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 ) |
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 ) |