dihjustlem

Description: Part of proof after Lemma N of Crawley p. 122 line 4, "the definition of phi(x) is independent of the atom q." (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	dihjustlem	$⊢ ((K \in HL \land W \in H) \land ((Q \in A \land \neg Q \underset{˙}{\leq} W) \land (R \in A \land \neg R \underset{˙}{\leq} W) \land X \in B) \land Q \underset{˙}{\lor} (X \underset{˙}{\land} W) = R \underset{˙}{\lor} (X \underset{˙}{\land} W)) \to J (Q) \underset{˙}{\oplus} I (X \underset{˙}{\land} W) \subseteq J (R) \underset{˙}{\oplus} I (X \underset{˙}{\land} W)$

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		simp1l	$⊢ ((K \in HL \land W \in H) \land ((Q \in A \land \neg Q \underset{˙}{\leq} W) \land (R \in A \land \neg R \underset{˙}{\leq} W) \land X \in B) \land Q \underset{˙}{\lor} (X \underset{˙}{\land} W) = R \underset{˙}{\lor} (X \underset{˙}{\land} W)) \to K \in HL$
12	11	hllatd	$⊢ ((K \in HL \land W \in H) \land ((Q \in A \land \neg Q \underset{˙}{\leq} W) \land (R \in A \land \neg R \underset{˙}{\leq} W) \land X \in B) \land Q \underset{˙}{\lor} (X \underset{˙}{\land} W) = R \underset{˙}{\lor} (X \underset{˙}{\land} W)) \to K \in Lat$
13		simp21l	$⊢ ((K \in HL \land W \in H) \land ((Q \in A \land \neg Q \underset{˙}{\leq} W) \land (R \in A \land \neg R \underset{˙}{\leq} W) \land X \in B) \land Q \underset{˙}{\lor} (X \underset{˙}{\land} W) = R \underset{˙}{\lor} (X \underset{˙}{\land} W)) \to Q \in A$
14	1 5	atbase	$⊢ Q \in A \to Q \in B$
15	13 14	syl	$⊢ ((K \in HL \land W \in H) \land ((Q \in A \land \neg Q \underset{˙}{\leq} W) \land (R \in A \land \neg R \underset{˙}{\leq} W) \land X \in B) \land Q \underset{˙}{\lor} (X \underset{˙}{\land} W) = R \underset{˙}{\lor} (X \underset{˙}{\land} W)) \to Q \in B$
16		simp23	$⊢ ((K \in HL \land W \in H) \land ((Q \in A \land \neg Q \underset{˙}{\leq} W) \land (R \in A \land \neg R \underset{˙}{\leq} W) \land X \in B) \land Q \underset{˙}{\lor} (X \underset{˙}{\land} W) = R \underset{˙}{\lor} (X \underset{˙}{\land} W)) \to X \in B$
17		simp1r	$⊢ ((K \in HL \land W \in H) \land ((Q \in A \land \neg Q \underset{˙}{\leq} W) \land (R \in A \land \neg R \underset{˙}{\leq} W) \land X \in B) \land Q \underset{˙}{\lor} (X \underset{˙}{\land} W) = R \underset{˙}{\lor} (X \underset{˙}{\land} W)) \to W \in H$
18	1 6	lhpbase	$⊢ W \in H \to W \in B$
19	17 18	syl	$⊢ ((K \in HL \land W \in H) \land ((Q \in A \land \neg Q \underset{˙}{\leq} W) \land (R \in A \land \neg R \underset{˙}{\leq} W) \land X \in B) \land Q \underset{˙}{\lor} (X \underset{˙}{\land} W) = R \underset{˙}{\lor} (X \underset{˙}{\land} W)) \to W \in B$
20	1 4	latmcl	$⊢ (K \in Lat \land X \in B \land W \in B) \to X \underset{˙}{\land} W \in B$
21	12 16 19 20	syl3anc	$⊢ ((K \in HL \land W \in H) \land ((Q \in A \land \neg Q \underset{˙}{\leq} W) \land (R \in A \land \neg R \underset{˙}{\leq} W) \land X \in B) \land Q \underset{˙}{\lor} (X \underset{˙}{\land} W) = R \underset{˙}{\lor} (X \underset{˙}{\land} W)) \to X \underset{˙}{\land} W \in B$
22	1 2 3	latlej1	$⊢ (K \in Lat \land Q \in B \land X \underset{˙}{\land} W \in B) \to Q \underset{˙}{\leq} (Q \underset{˙}{\lor} (X \underset{˙}{\land} W))$
