Theorem dibelval2nd

Description: Membership in value of the partial isomorphism B for a lattice K . (Contributed by NM, 13-Feb-2014)

	Ref	Expression
Hypotheses	dibelval2nd.b	$$⊢B={\mathrm{Base}}_K$$
	dibelval2nd.l	$$⊢\underset{\dot{}}{\le }={\le }_K$$
	dibelval2nd.h	$$⊢H=\mathrm{LHyp}\left(K\right)$$
	dibelval2nd.t	$$⊢T=\mathrm{LTrn}\left(K\right)\left(W\right)$$
	dibelval2nd.o	$$⊢\underset{\dot{}}{0}=\left(f\in T⟼{\mathrm{I}\upharpoonright }_B\right)$$
	dibelval2nd.i	$$⊢I=\mathrm{DIsoB}\left(K\right)\left(W\right)$$
Assertion	dibelval2nd	$$⊢\left(\left(K\in V\wedge W\in H\right)\wedge \left(X\in B\wedge X\underset{\dot{}}{\le }W\right)\wedge Y\in I\left(X\right)\right)\to 2^{nd}\left(Y\right)=\underset{\dot{}}{0}$$

Proof

Step	Hyp	Ref	Expression
1		dibelval2nd.b	$$⊢B={\mathrm{Base}}_K$$
2		dibelval2nd.l	$$⊢\underset{\dot{}}{\le }={\le }_K$$
3		dibelval2nd.h	$$⊢H=\mathrm{LHyp}\left(K\right)$$
4		dibelval2nd.t	$$⊢T=\mathrm{LTrn}\left(K\right)\left(W\right)$$
5		dibelval2nd.o	$$⊢\underset{\dot{}}{0}=\left(f\in T⟼{\mathrm{I}\upharpoonright }_B\right)$$
6		dibelval2nd.i	$$⊢I=\mathrm{DIsoB}\left(K\right)\left(W\right)$$
7		eqid	$$⊢\mathrm{DIsoA}\left(K\right)\left(W\right)=\mathrm{DIsoA}\left(K\right)\left(W\right)$$
8	1 2 3 4 5 7 6	dibval2	$$⊢\left(\left(K\in V\wedge W\in H\right)\wedge \left(X\in B\wedge X\underset{\dot{}}{\le }W\right)\right)\to I\left(X\right)=\mathrm{DIsoA}\left(K\right)\left(W\right)\left(X\right)\times \left\{\underset{\dot{}}{0}\right\}$$
9	8	eleq2d	$$⊢\left(\left(K\in V\wedge W\in H\right)\wedge \left(X\in B\wedge X\underset{\dot{}}{\le }W\right)\right)\to \left(Y\in I\left(X\right)\leftrightarrow Y\in \left(\mathrm{DIsoA}\left(K\right)\left(W\right)\left(X\right)\times \left\{\underset{\dot{}}{0}\right\}\right)\right)$$
10	9	biimp3a	$$⊢\left(\left(K\in V\wedge W\in H\right)\wedge \left(X\in B\wedge X\underset{\dot{}}{\le }W\right)\wedge Y\in I\left(X\right)\right)\to Y\in \left(\mathrm{DIsoA}\left(K\right)\left(W\right)\left(X\right)\times \left\{\underset{\dot{}}{0}\right\}\right)$$
11		xp2nd	$$⊢Y\in \left(\mathrm{DIsoA}\left(K\right)\left(W\right)\left(X\right)\times \left\{\underset{\dot{}}{0}\right\}\right)\to 2^{nd}\left(Y\right)\in \left\{\underset{\dot{}}{0}\right\}$$
12		elsni	$$⊢2^{nd}\left(Y\right)\in \left\{\underset{\dot{}}{0}\right\}\to 2^{nd}\left(Y\right)=\underset{\dot{}}{0}$$
13	10 11 12	3syl	$$⊢\left(\left(K\in V\wedge W\in H\right)\wedge \left(X\in B\wedge X\underset{\dot{}}{\le }W\right)\wedge Y\in I\left(X\right)\right)\to 2^{nd}\left(Y\right)=\underset{\dot{}}{0}$$