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Metamath Proof Explorer

Theorem subrgmre

Description: The subrings of a ring are a Moore system. (Contributed by Stefan O'Rear, 9-Mar-2015)

Ref Expression
Hypothesis subrgmre.b 
|- B = ( Base ` R )
Assertion subrgmre 
|- ( R e. Ring -> ( SubRing ` R ) e. ( Moore ` B ) )

Proof

Step Hyp Ref Expression
1 subrgmre.b 
 |-  B = ( Base ` R )
2 1 subrgss 
 |-  ( a e. ( SubRing ` R ) -> a C_ B )
3 velpw 
 |-  ( a e. ~P B <-> a C_ B )
4 2 3 sylibr 
 |-  ( a e. ( SubRing ` R ) -> a e. ~P B )
5 4 a1i 
 |-  ( R e. Ring -> ( a e. ( SubRing ` R ) -> a e. ~P B ) )
6 5 ssrdv 
 |-  ( R e. Ring -> ( SubRing ` R ) C_ ~P B )
7 1 subrgid 
 |-  ( R e. Ring -> B e. ( SubRing ` R ) )
8 subrgint 
 |-  ( ( a C_ ( SubRing ` R ) /\ a =/= (/) ) -> |^| a e. ( SubRing ` R ) )
9 8 3adant1 
 |-  ( ( R e. Ring /\ a C_ ( SubRing ` R ) /\ a =/= (/) ) -> |^| a e. ( SubRing ` R ) )
10 6 7 9 ismred 
 |-  ( R e. Ring -> ( SubRing ` R ) e. ( Moore ` B ) )