Metamath Proof Explorer

Theorem prjspnerlem

Description: A lemma showing that the equivalence relation used in prjspnval2 and the equivalence relation used in prjspval are equal, but only with the antecedent K e. DivRing . (Contributed by SN, 15-Jul-2023)

Ref Expression
Hypotheses prjspnerlem.e 
|- .~ = { <. x , y >. | ( ( x e. B /\ y e. B ) /\ E. l e. S x = ( l .x. y ) ) }
prjspnerlem.w 
|- W = ( K freeLMod ( 0 ... N ) )
prjspnerlem.b 
|- B = ( ( Base ` W ) \ { ( 0g ` W ) } )
prjspnerlem.s 
|- S = ( Base ` K )
prjspnerlem.x 
|- .x. = ( .s ` W )
Assertion prjspnerlem 
|- ( K e. DivRing -> .~ = { <. x , y >. | ( ( x e. B /\ y e. B ) /\ E. l e. ( Base ` ( Scalar ` W ) ) x = ( l .x. y ) ) } )

Proof

Step Hyp Ref Expression
1 prjspnerlem.e 
 |-  .~ = { <. x , y >. | ( ( x e. B /\ y e. B ) /\ E. l e. S x = ( l .x. y ) ) }
2 prjspnerlem.w 
 |-  W = ( K freeLMod ( 0 ... N ) )
3 prjspnerlem.b 
 |-  B = ( ( Base ` W ) \ { ( 0g ` W ) } )
4 prjspnerlem.s 
 |-  S = ( Base ` K )
5 prjspnerlem.x 
 |-  .x. = ( .s ` W )
6 ovex 
 |-  ( 0 ... N ) e. _V
7 2 frlmsca 
 |-  ( ( K e. DivRing /\ ( 0 ... N ) e. _V ) -> K = ( Scalar ` W ) )
8 6 7 mpan2 
 |-  ( K e. DivRing -> K = ( Scalar ` W ) )
9 8 fveq2d 
 |-  ( K e. DivRing -> ( Base ` K ) = ( Base ` ( Scalar ` W ) ) )
10 4 9 eqtrid 
 |-  ( K e. DivRing -> S = ( Base ` ( Scalar ` W ) ) )
11 10 rexeqdv 
 |-  ( K e. DivRing -> ( E. l e. S x = ( l .x. y ) <-> E. l e. ( Base ` ( Scalar ` W ) ) x = ( l .x. y ) ) )
12 11 anbi2d 
 |-  ( K e. DivRing -> ( ( ( x e. B /\ y e. B ) /\ E. l e. S x = ( l .x. y ) ) <-> ( ( x e. B /\ y e. B ) /\ E. l e. ( Base ` ( Scalar ` W ) ) x = ( l .x. y ) ) ) )
13 12 opabbidv 
 |-  ( K e. DivRing -> { <. x , y >. | ( ( x e. B /\ y e. B ) /\ E. l e. S x = ( l .x. y ) ) } = { <. x , y >. | ( ( x e. B /\ y e. B ) /\ E. l e. ( Base ` ( Scalar ` W ) ) x = ( l .x. y ) ) } )
14 1 13 eqtrid 
 |-  ( K e. DivRing -> .~ = { <. x , y >. | ( ( x e. B /\ y e. B ) /\ E. l e. ( Base ` ( Scalar ` W ) ) x = ( l .x. y ) ) } )