dihord1

Description: Part of proof after Lemma N of Crawley p. 122. Forward ordering property. TODO: change ( Q .\/ ( X ./\ W ) ) = X to Q .<_ X using lhpmcvr3 , here and all theorems below. (Contributed by NM, 2-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	$⊢ \underset{˙}{\oplus} = LSSum (U)$
Assertion	dihord1	$⊢ (((K \in HL \land W \in H) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land (R \in A \land \neg R \underset{˙}{\leq} W)) \land (X \in B \land Y \in B) \land (Q \underset{˙}{\lor} (X \underset{˙}{\land} W) = X \land R \underset{˙}{\lor} (Y \underset{˙}{\land} W) = Y \land X \underset{˙}{\leq} Y)) \to J (Q) \underset{˙}{\oplus} I (X \underset{˙}{\land} W) \subseteq J (R) \underset{˙}{\oplus} I (Y \underset{˙}{\land} W)$

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		simp11	$⊢ (((K \in HL \land W \in H) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land (R \in A \land \neg R \underset{˙}{\leq} W)) \land (X \in B \land Y \in B) \land (Q \underset{˙}{\lor} (X \underset{˙}{\land} W) = X \land R \underset{˙}{\lor} (Y \underset{˙}{\land} W) = Y \land X \underset{˙}{\leq} Y)) \to (K \in HL \land W \in H)$
12		simp13	$⊢ (((K \in HL \land W \in H) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land (R \in A \land \neg R \underset{˙}{\leq} W)) \land (X \in B \land Y \in B) \land (Q \underset{˙}{\lor} (X \underset{˙}{\land} W) = X \land R \underset{˙}{\lor} (Y \underset{˙}{\land} W) = Y \land X \underset{˙}{\leq} Y)) \to (R \in A \land \neg R \underset{˙}{\leq} W)$
13		simp12	$⊢ (((K \in HL \land W \in H) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land (R \in A \land \neg R \underset{˙}{\leq} W)) \land (X \in B \land Y \in B) \land (Q \underset{˙}{\lor} (X \underset{˙}{\land} W) = X \land R \underset{˙}{\lor} (Y \underset{˙}{\land} W) = Y \land X \underset{˙}{\leq} Y)) \to (Q \in A \land \neg Q \underset{˙}{\leq} W)$
14		simp11l	$⊢ (((K \in HL \land W \in H) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land (R \in A \land \neg R \underset{˙}{\leq} W)) \land (X \in B \land Y \in B) \land (Q \underset{˙}{\lor} (X \underset{˙}{\land} W) = X \land R \underset{˙}{\lor} (Y \underset{˙}{\land} W) = Y \land X \underset{˙}{\leq} Y)) \to K \in HL$
15	14	hllatd	$⊢ (((K \in HL \land W \in H) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land (R \in A \land \neg R \underset{˙}{\leq} W)) \land (X \in B \land Y \in B) \land (Q \underset{˙}{\lor} (X \underset{˙}{\land} W) = X \land R \underset{˙}{\lor} (Y \underset{˙}{\land} W) = Y \land X \underset{˙}{\leq} Y)) \to K \in Lat$
16		simp2r	$⊢ (((K \in HL \land W \in H) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land (R \in A \land \neg R \underset{˙}{\leq} W)) \land (X \in B \land Y \in B) \land (Q \underset{˙}{\lor} (X \underset{˙}{\land} W) = X \land R \underset{˙}{\lor} (Y \underset{˙}{\land} W) = Y \land X \underset{˙}{\leq} Y)) \to Y \in B$
