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Metamath Proof Explorer
Description: A monoid is a semigroup. (Contributed by FL, 2-Nov-2009) (Revised by AV, 6-Jan-2020) (Proof shortened by AV, 6-Feb-2020)
|
|
Ref |
Expression |
|
Assertion |
mndsgrp |
⊢ ( 𝐺 ∈ Mnd → 𝐺 ∈ Smgrp ) |
Proof
| Step |
Hyp |
Ref |
Expression |
| 1 |
|
eqid |
⊢ ( Base ‘ 𝐺 ) = ( Base ‘ 𝐺 ) |
| 2 |
|
eqid |
⊢ ( +g ‘ 𝐺 ) = ( +g ‘ 𝐺 ) |
| 3 |
1 2
|
ismnddef |
⊢ ( 𝐺 ∈ Mnd ↔ ( 𝐺 ∈ Smgrp ∧ ∃ 𝑒 ∈ ( Base ‘ 𝐺 ) ∀ 𝑥 ∈ ( Base ‘ 𝐺 ) ( ( 𝑒 ( +g ‘ 𝐺 ) 𝑥 ) = 𝑥 ∧ ( 𝑥 ( +g ‘ 𝐺 ) 𝑒 ) = 𝑥 ) ) ) |
| 4 |
3
|
simplbi |
⊢ ( 𝐺 ∈ Mnd → 𝐺 ∈ Smgrp ) |