Description: The unity element of a ring is unique. (Contributed by NM, 4-Apr-2009) (Revised by NM, 27-Aug-2011) (Revised by Mario Carneiro, 6-Jan-2015)
| Ref | Expression | ||
|---|---|---|---|
| Hypotheses | ringcl.b | ⊢ 𝐵 = ( Base ‘ 𝑅 ) | |
| ringcl.t | ⊢ · = ( .r ‘ 𝑅 ) | ||
| Assertion | ringideu | ⊢ ( 𝑅 ∈ Ring → ∃! 𝑢 ∈ 𝐵 ∀ 𝑥 ∈ 𝐵 ( ( 𝑢 · 𝑥 ) = 𝑥 ∧ ( 𝑥 · 𝑢 ) = 𝑥 ) ) |
| 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 | mndideu | ⊢ ( ( mulGrp ‘ 𝑅 ) ∈ Mnd → ∃! 𝑢 ∈ 𝐵 ∀ 𝑥 ∈ 𝐵 ( ( 𝑢 · 𝑥 ) = 𝑥 ∧ ( 𝑥 · 𝑢 ) = 𝑥 ) ) |
| 8 | 4 7 | syl | ⊢ ( 𝑅 ∈ Ring → ∃! 𝑢 ∈ 𝐵 ∀ 𝑥 ∈ 𝐵 ( ( 𝑢 · 𝑥 ) = 𝑥 ∧ ( 𝑥 · 𝑢 ) = 𝑥 ) ) |