cdlemk5a

Description: Part of proof of Lemma K of Crawley p. 118. (Contributed by NM, 3-Jul-2013)

	Ref	Expression
Hypotheses	cdlemk.b	$⊢ B = {Base}_{K}$
	cdlemk.l	$⊢ \underset{˙}{\leq} = \leq_{K}$
	cdlemk.j	$⊢ \underset{˙}{\lor} = join (K)$
	cdlemk.a	$⊢ A = Atoms (K)$
	cdlemk.h	$⊢ H = LHyp (K)$
	cdlemk.t	$⊢ T = LTrn (K) (W)$
	cdlemk.r	$⊢ R = trL (K) (W)$
	cdlemk.m	$⊢ \underset{˙}{\land} = meet (K)$
Assertion	cdlemk5a	$⊢ ((K \in HL \land W \in H) \land (F \in T \land G \in T \land X \in T) \land (R (G) \neq R (F) \land (F \neq {I ↾}_{B} \land G \neq {I ↾}_{B}) \land (P \in A \land \neg P \underset{˙}{\leq} W))) \to ((F (P) \underset{˙}{\lor} R (F)) \underset{˙}{\land} (F (P) \underset{˙}{\lor} R (G \circ F^{-1}))) \underset{˙}{\leq} (X (P) \underset{˙}{\lor} R (X \circ F^{-1}))$

Proof

Step	Hyp	Ref	Expression
1		cdlemk.b	$⊢ B = {Base}_{K}$
2		cdlemk.l	$⊢ \underset{˙}{\leq} = \leq_{K}$
3		cdlemk.j	$⊢ \underset{˙}{\lor} = join (K)$
4		cdlemk.a	$⊢ A = Atoms (K)$
5		cdlemk.h	$⊢ H = LHyp (K)$
6		cdlemk.t	$⊢ T = LTrn (K) (W)$
7		cdlemk.r	$⊢ R = trL (K) (W)$
8		cdlemk.m	$⊢ \underset{˙}{\land} = meet (K)$
9		simp1l	$⊢ ((K \in HL \land W \in H) \land (F \in T \land G \in T \land X \in T) \land (R (G) \neq R (F) \land (F \neq {I ↾}_{B} \land G \neq {I ↾}_{B}) \land (P \in A \land \neg P \underset{˙}{\leq} W))) \to K \in HL$
10		simp1r	$⊢ ((K \in HL \land W \in H) \land (F \in T \land G \in T \land X \in T) \land (R (G) \neq R (F) \land (F \neq {I ↾}_{B} \land G \neq {I ↾}_{B}) \land (P \in A \land \neg P \underset{˙}{\leq} W))) \to W \in H$
11		simp21	$⊢ ((K \in HL \land W \in H) \land (F \in T \land G \in T \land X \in T) \land (R (G) \neq R (F) \land (F \neq {I ↾}_{B} \land G \neq {I ↾}_{B}) \land (P \in A \land \neg P \underset{˙}{\leq} W))) \to F \in T$
12		simp22	$⊢ ((K \in HL \land W \in H) \land (F \in T \land G \in T \land X \in T) \land (R (G) \neq R (F) \land (F \neq {I ↾}_{B} \land G \neq {I ↾}_{B}) \land (P \in A \land \neg P \underset{˙}{\leq} W))) \to G \in T$
13		simp3	$⊢ ((K \in HL \land W \in H) \land (F \in T \land G \in T \land X \in T) \land (R (G) \neq R (F) \land (F \neq {I ↾}_{B} \land G \neq {I ↾}_{B}) \land (P \in A \land \neg P \underset{˙}{\leq} W))) \to (R (G) \neq R (F) \land (F \neq {I ↾}_{B} \land G \neq {I ↾}_{B}) \land (P \in A \land \neg P \underset{˙}{\leq} W))$
14	1 2 3 4 5 6 7 8	cdlemk3	$⊢ ((K \in HL \land W \in H) \land (F \in T \land G \in T) \land (R (G) \neq R (F) \land (F \neq {I ↾}_{B} \land G \neq {I ↾}_{B}) \land (P \in A \land \neg P \underset{˙}{\leq} W))) \to (F (P) \underset{˙}{\lor} R (F)) \underset{˙}{\land} (F (P) \underset{˙}{\lor} R (G \circ F^{-1})) = F (P)$
15	9 10 11 12 13 14	syl221anc	$⊢ ((K \in HL \land W \in H) \land (F \in T \land G \in T \land X \in T) \land (R (G) \neq R (F) \land (F \neq {I ↾}_{B} \land G \neq {I ↾}_{B}) \land (P \in A \land \neg P \underset{˙}{\leq} W))) \to (F (P) \underset{˙}{\lor} R (F)) \underset{˙}{\land} (F (P) \underset{˙}{\lor} R (G \circ F^{-1})) = F (P)$
16		simp23	$⊢ ((K \in HL \land W \in H) \land (F \in T \land G \in T \land X \in T) \land (R (G) \neq R (F) \land (F \neq {I ↾}_{B} \land G \neq {I ↾}_{B}) \land (P \in A \land \neg P \underset{˙}{\leq} W))) \to X \in T$
17		simp33l	$⊢ ((K \in HL \land W \in H) \land (F \in T \land G \in T \land X \in T) \land (R (G) \neq R (F) \land (F \neq {I ↾}_{B} \land G \neq {I ↾}_{B}) \land (P \in A \land \neg P \underset{˙}{\leq} W))) \to P \in A$
18		simp33r	$⊢ ((K \in HL \land W \in H) \land (F \in T \land G \in T \land X \in T) \land (R (G) \neq R (F) \land (F \neq {I ↾}_{B} \land G \neq {I ↾}_{B}) \land (P \in A \land \neg P \underset{˙}{\leq} W))) \to \neg P \underset{˙}{\leq} W$
19	1 2 3 4 5 6 7 8	cdlemk4	$⊢ ((K \in HL \land W \in H) \land (F \in T \land X \in T) \land (P \in A \land \neg P \underset{˙}{\leq} W)) \to F (P) \underset{˙}{\leq} (X (P) \underset{˙}{\lor} R (X \circ F^{-1}))$
20	9 10 11 16 17 18 19	syl222anc	$⊢ ((K \in HL \land W \in H) \land (F \in T \land G \in T \land X \in T) \land (R (G) \neq R (F) \land (F \neq {I ↾}_{B} \land G \neq {I ↾}_{B}) \land (P \in A \land \neg P \underset{˙}{\leq} W))) \to F (P) \underset{˙}{\leq} (X (P) \underset{˙}{\lor} R (X \circ F^{-1}))$
21	15 20	eqbrtrd	$⊢ ((K \in HL \land W \in H) \land (F \in T \land G \in T \land X \in T) \land (R (G) \neq R (F) \land (F \neq {I ↾}_{B} \land G \neq {I ↾}_{B}) \land (P \in A \land \neg P \underset{˙}{\leq} W))) \to ((F (P) \underset{˙}{\lor} R (F)) \underset{˙}{\land} (F (P) \underset{˙}{\lor} R (G \circ F^{-1}))) \underset{˙}{\leq} (X (P) \underset{˙}{\lor} R (X \circ F^{-1}))$

Metamath Proof Explorer

Theorem cdlemk5a

Proof