Description: A monoid has a two-sided identity element. (Contributed by NM, 16-Aug-2011)
Ref | Expression | ||
---|---|---|---|
Hypotheses | mndcl.b | ⊢ 𝐵 = ( Base ‘ 𝐺 ) | |
mndcl.p | ⊢ + = ( +g ‘ 𝐺 ) | ||
Assertion | mndid | ⊢ ( 𝐺 ∈ Mnd → ∃ 𝑢 ∈ 𝐵 ∀ 𝑥 ∈ 𝐵 ( ( 𝑢 + 𝑥 ) = 𝑥 ∧ ( 𝑥 + 𝑢 ) = 𝑥 ) ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mndcl.b | ⊢ 𝐵 = ( Base ‘ 𝐺 ) | |
2 | mndcl.p | ⊢ + = ( +g ‘ 𝐺 ) | |
3 | 1 2 | ismnd | ⊢ ( 𝐺 ∈ Mnd ↔ ( ∀ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐵 ( ( 𝑥 + 𝑦 ) ∈ 𝐵 ∧ ∀ 𝑧 ∈ 𝐵 ( ( 𝑥 + 𝑦 ) + 𝑧 ) = ( 𝑥 + ( 𝑦 + 𝑧 ) ) ) ∧ ∃ 𝑢 ∈ 𝐵 ∀ 𝑥 ∈ 𝐵 ( ( 𝑢 + 𝑥 ) = 𝑥 ∧ ( 𝑥 + 𝑢 ) = 𝑥 ) ) ) |
4 | 3 | simprbi | ⊢ ( 𝐺 ∈ Mnd → ∃ 𝑢 ∈ 𝐵 ∀ 𝑥 ∈ 𝐵 ( ( 𝑢 + 𝑥 ) = 𝑥 ∧ ( 𝑥 + 𝑢 ) = 𝑥 ) ) |