Theorem dihmeetlem8N

Description: Lemma for isomorphism H of a lattice meet. TODO: shorter proof if we change .\/ order of ( X ./\ Y ) .\/ p here and down? (Contributed by NM, 6-Apr-2014) (New usage is discouraged.)

	Ref	Expression
Hypotheses	dihmeetlem8.b	$$⊢B={\mathrm{Base}}_K$$
	dihmeetlem8.l	$$⊢\underset{\dot{}}{\le }={\le }_K$$
	dihmeetlem8.h	$$⊢H=\mathrm{LHyp}\left(K\right)$$
	dihmeetlem8.j	$$⊢\underset{\dot{}}{\vee }=\mathrm{join}\left(K\right)$$
	dihmeetlem8.m	$$⊢\underset{\dot{}}{\wedge }=\mathrm{meet}\left(K\right)$$
	dihmeetlem8.a	$$⊢A=\mathrm{Atoms}\left(K\right)$$
	dihmeetlem8.u	$$⊢U=\mathrm{DVecH}\left(K\right)\left(W\right)$$
	dihmeetlem8.s	$$⊢\underset{\dot{}}{\oplus }=\mathrm{LSSum}\left(U\right)$$
	dihmeetlem8.i	$$⊢I=\mathrm{DIsoH}\left(K\right)\left(W\right)$$
Assertion	dihmeetlem8N	$$⊢\left(\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge X\in B\wedge Y\in B\right)\wedge \left(p\in A\wedge \neg p\underset{\dot{}}{\le }W\right)\wedge \left(p\underset{\dot{}}{\le }X\wedge \left(X\underset{\dot{}}{\wedge }Y\right)\underset{\dot{}}{\le }W\right)\right)\to I\left(\left(X\underset{\dot{}}{\wedge }Y\right)\underset{\dot{}}{\vee }p\right)=I\left(p\right)\underset{\dot{}}{\oplus }I\left(X\underset{\dot{}}{\wedge }Y\right)$$

Proof

Step	Hyp	Ref	Expression
1		dihmeetlem8.b	$$⊢B={\mathrm{Base}}_K$$
2		dihmeetlem8.l	$$⊢\underset{\dot{}}{\le }={\le }_K$$
3		dihmeetlem8.h	$$⊢H=\mathrm{LHyp}\left(K\right)$$
4		dihmeetlem8.j	$$⊢\underset{\dot{}}{\vee }=\mathrm{join}\left(K\right)$$
5		dihmeetlem8.m	$$⊢\underset{\dot{}}{\wedge }=\mathrm{meet}\left(K\right)$$
6		dihmeetlem8.a	$$⊢A=\mathrm{Atoms}\left(K\right)$$
7		dihmeetlem8.u	$$⊢U=\mathrm{DVecH}\left(K\right)\left(W\right)$$
8		dihmeetlem8.s	$$⊢\underset{\dot{}}{\oplus }=\mathrm{LSSum}\left(U\right)$$
9		dihmeetlem8.i	$$⊢I=\mathrm{DIsoH}\left(K\right)\left(W\right)$$
10	1 2 3 4 5 6 7 8 9	dihjatc1	$$⊢\left(\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge X\in B\wedge Y\in B\right)\wedge \left(p\in A\wedge \neg p\underset{\dot{}}{\le }W\right)\wedge \left(p\underset{\dot{}}{\le }X\wedge \left(X\underset{\dot{}}{\wedge }Y\right)\underset{\dot{}}{\le }W\right)\right)\to I\left(\left(X\underset{\dot{}}{\wedge }Y\right)\underset{\dot{}}{\vee }p\right)=I\left(p\right)\underset{\dot{}}{\oplus }I\left(X\underset{\dot{}}{\wedge }Y\right)$$