}{\lor} V) \land F (G (P)) = P)) \to P \in A$
17	1 2 3 4	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	17	simpld	$⊢ ((K \in HL \land W \in H) \land G \in T \land (P \in A \land \neg P \underset{˙}{\leq} W)) \to G (P) \in A$
19	8 11 9 18	syl3anc	$⊢ ((K \in HL \land W \in H) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land F \in T) \land (G \in T \land \neg Q \underset{˙}{\leq} (P \underset{˙}{\lor} V) \land F (G (P)) = P)) \to G (P) \in A$
20	6 2	hlatjcom	$⊢ (K \in HL \land P \in A \land G (P) \in A) \to P \underset{˙}{\lor} G (P) = G (P) \underset{˙}{\lor} P$
21	15 16 19 20	syl3anc	$⊢ ((K \in HL \land W \in H) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land F \in T) \land (G \in T \land \neg Q \underset{˙}{\leq} (P \underset{˙}{\lor} V) \land F (G (P)) = P)) \to P \underset{˙}{\lor} G (P) = G (P) \underset{˙}{\lor} P$
22	1 2 3 4 5 6 7	cdlemg4b1	$⊢ ((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land G \in T) \to P \underset{˙}{\lor} V = P \underset{˙}{\lor} G (P)$
23	8 9 11 22	syl3anc	$⊢ ((K \in HL \land W \in H) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land F \in T) \land (G \in T \land \neg Q \underset{˙}{\leq} (P \underset{˙}{\lor} V) \land F (G (P)) = P)) \to P \underset{˙}{\lor} V = P \underset{˙}{\lor} G (P)$
24		simp33	$⊢ ((K \in HL \land W \in H) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land F \in T) \land (G \in T \land \neg Q \underset{˙}{\leq} (P \underset{˙}{\lor} V) \land F (G (P)) = P)) \to F (G (P)) = P$
25	24	oveq2d	$⊢ ((K \in HL \land W \in H) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land F \in T) \land (G \in T \land \neg Q \underset{˙}{\leq} (P \underset{˙}{\lor} V) \land F (G (P)) = P)) \to G (P) \underset{˙}{\lor} F (G (P)) = G (P) \underset{˙}{\lor} P$
26	21 23 25	3eqtr4rd	$⊢ ((K \in HL \land W \in H) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land F \in T) \land (G \in T \land \neg Q \underset{˙}{\leq} (P \underset{˙}{\lor} V) \land F (G (P)) = P)) \to G (P) \underset{˙}{\lor} F (G (P)) = P \underset{˙}{\lor} V$
27	26	breq2d	$⊢ ((K \in HL \land W \in H) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land F \in T) \land (G \in T \land \neg Q \underset{˙}{\leq} (P \underset{˙}{\lor} V) \land F (G (P)) = P)) \to (G (Q) \underset{˙}{\leq} (G (P) \underset{˙}{\lor} F (G (P))) \leftrightarrow G (Q) \underset{˙}{\leq} (P \underset{˙}{\lor} V))$
28	14 27	mtbird	$⊢ ((K \in HL \land W \in H) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land F \in T) \land (G \in T \land \neg Q \underset{˙}{\leq} (P \underset{˙}{\lor} V) \land F (G (P)) = P)) \to \neg G (Q) \underset{˙}{\leq} (G (P) \underset{˙}{\lor} F (G (P)))$

Metamath Proof Explorer

Theorem cdlemg4d

Proof