Metamath Proof Explorer


Theorem mgm2mgm

Description: Equivalence of the two definitions of a magma. (Contributed by AV, 16-Jan-2020)

Ref Expression
Assertion mgm2mgm ( 𝑀 ∈ MgmALT ↔ 𝑀 ∈ Mgm )

Proof

Step Hyp Ref Expression
1 eqid ( Base ‘ 𝑀 ) = ( Base ‘ 𝑀 )
2 eqid ( +g𝑀 ) = ( +g𝑀 )
3 1 2 ismgmALT ( 𝑀 ∈ MgmALT → ( 𝑀 ∈ MgmALT ↔ ( +g𝑀 ) clLaw ( Base ‘ 𝑀 ) ) )
4 fvex ( +g𝑀 ) ∈ V
5 fvex ( Base ‘ 𝑀 ) ∈ V
6 iscllaw ( ( ( +g𝑀 ) ∈ V ∧ ( Base ‘ 𝑀 ) ∈ V ) → ( ( +g𝑀 ) clLaw ( Base ‘ 𝑀 ) ↔ ∀ 𝑥 ∈ ( Base ‘ 𝑀 ) ∀ 𝑦 ∈ ( Base ‘ 𝑀 ) ( 𝑥 ( +g𝑀 ) 𝑦 ) ∈ ( Base ‘ 𝑀 ) ) )
7 4 5 6 mp2an ( ( +g𝑀 ) clLaw ( Base ‘ 𝑀 ) ↔ ∀ 𝑥 ∈ ( Base ‘ 𝑀 ) ∀ 𝑦 ∈ ( Base ‘ 𝑀 ) ( 𝑥 ( +g𝑀 ) 𝑦 ) ∈ ( Base ‘ 𝑀 ) )
8 1 2 ismgm ( 𝑀 ∈ MgmALT → ( 𝑀 ∈ Mgm ↔ ∀ 𝑥 ∈ ( Base ‘ 𝑀 ) ∀ 𝑦 ∈ ( Base ‘ 𝑀 ) ( 𝑥 ( +g𝑀 ) 𝑦 ) ∈ ( Base ‘ 𝑀 ) ) )
9 8 biimprd ( 𝑀 ∈ MgmALT → ( ∀ 𝑥 ∈ ( Base ‘ 𝑀 ) ∀ 𝑦 ∈ ( Base ‘ 𝑀 ) ( 𝑥 ( +g𝑀 ) 𝑦 ) ∈ ( Base ‘ 𝑀 ) → 𝑀 ∈ Mgm ) )
10 7 9 syl5bi ( 𝑀 ∈ MgmALT → ( ( +g𝑀 ) clLaw ( Base ‘ 𝑀 ) → 𝑀 ∈ Mgm ) )
11 3 10 sylbid ( 𝑀 ∈ MgmALT → ( 𝑀 ∈ MgmALT → 𝑀 ∈ Mgm ) )
12 11 pm2.43i ( 𝑀 ∈ MgmALT → 𝑀 ∈ Mgm )
13 mgmplusgiopALT ( 𝑀 ∈ Mgm → ( +g𝑀 ) clLaw ( Base ‘ 𝑀 ) )
14 1 2 ismgmALT ( 𝑀 ∈ Mgm → ( 𝑀 ∈ MgmALT ↔ ( +g𝑀 ) clLaw ( Base ‘ 𝑀 ) ) )
15 13 14 mpbird ( 𝑀 ∈ Mgm → 𝑀 ∈ MgmALT )
16 12 15 impbii ( 𝑀 ∈ MgmALT ↔ 𝑀 ∈ Mgm )