Theorem grplidd

Description: The identity element of a group is a left identity. Deduction associated with grplid . (Contributed by SN, 29-Jan-2025)

	Ref	Expression
Hypotheses	grpbn0.b	⊢ 𝐵 = ( Base ‘ 𝐺 )
	grplid.p	⊢ + = ( +g ‘ 𝐺 )
	grplid.o	⊢ 0 = ( 0g ‘ 𝐺 )
	grplidd.g	⊢ ( 𝜑 → 𝐺 ∈ Grp )
	grplidd.1	⊢ ( 𝜑 → 𝑋 ∈ 𝐵 )
Assertion	grplidd	⊢ ( 𝜑 → ( 0 + 𝑋 ) = 𝑋 )

Proof

Step	Hyp	Ref	Expression
1		grpbn0.b	⊢ 𝐵 = ( Base ‘ 𝐺 )
2		grplid.p	⊢ + = ( +g ‘ 𝐺 )
3		grplid.o	⊢ 0 = ( 0g ‘ 𝐺 )
4		grplidd.g	⊢ ( 𝜑 → 𝐺 ∈ Grp )
5		grplidd.1	⊢ ( 𝜑 → 𝑋 ∈ 𝐵 )
6	1 2 3	grplid	⊢ ( ( 𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵 ) → ( 0 + 𝑋 ) = 𝑋 )
7	4 5 6	syl2anc	⊢ ( 𝜑 → ( 0 + 𝑋 ) = 𝑋 )