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Metamath Proof Explorer

Theorem rng2idlsubgsubrng

Description: A two-sided ideal of a non-unital ring which is a subgroup of the ring is a subring of the ring. (Contributed by AV, 11-Mar-2025)

Ref Expression
Hypotheses rng2idlsubgsubrng.r 
|- ( ph -> R e. Rng )
rng2idlsubgsubrng.i 
|- ( ph -> I e. ( 2Ideal ` R ) )
rng2idlsubgsubrng.u 
|- ( ph -> I e. ( SubGrp ` R ) )
Assertion rng2idlsubgsubrng 
|- ( ph -> I e. ( SubRng ` R ) )

Proof

Step Hyp Ref Expression
1 rng2idlsubgsubrng.r 
 |-  ( ph -> R e. Rng )
2 rng2idlsubgsubrng.i 
 |-  ( ph -> I e. ( 2Ideal ` R ) )
3 rng2idlsubgsubrng.u 
 |-  ( ph -> I e. ( SubGrp ` R ) )
4 eqid 
 |-  ( LIdeal ` R ) = ( LIdeal ` R )
5 eqid 
 |-  ( oppR ` R ) = ( oppR ` R )
6 eqid 
 |-  ( LIdeal ` ( oppR ` R ) ) = ( LIdeal ` ( oppR ` R ) )
7 eqid 
 |-  ( 2Ideal ` R ) = ( 2Ideal ` R )
8 4 5 6 7 2idlelb 
 |-  ( I e. ( 2Ideal ` R ) <-> ( I e. ( LIdeal ` R ) /\ I e. ( LIdeal ` ( oppR ` R ) ) ) )
9 8 simplbi 
 |-  ( I e. ( 2Ideal ` R ) -> I e. ( LIdeal ` R ) )
10 2 9 syl 
 |-  ( ph -> I e. ( LIdeal ` R ) )
11 eqid 
 |-  ( R |`s I ) = ( R |`s I )
12 4 11 rnglidlrng 
 |-  ( ( R e. Rng /\ I e. ( LIdeal ` R ) /\ I e. ( SubGrp ` R ) ) -> ( R |`s I ) e. Rng )
13 1 10 3 12 syl3anc 
 |-  ( ph -> ( R |`s I ) e. Rng )
14 1 2 13 rng2idlsubrng 
 |-  ( ph -> I e. ( SubRng ` R ) )