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Metamath Proof Explorer

Theorem sspims

Description: The induced metric on a subspace is a restriction of the induced metric on the parent space. (Contributed by NM, 1-Feb-2008) (New usage is discouraged.)

Ref Expression
Hypotheses sspims.y 
|- Y = ( BaseSet ` W )
sspims.d 
|- D = ( IndMet ` U )
sspims.c 
|- C = ( IndMet ` W )
sspims.h 
|- H = ( SubSp ` U )
Assertion sspims 
|- ( ( U e. NrmCVec /\ W e. H ) -> C = ( D |` ( Y X. Y ) ) )

Proof

Step Hyp Ref Expression
1 sspims.y 
 |-  Y = ( BaseSet ` W )
2 sspims.d 
 |-  D = ( IndMet ` U )
3 sspims.c 
 |-  C = ( IndMet ` W )
4 sspims.h 
 |-  H = ( SubSp ` U )
5 1 2 3 4 sspimsval 
 |-  ( ( ( U e. NrmCVec /\ W e. H ) /\ ( x e. Y /\ y e. Y ) ) -> ( x C y ) = ( x D y ) )
6 1 3 imsdf 
 |-  ( W e. NrmCVec -> C : ( Y X. Y ) --> RR )
7 eqid 
 |-  ( BaseSet ` U ) = ( BaseSet ` U )
8 7 2 imsdf 
 |-  ( U e. NrmCVec -> D : ( ( BaseSet ` U ) X. ( BaseSet ` U ) ) --> RR )
9 1 4 5 6 8 sspmlem 
 |-  ( ( U e. NrmCVec /\ W e. H ) -> C = ( D |` ( Y X. Y ) ) )