This is an inofficial mirror of http://metamath.tirix.org for personal testing of a visualizer extension only.

Metamath Proof Explorer

Theorem grpinvinvd

Description: Double inverse law for groups. (Contributed by Thierry Arnoux, 15-Feb-2026)

Ref Expression
Hypotheses grpinvinvd.1 ⊢ 𝐵 = ( Base ‘ 𝐺 )
grpinvinvd.2 ⊢ 𝑁 = ( invg ‘ 𝐺 )
grpinvinvd.3 ⊢ ( 𝜑 → 𝐺 ∈ Grp )
grpinvinvd.4 ⊢ ( 𝜑 → 𝑋 ∈ 𝐵 )
Assertion grpinvinvd ( 𝜑 → ( 𝑁 ‘ ( 𝑁 ‘ 𝑋 ) ) = 𝑋 )

Proof

Step Hyp Ref Expression
1 grpinvinvd.1 ⊢ 𝐵 = ( Base ‘ 𝐺 )
2 grpinvinvd.2 ⊢ 𝑁 = ( invg ‘ 𝐺 )
3 grpinvinvd.3 ⊢ ( 𝜑 → 𝐺 ∈ Grp )
4 grpinvinvd.4 ⊢ ( 𝜑 → 𝑋 ∈ 𝐵 )
5 1 2 grpinvinv ⊢ ( ( 𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵 ) → ( 𝑁 ‘ ( 𝑁 ‘ 𝑋 ) ) = 𝑋 )
6 3 4 5 syl2anc ⊢ ( 𝜑 → ( 𝑁 ‘ ( 𝑁 ‘ 𝑋 ) ) = 𝑋 )