Theorem diass

Description: The value of the partial isomorphism A is a set of translations, i.e., a set of vectors. (Contributed by NM, 26-Nov-2013)

	Ref	Expression
Hypotheses	diass.b	$$⊢B={\mathrm{Base}}_K$$
	diass.l	$$⊢\underset{\dot{}}{\le }={\le }_K$$
	diass.h	$$⊢H=\mathrm{LHyp}\left(K\right)$$
	diass.t	$$⊢T=\mathrm{LTrn}\left(K\right)\left(W\right)$$
	diass.i	$$⊢I=\mathrm{DIsoA}\left(K\right)\left(W\right)$$
Assertion	diass	$$⊢\left(\left(K\in V\wedge W\in H\right)\wedge \left(X\in B\wedge X\underset{\dot{}}{\le }W\right)\right)\to I\left(X\right)\subseteq T$$

Proof

Step	Hyp	Ref	Expression
1		diass.b	$$⊢B={\mathrm{Base}}_K$$
2		diass.l	$$⊢\underset{\dot{}}{\le }={\le }_K$$
3		diass.h	$$⊢H=\mathrm{LHyp}\left(K\right)$$
4		diass.t	$$⊢T=\mathrm{LTrn}\left(K\right)\left(W\right)$$
5		diass.i	$$⊢I=\mathrm{DIsoA}\left(K\right)\left(W\right)$$
6		eqid	$$⊢\mathrm{trL}\left(K\right)\left(W\right)=\mathrm{trL}\left(K\right)\left(W\right)$$
7	1 2 3 4 6 5	diaval	$$⊢\left(\left(K\in V\wedge W\in H\right)\wedge \left(X\in B\wedge X\underset{\dot{}}{\le }W\right)\right)\to I\left(X\right)=\left\{f\in T|\mathrm{trL}\left(K\right)\left(W\right)\left(f\right)\underset{\dot{}}{\le }X\right\}$$
8		ssrab2	$$⊢\left\{f\in T|\mathrm{trL}\left(K\right)\left(W\right)\left(f\right)\underset{\dot{}}{\le }X\right\}\subseteq T$$
9	7 8	eqsstrdi	$$⊢\left(\left(K\in V\wedge W\in H\right)\wedge \left(X\in B\wedge X\underset{\dot{}}{\le }W\right)\right)\to I\left(X\right)\subseteq T$$