Description: A two-sided ideal of a non-unital ring which is a subgroup of the ring is a normal subgroup of the ring. (Contributed by AV, 20-Feb-2025)
Ref | Expression | ||
---|---|---|---|
Hypotheses | rng2idlsubgsubrng.r | ⊢ ( 𝜑 → 𝑅 ∈ Rng ) | |
rng2idlsubgsubrng.i | ⊢ ( 𝜑 → 𝐼 ∈ ( 2Ideal ‘ 𝑅 ) ) | ||
rng2idlsubgsubrng.u | ⊢ ( 𝜑 → 𝐼 ∈ ( SubGrp ‘ 𝑅 ) ) | ||
Assertion | rng2idlsubgnsg | ⊢ ( 𝜑 → 𝐼 ∈ ( NrmSGrp ‘ 𝑅 ) ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rng2idlsubgsubrng.r | ⊢ ( 𝜑 → 𝑅 ∈ Rng ) | |
2 | rng2idlsubgsubrng.i | ⊢ ( 𝜑 → 𝐼 ∈ ( 2Ideal ‘ 𝑅 ) ) | |
3 | rng2idlsubgsubrng.u | ⊢ ( 𝜑 → 𝐼 ∈ ( SubGrp ‘ 𝑅 ) ) | |
4 | 1 2 3 | rng2idlsubgsubrng | ⊢ ( 𝜑 → 𝐼 ∈ ( SubRng ‘ 𝑅 ) ) |
5 | subrngringnsg | ⊢ ( 𝐼 ∈ ( SubRng ‘ 𝑅 ) → 𝐼 ∈ ( NrmSGrp ‘ 𝑅 ) ) | |
6 | 4 5 | syl | ⊢ ( 𝜑 → 𝐼 ∈ ( NrmSGrp ‘ 𝑅 ) ) |