23	12 15 21 22	syl3anc	$⊢ ((K \in HL \land W \in H) \land ((Q \in A \land \neg Q \underset{˙}{\leq} W) \land (R \in A \land \neg R \underset{˙}{\leq} W) \land X \in B) \land Q \underset{˙}{\lor} (X \underset{˙}{\land} W) = R \underset{˙}{\lor} (X \underset{˙}{\land} W)) \to Q \underset{˙}{\leq} (Q \underset{˙}{\lor} (X \underset{˙}{\land} W))$
24		simp3	$⊢ ((K \in HL \land W \in H) \land ((Q \in A \land \neg Q \underset{˙}{\leq} W) \land (R \in A \land \neg R \underset{˙}{\leq} W) \land X \in B) \land Q \underset{˙}{\lor} (X \underset{˙}{\land} W) = R \underset{˙}{\lor} (X \underset{˙}{\land} W)) \to Q \underset{˙}{\lor} (X \underset{˙}{\land} W) = R \underset{˙}{\lor} (X \underset{˙}{\land} W)$
25	23 24	breqtrd	$⊢ ((K \in HL \land W \in H) \land ((Q \in A \land \neg Q \underset{˙}{\leq} W) \land (R \in A \land \neg R \underset{˙}{\leq} W) \land X \in B) \land Q \underset{˙}{\lor} (X \underset{˙}{\land} W) = R \underset{˙}{\lor} (X \underset{˙}{\land} W)) \to Q \underset{˙}{\leq} (R \underset{˙}{\lor} (X \underset{˙}{\land} W))$
26		simp1	$⊢ ((K \in HL \land W \in H) \land ((Q \in A \land \neg Q \underset{˙}{\leq} W) \land (R \in A \land \neg R \underset{˙}{\leq} W) \land X \in B) \land Q \underset{˙}{\lor} (X \underset{˙}{\land} W) = R \underset{˙}{\lor} (X \underset{˙}{\land} W)) \to (K \in HL \land W \in H)$
27		simp22	$⊢ ((K \in HL \land W \in H) \land ((Q \in A \land \neg Q \underset{˙}{\leq} W) \land (R \in A \land \neg R \underset{˙}{\leq} W) \land X \in B) \land Q \underset{˙}{\lor} (X \underset{˙}{\land} W) = R \underset{˙}{\lor} (X \underset{˙}{\land} W)) \to (R \in A \land \neg R \underset{˙}{\leq} W)$
28		simp21	$⊢ ((K \in HL \land W \in H) \land ((Q \in A \land \neg Q \underset{˙}{\leq} W) \land (R \in A \land \neg R \underset{˙}{\leq} W) \land X \in B) \land Q \underset{˙}{\lor} (X \underset{˙}{\land} W) = R \underset{˙}{\lor} (X \underset{˙}{\land} W)) \to (Q \in A \land \neg Q \underset{˙}{\leq} W)$
29	1 2 4	latmle2	$⊢ (K \in Lat \land X \in B \land W \in B) \to (X \underset{˙}{\land} W) \underset{˙}{\leq} W$
30	12 16 19 29	syl3anc	$⊢ ((K \in HL \land W \in H) \land ((Q \in A \land \neg Q \underset{˙}{\leq} W) \land (R \in A \land \neg R \underset{˙}{\leq} W) \land X \in B) \land Q \underset{˙}{\lor} (X \underset{˙}{\land} W) = R \underset{˙}{\lor} (X \underset{˙}{\land} W)) \to (X \underset{˙}{\land} W) \underset{˙}{\leq} W$
31	21 30	jca	$⊢ ((K \in HL \land W \in H) \land ((Q \in A \land \neg Q \underset{˙}{\leq} W) \land (R \in A \land \neg R \underset{˙}{\leq} W) \land X \in B) \land Q \underset{˙}{\lor} (X \underset{˙}{\land} W) = R \underset{˙}{\lor} (X \underset{˙}{\land} W)) \to (X \underset{˙}{\land} W \in B \land (X \underset{˙}{\land} W) \underset{˙}{\leq} W)$
32	1 2 3 5 6 7 8 9 10	cdlemn	$⊢ ((K \in HL \land W \in H) \land ((R \in A \land \neg R \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land (X \underset{˙}{\land} W \in B \land (X \underset{˙}{\land} W) \underset{˙}{\leq} W))) \to (Q \underset{˙}{\leq} (R \underset{˙}{\lor} (X \underset{˙}{\land} W)) \leftrightarrow J (Q) \subseteq J (R) \underset{˙}{\oplus} I (X \underset{˙}{\land} W))$
