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Metamath Proof Explorer

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

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		plusgid	$⊢ +_{𝑔} = Slot +_{ndx}$
5	1 2 3 4	2strop	$⊢ \underset{˙}{+} \in V \to \underset{˙}{+} = +_{G}$