dihjatc3

Description: Isomorphism H of join with an atom. (Contributed by NM, 26-Aug-2014)

	Ref	Expression
Hypotheses	dihjatc1.b	$⊢ B = {Base}_{K}$
	dihjatc1.l	$⊢ \underset{˙}{\leq} = \leq_{K}$
	dihjatc1.h	$⊢ H = LHyp (K)$
	dihjatc1.j	$⊢ \underset{˙}{\lor} = join (K)$
	dihjatc1.m	$⊢ \underset{˙}{\land} = meet (K)$
	dihjatc1.a	$⊢ A = Atoms (K)$
	dihjatc1.u	$⊢ U = DVecH (K) (W)$
	dihjatc1.s	$⊢ \underset{˙}{\oplus} = LSSum (U)$
	dihjatc1.i	$⊢ I = DIsoH (K) (W)$
Assertion	dihjatc3	$⊢ (((K \in HL \land W \in H) \land X \in B \land Y \in B) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land (Q \underset{˙}{\leq} X \land (X \underset{˙}{\land} Y) \underset{˙}{\leq} W)) \to I ((X \underset{˙}{\land} Y) \underset{˙}{\lor} Q) = I (X \underset{˙}{\land} Y) \underset{˙}{\oplus} I (Q)$

Proof

Step	Hyp	Ref	Expression
1		dihjatc1.b	$⊢ B = {Base}_{K}$
2		dihjatc1.l	$⊢ \underset{˙}{\leq} = \leq_{K}$
3		dihjatc1.h	$⊢ H = LHyp (K)$
4		dihjatc1.j	$⊢ \underset{˙}{\lor} = join (K)$
5		dihjatc1.m	$⊢ \underset{˙}{\land} = meet (K)$
6		dihjatc1.a	$⊢ A = Atoms (K)$
7		dihjatc1.u	$⊢ U = DVecH (K) (W)$
8		dihjatc1.s	$⊢ \underset{˙}{\oplus} = LSSum (U)$
9		dihjatc1.i	$⊢ I = DIsoH (K) (W)$
10	1 2 3 4 5 6 7 8 9	dihjatc1	$⊢ (((K \in HL \land W \in H) \land X \in B \land Y \in B) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land (Q \underset{˙}{\leq} X \land (X \underset{˙}{\land} Y) \underset{˙}{\leq} W)) \to I ((X \underset{˙}{\land} Y) \underset{˙}{\lor} Q) = I (Q) \underset{˙}{\oplus} I (X \underset{˙}{\land} Y)$
11		simp11	$⊢ (((K \in HL \land W \in H) \land X \in B \land Y \in B) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land (Q \underset{˙}{\leq} X \land (X \underset{˙}{\land} Y) \underset{˙}{\leq} W)) \to (K \in HL \land W \in H)$
12	3 7 11	dvhlmod	$⊢ (((K \in HL \land W \in H) \land X \in B \land Y \in B) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land (Q \underset{˙}{\leq} X \land (X \underset{˙}{\land} Y) \underset{˙}{\leq} W)) \to U \in LMod$
13		lmodabl	$⊢ U \in LMod \to U \in Abel$
14	12 13	syl	$⊢ (((K \in HL \land W \in H) \land X \in B \land Y \in B) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land (Q \underset{˙}{\leq} X \land (X \underset{˙}{\land} Y) \underset{˙}{\leq} W)) \to U \in Abel$
15		eqid	$⊢ LSubSp (U) = LSubSp (U)$
16	15	lsssssubg	$⊢ U \in LMod \to LSubSp (U) \subseteq SubGrp (U)$
17	12 16	syl	$⊢ (((K \in HL \land W \in H) \land X \in B \land Y \in B) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land (Q \underset{˙}{\leq} X \land (X \underset{˙}{\land} Y) \underset{˙}{\leq} W)) \to LSubSp (U) \subseteq SubGrp (U)$
18		simp11l	$⊢ (((K \in HL \land W \in H) \land X \in B \land Y \in B) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land (Q \underset{˙}{\leq} X \land (X \underset{˙}{\land} Y) \underset{˙}{\leq} W)) \to K \in HL$
19	18	hllatd	$⊢ (((K \in HL \land W \in H) \land X \in B \land Y \in B) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land (Q \underset{˙}{\leq} X \land (X \underset{˙}{\land} Y) \underset{˙}{\leq} W)) \to K \in Lat$
20		simp12	$⊢ (((K \in HL \land W \in H) \land X \in B \land Y \in B) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land (Q \underset{˙}{\leq} X \land (X \underset{˙}{\land} Y) \underset{˙}{\leq} W)) \to X \in B$
