Theorem dihmeetlem4N

Description: Lemma for isomorphism H of a lattice meet. (Contributed by NM, 30-Mar-2014) (New usage is discouraged.)

	Ref	Expression
Hypotheses	dihmeetlem4.b	$$⊢B={\mathrm{Base}}_K$$
	dihmeetlem4.l	$$⊢\underset{\dot{}}{\le }={\le }_K$$
	dihmeetlem4.m	$$⊢\underset{\dot{}}{\wedge }=\mathrm{meet}\left(K\right)$$
	dihmeetlem4.a	$$⊢A=\mathrm{Atoms}\left(K\right)$$
	dihmeetlem4.h	$$⊢H=\mathrm{LHyp}\left(K\right)$$
	dihmeetlem4.i	$$⊢I=\mathrm{DIsoH}\left(K\right)\left(W\right)$$
	dihmeetlem4.u	$$⊢U=\mathrm{DVecH}\left(K\right)\left(W\right)$$
	dihmeetlem4.z	$$⊢\underset{\dot{}}{0}=0_U$$
Assertion	dihmeetlem4N	$$⊢\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(X\in B\wedge \neg X\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\right)\to I\left(Q\right)\cap I\left(X\underset{\dot{}}{\wedge }W\right)=\left\{\underset{\dot{}}{0}\right\}$$

Proof

Step	Hyp	Ref	Expression
1		dihmeetlem4.b	$$⊢B={\mathrm{Base}}_K$$
2		dihmeetlem4.l	$$⊢\underset{\dot{}}{\le }={\le }_K$$
3		dihmeetlem4.m	$$⊢\underset{\dot{}}{\wedge }=\mathrm{meet}\left(K\right)$$
4		dihmeetlem4.a	$$⊢A=\mathrm{Atoms}\left(K\right)$$
5		dihmeetlem4.h	$$⊢H=\mathrm{LHyp}\left(K\right)$$
6		dihmeetlem4.i	$$⊢I=\mathrm{DIsoH}\left(K\right)\left(W\right)$$
7		dihmeetlem4.u	$$⊢U=\mathrm{DVecH}\left(K\right)\left(W\right)$$
8		dihmeetlem4.z	$$⊢\underset{\dot{}}{0}=0_U$$
9		eqid	$$⊢\left(\iota g\in \mathrm{LTrn}\left(K\right)\left(W\right)|g\left(\mathrm{oc}\left(K\right)\left(W\right)\right)=Q\right)=\left(\iota g\in \mathrm{LTrn}\left(K\right)\left(W\right)|g\left(\mathrm{oc}\left(K\right)\left(W\right)\right)=Q\right)$$
10		eqid	$$⊢\mathrm{oc}\left(K\right)\left(W\right)=\mathrm{oc}\left(K\right)\left(W\right)$$
11		eqid	$$⊢\mathrm{LTrn}\left(K\right)\left(W\right)=\mathrm{LTrn}\left(K\right)\left(W\right)$$
12		eqid	$$⊢\mathrm{trL}\left(K\right)\left(W\right)=\mathrm{trL}\left(K\right)\left(W\right)$$
13		eqid	$$⊢\mathrm{TEndo}\left(K\right)\left(W\right)=\mathrm{TEndo}\left(K\right)\left(W\right)$$
14		eqid	$$⊢\left(h\in \mathrm{LTrn}\left(K\right)\left(W\right)⟼{\mathrm{I}\upharpoonright }_B\right)=\left(h\in \mathrm{LTrn}\left(K\right)\left(W\right)⟼{\mathrm{I}\upharpoonright }_B\right)$$
15	1 2 3 4 5 6 7 8 9 10 11 12 13 14	dihmeetlem4preN	$$⊢\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(X\in B\wedge \neg X\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\right)\to I\left(Q\right)\cap I\left(X\underset{\dot{}}{\wedge }W\right)=\left\{\underset{\dot{}}{0}\right\}$$