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Metamath Proof Explorer

Theorem rspssp

Description: The ideal span of a set of elements in a ring is contained in any subring which contains those elements. (Contributed by Stefan O'Rear, 3-Jan-2015)

Ref Expression
Hypotheses rspcl.k 
|- K = ( RSpan ` R )
rspssp.u 
|- U = ( LIdeal ` R )
Assertion rspssp 
|- ( ( R e. Ring /\ I e. U /\ G C_ I ) -> ( K ` G ) C_ I )

Proof

Step Hyp Ref Expression
1 rspcl.k 
 |-  K = ( RSpan ` R )
2 rspssp.u 
 |-  U = ( LIdeal ` R )
3 rlmlmod 
 |-  ( R e. Ring -> ( ringLMod ` R ) e. LMod )
4 lidlval 
 |-  ( LIdeal ` R ) = ( LSubSp ` ( ringLMod ` R ) )
5 2 4 eqtri 
 |-  U = ( LSubSp ` ( ringLMod ` R ) )
6 rspval 
 |-  ( RSpan ` R ) = ( LSpan ` ( ringLMod ` R ) )
7 1 6 eqtri 
 |-  K = ( LSpan ` ( ringLMod ` R ) )
8 5 7 lspssp 
 |-  ( ( ( ringLMod ` R ) e. LMod /\ I e. U /\ G C_ I ) -> ( K ` G ) C_ I )
9 3 8 syl3an1 
 |-  ( ( R e. Ring /\ I e. U /\ G C_ I ) -> ( K ` G ) C_ I )