Description: The unit element of a ring belongs to the base set of the ring. (Contributed by NM, 27-Aug-2011) (Revised by Mario Carneiro, 27-Dec-2014)
| Ref | Expression | ||
|---|---|---|---|
| Hypotheses | ringidcl.b | ⊢ 𝐵 = ( Base ‘ 𝑅 ) | |
| ringidcl.u | ⊢ 1 = ( 1r ‘ 𝑅 ) | ||
| Assertion | ringidcl | ⊢ ( 𝑅 ∈ Ring → 1 ∈ 𝐵 ) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | ringidcl.b | ⊢ 𝐵 = ( Base ‘ 𝑅 ) | |
| 2 | ringidcl.u | ⊢ 1 = ( 1r ‘ 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 | ringidval | ⊢ 1 = ( 0g ‘ ( mulGrp ‘ 𝑅 ) ) |
| 7 | 5 6 | mndidcl | ⊢ ( ( mulGrp ‘ 𝑅 ) ∈ Mnd → 1 ∈ 𝐵 ) |
| 8 | 4 7 | syl | ⊢ ( 𝑅 ∈ Ring → 1 ∈ 𝐵 ) |