17		simp11r	$⊢ (((K \in HL \land W \in H) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land (R \in A \land \neg R \underset{˙}{\leq} W)) \land (X \in B \land Y \in B) \land (Q \underset{˙}{\lor} (X \underset{˙}{\land} W) = X \land R \underset{˙}{\lor} (Y \underset{˙}{\land} W) = Y \land X \underset{˙}{\leq} Y)) \to W \in H$
18	1 6	lhpbase	$⊢ W \in H \to W \in B$
19	17 18	syl	$⊢ (((K \in HL \land W \in H) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land (R \in A \land \neg R \underset{˙}{\leq} W)) \land (X \in B \land Y \in B) \land (Q \underset{˙}{\lor} (X \underset{˙}{\land} W) = X \land R \underset{˙}{\lor} (Y \underset{˙}{\land} W) = Y \land X \underset{˙}{\leq} Y)) \to W \in B$
20	1 4	latmcl	$⊢ (K \in Lat \land Y \in B \land W \in B) \to Y \underset{˙}{\land} W \in B$
21	15 16 19 20	syl3anc	$⊢ (((K \in HL \land W \in H) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land (R \in A \land \neg R \underset{˙}{\leq} W)) \land (X \in B \land Y \in B) \land (Q \underset{˙}{\lor} (X \underset{˙}{\land} W) = X \land R \underset{˙}{\lor} (Y \underset{˙}{\land} W) = Y \land X \underset{˙}{\leq} Y)) \to Y \underset{˙}{\land} W \in B$
22	1 2 4	latmle2	$⊢ (K \in Lat \land Y \in B \land W \in B) \to (Y \underset{˙}{\land} W) \underset{˙}{\leq} W$
23	15 16 19 22	syl3anc	$⊢ (((K \in HL \land W \in H) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land (R \in A \land \neg R \underset{˙}{\leq} W)) \land (X \in B \land Y \in B) \land (Q \underset{˙}{\lor} (X \underset{˙}{\land} W) = X \land R \underset{˙}{\lor} (Y \underset{˙}{\land} W) = Y \land X \underset{˙}{\leq} Y)) \to (Y \underset{˙}{\land} W) \underset{˙}{\leq} W$
24	21 23	jca	$⊢ (((K \in HL \land W \in H) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land (R \in A \land \neg R \underset{˙}{\leq} W)) \land (X \in B \land Y \in B) \land (Q \underset{˙}{\lor} (X \underset{˙}{\land} W) = X \land R \underset{˙}{\lor} (Y \underset{˙}{\land} W) = Y \land X \underset{˙}{\leq} Y)) \to (Y \underset{˙}{\land} W \in B \land (Y \underset{˙}{\land} W) \underset{˙}{\leq} W)$
25		simp12l	$⊢ (((K \in HL \land W \in H) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land (R \in A \land \neg R \underset{˙}{\leq} W)) \land (X \in B \land Y \in B) \land (Q \underset{˙}{\lor} (X \underset{˙}{\land} W) = X \land R \underset{˙}{\lor} (Y \underset{˙}{\land} W) = Y \land X \underset{˙}{\leq} Y)) \to Q \in A$
26	1 5	atbase	$⊢ Q \in A \to Q \in B$
27	25 26	syl	$⊢ (((K \in HL \land W \in H) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land (R \in A \land \neg R \underset{˙}{\leq} W)) \land (X \in B \land Y \in B) \land (Q \underset{˙}{\lor} (X \underset{˙}{\land} W) = X \land R \underset{˙}{\lor} (Y \underset{˙}{\land} W) = Y \land X \underset{˙}{\leq} Y)) \to Q \in B$
28		simp2l	$⊢ (((K \in HL \land W \in H) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land (R \in A \land \neg R \underset{˙}{\leq} W)) \land (X \in B \land Y \in B) \land (Q \underset{˙}{\lor} (X \underset{˙}{\land} W) = X \land R \underset{˙}{\lor} (Y \underset{˙}{\land} W) = Y \land X \underset{˙}{\leq} Y)) \to X \in B$
29	1 4	latmcl	$⊢ (K \in Lat \land X \in B \land W \in B) \to X \underset{˙}{\land} W \in B$
