dihord2pre2

Description: Part of proof after Lemma N of Crawley p. 122. Reverse ordering property. (Contributed by NM, 4-Mar-2014)

	Ref	Expression
Hypotheses	dihjust.b	$⊢ B = {Base}_{K}$
	dihjust.l	$⊢ \underset{˙}{\leq} = \leq_{K}$
	dihjust.j	$⊢ \underset{˙}{\lor} = join (K)$
	dihjust.m	$⊢ \underset{˙}{\land} = meet (K)$
	dihjust.a	$⊢ A = Atoms (K)$
	dihjust.h	$⊢ H = LHyp (K)$
	dihjust.i	$⊢ I = DIsoB (K) (W)$
	dihjust.J	$⊢ J = DIsoC (K) (W)$
	dihjust.u	$⊢ U = DVecH (K) (W)$
	dihjust.s	$⊢ \underset{˙}{\oplus} = LSSum (U)$
	dihord2c.t	$⊢ T = LTrn (K) (W)$
	dihord2c.r	$⊢ R = trL (K) (W)$
	dihord2c.o	$⊢ O = (h \in T ⟼ {I ↾}_{B})$
	dihord2.p	$⊢ P = oc (K) (W)$
	dihord2.e	$⊢ E = TEndo (K) (W)$
	dihord2.d	$⊢ \underset{˙}{+} = +_{U}$
	dihord2.g	$⊢ G = (ι h \in T \| h (P) = N)$
Assertion	dihord2pre2	$⊢ (((K \in HL \land W \in H) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land (N \in A \land \neg N \underset{˙}{\leq} W)) \land (X \in B \land Y \in B) \land (Q \underset{˙}{\lor} (X \underset{˙}{\land} W) = X \land N \underset{˙}{\lor} (Y \underset{˙}{\land} W) = Y \land J (Q) \underset{˙}{\oplus} I (X \underset{˙}{\land} W) \subseteq J (N) \underset{˙}{\oplus} I (Y \underset{˙}{\land} W))) \to (Q \underset{˙}{\lor} (X \underset{˙}{\land} W)) \underset{˙}{\leq} (N \underset{˙}{\lor} (Y \underset{˙}{\land} W))$

