Metamath Proof Explorer


Theorem resvid2

Description: General behavior of trivial restriction. (Contributed by Thierry Arnoux, 6-Sep-2018)

Ref Expression
Hypotheses resvsca.r
|- R = ( W |`v A )
resvsca.f
|- F = ( Scalar ` W )
resvsca.b
|- B = ( Base ` F )
Assertion resvid2
|- ( ( B C_ A /\ W e. X /\ A e. Y ) -> R = W )

Proof

Step Hyp Ref Expression
1 resvsca.r
 |-  R = ( W |`v A )
2 resvsca.f
 |-  F = ( Scalar ` W )
3 resvsca.b
 |-  B = ( Base ` F )
4 1 2 3 resvval
 |-  ( ( W e. X /\ A e. Y ) -> R = if ( B C_ A , W , ( W sSet <. ( Scalar ` ndx ) , ( F |`s A ) >. ) ) )
5 iftrue
 |-  ( B C_ A -> if ( B C_ A , W , ( W sSet <. ( Scalar ` ndx ) , ( F |`s A ) >. ) ) = W )
6 4 5 sylan9eqr
 |-  ( ( B C_ A /\ ( W e. X /\ A e. Y ) ) -> R = W )
7 6 3impb
 |-  ( ( B C_ A /\ W e. X /\ A e. Y ) -> R = W )