cdlemg36

Description: Use cdlemg35 to eliminate v from cdlemg34 . TODO: Fix comment. (Contributed by NM, 31-May-2013)

	Ref	Expression
Hypotheses	cdlemg35.l	$⊢ \underset{˙}{\leq} = \leq_{K}$
	cdlemg35.j	$⊢ \underset{˙}{\lor} = join (K)$
	cdlemg35.m	$⊢ \underset{˙}{\land} = meet (K)$
	cdlemg35.a	$⊢ A = Atoms (K)$
	cdlemg35.h	$⊢ H = LHyp (K)$
	cdlemg35.t	$⊢ T = LTrn (K) (W)$
	cdlemg35.r	$⊢ R = trL (K) (W)$
Assertion	cdlemg36	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (F \in T \land G \in T \land P \neq Q) \land ((F (P) \neq P \land G (P) \neq P) \land R (F) \neq R (G) \land \exists r \in A (\neg r \underset{˙}{\leq} W \land P \underset{˙}{\lor} r = Q \underset{˙}{\lor} r))) \to (P \underset{˙}{\lor} F (G (P))) \underset{˙}{\land} W = (Q \underset{˙}{\lor} F (G (Q))) \underset{˙}{\land} W$

Proof

Step	Hyp	Ref	Expression
1		cdlemg35.l	$⊢ \underset{˙}{\leq} = \leq_{K}$
2		cdlemg35.j	$⊢ \underset{˙}{\lor} = join (K)$
3		cdlemg35.m	$⊢ \underset{˙}{\land} = meet (K)$
4		cdlemg35.a	$⊢ A = Atoms (K)$
5		cdlemg35.h	$⊢ H = LHyp (K)$
6		cdlemg35.t	$⊢ T = LTrn (K) (W)$
7		cdlemg35.r	$⊢ R = trL (K) (W)$
8		simp11	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (F \in T \land G \in T \land P \neq Q) \land ((F (P) \neq P \land G (P) \neq P) \land R (F) \neq R (G) \land \exists r \in A (\neg r \underset{˙}{\leq} W \land P \underset{˙}{\lor} r = Q \underset{˙}{\lor} r))) \to (K \in HL \land W \in H)$
9		simp12	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (F \in T \land G \in T \land P \neq Q) \land ((F (P) \neq P \land G (P) \neq P) \land R (F) \neq R (G) \land \exists r \in A (\neg r \underset{˙}{\leq} W \land P \underset{˙}{\lor} r = Q \underset{˙}{\lor} r))) \to (P \in A \land \neg P \underset{˙}{\leq} W)$
10		simp21	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (F \in T \land G \in T \land P \neq Q) \land ((F (P) \neq P \land G (P) \neq P) \land R (F) \neq R (G) \land \exists r \in A (\neg r \underset{˙}{\leq} W \land P \underset{˙}{\lor} r = Q \underset{˙}{\lor} r))) \to F \in T$
11		simp22	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (F \in T \land G \in T \land P \neq Q) \land ((F (P) \neq P \land G (P) \neq P) \land R (F) \neq R (G) \land \exists r \in A (\neg r \underset{˙}{\leq} W \land P \underset{˙}{\lor} r = Q \underset{˙}{\lor} r))) \to G \in T$
12		simp31l	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (F \in T \land G \in T \land P \neq Q) \land ((F (P) \neq P \land G (P) \neq P) \land R (F) \neq R (G) \land \exists r \in A (\neg r \underset{˙}{\leq} W \land P \underset{˙}{\lor} r = Q \underset{˙}{\lor} r))) \to F (P) \neq P$
13		simp31r	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (F \in T \land G \in T \land P \neq Q) \land ((F (P) \neq P \land G (P) \neq P) \land R (F) \neq R (G) \land \exists r \in A (\neg r \underset{˙}{\leq} W \land P \underset{˙}{\lor} r = Q \underset{˙}{\lor} r))) \to G (P) \neq P$
14		simp32	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (F \in T \land G \in T \land P \neq Q) \land ((F (P) \neq P \land G (P) \neq P) \land R (F) \neq R (G) \land \exists r \in A (\neg r \underset{˙}{\leq} W \land P \underset{˙}{\lor} r = Q \underset{˙}{\lor} r))) \to R (F) \neq R (G)$
15	1 2 3 4 5 6 7	cdlemg35	$⊢ ((K \in HL \land W \in H) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land F \in T \land G \in T) \land (F (P) \neq P \land G (P) \neq P \land R (F) \neq R (G))) \to \exists v \in A (v \underset{˙}{\leq} W \land (v \neq R (F) \land v \neq R (G)))$
