Metamath Proof Explorer


Theorem dihcnvid1

Description: The converse isomorphism of an isomorphism. (Contributed by NM, 5-Aug-2014)

Ref Expression
Hypotheses dihcnvid1.b B=BaseK
dihcnvid1.h H=LHypK
dihcnvid1.i I=DIsoHKW
Assertion dihcnvid1 KHLWHXBI-1IX=X

Proof

Step Hyp Ref Expression
1 dihcnvid1.b B=BaseK
2 dihcnvid1.h H=LHypK
3 dihcnvid1.i I=DIsoHKW
4 eqid DVecHKW=DVecHKW
5 eqid LSubSpDVecHKW=LSubSpDVecHKW
6 1 2 3 4 5 dihf11 KHLWHI:B1-1LSubSpDVecHKW
7 f1f1orn I:B1-1LSubSpDVecHKWI:B1-1 ontoranI
8 6 7 syl KHLWHI:B1-1 ontoranI
9 f1ocnvfv1 I:B1-1 ontoranIXBI-1IX=X
10 8 9 sylan KHLWHXBI-1IX=X