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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 ⊢ + = ( +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 )