Proof

Step	Hyp	Ref	Expression
1		dihjust.b	$⊢ B = {Base}_{K}$
2		dihjust.l	$⊢ \underset{˙}{\leq} = \leq_{K}$
3		dihjust.j	$⊢ \underset{˙}{\lor} = join (K)$
4		dihjust.m	$⊢ \underset{˙}{\land} = meet (K)$
5		dihjust.a	$⊢ A = Atoms (K)$
6		dihjust.h	$⊢ H = LHyp (K)$
7		dihjust.i	$⊢ I = DIsoB (K) (W)$
8		dihjust.J	$⊢ J = DIsoC (K) (W)$
9		dihjust.u	$⊢ U = DVecH (K) (W)$
10		dihjust.s	$⊢ \underset{˙}{\oplus} = LSSum (U)$
11		dihord2c.t	$⊢ T = LTrn (K) (W)$
12		dihord2c.r	$⊢ R = trL (K) (W)$
13		dihord2c.o	$⊢ O = (h \in T ⟼ {I ↾}_{B})$
14		dihord2.p	$⊢ P = oc (K) (W)$
15		dihord2.e	$⊢ E = TEndo (K) (W)$
16		dihord2.d	$⊢ \underset{˙}{+} = +_{U}$
17		dihord2.g	$⊢ G = (ι h \in T \| h (P) = N)$
18	1 2 3 4 5 6 7 8 9 10	dihord2a	$⊢ (((K \in HL \land W \in H) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land (N \in A \land \neg N \underset{˙}{\leq} W)) \land (X \in B \land Y \in B) \land (Q \underset{˙}{\lor} (X \underset{˙}{\land} W) = X \land N \underset{˙}{\lor} (Y \underset{˙}{\land} W) = Y \land J (Q) \underset{˙}{\oplus} I (X \underset{˙}{\land} W) \subseteq J (N) \underset{˙}{\oplus} I (Y \underset{˙}{\land} W))) \to Q \underset{˙}{\leq} (N \underset{˙}{\lor} (Y \underset{˙}{\land} W))$
19		simp11l	$⊢ (((K \in HL \land W \in H) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land (N \in A \land \neg N \underset{˙}{\leq} W)) \land (X \in B \land Y \in B) \land (Q \underset{˙}{\lor} (X \underset{˙}{\land} W) = X \land N \underset{˙}{\lor} (Y \underset{˙}{\land} W) = Y \land J (Q) \underset{˙}{\oplus} I (X \underset{˙}{\land} W) \subseteq J (N) \underset{˙}{\oplus} I (Y \underset{˙}{\land} W))) \to K \in HL$
20	19	hllatd	$⊢ (((K \in HL \land W \in H) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land (N \in A \land \neg N \underset{˙}{\leq} W)) \land (X \in B \land Y \in B) \land (Q \underset{˙}{\lor} (X \underset{˙}{\land} W) = X \land N \underset{˙}{\lor} (Y \underset{˙}{\land} W) = Y \land J (Q) \underset{˙}{\oplus} I (X \underset{˙}{\land} W) \subseteq J (N) \underset{˙}{\oplus} I (Y \underset{˙}{\land} W))) \to K \in Lat$
21		simp2l	$⊢ (((K \in HL \land W \in H) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land (N \in A \land \neg N \underset{˙}{\leq} W)) \land (X \in B \land Y \in B) \land (Q \underset{˙}{\lor} (X \underset{˙}{\land} W) = X \land N \underset{˙}{\lor} (Y \underset{˙}{\land} W) = Y \land J (Q) \underset{˙}{\oplus} I (X \underset{˙}{\land} W) \subseteq J (N) \underset{˙}{\oplus} I (Y \underset{˙}{\land} W))) \to X \in B$
22		simp11r	$⊢ (((K \in HL \land W \in H) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land (N \in A \land \neg N \underset{˙}{\leq} W)) \land (X \in B \land Y \in B) \land (Q \underset{˙}{\lor} (X \underset{˙}{\land} W) = X \land N \underset{˙}{\lor} (Y \underset{˙}{\land} W) = Y \land J (Q) \underset{˙}{\oplus} I (X \underset{˙}{\land} W) \subseteq J (N) \underset{˙}{\oplus} I (Y \underset{˙}{\land} W))) \to W \in H$
23	1 6	lhpbase	$⊢ W \in H \to W \in B$
24	22 23	syl	$⊢ (((K \in HL \land W \in H) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land (N \in A \land \neg N \underset{˙}{\leq} W)) \land (X \in B \land Y \in B) \land (Q \underset{˙}{\lor} (X \underset{˙}{\land} W) = X \land N \underset{˙}{\lor} (Y \underset{˙}{\land} W) = Y \land J (Q) \underset{˙}{\oplus} I (X \underset{˙}{\land} W) \subseteq J (N) \underset{˙}{\oplus} I (Y \underset{˙}{\land} W))) \to W \in B$
25	1 4	latmcl	$⊢ (K \in Lat \land X \in B \land W \in B) \to X \underset{˙}{\land} W \in B$
26	20 21 24 25	syl3anc	$⊢ (((K \in HL \land W \in H) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land (N \in A \land \neg N \underset{˙}{\leq} W)) \land (X \in B \land Y \in B) \land (Q \underset{˙}{\lor} (X \underset{˙}{\land} W) = X \land N \underset{˙}{\lor} (Y \underset{˙}{\land} W) = Y \land J (Q) \underset{˙}{\oplus} I (X \underset{˙}{\land} W) \subseteq J (N) \underset{˙}{\oplus} I (Y \underset{˙}{\land} W))) \to X \underset{˙}{\land} W \in B$
