Theorem ringgrp

Description: A ring is a group. (Contributed by Steve Rodriguez, 9-Sep-2007) (Revised by NM, 15-Sep-2011)

	Ref	Expression
Assertion	ringgrp	⊢ ( 𝑅 ∈ Ring → 𝑅 ∈ Grp )

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	1 2 3 4	isring	⊢ ( 𝑅 ∈ Ring ↔ ( 𝑅 ∈ Grp ∧ ( 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 ) ) ) ) )
6	5	simp1bi	⊢ ( 𝑅 ∈ Ring → 𝑅 ∈ Grp )