Description: The ring unity of a two-sided ideal of a non-unital ring belongs to the base set of the ring. (Contributed by AV, 15-Mar-2025)
| Ref | Expression | ||
|---|---|---|---|
| Hypotheses | rng2idlring.r | ⊢ ( 𝜑 → 𝑅 ∈ Rng ) | |
| rng2idlring.i | ⊢ ( 𝜑 → 𝐼 ∈ ( 2Ideal ‘ 𝑅 ) ) | ||
| rng2idlring.j | ⊢ 𝐽 = ( 𝑅 ↾s 𝐼 ) | ||
| rng2idlring.u | ⊢ ( 𝜑 → 𝐽 ∈ Ring ) | ||
| rng2idlring.b | ⊢ 𝐵 = ( Base ‘ 𝑅 ) | ||
| rng2idlring.t | ⊢ · = ( .r ‘ 𝑅 ) | ||
| rng2idlring.1 | ⊢ 1 = ( 1r ‘ 𝐽 ) | ||
| Assertion | rngqiprng1elbas | ⊢ ( 𝜑 → 1 ∈ 𝐵 ) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | rng2idlring.r | ⊢ ( 𝜑 → 𝑅 ∈ Rng ) | |
| 2 | rng2idlring.i | ⊢ ( 𝜑 → 𝐼 ∈ ( 2Ideal ‘ 𝑅 ) ) | |
| 3 | rng2idlring.j | ⊢ 𝐽 = ( 𝑅 ↾s 𝐼 ) | |
| 4 | rng2idlring.u | ⊢ ( 𝜑 → 𝐽 ∈ Ring ) | |
| 5 | rng2idlring.b | ⊢ 𝐵 = ( Base ‘ 𝑅 ) | |
| 6 | rng2idlring.t | ⊢ · = ( .r ‘ 𝑅 ) | |
| 7 | rng2idlring.1 | ⊢ 1 = ( 1r ‘ 𝐽 ) | |
| 8 | 3 5 | ressbasss | ⊢ ( Base ‘ 𝐽 ) ⊆ 𝐵 |
| 9 | eqid | ⊢ ( Base ‘ 𝐽 ) = ( Base ‘ 𝐽 ) | |
| 10 | 9 7 | ringidcl | ⊢ ( 𝐽 ∈ Ring → 1 ∈ ( Base ‘ 𝐽 ) ) |
| 11 | 4 10 | syl | ⊢ ( 𝜑 → 1 ∈ ( Base ‘ 𝐽 ) ) |
| 12 | 8 11 | sselid | ⊢ ( 𝜑 → 1 ∈ 𝐵 ) |