Theorem dihfn

Description: Functionality and domain of isomorphism H. (Contributed by NM, 9-Mar-2014)

	Ref	Expression
Hypotheses	dihfn.b	$$⊢B={\mathrm{Base}}_K$$
	dihfn.h	$$⊢H=\mathrm{LHyp}\left(K\right)$$
	dihfn.i	$$⊢I=\mathrm{DIsoH}\left(K\right)\left(W\right)$$
Assertion	dihfn	$$⊢\left(K\in \mathrm{HL}\wedge W\in H\right)\to IFnB$$

Proof

Step	Hyp	Ref	Expression
1		dihfn.b	$$⊢B={\mathrm{Base}}_K$$
2		dihfn.h	$$⊢H=\mathrm{LHyp}\left(K\right)$$
3		dihfn.i	$$⊢I=\mathrm{DIsoH}\left(K\right)\left(W\right)$$
4		eqid	$$⊢\mathrm{DVecH}\left(K\right)\left(W\right)=\mathrm{DVecH}\left(K\right)\left(W\right)$$
5		eqid	$$⊢\mathrm{LSubSp}\left(\mathrm{DVecH}\left(K\right)\left(W\right)\right)=\mathrm{LSubSp}\left(\mathrm{DVecH}\left(K\right)\left(W\right)\right)$$
6	1 2 3 4 5	dihf11	$$⊢\left(K\in \mathrm{HL}\wedge W\in H\right)\to I:B\underset{1-1}{⟶}\mathrm{LSubSp}\left(\mathrm{DVecH}\left(K\right)\left(W\right)\right)$$
7		f1fn	$$⊢I:B\underset{1-1}{⟶}\mathrm{LSubSp}\left(\mathrm{DVecH}\left(K\right)\left(W\right)\right)\to IFnB$$
8	6 7	syl	$$⊢\left(K\in \mathrm{HL}\wedge W\in H\right)\to IFnB$$