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Database BASIC ALGEBRAIC STRUCTURES Groups Definition and basic properties grpideu
Description: The two-sided identity element of a group is unique. Lemma 2.2.1(a) of
Herstein p. 55. (Contributed by NM , 16-Aug-2011) (Revised by Mario
Carneiro , 8-Dec-2014)
Ref
Expression
Hypotheses
grpcl.b ⊢ 𝐵 = ( Base ‘ 𝐺 )
grpcl.p ⊢ + = ( +g ‘ 𝐺 )
grpinvex.p ⊢ 0 = ( 0g ‘ 𝐺 )
Assertion
grpideu ⊢ ( 𝐺 ∈ Grp → ∃! 𝑢 ∈ 𝐵 ∀ 𝑥 ∈ 𝐵 ( ( 𝑢 + 𝑥 ) = 𝑥 ∧ ( 𝑥 + 𝑢 ) = 𝑥 ) )
Proof
Step
Hyp
Ref
Expression
1
grpcl.b ⊢ 𝐵 = ( Base ‘ 𝐺 )
2
grpcl.p ⊢ + = ( +g ‘ 𝐺 )
3
grpinvex.p ⊢ 0 = ( 0g ‘ 𝐺 )
4
grpmnd ⊢ ( 𝐺 ∈ Grp → 𝐺 ∈ Mnd )
5
1 2
mndideu ⊢ ( 𝐺 ∈ Mnd → ∃! 𝑢 ∈ 𝐵 ∀ 𝑥 ∈ 𝐵 ( ( 𝑢 + 𝑥 ) = 𝑥 ∧ ( 𝑥 + 𝑢 ) = 𝑥 ) )
6
4 5
syl ⊢ ( 𝐺 ∈ Grp → ∃! 𝑢 ∈ 𝐵 ∀ 𝑥 ∈ 𝐵 ( ( 𝑢 + 𝑥 ) = 𝑥 ∧ ( 𝑥 + 𝑢 ) = 𝑥 ) )