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Metamath Proof Explorer

Theorem srgmgp

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

Ref Expression
Hypothesis srgmgp.g ⊢ 𝐺 = ( mulGrp ‘ 𝑅 )
Assertion srgmgp ( 𝑅 ∈ SRing → 𝐺 ∈ Mnd )

Proof

Step Hyp Ref Expression
1 srgmgp.g ⊢ 𝐺 = ( mulGrp ‘ 𝑅 )
2 eqid ⊢ ( Base ‘ 𝑅 ) = ( Base ‘ 𝑅 )
3 eqid ⊢ ( +g ‘ 𝑅 ) = ( +g ‘ 𝑅 )
4 eqid ⊢ ( .r ‘ 𝑅 ) = ( .r ‘ 𝑅 )
5 eqid ⊢ ( 0g ‘ 𝑅 ) = ( 0g ‘ 𝑅 )
6 2 1 3 4 5 issrg ⊢ ( 𝑅 ∈ SRing ↔ ( 𝑅 ∈ CMnd ∧ 𝐺 ∈ 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 simp2bi ⊢ ( 𝑅 ∈ SRing → 𝐺 ∈ Mnd )