30	15 28 19 29	syl3anc	$⊢ (((K \in HL \land W \in H) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land (R \in A \land \neg R \underset{˙}{\leq} W)) \land (X \in B \land Y \in B) \land (Q \underset{˙}{\lor} (X \underset{˙}{\land} W) = X \land R \underset{˙}{\lor} (Y \underset{˙}{\land} W) = Y \land X \underset{˙}{\leq} Y)) \to X \underset{˙}{\land} W \in B$
31	1 3	latjcl	$⊢ (K \in Lat \land Q \in B \land X \underset{˙}{\land} W \in B) \to Q \underset{˙}{\lor} (X \underset{˙}{\land} W) \in B$
32	15 27 30 31	syl3anc	$⊢ (((K \in HL \land W \in H) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land (R \in A \land \neg R \underset{˙}{\leq} W)) \land (X \in B \land Y \in B) \land (Q \underset{˙}{\lor} (X \underset{˙}{\land} W) = X \land R \underset{˙}{\lor} (Y \underset{˙}{\land} W) = Y \land X \underset{˙}{\leq} Y)) \to Q \underset{˙}{\lor} (X \underset{˙}{\land} W) \in B$
33	1 2 3	latlej1	$⊢ (K \in Lat \land Q \in B \land X \underset{˙}{\land} W \in B) \to Q \underset{˙}{\leq} (Q \underset{˙}{\lor} (X \underset{˙}{\land} W))$
34	15 27 30 33	syl3anc	$⊢ (((K \in HL \land W \in H) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land (R \in A \land \neg R \underset{˙}{\leq} W)) \land (X \in B \land Y \in B) \land (Q \underset{˙}{\lor} (X \underset{˙}{\land} W) = X \land R \underset{˙}{\lor} (Y \underset{˙}{\land} W) = Y \land X \underset{˙}{\leq} Y)) \to Q \underset{˙}{\leq} (Q \underset{˙}{\lor} (X \underset{˙}{\land} W))$
35		simp31	$⊢ (((K \in HL \land W \in H) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land (R \in A \land \neg R \underset{˙}{\leq} W)) \land (X \in B \land Y \in B) \land (Q \underset{˙}{\lor} (X \underset{˙}{\land} W) = X \land R \underset{˙}{\lor} (Y \underset{˙}{\land} W) = Y \land X \underset{˙}{\leq} Y)) \to Q \underset{˙}{\lor} (X \underset{˙}{\land} W) = X$
36		simp33	$⊢ (((K \in HL \land W \in H) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land (R \in A \land \neg R \underset{˙}{\leq} W)) \land (X \in B \land Y \in B) \land (Q \underset{˙}{\lor} (X \underset{˙}{\land} W) = X \land R \underset{˙}{\lor} (Y \underset{˙}{\land} W) = Y \land X \underset{˙}{\leq} Y)) \to X \underset{˙}{\leq} Y$
37	35 36	eqbrtrd	$⊢ (((K \in HL \land W \in H) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land (R \in A \land \neg R \underset{˙}{\leq} W)) \land (X \in B \land Y \in B) \land (Q \underset{˙}{\lor} (X \underset{˙}{\land} W) = X \land R \underset{˙}{\lor} (Y \underset{˙}{\land} W) = Y \land X \underset{˙}{\leq} Y)) \to (Q \underset{˙}{\lor} (X \underset{˙}{\land} W)) \underset{˙}{\leq} Y$
38	1 2 15 27 32 16 34 37	lattrd	$⊢ (((K \in HL \land W \in H) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land (R \in A \land \neg R \underset{˙}{\leq} W)) \land (X \in B \land Y \in B) \land (Q \underset{˙}{\lor} (X \underset{˙}{\land} W) = X \land R \underset{˙}{\lor} (Y \underset{˙}{\land} W) = Y \land X \underset{˙}{\leq} Y)) \to Q \underset{˙}{\leq} Y$
