cdlemk17

Description: Part of proof of Lemma K of Crawley p. 118. Line 21 on p. 119. O , D are k_1, f_1. (Contributed by NM, 1-Jul-2013)

	Ref	Expression
Hypotheses	cdlemk1.b	$⊢ B = {Base}_{K}$
	cdlemk1.l	$⊢ \underset{˙}{\leq} = \leq_{K}$
	cdlemk1.j	$⊢ \underset{˙}{\lor} = join (K)$
	cdlemk1.m	$⊢ \underset{˙}{\land} = meet (K)$
	cdlemk1.a	$⊢ A = Atoms (K)$
	cdlemk1.h	$⊢ H = LHyp (K)$
	cdlemk1.t	$⊢ T = LTrn (K) (W)$
	cdlemk1.r	$⊢ R = trL (K) (W)$
	cdlemk1.s	$⊢ S = (f \in T ⟼ (ι i \in T \| i (P) = (P \underset{˙}{\lor} R (f)) \underset{˙}{\land} (N (P) \underset{˙}{\lor} R (f \circ F^{-1}))))$
	cdlemk1.o	$⊢ O = S (D)$
Assertion	cdlemk17	$⊢ (((K \in HL \land W \in H) \land F \in T \land D \in T) \land (N \in T \land (P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N)) \land (F \neq {I ↾}_{B} \land D \neq {I ↾}_{B} \land R (D) \neq R (F))) \to N (P) = (P \underset{˙}{\lor} R (F)) \underset{˙}{\land} (O (P) \underset{˙}{\lor} R (F \circ D^{-1}))$

