Theorem erngbase-rN

Description: The base set of the division ring on trace-preserving endomorphisms is the set of all trace-preserving endomorphisms (for a fiducial co-atom W ). (Contributed by NM, 9-Jun-2013) (New usage is discouraged.)

	Ref	Expression
Hypotheses	erngset.h-r	|- H = ( LHyp ` K )
	erngset.t-r	|- T = ( ( LTrn ` K ) ` W )
	erngset.e-r	|- E = ( ( TEndo ` K ) ` W )
	erngset.d-r	|- D = ( ( EDRingR ` K ) ` W )
	erng.c-r	|- C = ( Base ` D )
Assertion	erngbase-rN	|- ( ( K e. V /\ W e. H ) -> C = E )

Proof

Step	Hyp	Ref	Expression
1		erngset.h-r	|- H = ( LHyp ` K )
2		erngset.t-r	|- T = ( ( LTrn ` K ) ` W )
3		erngset.e-r	|- E = ( ( TEndo ` K ) ` W )
4		erngset.d-r	|- D = ( ( EDRingR ` K ) ` W )
5		erng.c-r	|- C = ( Base ` D )
6	1 2 3 4	erngset-rN	|- ( ( K e. V /\ W e. H ) -> D = { <. ( Base ` ndx ) , E >. , <. ( +g ` ndx ) , ( s e. E , t e. E |-> ( f e. T |-> ( ( s ` f ) o. ( t ` f ) ) ) ) >. , <. ( .r ` ndx ) , ( s e. E , t e. E |-> ( t o. s ) ) >. } )
7	6	fveq2d	|- ( ( K e. V /\ W e. H ) -> ( Base ` D ) = ( Base ` { <. ( Base ` ndx ) , E >. , <. ( +g ` ndx ) , ( s e. E , t e. E |-> ( f e. T |-> ( ( s ` f ) o. ( t ` f ) ) ) ) >. , <. ( .r ` ndx ) , ( s e. E , t e. E |-> ( t o. s ) ) >. } ) )
8	3	fvexi	|- E e. _V
9		eqid	|- { <. ( Base ` ndx ) , E >. , <. ( +g ` ndx ) , ( s e. E , t e. E |-> ( f e. T |-> ( ( s ` f ) o. ( t ` f ) ) ) ) >. , <. ( .r ` ndx ) , ( s e. E , t e. E |-> ( t o. s ) ) >. } = { <. ( Base ` ndx ) , E >. , <. ( +g ` ndx ) , ( s e. E , t e. E |-> ( f e. T |-> ( ( s ` f ) o. ( t ` f ) ) ) ) >. , <. ( .r ` ndx ) , ( s e. E , t e. E |-> ( t o. s ) ) >. }
10	9	rngbase	|- ( E e. _V -> E = ( Base ` { <. ( Base ` ndx ) , E >. , <. ( +g ` ndx ) , ( s e. E , t e. E |-> ( f e. T |-> ( ( s ` f ) o. ( t ` f ) ) ) ) >. , <. ( .r ` ndx ) , ( s e. E , t e. E |-> ( t o. s ) ) >. } ) )
11	8 10	ax-mp	|- E = ( Base ` { <. ( Base ` ndx ) , E >. , <. ( +g ` ndx ) , ( s e. E , t e. E |-> ( f e. T |-> ( ( s ` f ) o. ( t ` f ) ) ) ) >. , <. ( .r ` ndx ) , ( s e. E , t e. E |-> ( t o. s ) ) >. } )
12	7 5 11	3eqtr4g	|- ( ( K e. V /\ W e. H ) -> C = E )