39		simp32	$⊢ (((K \in HL \land W \in H) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land (R \in A \land \neg R \underset{˙}{\leq} W)) \land (X \in B \land Y \in B) \land (Q \underset{˙}{\lor} (X \underset{˙}{\land} W) = X \land R \underset{˙}{\lor} (Y \underset{˙}{\land} W) = Y \land X \underset{˙}{\leq} Y)) \to R \underset{˙}{\lor} (Y \underset{˙}{\land} W) = Y$
40	38 39	breqtrrd	$⊢ (((K \in HL \land W \in H) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land (R \in A \land \neg R \underset{˙}{\leq} W)) \land (X \in B \land Y \in B) \land (Q \underset{˙}{\lor} (X \underset{˙}{\land} W) = X \land R \underset{˙}{\lor} (Y \underset{˙}{\land} W) = Y \land X \underset{˙}{\leq} Y)) \to Q \underset{˙}{\leq} (R \underset{˙}{\lor} (Y \underset{˙}{\land} W))$
41	1 2 3 5 6 9 10 7 8	cdlemn5	$⊢ ((K \in HL \land W \in H) \land ((R \in A \land \neg R \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land (Y \underset{˙}{\land} W \in B \land (Y \underset{˙}{\land} W) \underset{˙}{\leq} W)) \land Q \underset{˙}{\leq} (R \underset{˙}{\lor} (Y \underset{˙}{\land} W))) \to J (Q) \subseteq J (R) \underset{˙}{\oplus} I (Y \underset{˙}{\land} W)$
42	11 12 13 24 40 41	syl131anc	$⊢ (((K \in HL \land W \in H) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land (R \in A \land \neg R \underset{˙}{\leq} W)) \land (X \in B \land Y \in B) \land (Q \underset{˙}{\lor} (X \underset{˙}{\land} W) = X \land R \underset{˙}{\lor} (Y \underset{˙}{\land} W) = Y \land X \underset{˙}{\leq} Y)) \to J (Q) \subseteq J (R) \underset{˙}{\oplus} I (Y \underset{˙}{\land} W)$
43	1 2 4	latmlem1	$⊢ (K \in Lat \land (X \in B \land Y \in B \land W \in B)) \to (X \underset{˙}{\leq} Y \to (X \underset{˙}{\land} W) \underset{˙}{\leq} (Y \underset{˙}{\land} W))$
44	15 28 16 19 43	syl13anc	$⊢ (((K \in HL \land W \in H) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land (R \in A \land \neg R \underset{˙}{\leq} W)) \land (X \in B \land Y \in B) \land (Q \underset{˙}{\lor} (X \underset{˙}{\land} W) = X \land R \underset{˙}{\lor} (Y \underset{˙}{\land} W) = Y \land X \underset{˙}{\leq} Y)) \to (X \underset{˙}{\leq} Y \to (X \underset{˙}{\land} W) \underset{˙}{\leq} (Y \underset{˙}{\land} W))$
45	36 44	mpd	$⊢ (((K \in HL \land W \in H) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land (R \in A \land \neg R \underset{˙}{\leq} W)) \land (X \in B \land Y \in B) \land (Q \underset{˙}{\lor} (X \underset{˙}{\land} W) = X \land R \underset{˙}{\lor} (Y \underset{˙}{\land} W) = Y \land X \underset{˙}{\leq} Y)) \to (X \underset{˙}{\land} W) \underset{˙}{\leq} (Y \underset{˙}{\land} W)$
46	1 2 4	latmle2	$⊢ (K \in Lat \land X \in B \land W \in B) \to (X \underset{˙}{\land} W) \underset{˙}{\leq} W$
47	15 28 19 46	syl3anc	$⊢ (((K \in HL \land W \in H) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land (R \in A \land \neg R \underset{˙}{\leq} W)) \land (X \in B \land Y \in B) \land (Q \underset{˙}{\lor} (X \underset{˙}{\land} W) = X \land R \underset{˙}{\lor} (Y \underset{˙}{\land} W) = Y \land X \underset{˙}{\leq} Y)) \to (X \underset{˙}{\land} W) \underset{˙}{\leq} W$
