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Metamath Proof Explorer

Theorem isrrg

Description: Membership in the set of left-regular elements. (Contributed by Stefan O'Rear, 22-Mar-2015)

Ref Expression
Hypotheses rrgval.e 
|- E = ( RLReg ` R )
rrgval.b 
|- B = ( Base ` R )
rrgval.t 
|- .x. = ( .r ` R )
rrgval.z 
|- .0. = ( 0g ` R )
Assertion isrrg 
|- ( X e. E <-> ( X e. B /\ A. y e. B ( ( X .x. y ) = .0. -> y = .0. ) ) )

Proof

Step Hyp Ref Expression
1 rrgval.e 
 |-  E = ( RLReg ` R )
2 rrgval.b 
 |-  B = ( Base ` R )
3 rrgval.t 
 |-  .x. = ( .r ` R )
4 rrgval.z 
 |-  .0. = ( 0g ` R )
5 oveq1 
 |-  ( x = X -> ( x .x. y ) = ( X .x. y ) )
6 5 eqeq1d 
 |-  ( x = X -> ( ( x .x. y ) = .0. <-> ( X .x. y ) = .0. ) )
7 6 imbi1d 
 |-  ( x = X -> ( ( ( x .x. y ) = .0. -> y = .0. ) <-> ( ( X .x. y ) = .0. -> y = .0. ) ) )
8 7 ralbidv 
 |-  ( x = X -> ( A. y e. B ( ( x .x. y ) = .0. -> y = .0. ) <-> A. y e. B ( ( X .x. y ) = .0. -> y = .0. ) ) )
9 1 2 3 4 rrgval 
 |-  E = { x e. B | A. y e. B ( ( x .x. y ) = .0. -> y = .0. ) }
10 8 9 elrab2 
 |-  ( X e. E <-> ( X e. B /\ A. y e. B ( ( X .x. y ) = .0. -> y = .0. ) ) )