Theorem grpstr

Description: A constructed group is a structure. Version not depending on the implementation of the indices. (Contributed by AV, 27-Oct-2024)

	Ref	Expression
Hypothesis	grpfn.g	⊢ 𝐺 = { ⟨ ( Base ‘ ndx ) , 𝐵 ⟩ , ⟨ ( +g ‘ ndx ) , + ⟩ }
Assertion	grpstr	⊢ 𝐺 Struct ⟨ ( Base ‘ ndx ) , ( +g ‘ ndx ) ⟩

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	1 2 3	2strstr	⊢ 𝐺 Struct ⟨ ( Base ‘ ndx ) , ( +g ‘ ndx ) ⟩