Theorem grpbase

Description: The base set of a constructed group. (Contributed by Mario Carneiro, 2-Aug-2013) (Revised by Mario Carneiro, 30-Apr-2015)

	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		df-plusg	⊢ +g = Slot 2
3		1lt2	⊢ 1 < 2
4		2nn	⊢ 2 ∈ ℕ
5	1 2 3 4	2strbas	⊢ ( 𝐵 ∈ 𝑉 → 𝐵 = ( Base ‘ 𝐺 ) )