Theorem dibval3N

Description: Value of the partial isomorphism B for a lattice K . (Contributed by NM, 24-Feb-2014) (New usage is discouraged.)

	Ref	Expression
Hypotheses	dibval3.b	$$⊢B={\mathrm{Base}}_K$$
	dibval3.l	$$⊢\underset{\dot{}}{\le }={\le }_K$$
	dibval3.h	$$⊢H=\mathrm{LHyp}\left(K\right)$$
	dibval3.t	$$⊢T=\mathrm{LTrn}\left(K\right)\left(W\right)$$
	dibval3.r	$$⊢R=\mathrm{trL}\left(K\right)\left(W\right)$$
	dibval3.o	$$⊢\underset{\dot{}}{0}=\left(g\in T⟼{\mathrm{I}\upharpoonright }_B\right)$$
	dibval3.i	$$⊢I=\mathrm{DIsoB}\left(K\right)\left(W\right)$$
Assertion	dibval3N	$$⊢\left(\left(K\in V\wedge W\in H\right)\wedge \left(X\in B\wedge X\underset{\dot{}}{\le }W\right)\right)\to I\left(X\right)=\left\{f\in T|R\left(f\right)\underset{\dot{}}{\le }X\right\}\times \left\{\underset{\dot{}}{0}\right\}$$

Proof

Step	Hyp	Ref	Expression
1		dibval3.b	$$⊢B={\mathrm{Base}}_K$$
2		dibval3.l	$$⊢\underset{\dot{}}{\le }={\le }_K$$
3		dibval3.h	$$⊢H=\mathrm{LHyp}\left(K\right)$$
4		dibval3.t	$$⊢T=\mathrm{LTrn}\left(K\right)\left(W\right)$$
5		dibval3.r	$$⊢R=\mathrm{trL}\left(K\right)\left(W\right)$$
6		dibval3.o	$$⊢\underset{\dot{}}{0}=\left(g\in T⟼{\mathrm{I}\upharpoonright }_B\right)$$
7		dibval3.i	$$⊢I=\mathrm{DIsoB}\left(K\right)\left(W\right)$$
8		eqid	$$⊢\mathrm{DIsoA}\left(K\right)\left(W\right)=\mathrm{DIsoA}\left(K\right)\left(W\right)$$
9	1 2 3 4 6 8 7	dibval2	$$⊢\left(\left(K\in V\wedge W\in H\right)\wedge \left(X\in B\wedge X\underset{\dot{}}{\le }W\right)\right)\to I\left(X\right)=\mathrm{DIsoA}\left(K\right)\left(W\right)\left(X\right)\times \left\{\underset{\dot{}}{0}\right\}$$
10	1 2 3 4 5 8	diaval	$$⊢\left(\left(K\in V\wedge W\in H\right)\wedge \left(X\in B\wedge X\underset{\dot{}}{\le }W\right)\right)\to \mathrm{DIsoA}\left(K\right)\left(W\right)\left(X\right)=\left\{f\in T|R\left(f\right)\underset{\dot{}}{\le }X\right\}$$
11	10	xpeq1d	$$⊢\left(\left(K\in V\wedge W\in H\right)\wedge \left(X\in B\wedge X\underset{\dot{}}{\le }W\right)\right)\to \mathrm{DIsoA}\left(K\right)\left(W\right)\left(X\right)\times \left\{\underset{\dot{}}{0}\right\}=\left\{f\in T|R\left(f\right)\underset{\dot{}}{\le }X\right\}\times \left\{\underset{\dot{}}{0}\right\}$$
12	9 11	eqtrd	$$⊢\left(\left(K\in V\wedge W\in H\right)\wedge \left(X\in B\wedge X\underset{\dot{}}{\le }W\right)\right)\to I\left(X\right)=\left\{f\in T|R\left(f\right)\underset{\dot{}}{\le }X\right\}\times \left\{\underset{\dot{}}{0}\right\}$$