Theorem grpbase

Description: The base set 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	grpbase	⊢ ( 𝐵 ∈ 𝑉 → 𝐵 = ( Base ‘ 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	1 2 3	2strbas	⊢ ( 𝐵 ∈ 𝑉 → 𝐵 = ( Base ‘ 𝐺 ) )