Theorem eringring

Description: An endomorphism ring is a ring. TODO: fix comment. (Contributed by NM, 4-Aug-2013)

	Ref	Expression
Hypotheses	ernggrp.h	|- H = ( LHyp ` K )
	ernggrp.d	|- D = ( ( EDRing ` K ) ` W )
Assertion	eringring	|- ( ( K e. HL /\ W e. H ) -> D e. Ring )

Proof

Step	Hyp	Ref	Expression
1		ernggrp.h	|- H = ( LHyp ` K )
2		ernggrp.d	|- D = ( ( EDRing ` K ) ` W )
3		eqid	|- ( Base ` K ) = ( Base ` K )
4		eqid	|- ( ( LTrn ` K ) ` W ) = ( ( LTrn ` K ) ` W )
5		eqid	|- ( ( TEndo ` K ) ` W ) = ( ( TEndo ` K ) ` W )
6		eqid	|- ( a e. ( ( TEndo ` K ) ` W ) , b e. ( ( TEndo ` K ) ` W ) |-> ( f e. ( ( LTrn ` K ) ` W ) |-> ( ( a ` f ) o. ( b ` f ) ) ) ) = ( a e. ( ( TEndo ` K ) ` W ) , b e. ( ( TEndo ` K ) ` W ) |-> ( f e. ( ( LTrn ` K ) ` W ) |-> ( ( a ` f ) o. ( b ` f ) ) ) )
7		eqid	|- ( f e. ( ( LTrn ` K ) ` W ) |-> ( _I |` ( Base ` K ) ) ) = ( f e. ( ( LTrn ` K ) ` W ) |-> ( _I |` ( Base ` K ) ) )
8		eqid	|- ( a e. ( ( TEndo ` K ) ` W ) |-> ( f e. ( ( LTrn ` K ) ` W ) |-> `' ( a ` f ) ) ) = ( a e. ( ( TEndo ` K ) ` W ) |-> ( f e. ( ( LTrn ` K ) ` W ) |-> `' ( a ` f ) ) )
9		eqid	|- ( a e. ( ( TEndo ` K ) ` W ) , b e. ( ( TEndo ` K ) ` W ) |-> ( a o. b ) ) = ( a e. ( ( TEndo ` K ) ` W ) , b e. ( ( TEndo ` K ) ` W ) |-> ( a o. b ) )
10	1 2 3 4 5 6 7 8 9	erngdvlem3	|- ( ( K e. HL /\ W e. H ) -> D e. Ring )