48	1 2 6 7	dibord	$⊢ ((K \in HL \land W \in H) \land (X \underset{˙}{\land} W \in B \land (X \underset{˙}{\land} W) \underset{˙}{\leq} W) \land (Y \underset{˙}{\land} W \in B \land (Y \underset{˙}{\land} W) \underset{˙}{\leq} W)) \to (I (X \underset{˙}{\land} W) \subseteq I (Y \underset{˙}{\land} W) \leftrightarrow (X \underset{˙}{\land} W) \underset{˙}{\leq} (Y \underset{˙}{\land} W))$
49	11 30 47 21 23 48	syl122anc	$⊢ (((K \in HL \land W \in H) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land (R \in A \land \neg R \underset{˙}{\leq} W)) \land (X \in B \land Y \in B) \land (Q \underset{˙}{\lor} (X \underset{˙}{\land} W) = X \land R \underset{˙}{\lor} (Y \underset{˙}{\land} W) = Y \land X \underset{˙}{\leq} Y)) \to (I (X \underset{˙}{\land} W) \subseteq I (Y \underset{˙}{\land} W) \leftrightarrow (X \underset{˙}{\land} W) \underset{˙}{\leq} (Y \underset{˙}{\land} W))$
50	45 49	mpbird	$⊢ (((K \in HL \land W \in H) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land (R \in A \land \neg R \underset{˙}{\leq} W)) \land (X \in B \land Y \in B) \land (Q \underset{˙}{\lor} (X \underset{˙}{\land} W) = X \land R \underset{˙}{\lor} (Y \underset{˙}{\land} W) = Y \land X \underset{˙}{\leq} Y)) \to I (X \underset{˙}{\land} W) \subseteq I (Y \underset{˙}{\land} W)$
51	6 9 11	dvhlmod	$⊢ (((K \in HL \land W \in H) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land (R \in A \land \neg R \underset{˙}{\leq} W)) \land (X \in B \land Y \in B) \land (Q \underset{˙}{\lor} (X \underset{˙}{\land} W) = X \land R \underset{˙}{\lor} (Y \underset{˙}{\land} W) = Y \land X \underset{˙}{\leq} Y)) \to U \in LMod$
52		eqid	$⊢ LSubSp (U) = LSubSp (U)$
53	52	lsssssubg	$⊢ U \in LMod \to LSubSp (U) \subseteq SubGrp (U)$
54	51 53	syl	$⊢ (((K \in HL \land W \in H) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land (R \in A \land \neg R \underset{˙}{\leq} W)) \land (X \in B \land Y \in B) \land (Q \underset{˙}{\lor} (X \underset{˙}{\land} W) = X \land R \underset{˙}{\lor} (Y \underset{˙}{\land} W) = Y \land X \underset{˙}{\leq} Y)) \to LSubSp (U) \subseteq SubGrp (U)$
55	2 5 6 9 8 52	diclss	$⊢ ((K \in HL \land W \in H) \land (R \in A \land \neg R \underset{˙}{\leq} W)) \to J (R) \in LSubSp (U)$
56	11 12 55	syl2anc	$⊢ (((K \in HL \land W \in H) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land (R \in A \land \neg R \underset{˙}{\leq} W)) \land (X \in B \land Y \in B) \land (Q \underset{˙}{\lor} (X \underset{˙}{\land} W) = X \land R \underset{˙}{\lor} (Y \underset{˙}{\land} W) = Y \land X \underset{˙}{\leq} Y)) \to J (R) \in LSubSp (U)$
57	54 56	sseldd	$⊢ (((K \in HL \land W \in H) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land (R \in A \land \neg R \underset{˙}{\leq} W)) \land (X \in B \land Y \in B) \land (Q \underset{˙}{\lor} (X \underset{˙}{\land} W) = X \land R \underset{˙}{\lor} (Y \underset{˙}{\land} W) = Y \land X \underset{˙}{\leq} Y)) \to J (R) \in SubGrp (U)$
58	1 2 6 9 7 52	diblss	$⊢ ((K \in HL \land W \in H) \land (Y \underset{˙}{\land} W \in B \land (Y \underset{˙}{\land} W) \underset{˙}{\leq} W)) \to I (Y \underset{˙}{\land} W) \in LSubSp (U)$
