This is an inofficial mirror of http://metamath.tirix.org for personal testing of a visualizer extension only.

Metamath Proof Explorer

Theorem lidl0ALT

Description: Alternate proof for lidl0 not using rnglidl0 : Every ring contains a zero ideal. (Contributed by Stefan O'Rear, 3-Jan-2015) (Proof modification is discouraged.) (New usage is discouraged.)

Ref Expression
Hypotheses rnglidl0.u 
|- U = ( LIdeal ` R )
rnglidl0.z 
|- .0. = ( 0g ` R )
Assertion lidl0ALT 
|- ( R e. Ring -> { .0. } e. U )

Proof

Step Hyp Ref Expression
1 rnglidl0.u 
 |-  U = ( LIdeal ` R )
2 rnglidl0.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 )