Theorem mgmideud

Description: Uniqueness of the left and right identity element of a magma when it exists. (Contributed by FL, 12-Dec-2009) (Revised by Mario Carneiro, 22-Dec-2013) (Revised by AV, 18-Aug-2026)

	Ref	Expression
Hypothesis	mgmideud.e	⊢ ( 𝜑 → ∃ 𝑢 ∈ 𝐵 ∀ 𝑥 ∈ 𝐵 ( ( 𝑢 + 𝑥 ) = 𝑥 ∧ ( 𝑥 + 𝑢 ) = 𝑥 ) )
Assertion	mgmideud	⊢ ( 𝜑 → ∃! 𝑢 ∈ 𝐵 ∀ 𝑥 ∈ 𝐵 ( ( 𝑢 + 𝑥 ) = 𝑥 ∧ ( 𝑥 + 𝑢 ) = 𝑥 ) )

Proof

Step	Hyp	Ref	Expression
1		mgmideud.e	⊢ ( 𝜑 → ∃ 𝑢 ∈ 𝐵 ∀ 𝑥 ∈ 𝐵 ( ( 𝑢 + 𝑥 ) = 𝑥 ∧ ( 𝑥 + 𝑢 ) = 𝑥 ) )
2		mgmidmo	⊢ ∃* 𝑢 ∈ 𝐵 ∀ 𝑥 ∈ 𝐵 ( ( 𝑢 + 𝑥 ) = 𝑥 ∧ ( 𝑥 + 𝑢 ) = 𝑥 )
3		reu5	⊢ ( ∃! 𝑢 ∈ 𝐵 ∀ 𝑥 ∈ 𝐵 ( ( 𝑢 + 𝑥 ) = 𝑥 ∧ ( 𝑥 + 𝑢 ) = 𝑥 ) ↔ ( ∃ 𝑢 ∈ 𝐵 ∀ 𝑥 ∈ 𝐵 ( ( 𝑢 + 𝑥 ) = 𝑥 ∧ ( 𝑥 + 𝑢 ) = 𝑥 ) ∧ ∃* 𝑢 ∈ 𝐵 ∀ 𝑥 ∈ 𝐵 ( ( 𝑢 + 𝑥 ) = 𝑥 ∧ ( 𝑥 + 𝑢 ) = 𝑥 ) ) )
4	1 2 3	sylanblrc	⊢ ( 𝜑 → ∃! 𝑢 ∈ 𝐵 ∀ 𝑥 ∈ 𝐵 ( ( 𝑢 + 𝑥 ) = 𝑥 ∧ ( 𝑥 + 𝑢 ) = 𝑥 ) )