59	11 21 23 58	syl12anc	$⊢ (((K \in HL \land W \in H) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land (R \in A \land \neg R \underset{˙}{\leq} W)) \land (X \in B \land Y \in B) \land (Q \underset{˙}{\lor} (X \underset{˙}{\land} W) = X \land R \underset{˙}{\lor} (Y \underset{˙}{\land} W) = Y \land X \underset{˙}{\leq} Y)) \to I (Y \underset{˙}{\land} W) \in LSubSp (U)$
60	54 59	sseldd	$⊢ (((K \in HL \land W \in H) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land (R \in A \land \neg R \underset{˙}{\leq} W)) \land (X \in B \land Y \in B) \land (Q \underset{˙}{\lor} (X \underset{˙}{\land} W) = X \land R \underset{˙}{\lor} (Y \underset{˙}{\land} W) = Y \land X \underset{˙}{\leq} Y)) \to I (Y \underset{˙}{\land} W) \in SubGrp (U)$
61	10	lsmub2	$⊢ (J (R) \in SubGrp (U) \land I (Y \underset{˙}{\land} W) \in SubGrp (U)) \to I (Y \underset{˙}{\land} W) \subseteq J (R) \underset{˙}{\oplus} I (Y \underset{˙}{\land} W)$
62	57 60 61	syl2anc	$⊢ (((K \in HL \land W \in H) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land (R \in A \land \neg R \underset{˙}{\leq} W)) \land (X \in B \land Y \in B) \land (Q \underset{˙}{\lor} (X \underset{˙}{\land} W) = X \land R \underset{˙}{\lor} (Y \underset{˙}{\land} W) = Y \land X \underset{˙}{\leq} Y)) \to I (Y \underset{˙}{\land} W) \subseteq J (R) \underset{˙}{\oplus} I (Y \underset{˙}{\land} W)$
63	50 62	sstrd	$⊢ (((K \in HL \land W \in H) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land (R \in A \land \neg R \underset{˙}{\leq} W)) \land (X \in B \land Y \in B) \land (Q \underset{˙}{\lor} (X \underset{˙}{\land} W) = X \land R \underset{˙}{\lor} (Y \underset{˙}{\land} W) = Y \land X \underset{˙}{\leq} Y)) \to I (X \underset{˙}{\land} W) \subseteq J (R) \underset{˙}{\oplus} I (Y \underset{˙}{\land} W)$
64	2 5 6 9 8 52	diclss	$⊢ ((K \in HL \land W \in H) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \to J (Q) \in LSubSp (U)$
65	11 13 64	syl2anc	$⊢ (((K \in HL \land W \in H) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land (R \in A \land \neg R \underset{˙}{\leq} W)) \land (X \in B \land Y \in B) \land (Q \underset{˙}{\lor} (X \underset{˙}{\land} W) = X \land R \underset{˙}{\lor} (Y \underset{˙}{\land} W) = Y \land X \underset{˙}{\leq} Y)) \to J (Q) \in LSubSp (U)$
66	54 65	sseldd	$⊢ (((K \in HL \land W \in H) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land (R \in A \land \neg R \underset{˙}{\leq} W)) \land (X \in B \land Y \in B) \land (Q \underset{˙}{\lor} (X \underset{˙}{\land} W) = X \land R \underset{˙}{\lor} (Y \underset{˙}{\land} W) = Y \land X \underset{˙}{\leq} Y)) \to J (Q) \in SubGrp (U)$
67	1 2 6 9 7 52	diblss	$⊢ ((K \in HL \land W \in H) \land (X \underset{˙}{\land} W \in B \land (X \underset{˙}{\land} W) \underset{˙}{\leq} W)) \to I (X \underset{˙}{\land} W) \in LSubSp (U)$
68	11 30 47 67	syl12anc	$⊢ (((K \in HL \land W \in H) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land (R \in A \land \neg R \underset{˙}{\leq} W)) \land (X \in B \land Y \in B) \land (Q \underset{˙}{\lor} (X \underset{˙}{\land} W) = X \land R \underset{˙}{\lor} (Y \underset{˙}{\land} W) = Y \land X \underset{˙}{\leq} Y)) \to I (X \underset{˙}{\land} W) \in LSubSp (U)$