16	8 9 10 11 12 13 14 15	syl133anc	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (F \in T \land G \in T \land P \neq Q) \land ((F (P) \neq P \land G (P) \neq P) \land R (F) \neq R (G) \land \exists r \in A (\neg r \underset{˙}{\leq} W \land P \underset{˙}{\lor} r = Q \underset{˙}{\lor} r))) \to \exists v \in A (v \underset{˙}{\leq} W \land (v \neq R (F) \land v \neq R (G)))$
17		simp11	$⊢ ((((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (F \in T \land G \in T \land P \neq Q) \land ((F (P) \neq P \land G (P) \neq P) \land R (F) \neq R (G) \land \exists r \in A (\neg r \underset{˙}{\leq} W \land P \underset{˙}{\lor} r = Q \underset{˙}{\lor} r))) \land v \in A \land (v \underset{˙}{\leq} W \land (v \neq R (F) \land v \neq R (G)))) \to ((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W))$
18		simp2	$⊢ ((((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (F \in T \land G \in T \land P \neq Q) \land ((F (P) \neq P \land G (P) \neq P) \land R (F) \neq R (G) \land \exists r \in A (\neg r \underset{˙}{\leq} W \land P \underset{˙}{\lor} r = Q \underset{˙}{\lor} r))) \land v \in A \land (v \underset{˙}{\leq} W \land (v \neq R (F) \land v \neq R (G)))) \to v \in A$
19		simp3l	$⊢ ((((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (F \in T \land G \in T \land P \neq Q) \land ((F (P) \neq P \land G (P) \neq P) \land R (F) \neq R (G) \land \exists r \in A (\neg r \underset{˙}{\leq} W \land P \underset{˙}{\lor} r = Q \underset{˙}{\lor} r))) \land v \in A \land (v \underset{˙}{\leq} W \land (v \neq R (F) \land v \neq R (G)))) \to v \underset{˙}{\leq} W$
20	18 19	jca	$⊢ ((((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (F \in T \land G \in T \land P \neq Q) \land ((F (P) \neq P \land G (P) \neq P) \land R (F) \neq R (G) \land \exists r \in A (\neg r \underset{˙}{\leq} W \land P \underset{˙}{\lor} r = Q \underset{˙}{\lor} r))) \land v \in A \land (v \underset{˙}{\leq} W \land (v \neq R (F) \land v \neq R (G)))) \to (v \in A \land v \underset{˙}{\leq} W)$
21		simp121	$⊢ ((((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (F \in T \land G \in T \land P \neq Q) \land ((F (P) \neq P \land G (P) \neq P) \land R (F) \neq R (G) \land \exists r \in A (\neg r \underset{˙}{\leq} W \land P \underset{˙}{\lor} r = Q \underset{˙}{\lor} r))) \land v \in A \land (v \underset{˙}{\leq} W \land (v \neq R (F) \land v \neq R (G)))) \to F \in T$
22		simp122	$⊢ ((((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (F \in T \land G \in T \land P \neq Q) \land ((F (P) \neq P \land G (P) \neq P) \land R (F) \neq R (G) \land \exists r \in A (\neg r \underset{˙}{\leq} W \land P \underset{˙}{\lor} r = Q \underset{˙}{\lor} r))) \land v \in A \land (v \underset{˙}{\leq} W \land (v \neq R (F) \land v \neq R (G)))) \to G \in T$
23	21 22	jca	$⊢ ((((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (F \in T \land G \in T \land P \neq Q) \land ((F (P) \neq P \land G (P) \neq P) \land R (F) \neq R (G) \land \exists r \in A (\neg r \underset{˙}{\leq} W \land P \underset{˙}{\lor} r = Q \underset{˙}{\lor} r))) \land v \in A \land (v \underset{˙}{\leq} W \land (v \neq R (F) \land v \neq R (G)))) \to (F \in T \land G \in T)$
24		simp123	$⊢ ((((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (F \in T \land G \in T \land P \neq Q) \land ((F (P) \neq P \land G (P) \neq P) \land R (F) \neq R (G) \land \exists r \in A (\neg r \underset{˙}{\leq} W \land P \underset{˙}{\lor} r = Q \underset{˙}{\lor} r))) \land v \in A \land (v \underset{˙}{\leq} W \land (v \neq R (F) \land v \neq R (G)))) \to P \neq Q$
