Metamath Proof Explorer


Theorem lidl0

Description: Every ring contains a zero ideal. (Contributed by Stefan O'Rear, 3-Jan-2015)

Ref Expression
Hypotheses lidl0.u
|- U = ( LIdeal ` R )
lidl0.z
|- .0. = ( 0g ` R )
Assertion lidl0
|- ( R e. Ring -> { .0. } e. U )

Proof

Step Hyp Ref Expression
1 lidl0.u
 |-  U = ( LIdeal ` R )
2 lidl0.z
 |-  .0. = ( 0g ` R )
3 rlmlmod
 |-  ( R e. Ring -> ( ringLMod ` R ) e. LMod )
4 rlm0
 |-  ( 0g ` R ) = ( 0g ` ( ringLMod ` R ) )
5 2 4 eqtri
 |-  .0. = ( 0g ` ( ringLMod ` R ) )
6 eqid
 |-  ( LSubSp ` ( ringLMod ` R ) ) = ( LSubSp ` ( ringLMod ` R ) )
7 5 6 lsssn0
 |-  ( ( ringLMod ` R ) e. LMod -> { .0. } e. ( LSubSp ` ( ringLMod ` R ) ) )
8 3 7 syl
 |-  ( R e. Ring -> { .0. } e. ( LSubSp ` ( ringLMod ` R ) ) )
9 lidlval
 |-  ( LIdeal ` R ) = ( LSubSp ` ( ringLMod ` R ) )
10 1 9 eqtri
 |-  U = ( LSubSp ` ( ringLMod ` R ) )
11 8 10 eleqtrrdi
 |-  ( R e. Ring -> { .0. } e. U )