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Metamath Proof Explorer

Theorem df2idl2rng

Description: Alternate (the usual textbook) definition of a two-sided ideal of a non-unital ring to be a subgroup of the additive group of the ring which is closed under left- and right-multiplication by elements of the full ring. (Contributed by AV, 21-Mar-2025)

Ref Expression
Hypotheses df2idl2rng.u 
|- U = ( 2Ideal ` R )
df2idl2rng.b 
|- B = ( Base ` R )
df2idl2rng.t 
|- .x. = ( .r ` R )
Assertion df2idl2rng 
|- ( ( R e. Rng /\ I e. ( SubGrp ` R ) ) -> ( I e. U <-> A. x e. B A. y e. I ( ( x .x. y ) e. I /\ ( y .x. x ) e. I ) ) )

Proof

Step Hyp Ref Expression
1 df2idl2rng.u 
 |-  U = ( 2Ideal ` R )
2 df2idl2rng.b 
 |-  B = ( Base ` R )
3 df2idl2rng.t 
 |-  .x. = ( .r ` R )
4 eqid 
 |-  ( LIdeal ` R ) = ( LIdeal ` R )
5 4 2 3 dflidl2rng 
 |-  ( ( R e. Rng /\ I e. ( SubGrp ` R ) ) -> ( I e. ( LIdeal ` R ) <-> A. x e. B A. y e. I ( x .x. y ) e. I ) )
6 eqid 
 |-  ( LIdeal ` ( oppR ` R ) ) = ( LIdeal ` ( oppR ` R ) )
7 6 2 3 isridlrng 
 |-  ( ( R e. Rng /\ I e. ( SubGrp ` R ) ) -> ( I e. ( LIdeal ` ( oppR ` R ) ) <-> A. x e. B A. y e. I ( y .x. x ) e. I ) )
8 5 7 anbi12d 
 |-  ( ( R e. Rng /\ I e. ( SubGrp ` R ) ) -> ( ( I e. ( LIdeal ` R ) /\ I e. ( LIdeal ` ( oppR ` R ) ) ) <-> ( A. x e. B A. y e. I ( x .x. y ) e. I /\ A. x e. B A. y e. I ( y .x. x ) e. I ) ) )
9 eqid 
 |-  ( oppR ` R ) = ( oppR ` R )
10 4 9 6 1 2idlelb 
 |-  ( I e. U <-> ( I e. ( LIdeal ` R ) /\ I e. ( LIdeal ` ( oppR ` R ) ) ) )
11 r19.26-2 
 |-  ( A. x e. B A. y e. I ( ( x .x. y ) e. I /\ ( y .x. x ) e. I ) <-> ( A. x e. B A. y e. I ( x .x. y ) e. I /\ A. x e. B A. y e. I ( y .x. x ) e. I ) )
12 8 10 11 3bitr4g 
 |-  ( ( R e. Rng /\ I e. ( SubGrp ` R ) ) -> ( I e. U <-> A. x e. B A. y e. I ( ( x .x. y ) e. I /\ ( y .x. x ) e. I ) ) )