Theorem grpasscan2d

Description: An associative cancellation law for groups. (Contributed by SN, 29-Jan-2025)

	Ref	Expression
Hypotheses	grpasscan2d.b	⊢ 𝐵 = ( Base ‘ 𝐺 )
	grpasscan2d.p	⊢ + = ( +g ‘ 𝐺 )
	grpasscan2d.n	⊢ 𝑁 = ( invg ‘ 𝐺 )
	grpasscan2d.g	⊢ ( 𝜑 → 𝐺 ∈ Grp )
	grpasscan2d.1	⊢ ( 𝜑 → 𝑋 ∈ 𝐵 )
	grpasscan2d.2	⊢ ( 𝜑 → 𝑌 ∈ 𝐵 )
Assertion	grpasscan2d	⊢ ( 𝜑 → ( ( 𝑋 + ( 𝑁 ‘ 𝑌 ) ) + 𝑌 ) = 𝑋 )

Proof

Step	Hyp	Ref	Expression
1		grpasscan2d.b	⊢ 𝐵 = ( Base ‘ 𝐺 )
2		grpasscan2d.p	⊢ + = ( +g ‘ 𝐺 )
3		grpasscan2d.n	⊢ 𝑁 = ( invg ‘ 𝐺 )
4		grpasscan2d.g	⊢ ( 𝜑 → 𝐺 ∈ Grp )
5		grpasscan2d.1	⊢ ( 𝜑 → 𝑋 ∈ 𝐵 )
6		grpasscan2d.2	⊢ ( 𝜑 → 𝑌 ∈ 𝐵 )
7	1 2 3	grpasscan2	⊢ ( ( 𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( ( 𝑋 + ( 𝑁 ‘ 𝑌 ) ) + 𝑌 ) = 𝑋 )
8	4 5 6 7	syl3anc	⊢ ( 𝜑 → ( ( 𝑋 + ( 𝑁 ‘ 𝑌 ) ) + 𝑌 ) = 𝑋 )