Theorem ringideu

Description: The unit element of a ring is unique. (Contributed by NM, 27-Aug-2011) (Revised by Mario Carneiro, 6-Jan-2015)

	Ref	Expression
Hypotheses	ringcl.b	⊢ 𝐵 = ( Base ‘ 𝑅 )
	ringcl.t	⊢ · = ( .r ‘ 𝑅 )
Assertion	ringideu	⊢ ( 𝑅 ∈ Ring → ∃! 𝑢 ∈ 𝐵 ∀ 𝑥 ∈ 𝐵 ( ( 𝑢 · 𝑥 ) = 𝑥 ∧ ( 𝑥 · 𝑢 ) = 𝑥 ) )

Proof

Step	Hyp	Ref	Expression
1		ringcl.b	⊢ 𝐵 = ( Base ‘ 𝑅 )
2		ringcl.t	⊢ · = ( .r ‘ 𝑅 )
3		eqid	⊢ ( mulGrp ‘ 𝑅 ) = ( mulGrp ‘ 𝑅 )
4	3	ringmgp	⊢ ( 𝑅 ∈ Ring → ( mulGrp ‘ 𝑅 ) ∈ Mnd )
5	3 1	mgpbas	⊢ 𝐵 = ( Base ‘ ( mulGrp ‘ 𝑅 ) )
6	3 2	mgpplusg	⊢ · = ( +g ‘ ( mulGrp ‘ 𝑅 ) )
7	5 6	mndideu	⊢ ( ( mulGrp ‘ 𝑅 ) ∈ Mnd → ∃! 𝑢 ∈ 𝐵 ∀ 𝑥 ∈ 𝐵 ( ( 𝑢 · 𝑥 ) = 𝑥 ∧ ( 𝑥 · 𝑢 ) = 𝑥 ) )
8	4 7	syl	⊢ ( 𝑅 ∈ Ring → ∃! 𝑢 ∈ 𝐵 ∀ 𝑥 ∈ 𝐵 ( ( 𝑢 · 𝑥 ) = 𝑥 ∧ ( 𝑥 · 𝑢 ) = 𝑥 ) )