Metamath Proof Explorer


Theorem uvcvval

Description: Value of a unit vector coordinate in a free module. (Contributed by Stefan O'Rear, 3-Feb-2015)

Ref Expression
Hypotheses uvcfval.u
|- U = ( R unitVec I )
uvcfval.o
|- .1. = ( 1r ` R )
uvcfval.z
|- .0. = ( 0g ` R )
Assertion uvcvval
|- ( ( ( R e. V /\ I e. W /\ J e. I ) /\ K e. I ) -> ( ( U ` J ) ` K ) = if ( K = J , .1. , .0. ) )

Proof

Step Hyp Ref Expression
1 uvcfval.u
 |-  U = ( R unitVec I )
2 uvcfval.o
 |-  .1. = ( 1r ` R )
3 uvcfval.z
 |-  .0. = ( 0g ` R )
4 1 2 3 uvcval
 |-  ( ( R e. V /\ I e. W /\ J e. I ) -> ( U ` J ) = ( k e. I |-> if ( k = J , .1. , .0. ) ) )
5 4 fveq1d
 |-  ( ( R e. V /\ I e. W /\ J e. I ) -> ( ( U ` J ) ` K ) = ( ( k e. I |-> if ( k = J , .1. , .0. ) ) ` K ) )
6 5 adantr
 |-  ( ( ( R e. V /\ I e. W /\ J e. I ) /\ K e. I ) -> ( ( U ` J ) ` K ) = ( ( k e. I |-> if ( k = J , .1. , .0. ) ) ` K ) )
7 simpr
 |-  ( ( ( R e. V /\ I e. W /\ J e. I ) /\ K e. I ) -> K e. I )
8 2 fvexi
 |-  .1. e. _V
9 3 fvexi
 |-  .0. e. _V
10 8 9 ifex
 |-  if ( K = J , .1. , .0. ) e. _V
11 eqeq1
 |-  ( k = K -> ( k = J <-> K = J ) )
12 11 ifbid
 |-  ( k = K -> if ( k = J , .1. , .0. ) = if ( K = J , .1. , .0. ) )
13 eqid
 |-  ( k e. I |-> if ( k = J , .1. , .0. ) ) = ( k e. I |-> if ( k = J , .1. , .0. ) )
14 12 13 fvmptg
 |-  ( ( K e. I /\ if ( K = J , .1. , .0. ) e. _V ) -> ( ( k e. I |-> if ( k = J , .1. , .0. ) ) ` K ) = if ( K = J , .1. , .0. ) )
15 7 10 14 sylancl
 |-  ( ( ( R e. V /\ I e. W /\ J e. I ) /\ K e. I ) -> ( ( k e. I |-> if ( k = J , .1. , .0. ) ) ` K ) = if ( K = J , .1. , .0. ) )
16 6 15 eqtrd
 |-  ( ( ( R e. V /\ I e. W /\ J e. I ) /\ K e. I ) -> ( ( U ` J ) ` K ) = if ( K = J , .1. , .0. ) )