33	26 27 28 31 32	syl13anc	$⊢ ((K \in HL \land W \in H) \land ((Q \in A \land \neg Q \underset{˙}{\leq} W) \land (R \in A \land \neg R \underset{˙}{\leq} W) \land X \in B) \land Q \underset{˙}{\lor} (X \underset{˙}{\land} W) = R \underset{˙}{\lor} (X \underset{˙}{\land} W)) \to (Q \underset{˙}{\leq} (R \underset{˙}{\lor} (X \underset{˙}{\land} W)) \leftrightarrow J (Q) \subseteq J (R) \underset{˙}{\oplus} I (X \underset{˙}{\land} W))$
34	25 33	mpbid	$⊢ ((K \in HL \land W \in H) \land ((Q \in A \land \neg Q \underset{˙}{\leq} W) \land (R \in A \land \neg R \underset{˙}{\leq} W) \land X \in B) \land Q \underset{˙}{\lor} (X \underset{˙}{\land} W) = R \underset{˙}{\lor} (X \underset{˙}{\land} W)) \to J (Q) \subseteq J (R) \underset{˙}{\oplus} I (X \underset{˙}{\land} W)$
35	6 9 26	dvhlmod	$⊢ ((K \in HL \land W \in H) \land ((Q \in A \land \neg Q \underset{˙}{\leq} W) \land (R \in A \land \neg R \underset{˙}{\leq} W) \land X \in B) \land Q \underset{˙}{\lor} (X \underset{˙}{\land} W) = R \underset{˙}{\lor} (X \underset{˙}{\land} W)) \to U \in LMod$
36		eqid	$⊢ LSubSp (U) = LSubSp (U)$
37	36	lsssssubg	$⊢ U \in LMod \to LSubSp (U) \subseteq SubGrp (U)$
38	35 37	syl	$⊢ ((K \in HL \land W \in H) \land ((Q \in A \land \neg Q \underset{˙}{\leq} W) \land (R \in A \land \neg R \underset{˙}{\leq} W) \land X \in B) \land Q \underset{˙}{\lor} (X \underset{˙}{\land} W) = R \underset{˙}{\lor} (X \underset{˙}{\land} W)) \to LSubSp (U) \subseteq SubGrp (U)$
39	2 5 6 9 8 36	diclss	$⊢ ((K \in HL \land W \in H) \land (R \in A \land \neg R \underset{˙}{\leq} W)) \to J (R) \in LSubSp (U)$
40	26 27 39	syl2anc	$⊢ ((K \in HL \land W \in H) \land ((Q \in A \land \neg Q \underset{˙}{\leq} W) \land (R \in A \land \neg R \underset{˙}{\leq} W) \land X \in B) \land Q \underset{˙}{\lor} (X \underset{˙}{\land} W) = R \underset{˙}{\lor} (X \underset{˙}{\land} W)) \to J (R) \in LSubSp (U)$
41	38 40	sseldd	$⊢ ((K \in HL \land W \in H) \land ((Q \in A \land \neg Q \underset{˙}{\leq} W) \land (R \in A \land \neg R \underset{˙}{\leq} W) \land X \in B) \land Q \underset{˙}{\lor} (X \underset{˙}{\land} W) = R \underset{˙}{\lor} (X \underset{˙}{\land} W)) \to J (R) \in SubGrp (U)$
42	1 2 6 9 7 36	diblss	$⊢ ((K \in HL \land W \in H) \land (X \underset{˙}{\land} W \in B \land (X \underset{˙}{\land} W) \underset{˙}{\leq} W)) \to I (X \underset{˙}{\land} W) \in LSubSp (U)$
43	26 21 30 42	syl12anc	$⊢ ((K \in HL \land W \in H) \land ((Q \in A \land \neg Q \underset{˙}{\leq} W) \land (R \in A \land \neg R \underset{˙}{\leq} W) \land X \in B) \land Q \underset{˙}{\lor} (X \underset{˙}{\land} W) = R \underset{˙}{\lor} (X \underset{˙}{\land} W)) \to I (X \underset{˙}{\land} W) \in LSubSp (U)$
44	38 43	sseldd	$⊢ ((K \in HL \land W \in H) \land ((Q \in A \land \neg Q \underset{˙}{\leq} W) \land (R \in A \land \neg R \underset{˙}{\leq} W) \land X \in B) \land Q \underset{˙}{\lor} (X \underset{˙}{\land} W) = R \underset{˙}{\lor} (X \underset{˙}{\land} W)) \to I (X \underset{˙}{\land} W) \in SubGrp (U)$
45	10	lsmub2	$⊢ (J (R) \in SubGrp (U) \land I (X \underset{˙}{\land} W) \in SubGrp (U)) \to I (X \underset{˙}{\land} W) \subseteq J (R) \underset{˙}{\oplus} I (X \underset{˙}{\land} W)$
46	41 44 45	syl2anc	$⊢ ((K \in HL \land W \in H) \land ((Q \in A \land \neg Q \underset{˙}{\leq} W) \land (R \in A \land \neg R \underset{˙}{\leq} W) \land X \in B) \land Q \underset{˙}{\lor} (X \underset{˙}{\land} W) = R \underset{˙}{\lor} (X \underset{˙}{\land} W)) \to I (X \underset{˙}{\land} W) \subseteq J (R) \underset{˙}{\oplus} I (X \underset{˙}{\land} W)$
