Theorem grpplusg

Description: The operation of a constructed group. (Contributed by Mario Carneiro, 2-Aug-2013) (Revised by Mario Carneiro, 30-Apr-2015) (Revised by AV, 27-Oct-2024)

	Ref	Expression
Hypothesis	grpfn.g	⊢ 𝐺 = { ⟨ ( Base ‘ ndx ) , 𝐵 ⟩ , ⟨ ( +g ‘ ndx ) , + ⟩ }
Assertion	grpplusg	⊢ ( + ∈ 𝑉 → + = ( +g ‘ 𝐺 ) )

Proof

Step	Hyp	Ref	Expression
1		grpfn.g	⊢ 𝐺 = { ⟨ ( Base ‘ ndx ) , 𝐵 ⟩ , ⟨ ( +g ‘ ndx ) , + ⟩ }
2		basendxltplusgndx	⊢ ( Base ‘ ndx ) < ( +g ‘ ndx )
3		plusgndxnn	⊢ ( +g ‘ ndx ) ∈ ℕ
4		plusgid	⊢ +g = Slot ( +g ‘ ndx )
5	1 2 3 4	2strop	⊢ ( + ∈ 𝑉 → + = ( +g ‘ 𝐺 ) )