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Metamath Proof Explorer

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 biimtrid 
 |-  ( ( 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 )