Theorem diaelrnN

Description: Any value of the partial isomorphism A is a set of translations i.e. a set of vectors. (Contributed by NM, 26-Nov-2013) (New usage is discouraged.)

	Ref	Expression
Hypotheses	diaelrn.h	|- H = ( LHyp ` K )
	diaelrn.t	|- T = ( ( LTrn ` K ) ` W )
	diaelrn.i	|- I = ( ( DIsoA ` K ) ` W )
Assertion	diaelrnN	|- ( ( ( K e. V /\ W e. H ) /\ S e. ran I ) -> S C_ T )

Proof

Step	Hyp	Ref	Expression
1		diaelrn.h	|- H = ( LHyp ` K )
2		diaelrn.t	|- T = ( ( LTrn ` K ) ` W )
3		diaelrn.i	|- I = ( ( DIsoA ` K ) ` W )
4		eqid	|- ( Base ` K ) = ( Base ` K )
5		eqid	|- ( le ` K ) = ( le ` K )
6	4 5 1 3	diafn	|- ( ( K e. V /\ W e. H ) -> I Fn { y e. ( Base ` K ) | y ( le ` K ) W } )
7		fvelrnb	|- ( I Fn { y e. ( Base ` K ) | y ( le ` K ) W } -> ( S e. ran I <-> E. x e. { y e. ( Base ` K ) | y ( le ` K ) W } ( I ` x ) = S ) )
8	6 7	syl	|- ( ( K e. V /\ W e. H ) -> ( S e. ran I <-> E. x e. { y e. ( Base ` K ) | y ( le ` K ) W } ( I ` x ) = S ) )
9		breq1	|- ( y = x -> ( y ( le ` K ) W <-> x ( le ` K ) W ) )
10	9	elrab	|- ( x e. { y e. ( Base ` K ) | y ( le ` K ) W } <-> ( x e. ( Base ` K ) /\ x ( le ` K ) W ) )
11	4 5 1 2 3	diass	|- ( ( ( K e. V /\ W e. H ) /\ ( x e. ( Base ` K ) /\ x ( le ` K ) W ) ) -> ( I ` x ) C_ T )
12	11	ex	|- ( ( K e. V /\ W e. H ) -> ( ( x e. ( Base ` K ) /\ x ( le ` K ) W ) -> ( I ` x ) C_ T ) )
13		sseq1	|- ( ( I ` x ) = S -> ( ( I ` x ) C_ T <-> S C_ T ) )
14	13	biimpcd	|- ( ( I ` x ) C_ T -> ( ( I ` x ) = S -> S C_ T ) )
15	12 14	syl6	|- ( ( K e. V /\ W e. H ) -> ( ( x e. ( Base ` K ) /\ x ( le ` K ) W ) -> ( ( I ` x ) = S -> S C_ T ) ) )
16	10 15	syl5bi	|- ( ( K e. V /\ W e. H ) -> ( x e. { y e. ( Base ` K ) | y ( le ` K ) W } -> ( ( I ` x ) = S -> S C_ T ) ) )
17	16	rexlimdv	|- ( ( K e. V /\ W e. H ) -> ( E. x e. { y e. ( Base ` K ) | y ( le ` K ) W } ( I ` x ) = S -> S C_ T ) )
18	8 17	sylbid	|- ( ( K e. V /\ W e. H ) -> ( S e. ran I -> S C_ T ) )
19	18	imp	|- ( ( ( K e. V /\ W e. H ) /\ S e. ran I ) -> S C_ T )