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Metamath Proof Explorer

Theorem igenidl2

Description: The ideal generated by an ideal is that ideal. (Contributed by Jeff Madsen, 10-Jun-2010)

Ref Expression
Assertion igenidl2 
|- ( ( R e. RingOps /\ I e. ( Idl ` R ) ) -> ( R IdlGen I ) = I )

Proof

Step Hyp Ref Expression
1 eqid 
 |-  ( 1st ` R ) = ( 1st ` R )
2 eqid 
 |-  ran ( 1st ` R ) = ran ( 1st ` R )
3 1 2 idlss 
 |-  ( ( R e. RingOps /\ I e. ( Idl ` R ) ) -> I C_ ran ( 1st ` R ) )
4 1 2 igenval 
 |-  ( ( R e. RingOps /\ I C_ ran ( 1st ` R ) ) -> ( R IdlGen I ) = |^| { j e. ( Idl ` R ) | I C_ j } )
5 3 4 syldan 
 |-  ( ( R e. RingOps /\ I e. ( Idl ` R ) ) -> ( R IdlGen I ) = |^| { j e. ( Idl ` R ) | I C_ j } )
6 intmin 
 |-  ( I e. ( Idl ` R ) -> |^| { j e. ( Idl ` R ) | I C_ j } = I )
7 6 adantl 
 |-  ( ( R e. RingOps /\ I e. ( Idl ` R ) ) -> |^| { j e. ( Idl ` R ) | I C_ j } = I )
8 5 7 eqtrd 
 |-  ( ( R e. RingOps /\ I e. ( Idl ` R ) ) -> ( R IdlGen I ) = I )