27		simp2r	$⊢ (((K \in HL \land W \in H) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land (N \in A \land \neg N \underset{˙}{\leq} W)) \land (X \in B \land Y \in B) \land (Q \underset{˙}{\lor} (X \underset{˙}{\land} W) = X \land N \underset{˙}{\lor} (Y \underset{˙}{\land} W) = Y \land J (Q) \underset{˙}{\oplus} I (X \underset{˙}{\land} W) \subseteq J (N) \underset{˙}{\oplus} I (Y \underset{˙}{\land} W))) \to Y \in B$
28	1 4	latmcl	$⊢ (K \in Lat \land Y \in B \land W \in B) \to Y \underset{˙}{\land} W \in B$
29	20 27 24 28	syl3anc	$⊢ (((K \in HL \land W \in H) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land (N \in A \land \neg N \underset{˙}{\leq} W)) \land (X \in B \land Y \in B) \land (Q \underset{˙}{\lor} (X \underset{˙}{\land} W) = X \land N \underset{˙}{\lor} (Y \underset{˙}{\land} W) = Y \land J (Q) \underset{˙}{\oplus} I (X \underset{˙}{\land} W) \subseteq J (N) \underset{˙}{\oplus} I (Y \underset{˙}{\land} W))) \to Y \underset{˙}{\land} W \in B$
30		simp13l	$⊢ (((K \in HL \land W \in H) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land (N \in A \land \neg N \underset{˙}{\leq} W)) \land (X \in B \land Y \in B) \land (Q \underset{˙}{\lor} (X \underset{˙}{\land} W) = X \land N \underset{˙}{\lor} (Y \underset{˙}{\land} W) = Y \land J (Q) \underset{˙}{\oplus} I (X \underset{˙}{\land} W) \subseteq J (N) \underset{˙}{\oplus} I (Y \underset{˙}{\land} W))) \to N \in A$
31	1 5	atbase	$⊢ N \in A \to N \in B$
32	30 31	syl	$⊢ (((K \in HL \land W \in H) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land (N \in A \land \neg N \underset{˙}{\leq} W)) \land (X \in B \land Y \in B) \land (Q \underset{˙}{\lor} (X \underset{˙}{\land} W) = X \land N \underset{˙}{\lor} (Y \underset{˙}{\land} W) = Y \land J (Q) \underset{˙}{\oplus} I (X \underset{˙}{\land} W) \subseteq J (N) \underset{˙}{\oplus} I (Y \underset{˙}{\land} W))) \to N \in B$
33	1 3	latjcl	$⊢ (K \in Lat \land N \in B \land Y \underset{˙}{\land} W \in B) \to N \underset{˙}{\lor} (Y \underset{˙}{\land} W) \in B$
34	20 32 29 33	syl3anc	$⊢ (((K \in HL \land W \in H) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land (N \in A \land \neg N \underset{˙}{\leq} W)) \land (X \in B \land Y \in B) \land (Q \underset{˙}{\lor} (X \underset{˙}{\land} W) = X \land N \underset{˙}{\lor} (Y \underset{˙}{\land} W) = Y \land J (Q) \underset{˙}{\oplus} I (X \underset{˙}{\land} W) \subseteq J (N) \underset{˙}{\oplus} I (Y \underset{˙}{\land} W))) \to N \underset{˙}{\lor} (Y \underset{˙}{\land} W) \in B$
35		simp33	$⊢ (((K \in HL \land W \in H) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land (N \in A \land \neg N \underset{˙}{\leq} W)) \land (X \in B \land Y \in B) \land (Q \underset{˙}{\lor} (X \underset{˙}{\land} W) = X \land N \underset{˙}{\lor} (Y \underset{˙}{\land} W) = Y \land J (Q) \underset{˙}{\oplus} I (X \underset{˙}{\land} W) \subseteq J (N) \underset{˙}{\oplus} I (Y \underset{˙}{\land} W))) \to J (Q) \underset{˙}{\oplus} I (X \underset{˙}{\land} W) \subseteq J (N) \underset{˙}{\oplus} I (Y \underset{˙}{\land} W)$
36	1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17	dihord2pre	$⊢ (((K \in HL \land W \in H) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land (N \in A \land \neg N \underset{˙}{\leq} W)) \land (X \in B \land Y \in B) \land J (Q) \underset{˙}{\oplus} I (X \underset{˙}{\land} W) \subseteq J (N) \underset{˙}{\oplus} I (Y \underset{˙}{\land} W)) \to (X \underset{˙}{\land} W) \underset{˙}{\leq} (Y \underset{˙}{\land} W)$
37	35 36	syld3an3	$⊢ (((K \in HL \land W \in H) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land (N \in A \land \neg N \underset{˙}{\leq} W)) \land (X \in B \land Y \in B) \land (Q \underset{˙}{\lor} (X \underset{˙}{\land} W) = X \land N \underset{˙}{\lor} (Y \underset{˙}{\land} W) = Y \land J (Q) \underset{˙}{\oplus} I (X \underset{˙}{\land} W) \subseteq J (N) \underset{˙}{\oplus} I (Y \underset{˙}{\land} W))) \to (X \underset{˙}{\land} W) \underset{˙}{\leq} (Y \underset{˙}{\land} W)$
