Theorem dihcnvcl

Description: Closure of isomorphism H converse. (Contributed by NM, 8-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	dihcnvcl	$$⊢\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge X\in \mathrm{ran}I\right)\to I^{-1}\left(X\right)\in B$$

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		f1f1orn	$$⊢I:B\underset{1-1}{⟶}\mathrm{LSubSp}\left(\mathrm{DVecH}\left(K\right)\left(W\right)\right)\to I:B\underset{1-1\; onto}{⟶}\mathrm{ran}I$$
8	6 7	syl	$$⊢\left(K\in \mathrm{HL}\wedge W\in H\right)\to I:B\underset{1-1\; onto}{⟶}\mathrm{ran}I$$
9		f1ocnvdm	$$⊢\left(I:B\underset{1-1\; onto}{⟶}\mathrm{ran}I\wedge X\in \mathrm{ran}I\right)\to I^{-1}\left(X\right)\in B$$
10	8 9	sylan	$$⊢\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge X\in \mathrm{ran}I\right)\to I^{-1}\left(X\right)\in B$$