Metamath Proof Explorer


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 + = ( +g𝐺 )
grpasscan2d.n 𝑁 = ( invg𝐺 )
grpasscan2d.g ( 𝜑𝐺 ∈ Grp )
grpasscan2d.1 ( 𝜑𝑋𝐵 )
grpasscan2d.2 ( 𝜑𝑌𝐵 )
Assertion grpasscan2d ( 𝜑 → ( ( 𝑋 + ( 𝑁𝑌 ) ) + 𝑌 ) = 𝑋 )

Proof

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