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Metamath Proof Explorer

Theorem srgcmn

Description: A semiring is a commutative monoid. (Contributed by Thierry Arnoux, 21-Mar-2018)

Ref Expression
Assertion srgcmn ( 𝑅 ∈ SRing → 𝑅 ∈ CMnd )

Proof

Step Hyp Ref Expression
1 eqid ⊢ ( Base ‘ 𝑅 ) = ( Base ‘ 𝑅 )
2 eqid ⊢ ( mulGrp ‘ 𝑅 ) = ( mulGrp ‘ 𝑅 )
3 eqid ⊢ ( +g ‘ 𝑅 ) = ( +g ‘ 𝑅 )
4 eqid ⊢ ( .r ‘ 𝑅 ) = ( .r ‘ 𝑅 )
5 eqid ⊢ ( 0g ‘ 𝑅 ) = ( 0g ‘ 𝑅 )
6 1 2 3 4 5 issrg ⊢ ( 𝑅 ∈ SRing ↔ ( 𝑅 ∈ CMnd ∧ ( mulGrp ‘ 𝑅 ) ∈ Mnd ∧ ∀ 𝑥 ∈ ( Base ‘ 𝑅 ) ( ∀ 𝑦 ∈ ( Base ‘ 𝑅 ) ∀ 𝑧 ∈ ( Base ‘ 𝑅 ) ( ( 𝑥 ( .r ‘ 𝑅 ) ( 𝑦 ( +g ‘ 𝑅 ) 𝑧 ) ) = ( ( 𝑥 ( .r ‘ 𝑅 ) 𝑦 ) ( +g ‘ 𝑅 ) ( 𝑥 ( .r ‘ 𝑅 ) 𝑧 ) ) ∧ ( ( 𝑥 ( +g ‘ 𝑅 ) 𝑦 ) ( .r ‘ 𝑅 ) 𝑧 ) = ( ( 𝑥 ( .r ‘ 𝑅 ) 𝑧 ) ( +g ‘ 𝑅 ) ( 𝑦 ( .r ‘ 𝑅 ) 𝑧 ) ) ) ∧ ( ( ( 0g ‘ 𝑅 ) ( .r ‘ 𝑅 ) 𝑥 ) = ( 0g ‘ 𝑅 ) ∧ ( 𝑥 ( .r ‘ 𝑅 ) ( 0g ‘ 𝑅 ) ) = ( 0g ‘ 𝑅 ) ) ) ) )
7 6 simp1bi ⊢ ( 𝑅 ∈ SRing → 𝑅 ∈ CMnd )