Description: Closure of the multiplication operation of a ring. (Contributed by Steve Rodriguez, 9-Sep-2007) (Revised by NM, 26-Aug-2011) (Revised by Mario Carneiro, 6-Jan-2015)
| Ref | Expression | ||
|---|---|---|---|
| Hypotheses | ringcl.b | ⊢ 𝐵 = ( Base ‘ 𝑅 ) | |
| ringcl.t | ⊢ · = ( .r ‘ 𝑅 ) | ||
| Assertion | ringcl | ⊢ ( ( 𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( 𝑋 · 𝑌 ) ∈ 𝐵 ) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | ringcl.b | ⊢ 𝐵 = ( Base ‘ 𝑅 ) | |
| 2 | ringcl.t | ⊢ · = ( .r ‘ 𝑅 ) | |
| 3 | eqid | ⊢ ( mulGrp ‘ 𝑅 ) = ( mulGrp ‘ 𝑅 ) | |
| 4 | 3 | ringmgp | ⊢ ( 𝑅 ∈ Ring → ( mulGrp ‘ 𝑅 ) ∈ Mnd ) |
| 5 | 3 1 | mgpbas | ⊢ 𝐵 = ( Base ‘ ( mulGrp ‘ 𝑅 ) ) |
| 6 | 3 2 | mgpplusg | ⊢ · = ( +g ‘ ( mulGrp ‘ 𝑅 ) ) |
| 7 | 5 6 | mndcl | ⊢ ( ( ( mulGrp ‘ 𝑅 ) ∈ Mnd ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( 𝑋 · 𝑌 ) ∈ 𝐵 ) |
| 8 | 4 7 | syl3an1 | ⊢ ( ( 𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( 𝑋 · 𝑌 ) ∈ 𝐵 ) |