69	54 68	sseldd	$⊢ (((K \in HL \land W \in H) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land (R \in A \land \neg R \underset{˙}{\leq} W)) \land (X \in B \land Y \in B) \land (Q \underset{˙}{\lor} (X \underset{˙}{\land} W) = X \land R \underset{˙}{\lor} (Y \underset{˙}{\land} W) = Y \land X \underset{˙}{\leq} Y)) \to I (X \underset{˙}{\land} W) \in SubGrp (U)$
70	52 10	lsmcl	$⊢ (U \in LMod \land J (R) \in LSubSp (U) \land I (Y \underset{˙}{\land} W) \in LSubSp (U)) \to J (R) \underset{˙}{\oplus} I (Y \underset{˙}{\land} W) \in LSubSp (U)$
71	51 56 59 70	syl3anc	$⊢ (((K \in HL \land W \in H) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land (R \in A \land \neg R \underset{˙}{\leq} W)) \land (X \in B \land Y \in B) \land (Q \underset{˙}{\lor} (X \underset{˙}{\land} W) = X \land R \underset{˙}{\lor} (Y \underset{˙}{\land} W) = Y \land X \underset{˙}{\leq} Y)) \to J (R) \underset{˙}{\oplus} I (Y \underset{˙}{\land} W) \in LSubSp (U)$
72	54 71	sseldd	$⊢ (((K \in HL \land W \in H) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land (R \in A \land \neg R \underset{˙}{\leq} W)) \land (X \in B \land Y \in B) \land (Q \underset{˙}{\lor} (X \underset{˙}{\land} W) = X \land R \underset{˙}{\lor} (Y \underset{˙}{\land} W) = Y \land X \underset{˙}{\leq} Y)) \to J (R) \underset{˙}{\oplus} I (Y \underset{˙}{\land} W) \in SubGrp (U)$
73	10	lsmlub	$⊢ (J (Q) \in SubGrp (U) \land I (X \underset{˙}{\land} W) \in SubGrp (U) \land J (R) \underset{˙}{\oplus} I (Y \underset{˙}{\land} W) \in SubGrp (U)) \to ((J (Q) \subseteq J (R) \underset{˙}{\oplus} I (Y \underset{˙}{\land} W) \land I (X \underset{˙}{\land} W) \subseteq J (R) \underset{˙}{\oplus} I (Y \underset{˙}{\land} W)) \leftrightarrow J (Q) \underset{˙}{\oplus} I (X \underset{˙}{\land} W) \subseteq J (R) \underset{˙}{\oplus} I (Y \underset{˙}{\land} W))$
74	66 69 72 73	syl3anc	$⊢ (((K \in HL \land W \in H) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land (R \in A \land \neg R \underset{˙}{\leq} W)) \land (X \in B \land Y \in B) \land (Q \underset{˙}{\lor} (X \underset{˙}{\land} W) = X \land R \underset{˙}{\lor} (Y \underset{˙}{\land} W) = Y \land X \underset{˙}{\leq} Y)) \to ((J (Q) \subseteq J (R) \underset{˙}{\oplus} I (Y \underset{˙}{\land} W) \land I (X \underset{˙}{\land} W) \subseteq J (R) \underset{˙}{\oplus} I (Y \underset{˙}{\land} W)) \leftrightarrow J (Q) \underset{˙}{\oplus} I (X \underset{˙}{\land} W) \subseteq J (R) \underset{˙}{\oplus} I (Y \underset{˙}{\land} W))$
75	42 63 74	mpbi2and	$⊢ (((K \in HL \land W \in H) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land (R \in A \land \neg R \underset{˙}{\leq} W)) \land (X \in B \land Y \in B) \land (Q \underset{˙}{\lor} (X \underset{˙}{\land} W) = X \land R \underset{˙}{\lor} (Y \underset{˙}{\land} W) = Y \land X \underset{˙}{\leq} Y)) \to J (Q) \underset{˙}{\oplus} I (X \underset{˙}{\land} W) \subseteq J (R) \underset{˙}{\oplus} I (Y \underset{˙}{\land} W)$

Metamath Proof Explorer

Theorem dihord1

Proof