Theorem diael

Description: A member of the value of the partial isomorphism A is a translation, i.e., a vector. (Contributed by NM, 17-Jan-2014)

	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	diael	$$⊢\left(\left(K\in V\wedge W\in H\right)\wedge \left(X\in B\wedge X\underset{\dot{}}{\le }W\right)\wedge F\in I\left(X\right)\right)\to F\in T$$

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	1 2 3 4 5	diass	$$⊢\left(\left(K\in V\wedge W\in H\right)\wedge \left(X\in B\wedge X\underset{\dot{}}{\le }W\right)\right)\to I\left(X\right)\subseteq T$$
7	6	sseld	$$⊢\left(\left(K\in V\wedge W\in H\right)\wedge \left(X\in B\wedge X\underset{\dot{}}{\le }W\right)\right)\to \left(F\in I\left(X\right)\to F\in T\right)$$
8	7	3impia	$$⊢\left(\left(K\in V\wedge W\in H\right)\wedge \left(X\in B\wedge X\underset{\dot{}}{\le }W\right)\wedge F\in I\left(X\right)\right)\to F\in T$$