Proof

Step	Hyp	Ref	Expression
1		cdlemk1.b	$⊢ B = {Base}_{K}$
2		cdlemk1.l	$⊢ \underset{˙}{\leq} = \leq_{K}$
3		cdlemk1.j	$⊢ \underset{˙}{\lor} = join (K)$
4		cdlemk1.m	$⊢ \underset{˙}{\land} = meet (K)$
5		cdlemk1.a	$⊢ A = Atoms (K)$
6		cdlemk1.h	$⊢ H = LHyp (K)$
7		cdlemk1.t	$⊢ T = LTrn (K) (W)$
8		cdlemk1.r	$⊢ R = trL (K) (W)$
9		cdlemk1.s	$⊢ S = (f \in T ⟼ (ι i \in T \| i (P) = (P \underset{˙}{\lor} R (f)) \underset{˙}{\land} (N (P) \underset{˙}{\lor} R (f \circ F^{-1}))))$
10		cdlemk1.o	$⊢ O = S (D)$
11	1 2 3 4 5 6 7 8 9 10	cdlemk15	$⊢ (((K \in HL \land W \in H) \land F \in T \land D \in T) \land (N \in T \land (P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N)) \land (F \neq {I ↾}_{B} \land D \neq {I ↾}_{B} \land R (D) \neq R (F))) \to N (P) \underset{˙}{\leq} ((P \underset{˙}{\lor} R (F)) \underset{˙}{\land} (O (P) \underset{˙}{\lor} R (F \circ D^{-1})))$
12		simp11l	$⊢ (((K \in HL \land W \in H) \land F \in T \land D \in T) \land (N \in T \land (P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N)) \land (F \neq {I ↾}_{B} \land D \neq {I ↾}_{B} \land R (D) \neq R (F))) \to K \in HL$
13		hlatl	$⊢ K \in HL \to K \in AtLat$
14	12 13	syl	$⊢ (((K \in HL \land W \in H) \land F \in T \land D \in T) \land (N \in T \land (P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N)) \land (F \neq {I ↾}_{B} \land D \neq {I ↾}_{B} \land R (D) \neq R (F))) \to K \in AtLat$
15		simp11	$⊢ (((K \in HL \land W \in H) \land F \in T \land D \in T) \land (N \in T \land (P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N)) \land (F \neq {I ↾}_{B} \land D \neq {I ↾}_{B} \land R (D) \neq R (F))) \to (K \in HL \land W \in H)$
16		simp21	$⊢ (((K \in HL \land W \in H) \land F \in T \land D \in T) \land (N \in T \land (P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N)) \land (F \neq {I ↾}_{B} \land D \neq {I ↾}_{B} \land R (D) \neq R (F))) \to N \in T$
17		simp22l	$⊢ (((K \in HL \land W \in H) \land F \in T \land D \in T) \land (N \in T \land (P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N)) \land (F \neq {I ↾}_{B} \land D \neq {I ↾}_{B} \land R (D) \neq R (F))) \to P \in A$
18	2 5 6 7	ltrnat	$⊢ ((K \in HL \land W \in H) \land N \in T \land P \in A) \to N (P) \in A$
19	15 16 17 18	syl3anc	$⊢ (((K \in HL \land W \in H) \land F \in T \land D \in T) \land (N \in T \land (P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N)) \land (F \neq {I ↾}_{B} \land D \neq {I ↾}_{B} \land R (D) \neq R (F))) \to N (P) \in A$
20	1 2 3 4 5 6 7 8 9 10	cdlemk16	$⊢ (((K \in HL \land W \in H) \land F \in T \land D \in T) \land (N \in T \land (P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N)) \land (F \neq {I ↾}_{B} \land D \neq {I ↾}_{B} \land R (D) \neq R (F))) \to ((P \underset{˙}{\lor} R (F)) \underset{˙}{\land} (O (P) \underset{˙}{\lor} R (F \circ D^{-1})) \in A \land \neg ((P \underset{˙}{\lor} R (F)) \underset{˙}{\land} (O (P) \underset{˙}{\lor} R (F \circ D^{-1}))) \underset{˙}{\leq} W)$
21	20	simpld	$⊢ (((K \in HL \land W \in H) \land F \in T \land D \in T) \land (N \in T \land (P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N)) \land (F \neq {I ↾}_{B} \land D \neq {I ↾}_{B} \land R (D) \neq R (F))) \to (P \underset{˙}{\lor} R (F)) \underset{˙}{\land} (O (P) \underset{˙}{\lor} R (F \circ D^{-1})) \in A$
22	2 5	atcmp	$⊢ (K \in AtLat \land N (P) \in A \land (P \underset{˙}{\lor} R (F)) \underset{˙}{\land} (O (P) \underset{˙}{\lor} R (F \circ D^{-1})) \in A) \to (N (P) \underset{˙}{\leq} ((P \underset{˙}{\lor} R (F)) \underset{˙}{\land} (O (P) \underset{˙}{\lor} R (F \circ D^{-1}))) \leftrightarrow N (P) = (P \underset{˙}{\lor} R (F)) \underset{˙}{\land} (O (P) \underset{˙}{\lor} R (F \circ D^{-1})))$
23	14 19 21 22	syl3anc	$⊢ (((K \in HL \land W \in H) \land F \in T \land D \in T) \land (N \in T \land (P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N)) \land (F \neq {I ↾}_{B} \land D \neq {I ↾}_{B} \land R (D) \neq R (F))) \to (N (P) \underset{˙}{\leq} ((P \underset{˙}{\lor} R (F)) \underset{˙}{\land} (O (P) \underset{˙}{\lor} R (F \circ D^{-1}))) \leftrightarrow N (P) = (P \underset{˙}{\lor} R (F)) \underset{˙}{\land} (O (P) \underset{˙}{\lor} R (F \circ D^{-1})))$
24	11 23	mpbid	$⊢ (((K \in HL \land W \in H) \land F \in T \land D \in T) \land (N \in T \land (P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N)) \land (F \neq {I ↾}_{B} \land D \neq {I ↾}_{B} \land R (D) \neq R (F))) \to N (P) = (P \underset{˙}{\lor} R (F)) \underset{˙}{\land} (O (P) \underset{˙}{\lor} R (F \circ D^{-1}))$

Metamath Proof Explorer

Theorem cdlemk17

Proof