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Metamath Proof Explorer

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	$⊢ G = \{⟨{Base}_{ndx}, B⟩, ⟨+_{ndx}, \underset{˙}{+}⟩\}$
	Assertion	grpbase	$⊢ B \in V \to B = {Base}_{G}$

Proof

Step	Hyp	Ref	Expression
1		grpfn.g	$⊢ G = \{⟨{Base}_{ndx}, B⟩, ⟨+_{ndx}, \underset{˙}{+}⟩\}$
2		basendxltplusgndx	$⊢ {Base}_{ndx} < +_{ndx}$
3		plusgndxnn	$⊢ +_{ndx} \in ℕ$
4	1 2 3	2strbas	$⊢ B \in V \to B = {Base}_{G}$