dihjatc2N

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

	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	dihjatc2N	$⊢ (((K \in HL \land W \in H) \land X \in B \land Y \in B) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land (Q \underset{˙}{\leq} X \land (X \underset{˙}{\land} Y) \underset{˙}{\leq} W)) \to I (Q \underset{˙}{\lor} (X \underset{˙}{\land} Y)) = I (Q) \underset{˙}{\oplus} I (X \underset{˙}{\land} Y)$

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		simp11l	$⊢ (((K \in HL \land W \in H) \land X \in B \land Y \in B) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land (Q \underset{˙}{\leq} X \land (X \underset{˙}{\land} Y) \underset{˙}{\leq} W)) \to K \in HL$
11	10	hllatd	$⊢ (((K \in HL \land W \in H) \land X \in B \land Y \in B) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land (Q \underset{˙}{\leq} X \land (X \underset{˙}{\land} Y) \underset{˙}{\leq} W)) \to K \in Lat$
12		simp2l	$⊢ (((K \in HL \land W \in H) \land X \in B \land Y \in B) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land (Q \underset{˙}{\leq} X \land (X \underset{˙}{\land} Y) \underset{˙}{\leq} W)) \to Q \in A$
13	1 6	atbase	$⊢ Q \in A \to Q \in B$
14	12 13	syl	$⊢ (((K \in HL \land W \in H) \land X \in B \land Y \in B) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land (Q \underset{˙}{\leq} X \land (X \underset{˙}{\land} Y) \underset{˙}{\leq} W)) \to Q \in B$
15		simp12	$⊢ (((K \in HL \land W \in H) \land X \in B \land Y \in B) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land (Q \underset{˙}{\leq} X \land (X \underset{˙}{\land} Y) \underset{˙}{\leq} W)) \to X \in B$
16		simp13	$⊢ (((K \in HL \land W \in H) \land X \in B \land Y \in B) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land (Q \underset{˙}{\leq} X \land (X \underset{˙}{\land} Y) \underset{˙}{\leq} W)) \to Y \in B$
17	1 5	latmcl	$⊢ (K \in Lat \land X \in B \land Y \in B) \to X \underset{˙}{\land} Y \in B$
18	11 15 16 17	syl3anc	$⊢ (((K \in HL \land W \in H) \land X \in B \land Y \in B) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land (Q \underset{˙}{\leq} X \land (X \underset{˙}{\land} Y) \underset{˙}{\leq} W)) \to X \underset{˙}{\land} Y \in B$
19	1 4	latjcom	$⊢ (K \in Lat \land Q \in B \land X \underset{˙}{\land} Y \in B) \to Q \underset{˙}{\lor} (X \underset{˙}{\land} Y) = (X \underset{˙}{\land} Y) \underset{˙}{\lor} Q$
20	11 14 18 19	syl3anc	$⊢ (((K \in HL \land W \in H) \land X \in B \land Y \in B) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land (Q \underset{˙}{\leq} X \land (X \underset{˙}{\land} Y) \underset{˙}{\leq} W)) \to Q \underset{˙}{\lor} (X \underset{˙}{\land} Y) = (X \underset{˙}{\land} Y) \underset{˙}{\lor} Q$
21	20	fveq2d	$⊢ (((K \in HL \land W \in H) \land X \in B \land Y \in B) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land (Q \underset{˙}{\leq} X \land (X \underset{˙}{\land} Y) \underset{˙}{\leq} W)) \to I (Q \underset{˙}{\lor} (X \underset{˙}{\land} Y)) = I ((X \underset{˙}{\land} Y) \underset{˙}{\lor} Q)$
22	1 2 3 4 5 6 7 8 9	dihjatc1	$⊢ (((K \in HL \land W \in H) \land X \in B \land Y \in B) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land (Q \underset{˙}{\leq} X \land (X \underset{˙}{\land} Y) \underset{˙}{\leq} W)) \to I ((X \underset{˙}{\land} Y) \underset{˙}{\lor} Q) = I (Q) \underset{˙}{\oplus} I (X \underset{˙}{\land} Y)$
23	21 22	eqtrd	$⊢ (((K \in HL \land W \in H) \land X \in B \land Y \in B) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land (Q \underset{˙}{\leq} X \land (X \underset{˙}{\land} Y) \underset{˙}{\leq} W)) \to I (Q \underset{˙}{\lor} (X \underset{˙}{\land} Y)) = I (Q) \underset{˙}{\oplus} I (X \underset{˙}{\land} Y)$

Metamath Proof Explorer

Theorem dihjatc2N

Proof