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Metamath Proof Explorer

Theorem idlss

Description: An ideal of R is a subset of R . (Contributed by Jeff Madsen, 10-Jun-2010)

Ref Expression
Hypotheses idlss.1 
|- G = ( 1st ` R )
idlss.2 
|- X = ran G
Assertion idlss 
|- ( ( R e. RingOps /\ I e. ( Idl ` R ) ) -> I C_ X )

Proof

Step Hyp Ref Expression
1 idlss.1 
 |-  G = ( 1st ` R )
2 idlss.2 
 |-  X = ran G
3 eqid 
 |-  ( 2nd ` R ) = ( 2nd ` R )
4 eqid 
 |-  ( GId ` G ) = ( GId ` G )
5 1 3 2 4 isidl 
 |-  ( R e. RingOps -> ( I e. ( Idl ` R ) <-> ( I C_ X /\ ( GId ` G ) e. I /\ A. x e. I ( A. y e. I ( x G y ) e. I /\ A. z e. X ( ( z ( 2nd ` R ) x ) e. I /\ ( x ( 2nd ` R ) z ) e. I ) ) ) ) )
6 5 biimpa 
 |-  ( ( R e. RingOps /\ I e. ( Idl ` R ) ) -> ( I C_ X /\ ( GId ` G ) e. I /\ A. x e. I ( A. y e. I ( x G y ) e. I /\ A. z e. X ( ( z ( 2nd ` R ) x ) e. I /\ ( x ( 2nd ` R ) z ) e. I ) ) ) )
7 6 simp1d 
 |-  ( ( R e. RingOps /\ I e. ( Idl ` R ) ) -> I C_ X )