Metamath Proof Explorer


Definition df-mgm2

Description: Amagma is a set equipped with a closed operation. Definition 1 of BourbakiAlg1 p. 1, or definition of a groupoid in section I.1 of Bruck p. 1. Note: The term "groupoid" is now widely used to refer to other objects: (small) categories all of whose morphisms are invertible, or groups with a partial function replacing the binary operation. Therefore, we will only use the term "magma" for the present notion in set.mm. (Contributed by AV, 6-Jan-2020)

Ref Expression
Assertion df-mgm2 MgmALT = { 𝑚 ∣ ( +g𝑚 ) clLaw ( Base ‘ 𝑚 ) }

Detailed syntax breakdown

Step Hyp Ref Expression
0 cmgm2 MgmALT
1 vm 𝑚
2 cplusg +g
3 1 cv 𝑚
4 3 2 cfv ( +g𝑚 )
5 ccllaw clLaw
6 cbs Base
7 3 6 cfv ( Base ‘ 𝑚 )
8 4 7 5 wbr ( +g𝑚 ) clLaw ( Base ‘ 𝑚 )
9 8 1 cab { 𝑚 ∣ ( +g𝑚 ) clLaw ( Base ‘ 𝑚 ) }
10 0 9 wceq MgmALT = { 𝑚 ∣ ( +g𝑚 ) clLaw ( Base ‘ 𝑚 ) }