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Metamath Proof Explorer

Theorem cphqss

Description: The scalar field of a subcomplex pre-Hilbert space contains the rational numbers. (Contributed by Mario Carneiro, 15-Oct-2015)

Ref Expression
Hypotheses cphsca.f 
|- F = ( Scalar ` W )
cphsca.k 
|- K = ( Base ` F )
Assertion cphqss 
|- ( W e. CPreHil -> QQ C_ K )

Proof

Step Hyp Ref Expression
1 cphsca.f 
 |-  F = ( Scalar ` W )
2 cphsca.k 
 |-  K = ( Base ` F )
3 1 2 cphsubrg 
 |-  ( W e. CPreHil -> K e. ( SubRing ` CCfld ) )
4 1 2 cphsca 
 |-  ( W e. CPreHil -> F = ( CCfld |`s K ) )
5 cphlvec 
 |-  ( W e. CPreHil -> W e. LVec )
6 1 lvecdrng 
 |-  ( W e. LVec -> F e. DivRing )
7 5 6 syl 
 |-  ( W e. CPreHil -> F e. DivRing )
8 4 7 eqeltrrd 
 |-  ( W e. CPreHil -> ( CCfld |`s K ) e. DivRing )
9 qsssubdrg 
 |-  ( ( K e. ( SubRing ` CCfld ) /\ ( CCfld |`s K ) e. DivRing ) -> QQ C_ K )
10 3 8 9 syl2anc 
 |-  ( W e. CPreHil -> QQ C_ K )