47	2 5 6 9 8 36	diclss	$⊢ ((K \in HL \land W \in H) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \to J (Q) \in LSubSp (U)$
48	26 28 47	syl2anc	$⊢ ((K \in HL \land W \in H) \land ((Q \in A \land \neg Q \underset{˙}{\leq} W) \land (R \in A \land \neg R \underset{˙}{\leq} W) \land X \in B) \land Q \underset{˙}{\lor} (X \underset{˙}{\land} W) = R \underset{˙}{\lor} (X \underset{˙}{\land} W)) \to J (Q) \in LSubSp (U)$
49	38 48	sseldd	$⊢ ((K \in HL \land W \in H) \land ((Q \in A \land \neg Q \underset{˙}{\leq} W) \land (R \in A \land \neg R \underset{˙}{\leq} W) \land X \in B) \land Q \underset{˙}{\lor} (X \underset{˙}{\land} W) = R \underset{˙}{\lor} (X \underset{˙}{\land} W)) \to J (Q) \in SubGrp (U)$
50	36 10	lsmcl	$⊢ (U \in LMod \land J (R) \in LSubSp (U) \land I (X \underset{˙}{\land} W) \in LSubSp (U)) \to J (R) \underset{˙}{\oplus} I (X \underset{˙}{\land} W) \in LSubSp (U)$
51	35 40 43 50	syl3anc	$⊢ ((K \in HL \land W \in H) \land ((Q \in A \land \neg Q \underset{˙}{\leq} W) \land (R \in A \land \neg R \underset{˙}{\leq} W) \land X \in B) \land Q \underset{˙}{\lor} (X \underset{˙}{\land} W) = R \underset{˙}{\lor} (X \underset{˙}{\land} W)) \to J (R) \underset{˙}{\oplus} I (X \underset{˙}{\land} W) \in LSubSp (U)$
52	38 51	sseldd	$⊢ ((K \in HL \land W \in H) \land ((Q \in A \land \neg Q \underset{˙}{\leq} W) \land (R \in A \land \neg R \underset{˙}{\leq} W) \land X \in B) \land Q \underset{˙}{\lor} (X \underset{˙}{\land} W) = R \underset{˙}{\lor} (X \underset{˙}{\land} W)) \to J (R) \underset{˙}{\oplus} I (X \underset{˙}{\land} W) \in SubGrp (U)$
53	10	lsmlub	$⊢ (J (Q) \in SubGrp (U) \land I (X \underset{˙}{\land} W) \in SubGrp (U) \land J (R) \underset{˙}{\oplus} I (X \underset{˙}{\land} W) \in SubGrp (U)) \to ((J (Q) \subseteq J (R) \underset{˙}{\oplus} I (X \underset{˙}{\land} W) \land I (X \underset{˙}{\land} W) \subseteq J (R) \underset{˙}{\oplus} I (X \underset{˙}{\land} W)) \leftrightarrow J (Q) \underset{˙}{\oplus} I (X \underset{˙}{\land} W) \subseteq J (R) \underset{˙}{\oplus} I (X \underset{˙}{\land} W))$
54	49 44 52 53	syl3anc	$⊢ ((K \in HL \land W \in H) \land ((Q \in A \land \neg Q \underset{˙}{\leq} W) \land (R \in A \land \neg R \underset{˙}{\leq} W) \land X \in B) \land Q \underset{˙}{\lor} (X \underset{˙}{\land} W) = R \underset{˙}{\lor} (X \underset{˙}{\land} W)) \to ((J (Q) \subseteq J (R) \underset{˙}{\oplus} I (X \underset{˙}{\land} W) \land I (X \underset{˙}{\land} W) \subseteq J (R) \underset{˙}{\oplus} I (X \underset{˙}{\land} W)) \leftrightarrow J (Q) \underset{˙}{\oplus} I (X \underset{˙}{\land} W) \subseteq J (R) \underset{˙}{\oplus} I (X \underset{˙}{\land} W))$
55	34 46 54	mpbi2and	$⊢ ((K \in HL \land W \in H) \land ((Q \in A \land \neg Q \underset{˙}{\leq} W) \land (R \in A \land \neg R \underset{˙}{\leq} W) \land X \in B) \land Q \underset{˙}{\lor} (X \underset{˙}{\land} W) = R \underset{˙}{\lor} (X \underset{˙}{\land} W)) \to J (Q) \underset{˙}{\oplus} I (X \underset{˙}{\land} W) \subseteq J (R) \underset{˙}{\oplus} I (X \underset{˙}{\land} W)$

Metamath Proof Explorer

Theorem dihjustlem

Proof