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	$$⊢G=\left\{⟨{\mathrm{Base}}_{\mathrm{ndx}},B⟩,⟨{+}_{\mathrm{ndx}},\underset{\dot{}}{+}⟩\right\}$$
Assertion	grpbase	$$⊢B\in V\to B={\mathrm{Base}}_G$$

Proof

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