25		simp3rl	$⊢ ((((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (F \in T \land G \in T \land P \neq Q) \land ((F (P) \neq P \land G (P) \neq P) \land R (F) \neq R (G) \land \exists r \in A (\neg r \underset{˙}{\leq} W \land P \underset{˙}{\lor} r = Q \underset{˙}{\lor} r))) \land v \in A \land (v \underset{˙}{\leq} W \land (v \neq R (F) \land v \neq R (G)))) \to v \neq R (F)$
26		simp3rr	$⊢ ((((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (F \in T \land G \in T \land P \neq Q) \land ((F (P) \neq P \land G (P) \neq P) \land R (F) \neq R (G) \land \exists r \in A (\neg r \underset{˙}{\leq} W \land P \underset{˙}{\lor} r = Q \underset{˙}{\lor} r))) \land v \in A \land (v \underset{˙}{\leq} W \land (v \neq R (F) \land v \neq R (G)))) \to v \neq R (G)$
27		simp133	$⊢ ((((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (F \in T \land G \in T \land P \neq Q) \land ((F (P) \neq P \land G (P) \neq P) \land R (F) \neq R (G) \land \exists r \in A (\neg r \underset{˙}{\leq} W \land P \underset{˙}{\lor} r = Q \underset{˙}{\lor} r))) \land v \in A \land (v \underset{˙}{\leq} W \land (v \neq R (F) \land v \neq R (G)))) \to \exists r \in A (\neg r \underset{˙}{\leq} W \land P \underset{˙}{\lor} r = Q \underset{˙}{\lor} r)$
28		eqid	$⊢ (P \underset{˙}{\lor} v) \underset{˙}{\land} (Q \underset{˙}{\lor} R (F)) = (P \underset{˙}{\lor} v) \underset{˙}{\land} (Q \underset{˙}{\lor} R (F))$
29		eqid	$⊢ (P \underset{˙}{\lor} v) \underset{˙}{\land} (Q \underset{˙}{\lor} R (G)) = (P \underset{˙}{\lor} v) \underset{˙}{\land} (Q \underset{˙}{\lor} R (G))$
30	1 2 3 4 5 6 7 28 29	cdlemg34	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((v \in A \land v \underset{˙}{\leq} W) \land (F \in T \land G \in T) \land P \neq Q) \land (v \neq R (F) \land v \neq R (G) \land \exists r \in A (\neg r \underset{˙}{\leq} W \land P \underset{˙}{\lor} r = Q \underset{˙}{\lor} r))) \to (P \underset{˙}{\lor} F (G (P))) \underset{˙}{\land} W = (Q \underset{˙}{\lor} F (G (Q))) \underset{˙}{\land} W$
31	17 20 23 24 25 26 27 30	syl133anc	$⊢ ((((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (F \in T \land G \in T \land P \neq Q) \land ((F (P) \neq P \land G (P) \neq P) \land R (F) \neq R (G) \land \exists r \in A (\neg r \underset{˙}{\leq} W \land P \underset{˙}{\lor} r = Q \underset{˙}{\lor} r))) \land v \in A \land (v \underset{˙}{\leq} W \land (v \neq R (F) \land v \neq R (G)))) \to (P \underset{˙}{\lor} F (G (P))) \underset{˙}{\land} W = (Q \underset{˙}{\lor} F (G (Q))) \underset{˙}{\land} W$
32	31	rexlimdv3a	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (F \in T \land G \in T \land P \neq Q) \land ((F (P) \neq P \land G (P) \neq P) \land R (F) \neq R (G) \land \exists r \in A (\neg r \underset{˙}{\leq} W \land P \underset{˙}{\lor} r = Q \underset{˙}{\lor} r))) \to (\exists v \in A (v \underset{˙}{\leq} W \land (v \neq R (F) \land v \neq R (G))) \to (P \underset{˙}{\lor} F (G (P))) \underset{˙}{\land} W = (Q \underset{˙}{\lor} F (G (Q))) \underset{˙}{\land} W)$
33	16 32	mpd	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (F \in T \land G \in T \land P \neq Q) \land ((F (P) \neq P \land G (P) \neq P) \land R (F) \neq R (G) \land \exists r \in A (\neg r \underset{˙}{\leq} W \land P \underset{˙}{\lor} r = Q \underset{˙}{\lor} r))) \to (P \underset{˙}{\lor} F (G (P))) \underset{˙}{\land} W = (Q \underset{˙}{\lor} F (G (Q))) \underset{˙}{\land} W$

Metamath Proof Explorer

Theorem cdlemg36

Proof