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 )