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Metamath Proof Explorer

Theorem phlsrng

Description: The scalar ring of a pre-Hilbert space is a star ring. (Contributed by Mario Carneiro, 7-Oct-2015)

Ref Expression
Hypothesis phlsrng.f 
|- F = ( Scalar ` W )
Assertion phlsrng 
|- ( W e. PreHil -> F e. *Ring )

Proof

Step Hyp Ref Expression
1 phlsrng.f 
 |-  F = ( Scalar ` W )
2 eqid 
 |-  ( Base ` W ) = ( Base ` W )
3 eqid 
 |-  ( .i ` W ) = ( .i ` W )
4 eqid 
 |-  ( 0g ` W ) = ( 0g ` W )
5 eqid 
 |-  ( *r ` F ) = ( *r ` F )
6 eqid 
 |-  ( 0g ` F ) = ( 0g ` F )
7 2 1 3 4 5 6 isphl 
 |-  ( W e. PreHil <-> ( W e. LVec /\ F e. *Ring /\ A. x e. ( Base ` W ) ( ( y e. ( Base ` W ) |-> ( y ( .i ` W ) x ) ) e. ( W LMHom ( ringLMod ` F ) ) /\ ( ( x ( .i ` W ) x ) = ( 0g ` F ) -> x = ( 0g ` W ) ) /\ A. y e. ( Base ` W ) ( ( *r ` F ) ` ( x ( .i ` W ) y ) ) = ( y ( .i ` W ) x ) ) ) )
8 7 simp2bi 
 |-  ( W e. PreHil -> F e. *Ring )