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Metamath Proof Explorer

Theorem uvcval

Description: Value of a single unit vector 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 uvcval 
|- ( ( R e. V /\ I e. W /\ J e. I ) -> ( U ` J ) = ( k e. I |-> 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 uvcfval 
 |-  ( ( R e. V /\ I e. W ) -> U = ( j e. I |-> ( k e. I |-> if ( k = j , .1. , .0. ) ) ) )
5 4 fveq1d 
 |-  ( ( R e. V /\ I e. W ) -> ( U ` J ) = ( ( j e. I |-> ( k e. I |-> if ( k = j , .1. , .0. ) ) ) ` J ) )
6 5 3adant3 
 |-  ( ( R e. V /\ I e. W /\ J e. I ) -> ( U ` J ) = ( ( j e. I |-> ( k e. I |-> if ( k = j , .1. , .0. ) ) ) ` J ) )
7 eqid 
 |-  ( j e. I |-> ( k e. I |-> if ( k = j , .1. , .0. ) ) ) = ( j e. I |-> ( k e. I |-> if ( k = j , .1. , .0. ) ) )
8 eqeq2 
 |-  ( j = J -> ( k = j <-> k = J ) )
9 8 ifbid 
 |-  ( j = J -> if ( k = j , .1. , .0. ) = if ( k = J , .1. , .0. ) )
10 9 mpteq2dv 
 |-  ( j = J -> ( k e. I |-> if ( k = j , .1. , .0. ) ) = ( k e. I |-> if ( k = J , .1. , .0. ) ) )
11 simp3 
 |-  ( ( R e. V /\ I e. W /\ J e. I ) -> J e. I )
12 mptexg 
 |-  ( I e. W -> ( k e. I |-> if ( k = J , .1. , .0. ) ) e. _V )
13 12 3ad2ant2 
 |-  ( ( R e. V /\ I e. W /\ J e. I ) -> ( k e. I |-> if ( k = J , .1. , .0. ) ) e. _V )
14 7 10 11 13 fvmptd3 
 |-  ( ( R e. V /\ I e. W /\ J e. I ) -> ( ( j e. I |-> ( k e. I |-> if ( k = j , .1. , .0. ) ) ) ` J ) = ( k e. I |-> if ( k = J , .1. , .0. ) ) )
15 6 14 eqtrd 
 |-  ( ( R e. V /\ I e. W /\ J e. I ) -> ( U ` J ) = ( k e. I |-> if ( k = J , .1. , .0. ) ) )