38	1 2 3	latlej2	$⊢ (K \in Lat \land N \in B \land Y \underset{˙}{\land} W \in B) \to (Y \underset{˙}{\land} W) \underset{˙}{\leq} (N \underset{˙}{\lor} (Y \underset{˙}{\land} W))$
39	20 32 29 38	syl3anc	$⊢ (((K \in HL \land W \in H) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land (N \in A \land \neg N \underset{˙}{\leq} W)) \land (X \in B \land Y \in B) \land (Q \underset{˙}{\lor} (X \underset{˙}{\land} W) = X \land N \underset{˙}{\lor} (Y \underset{˙}{\land} W) = Y \land J (Q) \underset{˙}{\oplus} I (X \underset{˙}{\land} W) \subseteq J (N) \underset{˙}{\oplus} I (Y \underset{˙}{\land} W))) \to (Y \underset{˙}{\land} W) \underset{˙}{\leq} (N \underset{˙}{\lor} (Y \underset{˙}{\land} W))$
40	1 2 20 26 29 34 37 39	lattrd	$⊢ (((K \in HL \land W \in H) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land (N \in A \land \neg N \underset{˙}{\leq} W)) \land (X \in B \land Y \in B) \land (Q \underset{˙}{\lor} (X \underset{˙}{\land} W) = X \land N \underset{˙}{\lor} (Y \underset{˙}{\land} W) = Y \land J (Q) \underset{˙}{\oplus} I (X \underset{˙}{\land} W) \subseteq J (N) \underset{˙}{\oplus} I (Y \underset{˙}{\land} W))) \to (X \underset{˙}{\land} W) \underset{˙}{\leq} (N \underset{˙}{\lor} (Y \underset{˙}{\land} W))$
41		simp12l	$⊢ (((K \in HL \land W \in H) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land (N \in A \land \neg N \underset{˙}{\leq} W)) \land (X \in B \land Y \in B) \land (Q \underset{˙}{\lor} (X \underset{˙}{\land} W) = X \land N \underset{˙}{\lor} (Y \underset{˙}{\land} W) = Y \land J (Q) \underset{˙}{\oplus} I (X \underset{˙}{\land} W) \subseteq J (N) \underset{˙}{\oplus} I (Y \underset{˙}{\land} W))) \to Q \in A$
42	1 5	atbase	$⊢ Q \in A \to Q \in B$
43	41 42	syl	$⊢ (((K \in HL \land W \in H) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land (N \in A \land \neg N \underset{˙}{\leq} W)) \land (X \in B \land Y \in B) \land (Q \underset{˙}{\lor} (X \underset{˙}{\land} W) = X \land N \underset{˙}{\lor} (Y \underset{˙}{\land} W) = Y \land J (Q) \underset{˙}{\oplus} I (X \underset{˙}{\land} W) \subseteq J (N) \underset{˙}{\oplus} I (Y \underset{˙}{\land} W))) \to Q \in B$
44	1 2 3	latjle12	$⊢ (K \in Lat \land (Q \in B \land X \underset{˙}{\land} W \in B \land N \underset{˙}{\lor} (Y \underset{˙}{\land} W) \in B)) \to ((Q \underset{˙}{\leq} (N \underset{˙}{\lor} (Y \underset{˙}{\land} W)) \land (X \underset{˙}{\land} W) \underset{˙}{\leq} (N \underset{˙}{\lor} (Y \underset{˙}{\land} W))) \leftrightarrow (Q \underset{˙}{\lor} (X \underset{˙}{\land} W)) \underset{˙}{\leq} (N \underset{˙}{\lor} (Y \underset{˙}{\land} W)))$
45	20 43 26 34 44	syl13anc	$⊢ (((K \in HL \land W \in H) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land (N \in A \land \neg N \underset{˙}{\leq} W)) \land (X \in B \land Y \in B) \land (Q \underset{˙}{\lor} (X \underset{˙}{\land} W) = X \land N \underset{˙}{\lor} (Y \underset{˙}{\land} W) = Y \land J (Q) \underset{˙}{\oplus} I (X \underset{˙}{\land} W) \subseteq J (N) \underset{˙}{\oplus} I (Y \underset{˙}{\land} W))) \to ((Q \underset{˙}{\leq} (N \underset{˙}{\lor} (Y \underset{˙}{\land} W)) \land (X \underset{˙}{\land} W) \underset{˙}{\leq} (N \underset{˙}{\lor} (Y \underset{˙}{\land} W))) \leftrightarrow (Q \underset{˙}{\lor} (X \underset{˙}{\land} W)) \underset{˙}{\leq} (N \underset{˙}{\lor} (Y \underset{˙}{\land} W)))$
46	18 40 45	mpbi2and	$⊢ (((K \in HL \land W \in H) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land (N \in A \land \neg N \underset{˙}{\leq} W)) \land (X \in B \land Y \in B) \land (Q \underset{˙}{\lor} (X \underset{˙}{\land} W) = X \land N \underset{˙}{\lor} (Y \underset{˙}{\land} W) = Y \land J (Q) \underset{˙}{\oplus} I (X \underset{˙}{\land} W) \subseteq J (N) \underset{˙}{\oplus} I (Y \underset{˙}{\land} W))) \to (Q \underset{˙}{\lor} (X \underset{˙}{\land} W)) \underset{˙}{\leq} (N \underset{˙}{\lor} (Y \underset{˙}{\land} W))$

Metamath Proof Explorer

Theorem dihord2pre2

Proof