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Metamath Proof Explorer

Theorem rspcl

Description: The span of a set of ring elements is an ideal. (Contributed by Stefan O'Rear, 3-Jan-2015) (Revised by Mario Carneiro, 2-Oct-2015)

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

Proof

Step Hyp Ref Expression
1 rspcl.k 
 |-  K = ( RSpan ` R )
2 rspcl.b 
 |-  B = ( Base ` R )
3 rspcl.u 
 |-  U = ( LIdeal ` R )
4 rlmlmod 
 |-  ( R e. Ring -> ( ringLMod ` R ) e. LMod )
5 rlmbas 
 |-  ( Base ` R ) = ( Base ` ( ringLMod ` R ) )
6 2 5 eqtri 
 |-  B = ( Base ` ( ringLMod ` R ) )
7 lidlval 
 |-  ( LIdeal ` R ) = ( LSubSp ` ( ringLMod ` R ) )
8 3 7 eqtri 
 |-  U = ( LSubSp ` ( ringLMod ` R ) )
9 rspval 
 |-  ( RSpan ` R ) = ( LSpan ` ( ringLMod ` R ) )
10 1 9 eqtri 
 |-  K = ( LSpan ` ( ringLMod ` R ) )
11 6 8 10 lspcl 
 |-  ( ( ( ringLMod ` R ) e. LMod /\ G C_ B ) -> ( K ` G ) e. U )
12 4 11 sylan 
 |-  ( ( R e. Ring /\ G C_ B ) -> ( K ` G ) e. U )