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Metamath Proof Explorer

Theorem nlmnrg

Description: The scalar component of a left module is a normed ring. (Contributed by Mario Carneiro, 4-Oct-2015)

Ref Expression
Hypothesis nlmnrg.1 
|- F = ( Scalar ` W )
Assertion nlmnrg 
|- ( W e. NrmMod -> F e. NrmRing )

Proof

Step Hyp Ref Expression
1 nlmnrg.1 
 |-  F = ( Scalar ` W )
2 eqid 
 |-  ( Base ` W ) = ( Base ` W )
3 eqid 
 |-  ( norm ` W ) = ( norm ` W )
4 eqid 
 |-  ( .s ` W ) = ( .s ` W )
5 eqid 
 |-  ( Base ` F ) = ( Base ` F )
6 eqid 
 |-  ( norm ` F ) = ( norm ` F )
7 2 3 4 1 5 6 isnlm 
 |-  ( W e. NrmMod <-> ( ( W e. NrmGrp /\ W e. LMod /\ F e. NrmRing ) /\ A. x e. ( Base ` F ) A. y e. ( Base ` W ) ( ( norm ` W ) ` ( x ( .s ` W ) y ) ) = ( ( ( norm ` F ) ` x ) x. ( ( norm ` W ) ` y ) ) ) )
8 7 simplbi 
 |-  ( W e. NrmMod -> ( W e. NrmGrp /\ W e. LMod /\ F e. NrmRing ) )
9 8 simp3d 
 |-  ( W e. NrmMod -> F e. NrmRing )