21		simp13	$⊢ (((K \in HL \land W \in H) \land X \in B \land Y \in B) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land (Q \underset{˙}{\leq} X \land (X \underset{˙}{\land} Y) \underset{˙}{\leq} W)) \to Y \in B$
22	1 5	latmcl	$⊢ (K \in Lat \land X \in B \land Y \in B) \to X \underset{˙}{\land} Y \in B$
23	19 20 21 22	syl3anc	$⊢ (((K \in HL \land W \in H) \land X \in B \land Y \in B) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land (Q \underset{˙}{\leq} X \land (X \underset{˙}{\land} Y) \underset{˙}{\leq} W)) \to X \underset{˙}{\land} Y \in B$
24	1 3 9 7 15	dihlss	$⊢ ((K \in HL \land W \in H) \land X \underset{˙}{\land} Y \in B) \to I (X \underset{˙}{\land} Y) \in LSubSp (U)$
25	11 23 24	syl2anc	$⊢ (((K \in HL \land W \in H) \land X \in B \land Y \in B) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land (Q \underset{˙}{\leq} X \land (X \underset{˙}{\land} Y) \underset{˙}{\leq} W)) \to I (X \underset{˙}{\land} Y) \in LSubSp (U)$
26	17 25	sseldd	$⊢ (((K \in HL \land W \in H) \land X \in B \land Y \in B) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land (Q \underset{˙}{\leq} X \land (X \underset{˙}{\land} Y) \underset{˙}{\leq} W)) \to I (X \underset{˙}{\land} Y) \in SubGrp (U)$
27		simp2l	$⊢ (((K \in HL \land W \in H) \land X \in B \land Y \in B) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land (Q \underset{˙}{\leq} X \land (X \underset{˙}{\land} Y) \underset{˙}{\leq} W)) \to Q \in A$
28	1 6	atbase	$⊢ Q \in A \to Q \in B$
29	27 28	syl	$⊢ (((K \in HL \land W \in H) \land X \in B \land Y \in B) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land (Q \underset{˙}{\leq} X \land (X \underset{˙}{\land} Y) \underset{˙}{\leq} W)) \to Q \in B$
30	1 3 9 7 15	dihlss	$⊢ ((K \in HL \land W \in H) \land Q \in B) \to I (Q) \in LSubSp (U)$
31	11 29 30	syl2anc	$⊢ (((K \in HL \land W \in H) \land X \in B \land Y \in B) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land (Q \underset{˙}{\leq} X \land (X \underset{˙}{\land} Y) \underset{˙}{\leq} W)) \to I (Q) \in LSubSp (U)$
32	17 31	sseldd	$⊢ (((K \in HL \land W \in H) \land X \in B \land Y \in B) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land (Q \underset{˙}{\leq} X \land (X \underset{˙}{\land} Y) \underset{˙}{\leq} W)) \to I (Q) \in SubGrp (U)$
33	8	lsmcom	$⊢ (U \in Abel \land I (X \underset{˙}{\land} Y) \in SubGrp (U) \land I (Q) \in SubGrp (U)) \to I (X \underset{˙}{\land} Y) \underset{˙}{\oplus} I (Q) = I (Q) \underset{˙}{\oplus} I (X \underset{˙}{\land} Y)$
34	14 26 32 33	syl3anc	$⊢ (((K \in HL \land W \in H) \land X \in B \land Y \in B) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land (Q \underset{˙}{\leq} X \land (X \underset{˙}{\land} Y) \underset{˙}{\leq} W)) \to I (X \underset{˙}{\land} Y) \underset{˙}{\oplus} I (Q) = I (Q) \underset{˙}{\oplus} I (X \underset{˙}{\land} Y)$
35	10 34	eqtr4d	$⊢ (((K \in HL \land W \in H) \land X \in B \land Y \in B) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land (Q \underset{˙}{\leq} X \land (X \underset{˙}{\land} Y) \underset{˙}{\leq} W)) \to I ((X \underset{˙}{\land} Y) \underset{˙}{\lor} Q) = I (X \underset{˙}{\land} Y) \underset{˙}{\oplus} I (Q)$

Metamath Proof Explorer

Theorem dihjatc3

Proof