Metamath Proof Explorer


Theorem srgideu

Description: The unit element of a semiring is unique. (Contributed by NM, 27-Aug-2011) (Revised by Mario Carneiro, 6-Jan-2015) (Revised by Thierry Arnoux, 1-Apr-2018)

Ref Expression
Hypotheses srgcl.b 𝐵 = ( Base ‘ 𝑅 )
srgcl.t · = ( .r𝑅 )
Assertion srgideu ( 𝑅 ∈ SRing → ∃! 𝑢𝐵𝑥𝐵 ( ( 𝑢 · 𝑥 ) = 𝑥 ∧ ( 𝑥 · 𝑢 ) = 𝑥 ) )

Proof

Step Hyp Ref Expression
1 srgcl.b 𝐵 = ( Base ‘ 𝑅 )
2 srgcl.t · = ( .r𝑅 )
3 eqid ( mulGrp ‘ 𝑅 ) = ( mulGrp ‘ 𝑅 )
4 3 srgmgp ( 𝑅 ∈ SRing → ( 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 → ∃! 𝑢𝐵𝑥𝐵 ( ( 𝑢 · 𝑥 ) = 𝑥 ∧ ( 𝑥 · 𝑢 ) = 𝑥 ) )
8 4 7 syl ( 𝑅 ∈ SRing → ∃! 𝑢𝐵𝑥𝐵 ( ( 𝑢 · 𝑥 ) = 𝑥 ∧ ( 𝑥 · 𝑢 ) = 𝑥 ) )