Metamath Proof Explorer

Theorem cdlemg6e

Description: TODO: FIX COMMENT. (Contributed by NM, 27-Apr-2013)

		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	cdlemg6e	$⊢ ((K \in HL \land W \in H) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land F \in T) \land (G \in T \land Q \underset{˙}{\leq} (P \underset{˙}{\lor} V) \land F (G (P)) = P)) \to F (G (Q)) = Q$

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	$⊢ ((K \in HL \land W \in H) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land F \in T) \land (G \in T \land Q \underset{˙}{\leq} (P \underset{˙}{\lor} V) \land F (G (P)) = P)) \to (K \in HL \land W \in H)$
9		simp21	$⊢ ((K \in HL \land W \in H) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land F \in T) \land (G \in T \land Q \underset{˙}{\leq} (P \underset{˙}{\lor} V) \land F (G (P)) = P)) \to (P \in A \land \neg P \underset{˙}{\leq} W)$
10		simp31	$⊢ ((K \in HL \land W \in H) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land F \in T) \land (G \in T \land Q \underset{˙}{\leq} (P \underset{˙}{\lor} V) \land F (G (P)) = P)) \to G \in T$
11	1 2 3 4	ltrnel	$⊢ ((K \in HL \land W \in H) \land G \in T \land (P \in A \land \neg P \underset{˙}{\leq} W)) \to (G (P) \in A \land \neg G (P) \underset{˙}{\leq} W)$
12	8 10 9 11	syl3anc	$⊢ ((K \in HL \land W \in H) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land F \in T) \land (G \in T \land Q \underset{˙}{\leq} (P \underset{˙}{\lor} V) \land F (G (P)) = P)) \to (G (P) \in A \land \neg G (P) \underset{˙}{\leq} W)$
13	1 6 2 3	cdlemb3	$⊢ ((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (G (P) \in A \land \neg G (P) \underset{˙}{\leq} W)) \to \exists r \in A (\neg r \underset{˙}{\leq} W \land \neg r \underset{˙}{\leq} (P \underset{˙}{\lor} G (P)))$
14	8 9 12 13	syl3anc	$⊢ ((K \in HL \land W \in H) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land F \in T) \land (G \in T \land Q \underset{˙}{\leq} (P \underset{˙}{\lor} V) \land F (G (P)) = P)) \to \exists r \in A (\neg r \underset{˙}{\leq} W \land \neg r \underset{˙}{\leq} (P \underset{˙}{\lor} G (P)))$
15	1 2 3 4 5 6 7	cdlemg6d	$⊢ ((K \in HL \land W \in H) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land F \in T) \land (G \in T \land Q \underset{˙}{\leq} (P \underset{˙}{\lor} V) \land F (G (P)) = P)) \to (((r \in A \land \neg r \underset{˙}{\leq} W) \land \neg r \underset{˙}{\leq} (P \underset{˙}{\lor} G (P))) \to F (G (Q)) = Q)$
16	15	exp4c	$⊢ ((K \in HL \land W \in H) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land F \in T) \land (G \in T \land Q \underset{˙}{\leq} (P \underset{˙}{\lor} V) \land F (G (P)) = P)) \to (r \in A \to (\neg r \underset{˙}{\leq} W \to (\neg r \underset{˙}{\leq} (P \underset{˙}{\lor} G (P)) \to F (G (Q)) = Q)))$
17	16	imp4a	$⊢ ((K \in HL \land W \in H) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land F \in T) \land (G \in T \land Q \underset{˙}{\leq} (P \underset{˙}{\lor} V) \land F (G (P)) = P)) \to (r \in A \to ((\neg r \underset{˙}{\leq} W \land \neg r \underset{˙}{\leq} (P \underset{˙}{\lor} G (P))) \to F (G (Q)) = Q))$
18	17	rexlimdv	$⊢ ((K \in HL \land W \in H) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land F \in T) \land (G \in T \land Q \underset{˙}{\leq} (P \underset{˙}{\lor} V) \land F (G (P)) = P)) \to (\exists r \in A (\neg r \underset{˙}{\leq} W \land \neg r \underset{˙}{\leq} (P \underset{˙}{\lor} G (P))) \to F (G (Q)) = Q)$
19	14 18	mpd	$⊢ ((K \in HL \land W \in H) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land F \in T) \land (G \in T \land Q \underset{˙}{\leq} (P \underset{˙}{\lor} V) \land F (G (P)) = P)) \to F (G (Q)) = Q$