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Metamath Proof Explorer

Theorem isridl

Description: A right ideal is a left ideal of the opposite ring. This theorem shows that this definition corresponds to the usual textbook definition of a right ideal of a ring to be a subgroup of the additive group of the ring which is closed under right-multiplication by elements of the full ring. (Contributed by AV, 13-Feb-2025)

Ref Expression
Hypotheses isridl.u ⊢ 𝑈 = ( LIdeal ‘ ( oppr ‘ 𝑅 ) )
isridl.b ⊢ 𝐵 = ( Base ‘ 𝑅 )
isridl.t ⊢ · = ( .r ‘ 𝑅 )
Assertion isridl ( 𝑅 ∈ Ring → ( 𝐼 ∈ 𝑈 ↔ ( 𝐼 ∈ ( SubGrp ‘ 𝑅 ) ∧ ∀ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐼 ( 𝑦 · 𝑥 ) ∈ 𝐼 ) ) )

Proof

Step Hyp Ref Expression
1 isridl.u ⊢ 𝑈 = ( LIdeal ‘ ( oppr ‘ 𝑅 ) )
2 isridl.b ⊢ 𝐵 = ( Base ‘ 𝑅 )
3 isridl.t ⊢ · = ( .r ‘ 𝑅 )
4 eqid ⊢ ( oppr ‘ 𝑅 ) = ( oppr ‘ 𝑅 )
5 4 opprring ⊢ ( 𝑅 ∈ Ring → ( oppr ‘ 𝑅 ) ∈ Ring )
6 4 2 opprbas ⊢ 𝐵 = ( Base ‘ ( oppr ‘ 𝑅 ) )
7 eqid ⊢ ( .r ‘ ( oppr ‘ 𝑅 ) ) = ( .r ‘ ( oppr ‘ 𝑅 ) )
8 1 6 7 dflidl2 ⊢ ( ( oppr ‘ 𝑅 ) ∈ Ring → ( 𝐼 ∈ 𝑈 ↔ ( 𝐼 ∈ ( SubGrp ‘ ( oppr ‘ 𝑅 ) ) ∧ ∀ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐼 ( 𝑥 ( .r ‘ ( oppr ‘ 𝑅 ) ) 𝑦 ) ∈ 𝐼 ) ) )
9 5 8 syl ⊢ ( 𝑅 ∈ Ring → ( 𝐼 ∈ 𝑈 ↔ ( 𝐼 ∈ ( SubGrp ‘ ( oppr ‘ 𝑅 ) ) ∧ ∀ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐼 ( 𝑥 ( .r ‘ ( oppr ‘ 𝑅 ) ) 𝑦 ) ∈ 𝐼 ) ) )
10 4 opprsubg ⊢ ( SubGrp ‘ 𝑅 ) = ( SubGrp ‘ ( oppr ‘ 𝑅 ) )
11 10 eqcomi ⊢ ( SubGrp ‘ ( oppr ‘ 𝑅 ) ) = ( SubGrp ‘ 𝑅 )
12 11 a1i ⊢ ( 𝑅 ∈ Ring → ( SubGrp ‘ ( oppr ‘ 𝑅 ) ) = ( SubGrp ‘ 𝑅 ) )
13 12 eleq2d ⊢ ( 𝑅 ∈ Ring → ( 𝐼 ∈ ( SubGrp ‘ ( oppr ‘ 𝑅 ) ) ↔ 𝐼 ∈ ( SubGrp ‘ 𝑅 ) ) )
14 2 3 4 7 opprmul ⊢ ( 𝑥 ( .r ‘ ( oppr ‘ 𝑅 ) ) 𝑦 ) = ( 𝑦 · 𝑥 )
15 14 eleq1i ⊢ ( ( 𝑥 ( .r ‘ ( oppr ‘ 𝑅 ) ) 𝑦 ) ∈ 𝐼 ↔ ( 𝑦 · 𝑥 ) ∈ 𝐼 )
16 15 a1i ⊢ ( ( ( 𝑅 ∈ Ring ∧ 𝑥 ∈ 𝐵 ) ∧ 𝑦 ∈ 𝐼 ) → ( ( 𝑥 ( .r ‘ ( oppr ‘ 𝑅 ) ) 𝑦 ) ∈ 𝐼 ↔ ( 𝑦 · 𝑥 ) ∈ 𝐼 ) )
17 16 ralbidva ⊢ ( ( 𝑅 ∈ Ring ∧ 𝑥 ∈ 𝐵 ) → ( ∀ 𝑦 ∈ 𝐼 ( 𝑥 ( .r ‘ ( oppr ‘ 𝑅 ) ) 𝑦 ) ∈ 𝐼 ↔ ∀ 𝑦 ∈ 𝐼 ( 𝑦 · 𝑥 ) ∈ 𝐼 ) )
18 17 ralbidva ⊢ ( 𝑅 ∈ Ring → ( ∀ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐼 ( 𝑥 ( .r ‘ ( oppr ‘ 𝑅 ) ) 𝑦 ) ∈ 𝐼 ↔ ∀ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐼 ( 𝑦 · 𝑥 ) ∈ 𝐼 ) )
19 13 18 anbi12d ⊢ ( 𝑅 ∈ Ring → ( ( 𝐼 ∈ ( SubGrp ‘ ( oppr ‘ 𝑅 ) ) ∧ ∀ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐼 ( 𝑥 ( .r ‘ ( oppr ‘ 𝑅 ) ) 𝑦 ) ∈ 𝐼 ) ↔ ( 𝐼 ∈ ( SubGrp ‘ 𝑅 ) ∧ ∀ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐼 ( 𝑦 · 𝑥 ) ∈ 𝐼 ) ) )
20 9 19 bitrd ⊢ ( 𝑅 ∈ Ring → ( 𝐼 ∈ 𝑈 ↔ ( 𝐼 ∈ ( SubGrp ‘ 𝑅 ) ∧ ∀ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐼 ( 𝑦 · 𝑥 ) ∈ 𝐼 ) ) )